1. Introduction
Let 
$p$ be a rational prime, 
$f$ be a positive integer, and let 
$L$ be the unramified extension of 
$\mathbf {Q}_p$ of degree 
$f$. Let 
$k_E$ be a finite extension of the residue field 
$k_L$ of 
$L$. For a field 
$k$, we implicitly choose a separable closure 
$k^{\mathrm {sep}}/k$ and set 
$G_k = \mathop {Gal}(k^{\mathrm {sep}}/k)$. A perhaps optimistic hope is that a mod 
$p$ local Langlands correspondence attaches to every continuous Galois representation 
${\bar {\rho }}: G_L \rightarrow \mathrm {GL}_2(k_E)$ a 
$\mathrm {GL}_2(L)$-representation 
$\pi ({\bar {\rho }})$ over 
$k_E$ satisfying several properties including compatibility with the classical local Langlands under reduction and local–global compatibility (more on this later). Such a correspondence has been established for 
$\mathrm {GL}_2(\mathbf {Q}_p)$ in [Reference BreuilBre03, Reference ColmezCol10, Reference PaškūnasPaš13, Reference EmertonEme11], but appears to be significantly more complex when 
$f>1$ or 
$\mathrm {GL}_2$ is replaced by a higher-rank group. One of the obstacles in realizing, or even formulating, this correspondence is that supersingular irreducible 
$k_E[\mathrm {GL}_2(L)]$-representations are not classified, and seem to be very difficult to even construct.
The classical way of studying representations of 
$p$-adic (or Lie) groups is first to study their restriction to a compact subgroup, and then to study residual symmetries, for example the algebra of Hecke operators. In this way, infinite-dimensional representations are replaced with combinatorial objects. The module theory of the pro-
$p$ Iwahori Hecke algebra captures much of the mod 
$p$ representation theory in the case of 
$\mathrm {GL}_2(\mathbf {Q}_p)$, but less in any more-complicated case. In [Reference Breuil and PaškūnasBP12, § 9], Breuil and Paškūnas instead consider a refinement of Hecke modules, which they call (basic) 
$0$-diagrams. For our purposes, a 
$0$-diagram 
$\mathcal {D}$ is a 
$\mathrm {GL}_2(k_L)$-representation 
$D_0$ with an automorphism 
$\Pi$ of 
$D_0^{I_1}$ whose square acts by a nonzero scalar, where 
$I_1 \subset \mathrm {GL}_2(k_L)$ is the subgroup of unipotent upper triangular matrices. From a 
$\mathrm {GL}_2(L)$-representation 
$\pi$ with central character, one obtains a 
$0$-diagram 
$\mathcal {D}(\pi )$ by taking 
$D_0$ to be 
$\pi ^{K_1}$ and 
$\Pi$ to be given by the action of 
$\big (\begin {smallmatrix} 0 & 1 \\ p & 0 \end {smallmatrix}\big )$, where 
$K_1$ is the kernel of the reduction map 
$\mathrm {GL}_2(\mathcal {O}_L) \rightarrow \mathrm {GL}_2(k_L)$. Note that if 
$\pi$ is nonzero and of characteristic 
$p$, then so is 
$\pi ^{K_1}$ since 
$K_1$ is a pro-
$p$ group. In [Reference Breuil and PaškūnasBP12, § 13], they constructed a family of mod 
$p$ 
$0$-diagrams for each generic 
${\bar {\rho }}$. From each of these 
$0$-diagrams 
$\mathcal {D}$, in [Reference Breuil and PaškūnasBP12, § 9] they constructed a family of mod 
$p$ 
$\mathrm {GL}_2(L)$-representations 
$\pi$ such that 
$\mathcal {D}$ is a subdiagram of 
$\mathcal {D}(\pi )$ in the obvious sense. For each generic 
${\bar {\rho }}$, the family of 
$0$-diagrams has size 
$1$ if 
$f=1$, but is infinite otherwise; moreover, for each basic 
$0$-diagram, the family of mod 
$p$ 
$\mathrm {GL}_2(L)$-representations has size 
$1$ if 
$f=1$, but is infinite if 
$f>1$ (see [Reference HuHu10]): this is a long way from the hoped-for correspondence when 
$f>1$.
Local–global compatibility gives another approach to the mod 
$p$ local Langlands correspondence. The idea is that a global mod 
$p$ Langlands correspondence (closely related to generalizations of Serre's conjecture) is often easy to formulate, if difficult to prove. Namely, as in the classical picture, it suffices to use the Satake parameterization to define an unramified mod 
$p$ local Langlands correspondence away from 
$p$. Mod 
$p$ local–global compatibility at 
$p$ states that the restriction of the global correspondence to places dividing 
$p$ should recover the local correspondence. Thus, one way to proceed is to globalize 
${\bar {\rho }}$ to a Galois representation 
$\bar {r}$ which is known to correspond to a space of mod 
$p$ automorphic forms on a quaternion algebra, and then to restrict this representation to get a candidate 
$\mathrm {GL}_2(L)$-representation 
$\pi _{\mathrm {glob}}({\bar {\rho }})$ (see Definition 4.9 and § 6.1). A drawback of this construction is that it is completely unclear if 
$\pi _{\mathrm {glob}}({\bar {\rho }})$ is independent of the various global choices. The representation 
$\pi _{\mathrm {glob}}({\bar {\rho }})$ is best understood when 
$\bar {r}$ satisfies favorable conditions such as the Taylor–Wiles hypothesis. We assume this to be the case in the rest of this section (see § 6.1 for more details).
Theorem 1.1 [Reference BreuilBre14, Reference Emerton, Gee and SavittEGS15, Reference Hu and WangHW18, Reference Le, Morra and SchraenLMS20, Reference LeLe19]
If 
${\bar {\rho }}$ is generic, 
$\mathcal {D}(\pi _{\mathrm {glob}}({\bar {\rho }}))$ is isomorphic to one of the 
$0$-diagrams attached to 
${\bar {\rho }}$ by [Reference Breuil and PaškūnasBP12].
More precisely, Breuil [Reference BreuilBre14] showed that 
$\mathcal {D}(\pi _{\mathrm {glob}}({\bar {\rho }}))$ contains one of the 
$0$-diagrams attached to 
${\bar {\rho }}$ conditional on a conjecture later established in [Reference Emerton, Gee and SavittEGS15] (under a slightly stronger genericity hypothesis). Building on this, the sequence of works [Reference Hu and WangHW18, Reference Le, Morra and SchraenLMS20, Reference LeLe19] showed that this inclusion is, in fact, an isomorphism (subject to still stronger genericity hypotheses). However, because this family of 
$0$-diagrams is infinite when 
$f>1$, one may still ask whether 
$\mathcal {D}(\pi _{\mathrm {glob}}({\bar {\rho }}))$ depends on global choices. The following is our first main result (combining Theorems 4.11 and 6.2).
Theorem 1.2 If 
${\bar {\rho }}$ is generic, 
$\mathcal {D}(\pi _{\mathrm {glob}}({\bar {\rho }}))$ depends only on 
${\bar {\rho }}$.
See § 3.2.2 for the precise genericity conditions we need: they are slightly more restrictive than those in [Reference Breuil and PaškūnasBP12]. Theorem 1.2 narrows the family of diagrams constructed by [Reference Breuil and PaškūnasBP12] to a single one for each generic 
${\bar {\rho }}$. In fact, our proof can be made completely explicit, which we do to some extent in the case when 
${\bar {\rho }}$ is semisimple (see Theorem 1.3).
The mod 
$p$ (and 
$p$-adic) local Langlands correspondence for 
$\mathrm {GL}_2(\mathbf {Q}_p)$ (see [Reference ColmezCol10, Reference KisinKis10, Reference PaškūnasPaš13, Reference Colmez, Dospinescu and PaškūnasCDP14]) is realized by Colmez's functor. In [Reference BreuilBre11], Breuil observes that this functor is rather combinatorial if 
${\bar {\rho }}$ is semisimple and can be generalized to the 
$\mathrm {GL}_2(L)$-case. Namely, to a Diamond 
$0$-diagram 
$\mathcal {D}$ (a 
$0$-diagram of a certain form, see § 4.2) he attaches a 
$(\varphi ,\Gamma )$-module 
$M(\mathcal {D})$, which recovers Colmez's functor in the 
$\mathrm {GL}_2(\mathbf {Q}_p)$ case when 
${\bar {\rho }}$ is semisimple (recall that the family of 
$0$-diagrams in this case is a singleton). In other words, if 
$f$ is 
$1$ and 
${\bar {\rho }}$ is semisimple, then 
$M(\mathcal {D}({\bar {\rho }}))$ corresponds to 
${\bar {\rho }}^\vee$ under Fontaine's equivalence (see [Reference BreuilBre11, § 1]). Our other main result generalizes this to arbitrary 
$f$.
Theorem 1.3 If 
${\bar {\rho }}:G_L \rightarrow \mathrm {GL}_2(k_E)$ is generic and semisimple, then 
$M(\mathcal {D}(\pi _{\mathrm {glob}}({\bar {\rho }})))$ corresponds under Fontaine's equivalence to the tensor induction 
$\mathrm {ind}_L^{\otimes \mathbf {Q}_p} {\bar {\rho }}^\vee$ from 
$G_L$ to 
$G_{\mathbf {Q}_p}$.
By [Reference BreuilBre11, Théorème 6.4], this theorem is equivalent to a determination of certain parameters of the diagram, which appears in Theorem 5.36 in the case 
$\det ({\bar {\rho }}) \circ \mathrm {Art}_L(p) = 1$ (the other cases follow by twisting). This gives a positive answer to [Reference BreuilBre14, Question 9.5] when 
${\bar {\rho }}$ is generic.
Remark 1.4 Hu [Reference HuHu16] obtained an analogue of this question in the setting of [Reference Breuil and DiamondBD14] where 
${\bar {\rho }}$ is maximally nonsplit. We repeat certain calculations from [Reference HuHu16] in slightly different contexts for the sake of completeness. (See Remark 5.20 for an instance of this.)
We now explain these parameters further. By Theorem 1.1, the 
$\mathrm {GL}_2(k_L)$-action on 
$\mathcal {D}(\pi _{\mathrm {glob}}({\bar {\rho }}))$ is known (by construction, the 
$\mathrm {GL}_2(k_L)$-actions within a single family are isomorphic). To prove Theorems 1.2 and 1.3, we must understand how the actions of 
$\mathrm {GL}_2(k_L)$ and 
$\Pi$ interact. We show that this interaction corresponds exactly to a character of a certain finite groupoid 
$\mathcal {G}$ with generators indexed by 
$I$-isotypic lines in 
$D_0^{I_1}$. It suffices to show that the action of group elements of 
$\mathcal {G}$ depend only on 
${\bar {\rho }}$. There has been a body of work studying the action of group elements which are themselves generators (see [Reference Breuil and DiamondBD14, Reference HuHu16, Reference Herzig, Le and MorraHLM17, Reference Le, Morra and ParkLMP18, Reference EnnsEnn18, Reference Park and QianPQ18]), often with the goal of showing that 
${\bar {\rho }}$ can be recovered from 
$\pi _{\mathrm {glob}}({\bar {\rho }})$. The methods in these works have a common theme. The study of 
$\Pi$ can be reduced to the study of the action of a Hecke operator 
$U_p$ and a certain element of the group algebra of 
$\mathrm {GL}_2(k_L)$. The action of 
$U_p$ can then be studied by using the classical local Langlands correspondence on a characteristic zero lift of 
$\pi _{\mathrm {glob}}({\bar {\rho }})$ and then taking the reduction mod 
$p$. This method cannot work directly because as soon as one considers a product of generators, the lifts one considers are typically distinct and not directly comparable. Thus, we need another tool.
The reason for imposing the Taylor–Wiles hypothesis is that we would like to use the Taylor–Wiles patching method. In fact, our results apply whenever 
$\pi _{\mathrm {glob}}({\bar {\rho }})$ satisfies certain axioms, namely it comes from a patched module with an arithmetic action. The study of Colmez's functor led to a proof of many cases of the Fontaine–Mazur conjecture in [Reference KisinKis09a], through the construction of restricted local deformation rings at 
$p$ (see [Reference KisinKis08]), a classical inertial local Langlands correspondence relating representations of compact open subgroups and representations of the inertia subgroups of local Galois groups, and the Taylor–Wiles patching method. In the other direction, the techniques introduced by Kisin have led to a number of developments in the inertial mod 
$p$ and 
$p$-adic Langlands correspondences beyond 
$\mathrm {GL}_2(\mathbf {Q}_p)$ (see, e.g., [Reference Gee, Liu and SavittGLS15, Reference Emerton, Gee and SavittEGS15, Reference Le, Le Hung, Levin and MorraLLLM18, Reference Le, Le Hung, Levin and MorraLLLM20, Reference Le, Le Hung and LevinLLL19]). Another approach to the Fontaine–Mazur conjecture is to prove local–global compatibility (see [Reference EmertonEme11, Reference Caraiani, Emerton, Gee, Geraghty, Paškūnas and ShinCEG+18]), where Hecke operators must play a crucial role. Our approach is similar to [Reference Caraiani, Emerton, Gee, Geraghty, Paškūnas and ShinCEG+18] in that we use the interpolation from [Reference Caraiani, Emerton, Gee, Geraghty, Paškūnas and ShinCEG+16] of the action of Hecke operators on Taylor–Wiles patched modules, but we further interpolate between different deformation rings.
Although the various characteristic zero lifts of spaces of mod 
$p$ automorphic forms one might consider are incomparable, they can be interpolated using the Taylor–Wiles patching method. The corresponding Hecke operators can then be identified with elements of various tamely potentially Barsotti–Tate deformation rings. In turn, these elements can be compared with elements of a single tamely potentially Barsotti–Tate deformation ring, corresponding to what we call the central inertial type. For this comparison, it is essential that we are working with patched modules which are maximal Cohen–Macaulay on deformation spaces. In this way, understanding the action of group elements of 
$\mathcal {G}$ is reduced to understanding a certain product of elements of a tamely potentially Barsotti–Tate deformation ring and the action of (a product of) elements of the group algebra of 
$\mathrm {GL}_2(k_L)$ on the Deligne–Lusztig representation corresponding to the central inertial type by inertial local Langlands. For the reader only interested in the proof of Theorem 1.2 in an axiomatic context (see Theorem 4.11), a proof with minimal reference to previous sections is given in § 4. To prove Theorem 1.3, we explicitly calculate these products in § 5 using §§ 2–3. Using the Taylor–Wiles method, § 6 gives examples of global contexts satisfying the axioms of § 4.
1.1 Notation and conventions
Throughout, 
$p > 5$ is a prime number (if 
$p \leq 5$ then there are no generic representations in the sense of § 3.2.2). Let 
$L = \mathbf {Q}_{p^f}$, write 
$\mathcal {O}_L$ for the ring of integers, 
$k_L$ for the residue field, and 
$q = p^f$, and fix an algebraic closure 
$\bar {L} / L$. We fix a coefficient field 
$E/ \mathbf {Q}_p$ (a finite extension) with ring of integers 
$\mathcal {O}_E$, uniformizer 
$\varpi _E$, and residue field 
$k_E$. By taking a further extension, we assume that 
$E$ is large enough for our purposes. In particular, we assume that every irreducible 
$\bar {E}$-representation of 
$\mathrm {GL}_2(k_L)$ is defined over 
$E$. We assume there exist embeddings of 
$k_L$ in 
$k_E$ and we fix one of them, denoted 
$\sigma _0 : k_L \to k_E$. We number embeddings as 
$\sigma _j = \sigma _0 \circ \varphi ^{-j}$, where 
$\varphi : x \mapsto x^p$.
For 
$\lambda \in k_L$ we write 
$[\lambda ] \in \mathcal {O}_E$ for the Teichmüller lift arising from the induced embedding 
$\sigma _0: W(k_L) \to \mathcal {O}_E$.
The group 
$\mathrm {GL}_2(\mathcal {O}_L)$ is denoted 
$K$, and the first congruence subgroup is
\[ K_1 = \ker(\mathrm{GL}_2(\mathcal{O}_L) \to \mathrm{GL}_2(k_L)). \]
According to the context, we write 
$H$ for the diagonally embedded 
$\mathcal {O}_L^\times \times \mathcal {O}_L^\times$ in 
$\mathrm {GL}_2(\mathcal {O}_L)$ or its image in 
$\mathrm {GL}_2(k_L)$, which is the diagonal torus. Similarly, 
$I$ denotes the Iwahori subgroup of 
$\mathrm {GL}_2(\mathcal {O}_L)$ or the upper triangular Borel subgroup of 
$\mathrm {GL}_2(k_L)$, and 
$I_1$ denotes the Sylow pro-
$p$ subgroup of 
$I$ or the upper unipotent subgroup of 
$\mathrm {GL}_2(k_L)$. We write 
$\alpha$ for the 
$H$-character 
$\big (\begin {smallmatrix} a & 0 \\ 0 & d \end {smallmatrix}\big ) \mapsto [ad^{-1}]$. If 
$\chi$ is a character of 
$H$, then 
$\chi ^s$ is its conjugate under the nontrivial element 
$s$ of the Weyl group 
$S_2$. Hence, if 
$\chi : \big (\begin {smallmatrix} a & 0 \\ 0 & d \end {smallmatrix}\big ) \mapsto [a]^r[ad]^s$, then 
$\chi ^s = \chi \alpha ^{-r}$. If 
$x \in k_E^\times$, we write 
$\mathrm {nr}_x$ for the unramified character 
$G_{\mathbf {Q}_p} \to k_E^\times$ that sends geometric Frobenius elements to 
$x$.
The determinant character of 
$\mathrm {GL}_2(k_L)$ is naturally valued in 
$k_L^\times$, and we obtain a 
$k_E^\times$-valued character through our fixed embedding 
$\sigma _0: k_L \to k_E$. We usually denote 
$\sigma _0 \circ \det$ by 
$\det$, and write 
$\det _E$ if we need to distinguish between the two.
We define two maps, 
$\delta _{\mathrm {red}}$ and 
$\delta _{\mathrm {irr}}$, from the set of subsets of 
$\{0, \ldots , f-1 \}$ to itself by
\begin{gather*} j \in \delta_{\mathrm{red}}(J) \text{ if and only if } j+1 \in J,\\ \begin{cases} j< f-1 : j \in \delta_{\mathrm{irr}}(J) \text{ if and only if } j+1 \in J,\\ f-1 \in \delta_{\mathrm{irr}}(J) \text{ if and only if } 0 \not \in J \end{cases} \end{gather*}
so that 
$\delta _{\mathrm {red}}$ is a left shift. The following lemma follows from the definitions.
Lemma 1.5 For all 
$\delta \in \{\delta _\mathrm {red}, \delta _\mathrm {irr}\}$, we have equalities 
$\delta (J^\textrm {c}) = (\delta J)^\textrm {c}$, 
$J \triangle \delta J = J^\textrm {c} \triangle \delta J^\textrm {c}$ and 
$\delta _\mathrm {red}(J \triangle \delta J) = \delta J \triangle \delta ^2 J$.
For a subset 
$J$ of 
$\mathbf {Z}/f = \{0, \ldots , f-1\}$, we write 
$\operatorname {ch}_J$ for the characteristic function of 
$J$. We sometimes employ the following shorthand in conjunction with 
$\delta _\mathrm {red}$: if 
$J \subset \{0, \ldots , f-1\}$, we write 
$\mathbf {r}(j, J) = \operatorname {ch}_J(j-1, j, j+1)$.
If 
$x$ is a positive integer, we define numbers 
$x_j \in [0, p-1]$ by writing 
$x = \sum _{j=0}^{f-1} x_ip^i + Q(q-1)$ for some 
$Q > 0$. (The only possible ambiguity occurs when 
$q-1|x$, in which case we put 
$x_j=0$ for all 
$j$.)
If 
$F$ is a local or global field, we write 
$\mathrm {Art}_F$ for the Artin reciprocity map, normalized so that the global map is compatible with the local maps, and the local map takes a uniformizer to a geometric Frobenius element. If 
$F$ is a number field and 
$w$ is a finite place of 
$F$, we write 
$\mathrm {Frob}_w$ for a geometric Frobenius element at 
$w$. We let 
$\varepsilon : G_L \rightarrow \mathbf {Z}_p^\times \rightarrow \mathcal {O}_E^\times$ denote the cyclotomic character and normalize Hodge–Tate weights so that 
$\mathrm {HT}_\kappa (\varepsilon ) = 1$ for all embeddings 
$\kappa : L \rightarrow E$.
2. Preliminaries in representation theory
2.1 
$S$-operators and Jacobi sums
Throughout, for 
$0 < i \leq q-1$ we write
\begin{align*} S_i &= \sum_{\lambda \in \mathbf{F}_q^\times} [\lambda]^i \begin{pmatrix} \lambda & 1 \\ 1 & 0 \end{pmatrix},\\ S_i^+ &= \sum_{\lambda \in \mathbf{F}_q^\times} [\lambda]^i \begin{pmatrix} 1 & 0 \\ \lambda & 1 \end{pmatrix} \end{align*}
and we put 
$S_0 = \big (\begin {smallmatrix} 0 & 1 \\ 1 & 0 \end {smallmatrix}\big ) + S_{q-1}, S_0^+ = \big (\begin {smallmatrix} 1 & 0 \\ 0 & 1 \end {smallmatrix}\big ) + S_{q-1}^+$. These are elements of the group algebra 
$\mathcal {O}_E[\mathrm {GL}_2(k_L)]$, where the Teichmüller lift is taken in 
$W(k_L) \subseteq W(k_E)$.
The operators 
$S_i$ play a prominent role in [Reference Breuil and PaškūnasBP12], whereas it is sometimes simpler to compute with the 
$S_i^+$. If 
$r$ is an integer, we will write 
$S_r, S_r^+$ for the operators indexed by a number in 
$\{0, \ldots , q-1\}$ congruent to 
$r$ mod 
$q-1$, and we explicitly say which one we mean for 
$0, q-1$ on a case-by-case basis.
For 
$0 \leq a, b \leq q-1$, introduce the Jacobi sum
\[ \mathbf{J}(a, b) = \sum_{\alpha \in \mathbf{F}_q} [\alpha]^a [1-\alpha]^b. \]
Usually, the convention is that all powers of 
$[0]$ are equal to 
$0$ except the 
$0$th power, which is equal to 
$1$. This is reflected in the definition of the 
$S$-operators, and it results in 
$\mathbf {J}(a, 0) = \sum _{\alpha \in \mathbf {F}_q} [\alpha ]^a$, which vanishes unless 
$a = 0, q-1$. Our computations will give rise to nonzero constants, and so will never contain Jacobi sums of this type. For this reason and not to disrupt convention, we introduce the sum 
$\mathbf {J}_0(a, b)$ that is defined in the same way, but with the assumption that 
$[0]^0 = 0$. Recall the following results.
Theorem 2.1 (Stickelberger) Let 
$0 < a, b < q-1$ be two integers such that 
$a + b \not = q-1$. Write their 
$p$-adic expansions as 
$a = \sum _i a_i p^i, b = \sum _j b_j p^j$, and similarly for 
$a+b = \sum _{k = 0}^{f-1} (a+b)_k p^k + (q-1)Q$. Then in 
$\mathcal {O}_E$ we have the equality
\[ \mathbf{J}(a, b) = U(a, b)p^{u(a, b)} + C \]
with 
$v_p(C) > u(a,b)$ and
\begin{gather*} u(a, b) = \frac{1}{p-1}\bigg(\sum_{j=0}^{f-1} p-1- (a_j + b_j -(a+b)_j)\bigg),\\ U(a, b) = ({-}1)^{f-1+u(a, b)} \frac{\prod_j a_j! b_j!}{\prod_j (a+b)_j!} \in \mathbf{Z}_p^\times. \end{gather*}Lemma 2.2 In the situation of the theorem, assume 
$a+b = q-1$. Then 
$\mathbf {J}(a, b) = (-1)^{a+1}$.
Proof. Write 
$\mathbf {J}(a, b) = \sum _{\alpha \in \mathbf {F}_q \backslash \{0, 1 \}}[{\alpha }/({1-\alpha })]^a = \sum _{x \in \mathbf {F}_q \backslash \{0, -1 \}} [x]^a = -(-1)^a$ because 
$\alpha \mapsto [\alpha ]^a$ is a nontrivial character of 
$\mathbf {F}_q^\times$ (see [Reference Ireland and RosenIR90, Theorem 8.1]).
Lemma 2.3 Let 
$0 < i, j < q-1$, so that 
$\mathbf {J}_0(i, j) = \mathbf {J}(i, j)$ is not zero by Theorem 2.1 and Lemma 2.2. Then
\[ S_i^+ S_j^+=\begin{cases} \mathbf{J}(i, j) S_{i+j}^+ &\mbox{if } i+j \neq q-1, \\ ({-}1)^{i+1}S_0^++ ({-}1)^iq \begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix} &\mbox{if } i+j = q-1. \end{cases} \]Proof. We work in this proof with 
$[0]^n = 0$ for all 
$n \in \mathbf {Z}$. Assume that 
$q-1$ does not divide 
$i+j$. We expand
\begin{align*} S_i^+S_j^+ &= \sum_{\lambda, \mu \in \mathbf{F}_q^\times}[\lambda]^i[\mu]^j\begin{pmatrix} 1 & 0 \\ \lambda + \mu & 1 \end{pmatrix} \\ &= \sum_{\lambda \in \mathbf{F}_q^\times, \nu \in \mathbf{F}_q \backslash \{\lambda \}} [\lambda]^i [\nu - \lambda]^j \begin{pmatrix} 1 & 0 \\ \nu & 1 \end{pmatrix} = \sum_{\lambda \in \mathbf{F}_q^\times, \nu \in \mathbf{F}_q} [\lambda]^i [\nu - \lambda]^j \begin{pmatrix} 1 & 0 \\ \nu & 1 \end{pmatrix}\\ &= \sum_{\lambda, \nu \in \mathbf{F}_q^\times}[\nu^{{-}1}\lambda]^i[1-\nu^{{-}1}\lambda]^j[\nu]^{i+j} \begin{pmatrix} 1 & 0 \\ \nu & 1 \end{pmatrix} + ({-}1)^j \bigg(\sum_{\lambda \in \mathbf{F}_q^\times}[\lambda]^{i+j}\bigg) \begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix}\\ &= \sum_{\nu \in \mathbf{F}_q^\times} \bigg(\sum_{\alpha \in \mathbf{F}_q^\times}[\alpha]^i[1-\alpha]^j\bigg)[\nu]^{i+j} \begin{pmatrix} 1 & 0 \\ \nu & 1 \end{pmatrix} + 0 = \bigg(\sum_{\alpha \in \mathbf{F}_q^\times}[\alpha]^i[1-\alpha]^j\bigg)S_{i+j}^+. \end{align*}
The vanishing of 
$(\sum _{\lambda \in \mathbf {F}_q^\times }[\lambda ]^{i+j})$ occurs because, by assumption, 
$[\lambda ]^{i+j}$ is a nontrivial character of 
$\mathbf {F}_q^\times$.
Otherwise, 
$i+j = q-1$ and the second summand equals 
$q-1$, so
\[ S_i^+ S_{q-1-i}^+= \bigg(\sum_{\alpha \in \mathbf{F}_q^\times}[\alpha]^i[1-\alpha]^{q-1-i} \bigg) S_{q-1}^++ ({-}1)^{q-1-i}(q-1)\begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix} \]
and the claim follows from Lemma 2.2 because 
$\mathbf {J}(i, q-1-i) = \mathbf {J}_0(i, q-1-i)$.
Lemma 2.4 Let 
$i$ and 
$j$ be as in Lemma 2.3. Then
\[ S_i S_j^+= \begin{cases} \mathbf{J}(i, j) S_{i+j} &\mbox{if } i+j \neq q-1, \\ ({-}1)^{i+1}S_0 + ({-}1)^iq \begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix} &\mbox{if } i+j = q-1. \end{cases} \]Proof. This follows from Lemma 2.3 and the identities 
$S_i = \big (\begin {smallmatrix} 0 & 1 \\ 1 & 0 \end {smallmatrix}\big ) S_i^+$ for all 
$0 \leq i \leq q-1$.
Now we extend Lemma 2.3 to more general products. The next proposition will be applied together with Lemma 2.9, and the reduction mod 
$p$ of 
$\beta$ will turn out to be closely related to Breuil's constants.
Proposition 2.5 Let 
$k \geq 2$ and suppose that 
$0< i_1,\ldots , i_k < q-1$ are integers such that 
$q-1$ divides 
$\mathcal {I}_t = \sum _{m=1}^t i_m$ if and only if 
$t=k$. Then there exist nonzero 
$\alpha , \beta \in W(k_L)$ such that
\begin{gather*} S^+_{i_1} S^+_{i_2} \cdots S^+_{i_k} = \alpha S_0^++ \beta \begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix},\\ v_p(\beta) = \frac{1}{p-1} \bigg(\sum_{\substack{1 \leq t \leq k \\ 0 \leq j \leq f-1}}p - 1 - i_{t, j}\bigg),\\ v_p(\alpha) = v_p(\beta) - f, \end{gather*}
the leading term of 
$\beta$ is the Teichmüller lift of
\begin{equation} c(\beta) = ({-}1)^{v_p(\beta) + (k-2)(f-1)} \prod_{\substack{1 \leq t \leq k\\0 \leq j \leq f-1}} i_{t, j}! \end{equation}
and the leading term of 
$\alpha$ is the Teichmüller lift of 
$-c(\beta )$.
Proof. First we record a lemma.
Lemma 2.6 For all 
$j$, the sum 
$\mathcal {I}_{k-1, j} + i_{k, j}$ is equal to 
$p-1$.
Proof. By definition, 
$\mathcal {I}_{k-1} = \sum _{j=0}^{f-1} \mathcal {I}_{k-1, j}p^j + Q(q-1)$ for some 
$Q \geq 0$. We are assuming that 
$q-1$ divides 
$\mathcal {I}_{k-1} + i_{k}$, hence 
$i_k + \sum _{j=0}^{f-1} \mathcal {I}_{k-1, j}p^j$ is divisible by 
$q-1$. However, both summands are contained in 
$[1, q-2]$, hence their sum must be equal to 
$q-1$. The lemma follows because 
$q -1 = \sum _{j=0}^{f-1} (p-1)p^j$ and 
$i_{k, j}, \mathcal {I}_{k, j} \in [0, p-2]$ for all 
$j$.
Now we induct on 
$k$. When 
$k =2$, Lemma 2.3 states
\begin{equation} S_{i_1}^+S_{i_2}^+= ({-}1)^{i_2+1}S_0^++ ({-}1)^{i_2}q \begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix}. \end{equation}
By Lemma 2.6 we have 
$i_{1, j} + i_{2, j} = p-1$, which implies the claim on 
$v_p(\alpha )$ and 
$v_p(\beta )$. For the rest, we recall the identity 
$x!(p-1-x)! \equiv (-1)^{x+1} \mod p$, valid for integers 
$x \in [0, p-2]$. It implies that the product term in (1) is 
$(-1)^{i_{2, 0} + \cdots + i_{2, j-1} + f}$, which equals 
$(-1)^{i_2 + f}$ because 
$p$ is odd, and we are done because 
$k = 2$ is even.
Assuming the proposition for 
$k$, we apply Lemma 2.3 and we deduce
\[ S^+_{i_1} S^+_{i_2} \cdots S^+_{i_{k+1}} = \mathbf{J}(i_1, i_2) S^+_{\mathcal{I}_2}S^+_{i_3}\cdots S^+_{i_{k+1}}. \]
The inductive assumption applies to the product at the right-hand side (where 
$\mathcal {I}_2$ is identified with its unique representative in 
$[1, q-2]$), and because 
$\mathbf {J}(i_1, i_2) \ne 0$ we see that the existence of 
$\alpha , \beta$ already follows by induction.
To compute their valuation and leading term, we apply Theorem 2.1 and write
\[ v_p\,\mathbf{J}(i_1, i_2) = \frac 1 {p-1} \bigg(\sum_{j=0}^{f-1} p-1-i_{1, j} - i_{2, j} + \mathcal{I}_{2, j}\bigg) \]
and the leading term of 
$\mathbf {J}(i_1, i_2)$ as
\[ ({-}1)^{f-1 + v_p \mathbf{J}(i_1, i_2)} \frac{\prod_j i_{1, j}!i_{2, j}!}{\prod_j \mathcal{I}_{2, j}!}. \]The claim follows by induction.
We now study the action of these 
$S^+$-operators on 
$\mathcal {O}_E[\mathrm {GL}_2(k_L)]$-modules with the goal of proving analogous results involving 
$S$-operators. They will apply with less generally than Lemma 2.3, in that we will not obtain an identity in the group algebra 
$\mathcal {O}_E[\mathrm {GL}_2(k_L)]$. In the following section, we introduce the special case of 
$\mathcal {O}_E[\mathrm {GL}_2(k_L)]$-modules coming from 
$\mathcal {O}_E$-lattices in tame types.
Lemma 2.7 Let 
$v$ be an 
$H$-eigenvector in an 
$\mathcal {O}_E[\mathrm {GL}_2(k_L)]$-module, of eigencharacter
\[ \chi: \begin{pmatrix} a & 0 \\ 0 & d \end{pmatrix} \mapsto [a]^r \eta(ad). \]
Then 
$S_i v$ has eigencharacter 
$\chi ^s \alpha ^{-i} = \chi \alpha ^{-r-i}$ and 
$S_i^+ v$ has eigencharacter 
$\chi \alpha ^{i}$.
Proof. This follows from an explicit computation based on the equalities
\begin{gather*} \begin{pmatrix} a & 0 \\ 0 & d \end{pmatrix}\begin{pmatrix} \lambda & 1 \\ 1 & 0 \end{pmatrix} = \begin{pmatrix} ad^{{-}1}\lambda & 1 \\ 1 & 0 \end{pmatrix}\begin{pmatrix} d & 0 \\ 0 & a \end{pmatrix},\\ \begin{pmatrix} a & 0 \\ 0 & d \end{pmatrix}\begin{pmatrix} 1 & 0 \\ \lambda & 1 \end{pmatrix} = \begin{pmatrix} 1 & 0 \\ a^{{-}1}d\lambda & 1 \end{pmatrix}\begin{pmatrix} a & 0 \\ 0 & d \end{pmatrix}. \end{gather*} If 
$v$ is as in Lemma 2.7, it follows that 
$S_i S_j v$ has 
$H$-eigencharacter 
$\chi ^s \alpha ^{-(i-j-r)}$. The following lemma is proved in [Reference Breuil and DiamondBD14, Théorème 2.5.2].
Lemma 2.8 Let 
$v$ be an 
$I$-eigenvector in an 
$\mathcal {O}_E[\mathrm {GL}_2(k_L)]$-module such that 
$I$ acts on 
$v$ through the 
$H$-eigencharacter 
$\chi : \big (\begin {smallmatrix} a & 0 \\ 0 & d \end {smallmatrix}\big ) \mapsto [a]^r\eta (ad)$. Assume that 
$i \not = 0$ and that 
$q-1$ does not divide neither 
$i-j-r$ nor 
$i-j$. Then
\[ S_i S_j v = \eta({-}1)\mathbf{J}_0(i, -j - r) S_{i-j-r} v. \]Proof. Throughout this proof, we use the convention that 
$[0]^n = 0$ for all 
$n \in \mathbf {Z}$. First, we note that for 
$s, t \in \mathbf {F}_q$ we have
\[ \begin{pmatrix} s & 1 \\ 1 & 0 \end{pmatrix}\begin{pmatrix} t & 1 \\ 1 & 0 \end{pmatrix} = \begin{cases} \begin{pmatrix} 1 & s \\ 0 & 1 \end{pmatrix} &\mbox{if } t=0, \\ \begin{pmatrix} s+t^{{-}1} & 1 \\ 1 & 0 \end{pmatrix}\begin{pmatrix} t & 1 \\ 0 & -t^{{-}1} \end{pmatrix} &\mbox{if } t \not = 0. \end{cases} \]
Assuming in addition that 
$j \not = 0$, we compute 
$S_i S_j v$ as follows:
\begin{align*} S_i S_j v &= \sum_{s, t \in \mathbf{F}_q^\times}[s]^i[t]^{{-}j}\begin{pmatrix} s+t & 1 \\ 1 & 0 \end{pmatrix} \begin{pmatrix} t^{{-}1} & 1 \\ 0 & -t \end{pmatrix}v = \eta({-}1) \sum_{s, t \in \mathbf{F}_q^\times}[s]^i[t]^{{-}j-r} \begin{pmatrix} s+t & 1 \\ 1 & 0 \end{pmatrix}v\\ &= \eta({-}1)\sum_{s \in \mathbf{F}_q^\times} \sum_{x \in \mathbf{F}_q \backslash \{s \}}[s]^i[x-s]^{{-}j-r} \begin{pmatrix} x & 1 \\ 1 & 0 \end{pmatrix} v = \eta({-}1)\sum_{s, x \in \mathbf{F}_q}[s]^i[x-s]^{{-}j-r}\begin{pmatrix} x & 1 \\ 1 & 0 \end{pmatrix} v \\ &= \eta({-}1)\sum_{x \in \mathbf{F}_q^\times}[x]^{i-j-r}\bigg(\sum_{s \in \mathbf{F}_q} [x^{{-}1}s]^i[1-x^{{-}1}s]^{{-}j-r} \bigg) \begin{pmatrix} x & 1 \\ 1 & 0 \end{pmatrix}v \\ &\quad + \eta({-}1)({-}1)^{{-}j-r}\sum_{s \in \mathbf{F}_q}[s]^{i-j-r}\begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix}v. \end{align*}
As 
$q-1$ does not divide 
$i-j-r$, the sum 
$\sum _{s \in \mathbf {F}_q}[s]^{i-j-r} = 0$ and
\[ S_i S_j v = \eta({-}1) \mathbf{J}_0(i, -j-r) S_{i-j-r} v. \]
When 
$j = 0$, we have an extra term
\[ S_i S_0 v = \sum_{s, t \in \mathbf{F}_q^\times}[s]^i\begin{pmatrix} s+t & 1 \\ 1 & 0 \end{pmatrix}\begin{pmatrix} t^{{-}1} & 1 \\ 0 & -t \end{pmatrix}v + \sum_{s \in \mathbf{F}_q} [s]^i \begin{pmatrix} 1 & s \\ 0 & 1 \end{pmatrix}v \]
but the second sum vanishes, because 
$I_1$ fixes 
$v$ and 
$q-1$ does not divide 
$i-j = i$.
To explain the hypothesis in the above lemma, we remark that we apply it to an 
$I$-eigenvector 
$v$ with eigencharacter 
$\chi$ in the principal series type 
$\mathrm {Ind}_I^{\mathrm {GL}_2(\mathcal {O}_L)} \chi$. Then the assumptions of the lemma are satisfied when the 
$H$-eigencharacter of 
$S_i S_j v$ has multiplicity one in the type, because the only characters with multiplicity two are 
$\chi ^s$ and 
$\chi = \chi ^s \alpha ^r$ (see Lemma 2.10), and 
$S_i S_j v$ has eigencharacter 
$\chi ^s \alpha ^{-(i-j-r)}$.
2.2 Lattices in tame $\mathrm {GL}_2(L)$-types
We only consider nonscalar tame 
$K$-types for 
$\mathrm {GL}_2(L)$. By the tameness assumption, we can also consider these types as 
$\mathrm {GL}_2(k_L)$-modules. Note that these types satisfy mod 
$p$ multiplicity one. The symbol 
$\theta$ will usually denote an 
$\mathcal {O}_E$-lattice in a tame type 
$\theta [1/p]$.
We are going to study the action of 
$S$-operators on tame types, recalling and generalizing results of [Reference BreuilBre14, Reference Breuil and DiamondBD14]. If 
$\theta$ is an 
$\mathcal {O}_E$-lattice in a tame type, then the restriction 
$\theta |_H$ splits into a direct sum of 
$H$-eigenspaces (because the order of the diagonal torus 
$H$ is coprime to 
$p$), and we begin with some general results about the action of 
$S$-operators on 
$H$-eigenvectors.
Lemma 2.9 Let 
$\theta$ be an 
$\mathcal {O}_E$-lattice in a tame type for 
$\mathrm {GL}_2(L)$ over 
$E$. If 
$\chi : H \to \mathcal {O}_E^\times$ occurs in 
$\theta$ with multiplicity one and 
$x_\chi$ is an 
$H$-eigenvector of eigenvalue 
$\chi$, then 
$S_0^+ x_{\chi } =~0$.
Proof. The restriction of 
$\theta [1/p]$ to the lower mirabolic subgroup is the induction of a regular character of the lower unipotent (if 
$\theta [1/p]$ is cuspidal) or the direct sum of such an induction and two characters (if 
$\theta [1/p]$ is principal series). As the center is acting by a character, this decomposition is stable under the lower-triangular Borel subgroup, and because 
$\chi$ has multiplicity one we see that 
$x_{\chi }$ is contained in the direct summand that is nontrivial under the mirabolic subgroup. Then the claim follows from the definition of 
$S_0^+$ because the restriction of this summand to the lower-triangular unipotent subgroup is a direct sum of nontrivial characters.
Now we construct two kinds of relations between 
$H$-eigenvectors in 
$\theta [1/p]$ depending on whether 
$\theta [1/p]$ is a principal series or cuspidal representation. The reason we need to treat these cases separately is that the action of 
$S_i^+$-operators is not transitive on a basis of 
$H$-eigencharacters of a principal series type.
Lemma 2.10 There are 
$q-1$ different 
$H$-eigencharacters in 
$\theta$. If 
$\theta [1/p]$ is cuspidal they all have multiplicity one, and if 
$\theta [1/p] \cong \mathrm {Ind}_{I}^{\mathrm {GL}_2(\mathcal {O}_L)}\chi$ they all have multiplicity one except 
$\chi$ and 
$\chi ^s$ which have each multiplicity two. There is an 
$\mathcal {O}_E$-basis 
$\{ x_\xi \}_\xi$ of 
$\theta$ such that:
(i)
$x_\xi$ is an eigenvector of $H$ with eigencharacter $\xi$;(ii) if
$\theta [1/p]$ is a cuspidal type and $\xi _1 \not = \xi _2$, then there exist an integer $0< i(\xi _1, \xi _2)^+ < q-1$ and a scalar $\alpha _{\xi _1, \xi _2}^+ \in E^\times$ such that
\[ x_{\xi_2} = \alpha_{\xi_1, \xi_2}^+ S_{i(\xi_1, \xi_2)^+}^+ x_{\xi_1}; \](iii) if
$\theta [1/p]$ is a principal series type, $\xi _1 \not = \xi _2$, and either $\xi _2$ has multiplicity one or both $x_{\xi _1}, x_{\xi _2}$ are $I_1$-fixed, then there exist an integer $0\leq i(\xi _1, \xi _2) \leq q-1$ and a scalar $\alpha _{\xi _1, \xi _2} \in E^\times$ such that
\[ x_{\xi_2} = \alpha_{\xi_1, \xi_2} S_{i(\xi_1, \xi_2)} x_{\xi_1}. \]
Proof. In the cuspidal case, begin with an arbitrary 
$H$-eigenvector 
$x_\xi$. The span of the vectors 
$\big (\begin {smallmatrix} 1 & 0 \\ \lambda & 1 \end {smallmatrix}\big )x_\xi$ for 
$\lambda \in \mathbf {F}_q$ is stable under 
$H$ and under the lower unipotent subgroup, hence under the lower mirabolic subgroup of 
$\mathrm {GL}_2(k_L)$. As 
$\theta$ is a lattice in a cuspidal type, the proof of Lemma 2.9 shows that these vectors span 
$\theta [1/p]$ (because the induction to the mirabolic of a regular character is irreducible). By Lemma 2.9, 
$S_0^+$ acts by 
$0$ on 
$\theta [1/p]$. Hence, the vectors 
$\big (\begin {smallmatrix} 1 & 0 \\ \lambda & 1 \end {smallmatrix}\big ) x_{\xi }$ for 
$\lambda \in \mathbf {F}_q^\times$ already span 
$\theta [1/p]$, and because 
$\dim _E \theta [1/p] = q-1$ they form a basis of 
$\theta [1/p]$ over 
$E$. Applying a Vandermonde matrix to this basis, we find that the 
$S_i^+ x_\xi$ for 
$1 \leq i \leq q-1$ form a basis of 
$\theta [1/p]$ over 
$E$. The result follows by scaling because 
$\theta$ has a basis of 
$H$-eigenvectors. We remark that 
$S_{q-1}^+ x_{\xi } = -x_{\xi }$, and then the existence of 
$\alpha _{\xi _1, \xi _2}^+$ follows from Lemma 2.3.
In the principal series case, let 
$\varphi$ be a generator of an 
$I_1$-stable 
$\mathcal {O}_E$-line in 
$\theta$ and write the 
$H$-character of 
$\varphi$ as 
$\big (\begin {smallmatrix} a & 0 \\ 0 & d \end {smallmatrix}\big ) \mapsto a^r\eta (ad)$. By the discussion in [Reference BreuilBre14, § 2] certain scalar multiples of the 
$S_i \varphi$ for 
$0 \leq i \leq q-1$ form a basis of 
$\theta$ over 
$\mathcal {O}_E$ together with 
$\varphi$. Let 
$\{x_\xi \}$ be this basis, so that for each 
$\xi$ either 
$x_\xi = \varphi$ or there exists 
$i(\xi )$ such that 
$x_{\xi }$ is a multiple of 
$S_{i(\xi )}\varphi$.
We now consider the claim in part (iii). If 
$x_{\xi _1} = \varphi$, then the claim holds by construction. Now note that if 
$\xi _2$ has multiplicity one, then we have an equation
\[ S_{i(\xi_2) + i(\xi_1) + r} S_{i(\xi_1)} \varphi = \eta({-}1)\mathbf{J}_0(i(\xi_2) + i(\xi_1) + r, -i(\xi_1)-r) S_{i(\xi_2)} \varphi \]
by Lemma 2.8, which applies because 
$q-1$ does not divide any of 
$i(\xi _2)$ and 
$i(\xi _2) + r$. (If it did, then 
$x_{\xi _2}$ would have eigencharacter 
$\chi ^s\alpha ^{-i(\xi _2)} = \chi ^s \text { or } \chi$, respectively, contradicting multiplicity one.) In the case 
$i(\xi _2) + i(\xi _1) + r$ is divisible by 
$q-1$, we take 
$i(\xi _1,\xi _2) = q-1$.
Finally, in the case where both 
$x_{\xi _1}$ and 
$x_{\xi _2}$ are fixed by 
$I_1$ the claim follows from the fact that 
$S_0$ sends a nonzero 
$I_1$-fixed vector to a nonzero 
$I_1$-fixed vector of the opposite 
$H$-eigencharacter.
Lemma 2.11 Let 
$\{x_\xi \}$ be a basis as in Lemma 2.10. If 
$\xi _1 \not = \xi _2$, and 
$\xi _2$ appears with multiplicity one in 
$\theta$, then one can find a relation
\[ x_{\xi_2} = \alpha_{\xi_1, \xi_2}^+ S_{i(\xi_1, \xi_2)^+}^+ x_{\xi_1} \]
with 
$\alpha _{\xi _1, \xi _2}^+ \in E^\times$ and 
$0< i(\xi _1, \xi _2)^+ < q-1$, even if 
$\theta [1/p]$ is a principal series type.
Proof. By Lemma 2.10, we only need to consider the case when 
$\theta [1/p]$ is a principal series representation. By Lemma 2.7, such an integer 
$i(\xi _1, \xi _2)^+$ is uniquely determined because 
$\xi _1 \not = \xi _2$ and the operator 
$S_j^+$ multiplies the character by 
$\alpha ^{j}$. By the multiplicity one assumption, it suffices to prove that 
$S_{i(\xi _1, \xi _2)^+}^+ x_{\xi _1}$ is nonzero. For this, the matrix identity
\[ \begin{pmatrix} \lambda & 1 \\ 1 & 0 \end{pmatrix}\begin{pmatrix} 1 & 0 \\ \mu & 1 \end{pmatrix} = \begin{pmatrix} \lambda+\mu & 1 \\ 1 & 0 \end{pmatrix} \]and the same computation as Lemma 2.3 prove that there is a relation
\[ S_j S_{i(\xi_1, \xi_2)^+}^+= \mathbf{J}(j, i(\xi_1, \xi_2)^+) S_{i(\xi_1, \xi_2)^++j} \]
whenever 
$0 < j < q-1$ and 
$i(\xi _1, \xi _2)^+ + j \not = q-1$.
The type 
$\theta [1/p]$ contains 
$q-3$ eigencharacters with multiplicity one. If 
$S_{i(\xi _1, \xi _2)^+}x_{\xi _1} \not = 0$, then we left multiply by 
$\big (\begin {smallmatrix} 0 & 1 \\ 1 & 0 \end {smallmatrix}\big )$ and deduce the claim that 
$S_{i(\xi _1, \xi _2)^+}^+ x_{\xi _1}$ is nonzero. Otherwise, because 
$p \geq 5$, we can take a 
$0< j< q-1$ not equal to 
$q-1-i(\xi _1, \xi _2)^+$ so that the (unambiguously defined) vector 
$S_{i(\xi _1, \xi _2)^++j}x_{\xi _1}$ does not vanish, by Lemma 2.10. The claim follows as the Jacobi sum 
$\mathbf {J}(j, i(\xi _1, \xi _2)^+)$ does not vanish.
We say that an inclusion of 
$\mathcal {O}_E$-lattices 
$\theta ^1 \to \theta ^2$ in a tame type 
$\theta [1/p]$ is saturated if the induced map 
$\theta ^1 \to \bar {\theta }^2$ is nonzero. By [Reference Emerton, Gee and SavittEGS15, Lemma 4.1.1], if 
$\sigma \in \mathrm {JH}(\bar {\theta })$ then there exists a unique homothety class of lattices in 
$\theta [1/p]$ with irreducible cosocle 
$\sigma$. We denote a representative for this class by 
$\theta ^\sigma$. We often choose representatives so that certain inclusions are saturated. If we fix two representatives 
$\theta ^{\sigma _1}$ and 
$\theta ^{\sigma _2}$ and a saturated inclusion 
$\theta ^{\sigma _1} \subset \theta ^{\sigma _2}$ as before, then there exists a unique 
$n \in \mathbf {N}$ such that 
$p^n \theta ^{\sigma _2} \subset \theta ^{\sigma _1}$ and the inclusion is saturated (because 
$\theta [1/p]$ and, thus, the 
$\mathcal {O}_E$-lattices 
$\theta ^{\sigma _1}$ and 
$\theta ^{\sigma _2}$ are defined over 
$W(k_E)[1/p]$ and 
$W(k_E)$, respectively). When we speak of a saturated inclusion 
$\theta ^{\sigma _2} \to \theta ^{\sigma _1}$, we mean the map
\[ \theta^{\sigma_2} \xrightarrow{p^n} p^n\theta^{\sigma_2} \subset \theta^{\sigma_1}. \]Definition 2.12 Let 
$\chi$ be an 
$\mathcal {O}_E^\times$-valued character of 
$H$. Let 
$\theta (\chi )$ be the 
$\mathcal {O}_E$-lattice 
$\mathrm {Ind}_I^K \chi$ in a principal series tame type, realized as a space of functions on 
$K$. Let 
$\varphi ^\chi$ be the unique element of 
$\theta (\chi )$ supported on 
$I$ such that 
$\varphi ^\chi (1) = 1$. Let 
$\sigma (\chi )$ be the (irreducible) cosocle of 
$\theta (\chi )$, and denote the image of 
$\varphi ^\chi$ in 
$\sigma (\chi )$ by 
$\varphi ^\chi$ as well. For a type 
$\theta$ with Jordan–Hölder factor isomorphic to 
$\sigma (\chi )$, we write 
$\theta ^\chi$ for short rather than 
$\theta ^{\sigma (\chi )}$.
Lemma 2.13 The representation
\[ \langle \mathcal{O}_E[\mathrm{GL}_2(k_L)] \cdot S_0\varphi^{\chi^s} \rangle \subset \theta(\chi^s) \]
is isomorphic to 
$\theta (\chi ^s)^\chi$, and the inclusion is saturated. If 
$\xi$ is neither 
$\chi$ nor 
$\chi ^s$, then both
\[ \langle \mathcal{O}_E[\mathrm{GL}_2(k_L)]\cdot S_{i(\chi^s,\xi)} \varphi^{\chi^s} \rangle,\quad \langle \mathcal{O}_E[\mathrm{GL}_2(k_L)]\cdot S^+_{i(\chi^s,\xi)^+} \varphi^{\chi^s} \rangle \subset \theta(\chi^s) \]
are isomorphic to 
$\theta (\chi ^s)^\xi$, and the inclusion is saturated.
Proof. For the first part, because 
$S_0 \varphi ^{\chi ^s}$ is an 
$I$-eigenvector with character 
$\chi$, it generates a sublattice isomorphic to 
$\theta (\chi )$. Moreover, 
$S_0 \varphi ^{\chi ^s}$ is nonzero in 
$\theta (\chi ^s)/\varpi _E$.
Let 
$\theta (\chi ^s)^\xi \subset \theta (\chi ^s)$ be a saturated inclusion. The image of 
$\theta (\chi ^s)^\xi$ in 
$\theta (\chi ^s)/\varpi _E$ is nonzero, and is characterized as being the only 
$k_E[\mathrm {GL}_2(k_L)]$-subrepresentation of 
$\bar {\theta }(\chi ^s)$ with irreducible cosocle isomorphic to 
$\sigma (\xi )$. Equivalently, the image is characterized as being the minimal subrepresentation of 
$\bar {\theta }(\chi ^s)$ containing 
$\sigma (\xi )$ as a Jordan–Hölder factor, or as the minimal subrepresentation containing 
$\xi$ as an 
$H$-eigencharacter (using that the 
$H$-character 
$\xi$ has multiplicity one in 
$\theta (\chi ^s)$ by Lemma 2.10). In particular, this image is generated over 
$k_E[\mathrm {GL}_2(k_L)]$ by the image of both 
$S_{i(\chi ^s,\xi )}\varphi ^{\chi ^s}$ and 
$S^+_{i(\chi ^s,\xi )^+} \varphi ^{\chi ^s}$ by Lemmas 2.10 and 2.11. As 
$H$ has order coprime to 
$p$, there exists an 
$H$-eigenvectors in 
$\theta (\chi ^s)^\xi$ lifting the images of 
$S_{i(\chi ^s,\xi )}\varphi ^{\chi ^s}$ and 
$S^+_{i(\chi ^s,\xi )^+} \varphi ^{\chi ^s}$. Since 
$\xi$ has multiplicity one, these lifts are multiples of 
$S_{i(\chi ^s,\xi )}\varphi ^{\chi ^s}$ and 
$S^+_{i(\chi ^s,\xi )^+} \varphi ^{\chi ^s}$, which therefore are contained in 
$\theta (\chi ^s)^\xi$. We conclude by noting that both 
$S_{i(\chi ^s,\xi )}\varphi ^{\chi ^s}$ and 
$S^+_{i(\chi ^s,\xi )^+} \varphi ^{\chi ^s}$ generate 
$\theta (\chi ^s)^\xi$ over 
$\mathcal {O}_E[\mathrm {GL}_2(k_L)]$ because their images each generate 
$\bar {\theta }(\chi ^s)^\xi$ over 
$k_E[\mathrm {GL}_2(k_L)]$.
We end this section by recalling a relation between 
$S_0$, 
$\Pi = \big (\begin {smallmatrix} 0 & 1 \\ p & 0 \end {smallmatrix}\big )$, and the Hecke operator 
$U_p$ on a principal series type.
Lemma 2.14 Let 
$\pi = \mathrm {Ind}_{B(L)}^{\mathrm {GL}_2(L)}(\chi _1| \cdot | \otimes \chi _2)$ for tamely ramified characters 
$\chi _i: L^\times \to E^\times$ (this is a smooth unnormalized parabolic induction) with different restrictions to 
$\mathcal {O}_L^\times$. Let 
$\varphi \in \pi$ be an 
$I_1$-fixed vector with 
$H$-eigencharacter 
$\chi _1|_{\mathcal {O}_L^\times } \otimes \chi _2|_{\mathcal {O}_L^\times }$. Then
\[ \Pi \varphi = q^{{-}1} \chi_1(p) S_0 \varphi. \]Proof. Compare with [Reference Breuil and DiamondBD14, Théorème 2.5.2]. The absolute value 
$|\cdot |$ is normalized for 
$L$, so that 
$|p| = q^{-1}$. By the Iwasawa decomposition, the space of 
$I_1$-fixed vectors in 
$\pi$ is two-dimensional and contains two different 
$H$-eigencharacters. An explicit computation gives
\[ S_0 \Pi = U_p = \sum_{\lambda \in \mathbf{F}_q} \begin{pmatrix} p & [\lambda] \\ 0 & 1 \end{pmatrix}. \]
This preserves the 
$H$-eigencharacter, hence 
$U_p \varphi = \alpha \varphi$ for some scalar 
$\alpha$. Evaluating the function 
$\alpha \varphi$ at the identity of 
$\mathrm {GL}_2(L)$, we obtain 
$\alpha \varphi (1) = q \cdot \chi _1(p)|p| \varphi (1)$, hence 
$\alpha = \chi _1(p)$ because 
$\varphi (1) \not = 0$ as 
$\varphi$ is supported in 
$B I_1$. Now 
$\pi$ is isomorphic to 
$\mathrm {Ind}_{B(L)}^{\mathrm {GL}_2(L)}(\chi _2| \cdot | \otimes \chi _1)$, as it is the normalized induction that is Weyl-invariant, and 
$\pi$ is the normalized induction of 
$\chi _1|\cdot |^{1/2} \otimes \chi _2|\cdot |^{1/2}$. Hence, it is also true that 
$U_p \Pi \varphi = \chi _2(p) \Pi \varphi$. Substituting 
$S_0 \Pi$ for 
$U_p$, we obtain
\[ q^{{-}1}\chi_1(p)\chi_2(p) S_0 \varphi = \chi_2(p) \Pi \varphi \]and the claim follows.
2.3 Tame types and inertial local Langlands
In this section, we parametrize tame 
$\mathrm {GL}_n(L)$ types and describe the generic tame inertial local Langlands correspondence.
2.3.1 Deligne–Lusztig induction
We recall a description of Deligne–Lusztig induction for the group 
$\mathbf {G}_0 = \mathrm {Res}_{k_L / \mathbf {F}_p} \mathrm {GL}_n$. We eventually take 
$n$ to be 
$2$, but we take 
$n$ to be general in this section and in parts of § 3. Let 
$F$ be the 
$p$th power, relative Frobenius endomorphism of 
$\mathbf {G}_0$ (relative and absolute Frobenius morphisms coincide here). The induced endomorphism of 
$\mathbf {G}_0(\bar {\mathbf {F}}_p)$ coincides with the endomorphism coming from 
$p$-power Frobenius endomorphism on 
$\bar {\mathbf {F}}_p$. Let 
$\mathbf {G}$ be 
$\mathbf {G}_0 \times _{\mathbf {F}_p} \bar {\mathbf {F}}_p$. Under the isomorphism
\begin{equation} \mathbf{G} \cong \prod_{\iota:k_L \hookrightarrow \bar{\mathbf{F}}_p} \mathrm{GL}_{n, \bar{\mathbf{F}}_p} \cong\prod_{i=0}^{f-1} \mathrm{GL}_{n, \bar{\mathbf{F}}_p} \end{equation}
given by the bijection 
$i \leftrightarrow \iota _{-i}$ for 
$i\in \mathbf {Z}/f$, the Frobenius endomorphism 
$F$ of the group acts on points by 
$(A_0, \ldots , A_{f-1}) \mapsto (A_{f-1}^{(p)}, A_0^{(p)}, \ldots , A_{f-2}^{(p)})$.
There is a diagonal torus 
$\mathbf {T}_0$ isomorphic to 
$\mathrm {Res}_{k_L / \mathbf {F}_p}(\mathbf {G}_m)^n$, and an upper-triangular Borel subgroup. The Weyl group 
$W = W(\mathbf {G}_0, \mathbf {T}_0)$ is isomorphic to 
$S_n^{\times f}$, and the 
$F$-conjugacy classes in 
$W$ are represented by 
$s= (s, 1, \ldots , 1)$ where 
$s$ represents conjugacy classes of the symmetric group 
$S_n$. Hence, the conjugacy classes of rational maximal tori are represented by 
$\mathrm {Res}_{k_L/\mathbf {F}_p}(\mathbf {T}_0')$, where 
$\mathbf {T}_0'$ runs over conjugacy classes of rational maximal tori of 
${\mathrm {GL}_n}_{/k_L}$. We write 
$\mathbf {T} = \mathbf {T}_0 \times _{\mathbf {F}_p} \bar {\mathbf {F}}_p$ and we identify the character lattice 
$X(\mathbf {T})$ with 
$(\mathbf {Z}^n)^{\oplus f}$ in the usual way using (3). The group 
$\mathrm {Gal}(\bar {\mathbf {F}}_p/\mathbf {F}_p)$ acts on the character lattice of 
$\mathbf {T}$ and we denote by 
$\pi$ the action of the arithmetic Frobenius element 
$x \mapsto x^p$. There is also an action of 
$F$ by 
$F(\chi ) = \chi \circ F$. One has the equality 
$(\pi \chi )(F(t)) = \chi (t)^p$, hence 
$\pi (\chi _0, \ldots , \chi _{f-1})= (\chi _{f-1}, \chi _0, \ldots , \chi _{f-1})$. Let 
$\Lambda _R \subset X(\mathbf {T})$ be the root lattice.
We let 
$\mathbf {G}_0^{\mathrm {der}} \subset \mathbf {G}_0$ be the derived subgroup 
$\mathrm {Res}_{k_L / \mathbf {F}_p} \mathrm {SL}_n$. We similarly define 
$\mathbf {G}^{\mathrm {der}}$, 
$\mathbf {T}_0^{\mathrm {der}}$, and 
$\mathbf {T}^{\mathrm {der}}$. We write 
$\Lambda _W$ for 
$X(\mathbf {T}^{\mathrm {der}})$. The root lattice 
$\Lambda _R$ in 
$X(\mathbf {T})$ is canonically identified with the root lattice in 
$\Lambda _W$. Let 
$X^0(\mathbf {T})$ denote the kernel of the natural restriction map 
$X(\mathbf {T}) \rightarrow X(\mathbf {T}^{\mathrm {der}}) = \Lambda _W$. Under the identification 
$X(\mathbf {T}) \cong (\mathbf {Z}^n)^{\oplus f}$, 
$X^0(\mathbf {T})$ is identified with 
$\mathbf {Z}^{\oplus f}$ where 
$\mathbf {Z} \hookrightarrow \mathbf {Z}^n$ is embedded diagonally.
If 
$\mu$ is a character of 
$\mathbf {T}$ and 
$w$ is an element of 
$W$, then there is an associated virtual representation 
$R_w(\mu )$ of 
$\mathrm {GL}_n(k_L)$ over 
$E$. There is an action of 
$W \ltimes X(\mathbf {T})$ (the semidirect product defined by the action 
$w \cdot \mu = F(w)\mu$ for 
$w\in W$ and 
$\mu \in X(\mathbf {T})$) on 
$W \times X(\mathbf {T})$ by
\[ (w_1, \mu_1) \cdot (w_2, \mu_2) = (w_1 w_2 F(w_1)^{{-}1}, w_1(\mu_2) + (F - w_1 w_2 F(w_1)^{{-}1})\mu_1). \]
The fibers of the map 
$(w, \mu ) \mapsto R_w(\mu )$ are precisely the orbits of this action, by [Reference HerzigHer09, Lemma 4.2].
Rather than giving the definition of 
$R_w(\mu )$ in full, we spell it out in the main case of interest when 
$n=2$. Let 
$1, s \in W$ be the identity and the element 
$s = (s, 1, \ldots , 1)$ (nontrivial in the first component). We identify 
$\mu$ with an 
$f$-tuple 
$(\mu _1, \mu _2) = ((\mu _{1, 0}, \mu _{2, 0}), \ldots , (\mu _{1, f-1}, \mu _{2, f-1})) \in (\mathbf {Z}^2)^{\oplus f}$. Then
\[ R_1(\mu) \cong \mathrm{Ind}_{B_2(k_L)}^{\mathrm{GL}_2(k_L)}\Big([-]^{\sum_{t=0}^{f-1}\mu_{1, t}p^t} \oplus [-]^{\sum_{t=0}^{f-1}\mu_{2, t}p^t}\Big) \]
whereas 
$R_s(\mu )$ is the negative of the Deligne–Lusztig induction of the character
\[ [-]^{\sum_{t=0}^{f-1}\mu_{1, t}p^t + p^f\sum_{t=0}^{f-1}\mu_{2, t}p^{t}} \]
of 
$k_{L, 2}^\times$ to 
$\mathrm {GL}_2(k_L)$.
2.3.2 Tame inertial types
Let 
$d$ be a natural number, and set 
$e'$ to be 
$p^{df}-1$. Let 
$\pi ' \in \bar{E}$ be a root of 
$u^{e'}+p$. We define the character
\begin{align*} [\omega_{df}]: I_{\mathbf{Q}_p} &\rightarrow \overline{\mathcal{O}}_E^\times \\ g &\mapsto \frac{g(\pi')}{\pi'}. \end{align*}
This definition does not depend on the choice of root 
$\pi '$. We assume that 
$E$ is sufficiently large so that this character is valued in 
$\mathcal {O}_E^\times$, and let 
$\omega _{df}$ be its reduction modulo 
$\varpi _E$.
For 
$(w,\mu ) \in W\times X(\mathbf {T})$, there is an 
$s \in W$ of the form 
$(s,1,\ldots ,1)$ such that 
$R_w(\mu ) \cong R_s(\mathbf {a})$ for some 
$\mathbf {a} \in X(\mathbf {T})$, as explained in § 2.3.1. Let 
$d$ be the order of 
$s$ and suppose that 
$\mathbf {a}$ maps to 
$(a_{k,j})_{k,j}$, with 
$1\leq k\leq n$ and 
$0 \leq j\leq f-1$, under the identification 
$X(\mathbf {T}) \cong (\mathbf {Z}^n)^{\oplus f}$. Let
\[ \mathbf{a}_k = \sum_{i=0}^{f-1} a_{k,i}p^i. \]
Then we define 
$\tau (w,\mu ):I_L \rightarrow \mathrm {GL}_n(\mathcal {O}_E)$ to be
\[ \bigoplus_{k=1}^n [\omega_{df}]^{\sum_{i=0}^{d-1} \mathbf{a}_{s^i(k)}}p^{if}. \]
Note that this does not depend on the choice of 
$s$ and 
$\mathbf {a}$ above. Moreover, 
$\tau (w,\mu )$ is an inertial type for the Weil group 
$W_L$, that is, it admits an extension to 
$W_L$.
If 
$(w,\mu )$ is a good pair in the sense of [Reference Le, Le Hung and LevinLLL19, § 2.2], then 
$\tau (w,\mu ) \cong \tau (w',\mu ')$ implies that 
$R_w(\mu ) \cong R_{w'}(\mu ')$, and moreover this Deligne–Lusztig representation is irreducible (see [Reference Le, Le Hung and LevinLLL19, Proposition 2.2.4]). If 
$(w,\mu )$ is a good pair, we set 
$\sigma (\tau (w,\mu ))$ to be 
$R_w(\mu )$ and note that this gives a well-defined injective map from a subset of tame inertial types to tame types for 
$K$ or, equivalently, irreducible representations of 
$\mathrm {GL}_n(\mathbf {F}_q)$. This defines a tame inertial local Langlands (see [Reference Emerton, Gee and HerzigEGH13, Proposition 2.4.1(i)]).
Noting, for any tame inertial type 
$\tau$, the reduction 
$\bar {\tau }$ is a sum of characters whose Teichmüller lift is 
$\tau$, we let 
$V(\bar {\tau }(w,\mu ))$ be 
$R_w(\mu )$ for 
$(w,\mu )$ a good pair as above. The map 
$V$ is closely related to the maps 
$V$ and 
$V_\phi$ that appear in [Reference HerzigHer09, Reference Le, Le Hung and LevinLLL19] and [Reference Gee, Herzig and SavittGHS18], respectively, but we do not need this in the following.
Again, we make the above explicit in the case 
$n=2$. Let 
${\bar{\rho }}:G_L \rightarrow \mathrm {GL}_2(k_E)$ be a continuous Galois representation. There are two possibilities for the semisimplification of 
${\bar {\rho }}|_{I_L}$, namely
$$\begin{gather} \begin{pmatrix} \omega_f^{\sum_{t=0}^{f-1}\mu_{1, t}p^t} & 0 \\ 0 & \omega_f^{\sum_{t=0}^{f-1} \mu_{2, t}p^t} \end{pmatrix} \text{ in the reducible case} \end{gather}$$
$$\begin{gather}\begin{pmatrix} \omega_{2f}^{\sum_{t=0}^{f-1}\mu_{1, t}p^t + p^f\sum_{t=0}^{f-1} \mu_{2, t}p^t} & 0 \\ 0 & \omega_{2f}^{\sum_{t=0}^{f-1}\mu_{2, t}p^t+ p^f \sum_{t=0}^{f-1}\mu_{1, t}p^t} \end{pmatrix} \text{in the irreducible case}, \end{gather}$$
where 
$\mu _{k,t} \in \mathbf {Z}$ for each 
$k \in \{1, 2\}$ and 
$0 \leq t \leq f-1$. We let 
$s_{{\bar {\rho }}} \in S_2$ be 
$1$ in the first case and the nontrivial element in the second, and we write 
$\mu = (\mu _1, \mu _2)$. Then the definitions above specialize to 
$V({\bar {\rho }}^{\mathrm {ss}}|_{I_L}) = R_{s_{{\bar {\rho }}}}(\mu )$.
2.4 Serre weights
Recall that a Serre weight for 
$\mathrm {GL}_2(k_L)$ is (the isomorphism class of) an irreducible 
$\mathrm {GL}_2(k_L)$-representation over 
$k_E$, or equivalently an irreducible 
$\mathrm {GL}_2(\mathcal {O}_L)$-representation over 
$k_E$. Every Serre weight is isomorphic to a representation of the form
\begin{equation} (\eta \circ \det) \otimes \bigotimes_{j\in \mathbf{Z}/f} \mathrm{Sym}^{r_j} (k_L^2 \otimes_{\sigma_j,k_L} k_E), \end{equation}
where 
$0 \leq r_j \leq p-1$ for all 
$j\in \mathbf {Z}/f$ and 
$\eta : k_L^\times \rightarrow k_E^\times$ is a character. We will use the shorthand 
$(r_0,\ldots ,r_{f-1})\otimes \eta$ for the Serre weight in (6).
We also use the following description of Serre weights. We use the notation of § 2.3.1 with 
$n=2$. Then 
$\mathbf {G}_0(\mathbf {F}_p) = \mathrm {GL}_2(k_L)$. If 
$\mu \in X(\mathbf {T})$ is dominant and 
$0 \leq \langle \alpha ,\mu \rangle \leq p-1$ for all positive roots 
$\alpha$ of 
$\mathbf {G}$, then we let 
$L(\mu )$ be the (unique up to isomorphism) irreducible representation of 
$\mathbf {G}$ with highest weight 
$\mu$. Let 
$F(\mu )$ be the restriction of 
$L(\mu )$ to 
$\mathrm {GL}_2(k_L)$. Then every Serre weight is isomorphic to 
$F(\mu )$ for some 
$\mu$ as above. Moreover, 
$F(\mu )$ and 
$F(\lambda )$ are isomorphic if and only if 
$\mu -\lambda \in (F-1)X^0(\mathbf {T}) = (p-\pi )X^0(\mathbf {T})$ (see [Reference Gee, Herzig and SavittGHS18, Lemma 9.2.4]). If 
$\mu$ is 
$(\mu _{1,j},\mu _{2,j})_{j\in \mathbf {Z}/f}$, then 
$F(\mu )$ is isomorphic to the representation
\[ (\mu_{1,j} - \mu_{2,j})_{j\in \mathbf{Z}/f} \otimes \sigma_0^{\sum_{j = 0}^{f-1} p^j \mu_{2,j}} \]
in the notation introduced previously. (Recall that 
$\sigma _0: k_L \to k_E$ is our fixed choice of embedding.)
Extension graph. In this section, 
$n=2$ so that in the notation of § 2.3.1, 
$\mathbf {G}_0$ is the group 
$\mathrm {Res}_{k_L / \mathbf {F}_p} \mathrm {GL}_2$. Let 
$\mu \in X(\mathbf {T})$ be dominant such that 
$0 \leq \langle \alpha ,\mu \rangle \leq p-1$ for all positive roots 
$\alpha$ of 
$\mathbf {G}$. Recall that 
$\Lambda _W$ is 
$X(\mathbf {G}_0^{\mathrm {der}})$ where 
$\mathbf {G}_0^{\mathrm {der}}\subset \mathbf {G}_0$ is the subgroup 
$\mathrm {Res}_{k_L / \mathbf {F}_p} \mathrm {SL}_2$. Let 
$\Lambda _W^\mu \subset \Lambda _W$ be the subset
\[ \Lambda_W^\mu = \{\omega \in \Lambda_W: 0 \leq \langle \alpha, \mu + \omega' \rangle \leq p-1 \text{ for all positive roots~$\alpha$ of~$\mathbf{G}$}\} \]
where 
$\omega ' \in X(\mathbf {T})$ is any lift of 
$\omega$. We now give an equivalent definition, based on [Reference Le, Le Hung, Levin and MorraLLLM20, § 2], of an injective map
\[ \mathfrak{t}_\mu: \Lambda_W^\mu \rightarrow X(\mathbf{T})/(F-1)X^0(\mathbf{T}) \]from [Reference Le, Morra and SchraenLMS20, § 2].
We first define a map 
$\mathfrak {t}_\mu ': X(\mathbf {T}) \rightarrow X(\mathbf {T})/(F-1)X^0(\mathbf {T})$ as follows. Let 
$W_a$ and 
$\widetilde {W}$ be the affine Weyl group and the extended affine Weyl group of 
$\mathbf {G}$, respectively. The natural inclusion 
$X(\mathbf {T}) \rightarrow \widetilde {W}$ induces an isomorphism 
$X(\mathbf {T})/\Lambda _R \rightarrow \widetilde {W}/W_a$. Let 
$\Omega \subset \widetilde {W}$ be the stabilizer of the dominant base (
$1$-)alcove for 
$\mathbf {G}$. Then we have a splitting 
$\widetilde {W} = \Omega \ltimes W_a$ of the inclusion 
$W_a \subset \widetilde {W}$.
From these considerations, for 
$\omega '\in X(\mathbf {T})$ there exists a unique 
$\widetilde {w}'\in \Omega$ such that 
$\widetilde {w}'t_{\pi ^{-1}(\omega ')} \in W_a$. We then define 
$\mathfrak {t}_\mu '(\omega ')$ to be
\[ \widetilde{w}'\cdot (\mu- \eta + \omega') \pmod{(F-1)X^0(\mathbf{T})}, \]
where 
$\eta _j = (1, 0)_j$ and the action of 
$\widetilde {w}'$ is by the 
$p$-dot action. It is easy to check that 
$\mathfrak {t}_\mu '$ is well-defined and factors through the natural quotient map 
$X(\mathbf {T}) \rightarrow \Lambda _W$. Then 
$\mathfrak {t}_\mu$ is defined to be the restriction of this map to 
$\Lambda _W^\mu \subset \Lambda _W$. For 
$\omega \in \Lambda _W^\mu$, the isomorphism class of 
$F(\mathfrak {t}_\mu (\omega ))$ is well-defined.
Now let 
$\omega _j \in \Lambda _W$ be the fundamental dominant weight which is nonzero (only) in embedding 
$j$. For a subset 
$J\subset \mathbf {Z}/f$, let 
$\omega _J$ be 
$\sum _{j\in J} \omega _j$. There is a unique element 
$w_J \in W$ such that 
$w_Jt_{-\pi ^{-1}(\omega _J)}$ fixes the dominant base (
$1$-)alcove for 
$\mathbf {G}^{{\mathrm {der}}}$. Explicitly, 
$w_J \in W$ is the element that is nontrivial on exactly 
$\delta _\mathrm {red}(J)$. Let 
$\widetilde {w}_J$ be 
$w_Jt_{-\pi ^{-1}(\omega _J)}$. (Recall that 
$\pi ^{-1}$ is a left shift on 
$X(\mathbf {T})$.)
Proposition 2.15 Let 
$w \in W$ and 
$\nu \in \Lambda _R$ such that 
$-\eta + \nu + w\omega _J \in \Lambda _W^\mu$ for all subsets 
$J \subset \mathbf {Z}/f$. Then the set 
$\mathrm {JH}(\overline {R_w(\mu -\eta +\nu )})$ is given by
\[ \{F(\mathfrak{t}_\mu(-\eta + \nu+ w\omega_J))\}_{J\subset \mathbf{Z}/f} \]
(where, by an abuse of notation, 
$\eta$ denotes the image of 
$\eta$ in 
$\Lambda _W$).
Proof. This follows from [Reference HerzigHer09, § A, Theorem 3.4] applied to the 
$\mathbf {F}_p$-form 
$\mathrm {Res}_{k_L / \mathbf {F}_p} \mathrm {GL}_2$ of 
$\mathrm {GL}_2^{\times f}$. Note that in the notation of [Reference HerzigHer09], 
$\gamma '_{w_1,w_2}$ is the Kronecker symbol 
$\delta _{w_1,w_2}$ for 
$w_1$ and 
$w_2\in W$. Moreover, the highest weight of each character that appears in the formula for 
$X_w'(1,\mu -\eta +\nu )$ is in the unique 
$p$-restricted dominant (
$p$-)alcove, and is thus the highest weight of a Serre weight. The proposition then follows from checking that the 
$2^f$ highest weights coincide with 
$\mathfrak {t}_\mu (-\eta +\nu +w\omega _J)$ for 
$J\subset \mathbf {Z}/f$ (modulo 
$(p-\pi )X^0(\mathbf {T})$).
Indeed, let 
$s\in W$ and 
$\epsilon \in X(\mathbf {T})$ be such that 
$\widetilde {s}=st_\epsilon$ is in 
$\Omega$. Let 
$\widetilde {w}_h$ be 
$w_0 t_{-\eta }$. Then 
$\widetilde {w}_h^{-1} \widetilde {s} t_{\pi ^{-1}(-\eta +\nu -w\pi \epsilon )} \in W_a$ so that
\begin{equation} \mathfrak{t}_\mu(-\eta+\nu-w\pi\epsilon) = \widetilde{w}_h^{{-}1}\widetilde{s}\cdot(\mu-2\eta+\nu-w\pi\epsilon) \pmod{(p-\pi)X^0(\mathbf{T})}. \end{equation}
If we let 
$w_1$ be 
$w_0 s$, then 
$\epsilon '_{w_0w_1}$ can be taken to be 
$\epsilon$ and 
$\rho '_{w_1}$ can be taken so that 
$\widetilde {w}_h^{-1} \widetilde {s} = t_{\rho '_{w_1}}w_1$ in the notation of [Reference HerzigHer09, § 5]. Then (7) becomes 
$w_1\cdot (\mu -\eta +\nu -w\pi \epsilon '_{w_0w_1} - \eta ) + p \rho '_{w_1}$ as in [Reference HerzigHer09, § A, Theorem 3.4]. We conclude by noting that the maps 
$s \mapsto -\pi \epsilon \pmod {X^0(\mathbf {T})}$ and 
$s \mapsto w_1 = w_0 s$ defined previously give bijections 
$W \rightarrow \{\omega _J\}_{J\subset \mathbf {Z}/f}$ and 
$W \rightarrow W$, respectively.
3. Galois deformation rings and Frobenius eigenvalues
3.1 Kisin modules
Notation. Recall that 
$L$ is 
$\mathbf {Q}_{p^f}$. Let 
$e$ be 
$p^f-1$ and 
$\pi$ be a root of 
$E(u) = u^e+p$ in our fixed algebraic closure of 
$L$. Denote by 
$K$ the extension 
$L(\pi )$ and by 
$\Delta$ the Galois group 
$\mathrm {Gal}(K/L)$. Then the character 
$[\omega _f]$ factors through 
$\Delta$.
Let 
$\mathbf {a} \in X(\mathbf {T})$ correspond to the tuple of integers 
$(a_{k,j})_j$ for 
$1\leq k\leq n$ and 
$0\leq j\leq f-1$. Let
\begin{gather*}\mathbf{a}_k = \sum_{i=0}^{f-1} a_{k,i}p^i \quad\text{and}\quad \mathbf{a}_k^{(j)} = \sum_{i=0}^{f-1} a_{k,i-j}p^i,\end{gather*}
so that 
$\mathbf {a}_k^{(j)} \equiv p^j\mathbf {a}_k \pmod {q-1}$. Write 
$\eta _k$ for the character 
$[\omega _f]^{-\mathbf {a}_k}$. Finally, let 
$\tau$ be the principal series tame inertial type 
$\tau (1,\mathbf {a})$ for 
$W_L$. In other words, the 
$\mathcal {O}_E$-dual 
$\tau ^\mathrm {d}$ is isomorphic to 
$\oplus _{k=1}^n \eta _k|_{I_L}$.
If 
$R$ is an 
$\mathcal {O}_E$-algebra, we give 
$(\mathcal {O}_L \otimes _{\mathbf {Z}_p} R)[\![u]\!]$ an action of 
$\Delta$ by 
$\mathcal {O}_L \otimes _{\mathbf {Z}_p} R$-linear ring automorphisms, by 
$\hat {g}: u \mapsto ([\omega _{f,L}](g) \otimes 1)u$ (where 
$[\omega _{f,L}]$ is the 
$\mathcal {O}_L$-valued character such that 
$[\omega _f] = \sigma _0 \circ [\omega _{f,L}]$). There is also a Frobenius homomorphism 
$\varphi : (\mathcal {O}_L \otimes _{\mathbf {Z}_p} R)[\![u]\!] \to (\mathcal {O}_L \otimes _{\mathbf {Z}_p} R)[\![u]\!]$ which is 
$\varphi \otimes 1$ on 
$\mathcal {O}_L \otimes _{\mathbf {Z}_p} R$ and sends 
$u$ to 
$u^p$.
Definition 3.1 Let 
$R$ be an 
$\mathcal {O}_E$-algebra. A Kisin module over 
$R$ is a finitely generated projective 
$(\mathcal {O}_L \otimes _{\mathbf {Z}_p} R)[\![u]\!]$-module 
$\mathfrak {M}_R$ together with a Frobenius map 
$\phi _{\mathfrak {M}_R} : \varphi ^*(\mathfrak {M}_R) \rightarrow \mathfrak {M}_R$ whose cokernel is annihilated by some power of 
$E(u)$. A Kisin module over 
$R$ with descent datum is a Kisin module over 
$R$ with a semilinear action of 
$\Delta$ which commutes with 
$\phi _{\mathfrak {M}_R}$. We say that the descent datum is of type 
$\tau$ if 
$\mathfrak {M}_R/u$ is isomorphic to 
$(\mathcal {O}_L \otimes _{\mathbf {Z}_p} R) \otimes _{\mathcal {O}_E} \tau ^\mathrm {d} \cong (\tau ^\mathrm {d}\otimes _{\mathcal {O}_E} R)^f$.
Remark 3.2 The dual in Definition 3.1 appears because we use contravariant functors to Galois representations in § 3.2.
In what follows, we work with Kisin modules whose 
$(\mathcal {O}_L \otimes _{\mathbf {Z}_p} R)[\![u]\!]$-rank is equal to 
$n$, though in applications 
$n$ will be 
$2$. The decomposition
\[ \mathcal{O}_L\otimes_{\mathbf{Z}_p} \mathcal{O}_E \cong \prod_{\iota: k_E \hookrightarrow k_E}\mathcal{O}_E \cong \prod_{j=0}^{f-1} \mathcal{O}_E \]
given by the bijection 
$j \leftrightarrow \iota _j$ gives a decomposition of a Kisin module 
$\mathfrak {M}_R$ as 
$\oplus _{j=0}^{f-1} \mathfrak {M}_R^{(j)}$, where 
$\mathfrak {M}_R^{(j)}$ is a projective 
$R[\![u]\!]$-module of rank 
$n$ for all 
$j$.
Eigenbases. Suppose now that 
$\tau$ is a principal series tame inertial type and that 
$\mathfrak {M}_R$ is a Kisin module over 
$R$ with descent datum 
$\tau$. Let 
$v$ be 
$u^e$. For each 
$1\leq k\leq n$, let 
$\mathfrak {M}^{(j)}_{R,k}$ be the 
$R[\![v]\!]$-submodule of 
$\mathfrak {M}_R^{(j)}$ on which 
$\Delta$ acts by 
$\eta _k$, that is,
\[ \mathfrak{M}_{R,k}^{(j)} = \{ m\in \mathfrak{M}_R^{(j)}\mid \hat{g}(m) = \eta_k(g) m\}. \]
Then 
$\mathfrak {M}_{R,k}^{(j)}$ is a projective 
$R[\![v]\!]$-module of rank 
$n$.
Suppose that the characters 
$\eta _k$ are distinct. An eigenbasis 
$\beta$ of a Kisin module 
$\mathfrak {M}_R$ over 
$R$ with descent datum of type 
$\tau$ is a tuple 
$(f_k^{(j)})_{k,j}$ where each 
$(f^{(j)}_1, \ldots , f^{(j)}_n)$ is an 
$R[\![u]\!]$-basis of 
$\mathfrak {M}_R^{(j)}$ such that 
$f_k^{(j)} \in \mathfrak {M}_{R,k}^{(j)}$. Note that this notion depends implicitly on a choice of ordering of the characters of 
$\tau$. Let 
$\beta ^{(j)}$ be the tuple 
$(f_k^{(j)})_k$.
Given an eigenbasis 
$\beta$ of a Kisin module 
$\mathfrak {M}_R$ over 
$R$ with descent datum of type 
$\tau$, for each 
$j$ let 
$C^{(j)}\in \mathrm {Mat}_n(R[\![u]\!])$ be the matrix such that
\[ \phi_{\mathfrak{M}}^{(j)}(\varphi^*(\beta^{(j)})) = \beta^{(j+1)} C^{(j)}. \]
In other words, the Frobenius map 
$\phi _{\mathfrak {M}}$ induces a semilinear map 
$\mathfrak {M}_R^{(j)} \to \mathfrak {M}_R^{(j+1)}$, and 
$C^{(j)}$ is a matrix over 
$R[\![u]\!]$ expressing the image of 
$\beta ^{(j)}$ in terms of 
$\beta ^{(j+1)}$.
3.1.1 Kisin modules and base change
Suppose that 
$\tau = \oplus _{k=1}^n \eta _k$ is an 
$n$-dimensional tame inertial type for 
$W_L$. There is a permutation 
$s_\tau \in S_n$ such that 
$\eta _k^q = \eta _{s_\tau ^{-1}(k)}$ for all 
$k$. Let 
$d$ be the order of 
$s_\tau$ and let 
$L'$ be the unramified extension of 
$L$ of degree 
$d$. Then the base change to 
$L'$ of 
$\tau$ is a principal series tame inertial type 
$\tau '$ for 
$W_{L'}$ (in this context, ‘base change’ amounts to regarding 
$\tau$ as a representation of 
$I_{L} = I_{L'}$).
Let 
$R$ be an 
$\mathcal {O}_E$-algebra. Let 
$\sigma :\mathcal {O}_{L'}[\![u]\!] \rightarrow \mathcal {O}_{L'}[\![u]\!]$ be the automorphism which acts trivially on 
$u$ and acts on 
$\mathcal {O}_{L'}$ by the action induced by the 
$q$ th power automorphism of 
$k_{L'}$. Then we define a Kisin module over 
$R$ with descent datum of type 
$\tau$ to be a pair 
$(\mathfrak {M}_R, \iota )$ where 
$\mathfrak {M}_R$ is a Kisin module over 
$R$ with descent datum of type 
$\tau '$ and 
$\iota : \sigma ^*(\mathfrak {M}_R) \rightarrow \mathfrak {M}_R$ is an isomorphism such that 
$\iota \circ \sigma ^*(\iota ) \circ \cdots \circ (\sigma ^{d-1})^*(\iota )$ is the identity morphism. An eigenbasis of 
$(\mathfrak {M}_R, \iota )$ is then defined to be an eigenbasis 
$\beta$ of 
$\mathfrak {M}_R$ such that 
$\iota (\sigma ^*(\beta ))$ is a permutation of the basis 
$\beta$. As 
$\iota$ can be identified with a 
$\sigma$-semilinear map 
$\mathfrak {M}_R \to \mathfrak {M}_R$, this means that 
$\iota$ permutes the vectors in 
$\beta$.
3.2 Étale $\varphi$-modules and Galois representations
3.2.1 Generalities
Recall from § 3.1 the definitions of 
$K$ and 
$L'$. Let 
$L_\infty$ be an infinite extension obtained by adjoining to 
$L$ compatible 
$p$-power roots of 
$-p$. Let 
$K'$ be the compositum of the fields 
$K$ and 
$L'$ and let 
$K_\infty '$ be the compositum of 
$K'$ and 
$L_\infty$. Let 
$\Delta '$ be the Galois group 
$\mathrm {Gal}(K'/L)$: it is isomorphic to the product of 
$\Delta$ with 
$\mathrm {Gal}(L'/L)\cong \mathbf {Z}/d$.
Equivalence of categories. Let 
$\mathcal {O}_{\mathcal {E},L}$ denote the 
$p$-adic completion of 
$\mathcal {O}_L(\!(v)\!)$, and let 
$\mathcal {O}_{\mathcal {E}^{\mathrm {un}},L}$ denote the 
$p$-adic completion of a maximal connected étale extension of 
$\mathcal {O}_{\mathcal {E},L}$. We similarly let 
$\mathcal {O}_{\mathcal {E},K'}$ denote the 
$p$-adic completion of 
$\mathcal {O}_{L'}(\!(u)\!)$, and let 
$\mathcal {O}_{\mathcal {E}^{\mathrm {un}},K'}$ denote the 
$p$-adic completion of a maximal connected étale extension of 
$\mathcal {O}_{\mathcal {E},K'}$. Then 
$\mathcal {O}_{\mathcal {E}, K'}$ is naturally a Galois finite étale extension of 
$\mathcal {O}_{\mathcal {E}, L}$ by putting 
$v = u^e$. This identifies 
$\mathcal {O}_{\mathcal {E}^{\mathrm {un}},L}$ with 
$\mathcal {O}_{\mathcal {E}^{\mathrm {un}},K'}$. Moreover, classical work of Fontaine (see [Reference FontaineFon90]) identifies 
$G_{\mathcal {O}_{\mathcal {E}, L}}$ and 
$G_{\mathcal {O}_{\mathcal {E}, K'}}$ with 
$G_{L_\infty }$ and 
$G_{K'_\infty }$, respectively. This identifies 
$\mathrm {Gal}(\mathcal {O}_{\mathcal {E}, K'}/\mathcal {O}_{\mathcal {E}, L})$ with 
$\mathrm {Gal}(K'_\infty /L_\infty )$, which is isomorphic to 
$\Delta '$. Under this isomorphism, 
$\Delta '$ acts on 
$\mathcal {O}_{\mathcal {E}, K'}$ by its natural action on 
$\mathcal {O}_{L'}$ and by 
$[\omega _{f',L'}]$ on 
$u$ (cf. § 3.1).
For 
$R$ a complete local Noetherian 
$\mathcal {O}_E$-algebra, let 
$\Phi \textrm {-}\mathrm {Mod}^{\textrm {\unicode{x00E9}}}(R)$ be the category of étale 
$\varphi$-modules over 
$\mathcal {O}_{\mathcal {E},L} \widehat {\otimes }_{\mathbf {Z}_p} R$, such that the underlying 
$\mathcal {O}_{\mathcal {E},L} \widehat {\otimes }_{\mathbf {Z}_p} R$-module is finite free. Similarly, let 
$\Phi \textrm {-}\mathrm {Mod}^{\textrm {\unicode{x00E9}}}_{\mathrm {dd}}(R)$ be the category of étale 
$\varphi$-modules over 
$\mathcal {O}_{\mathcal {E},K'}\widehat {\otimes }_{\mathbf {Z}_p} R$ with a semilinear action of 
$\Delta '$. Let 
$\mathrm {Rep}_{G_{L_{\infty }}}(R)$ be the category of (continuous) representations of 
$G_{L_{\infty }}$ over finite free 
$R$-modules. Fontaine [Reference FontaineFon90] gave an exact anti-equivalence of tensor categories
\begin{align*} \mathbf{V}^*: \Phi\textrm{-}\mathrm{Mod}^{\textrm{\unicode{x00E9}}}(R) &\rightarrow \mathrm{Rep}_{G_{L_{\infty}}}(R)\\ \mathcal{M} &\mapsto ((\mathcal{M}\otimes_{\mathcal{O}_{\mathcal{E},L}} \mathcal{O}_{\mathcal{E}^{\mathrm{un}},L})^{\varphi=1})^*, \end{align*}
where the action of 
$G_{L_\infty } \cong G_{\mathcal {O}_{\mathcal {E}, L}}$ comes from 
$\mathcal {O}_{\mathcal {E}^{\mathrm {un}},L}$ and 
$(-)^*$ denotes the contragredient representation.
A slight extension of the above gives an exact anti-equivalence of tensor categories
\begin{align*} \mathbf{V}^*_{\mathrm{dd}}: \Phi\textrm{-}\mathrm{Mod}^{\textrm{\unicode{x00E9}}}_{\mathrm{dd}}(R) &\rightarrow \mathrm{Rep}_{G_{L_{\infty}}}(R)\\ \mathcal{M} &\mapsto ((\mathcal{M} \otimes_{\mathcal{O}_{\mathcal{E},K'}} \mathcal{O}_{\mathcal{E}^{\mathrm{un}},L})^{\varphi = 1})^*, \end{align*}
where 
$G_{L_\infty }$ acts on 
$\mathcal {O}_{\mathcal {E}^{\mathrm {un}},L}$ as above and on 
$\mathcal {M}$ through the quotient 
$G_{L_\infty }/G_{K'_\infty } \cong \Delta '$. Moreover, by [Reference Caruso, David and MézardCDM18, Theorem 2.1.6 and (12)], there is an equivalence of categories
\begin{align} \Phi\textrm{-}\mathrm{Mod}^{\textrm{\unicode{x00E9}}}(R) &\rightarrow \Phi\textrm{-}\mathrm{Mod}^{\textrm{\unicode{x00E9}}}_{\mathrm{dd}}(R) \nonumber\\ \mathcal{M} &\mapsto \mathcal{M} \otimes_{\mathcal{O}_{\mathcal{E},L}} \mathcal{O}_{\mathcal{E},K'}, \nonumber \end{align}
which is compatible with Fontaine's functors above. A quasi-inverse of (8) is given by taking invariants under 
$\Delta '$.
Matrices of the action. We can write 
$\mathcal {O}_{\mathcal {E}, L} \cong \mathcal {O}_{\mathcal {E}} \otimes _{\mathbf {Z}_p} \mathcal {O}_L$, where we write 
$\mathcal {O}_{\mathcal {E}}$ for 
$\mathcal {O}_{\mathcal {E},\mathbf {Q}_p}$. As with Kisin modules, the decomposition 
$\mathcal {O}_L\otimes _{\mathbf {Z}_p} \mathcal {O}_{\mathcal {E}}$ as 
$\prod _{j=0}^{f-1} \mathcal {O}_{\mathcal {E}}$ (numbering embeddings via 
$j \leftrightarrow \sigma _j$) gives a decomposition of an element 
$\mathcal {M}_R\in \Phi \textrm {-}\mathrm {Mod}^{\textrm {\unicode{x00E9}}}(R)$ as 
$\oplus _{j=0}^{f-1} \mathcal {M}_R^{(j)}$, where each summand is a finite free 
$\mathcal {O}_{\mathcal {E}} \widehat {\otimes }_{\mathbf {Z}_p} R \cong \mathcal {O}_{\mathcal {E}, L} \widehat \otimes _{\mathcal {O}_L} R$-module of the same rank as that of 
$\mathcal {M}_R$ over 
$\mathcal {O}_{\mathcal {E}, L} \widehat {\otimes }_{\mathbf {Z}_p} R$. If 
$\gamma ^{(j)}$ is an 
$\mathcal {O}_{\mathcal {E}} \widehat {\otimes }_{\mathbf {Z}_p} R$-basis of 
$\mathcal {M}_R^{(j)}$ for all 
$j$, we let 
$B^{(j)}$ be the matrix such that
\[ \varphi_{\mathcal{M}}^{(j)}(\varphi^*(\gamma^{(j)})) = \gamma^{(j+1)} B^{(j)}. \]3.2.2 Galois representations
In this section, we state our assumptions on mod 
$p$ Galois representations and compute the 
$\varphi$-module of their restrictions to 
$G_{L_\infty }$. We specialize the discussion to 
$n = 2$.
Genericity. Let 
${\bar {\rho }}: G_L \rightarrow \mathrm {GL}_2(k_E)$ be a continuous representation, and write 
$V({\bar {\rho }}^{\mathrm {ss}}|_{I_L}) \cong R_{s_{\bar {\rho }}}(\mu )$, with 
$s_{\bar {\rho }} = (s_{\bar {\rho }},1,\ldots ,1)$ such that 
$s_{{\bar {\rho }}}$ is nontrivial if and only if 
${\bar {\rho }}$ is irreducible. Write 
$\mu$ as 
$(\mu _{1,j},\mu _{2,j})_j$. We assume that 
${\bar {\rho }}$ is generic, which will mean that
\[ 2 < \langle \alpha^{(j)},\mu \rangle < p-3 \]
for all 
$j\in \mathbf {Z}/f$. (Here 
$\alpha ^{(j)}$ is the positive root in embedding 
$j$, so that 
$\langle \alpha ^{(j)},\mu \rangle = \mu _{1, j} - \mu _{2, j}$). Note that this inequality can hold only if 
$p>5$.
With these assumptions, if 
${\bar {\rho }}$ is reducible, then 
${\bar {\rho }}$ is isomorphic to an extension of
\[ \mathrm{nr}_{\alpha'} \omega_f^{\sum_{j=0}^{f-1} \mu_{2,j}p^j} \textrm{ by } \mathrm{nr}_\alpha \omega_f^{\sum_{j=0}^{f-1} \mu_{1,j}p^j} \]
for some 
$\alpha$ and 
$\alpha ' \in k_E^\times$. Otherwise, 
${\bar {\rho }}$ is isomorphic to an induction
\[ \mathrm{Ind}_{G_{L_2}}^{G_L} \Big(nr_\beta \omega_{2f}^{\sum_{j=0}^{f-1} \mu_{1,j}p^j+p^f\sum_{j=0}^{f-1} \mu_{2,j}p^j}\Big) \]
for some 
$\beta \in k_E^\times$, where 
$L_2$ denotes the unramified quadratic extension of 
$L$. In this case, we denote by 
$\alpha$ the value at 
$p$ of the determinant of 
${\bar {\rho }}$, so that 
$\alpha = - \beta$, and we set 
$\alpha ' = -1$. The reason for doing so will become apparent when computing with 
$\varphi$-modules.
When working with modules over 
$k_L(\!(v)\!)$, we identify 
$s_{{\bar {\rho }}, j}$ with a permutation matrix and write 
$v^{\mu _j}$ for the image of 
$v$ under the cocharacter 
$\mu _j : \mathbf {G}_m \to \mathrm {GL}_2$, so that 
$v^{\mu _j} = {\mathrm {Diag}}(v^{\mu _{1,j}}, v^{\mu _{2,j}})$.
Proposition 3.3 There is an 
$\mathcal {M}_{k_E} \in \Phi \textrm {-}\mathrm {Mod}^{\textrm {\unicode{x00E9}}}(k_E)$ with 
$\mathbf {V}^*(\mathcal {M}_{k_E}) \cong {\bar {\rho }}|_{G_{L_\infty }}$ and a basis 
$(\gamma ^{(j)})_j$ of 
$\mathcal {M}_{k_E}$ such that
\begin{equation} B^{(j)} = U^{(j)} s^{{-}1}_{{\bar{\rho}},f-1-j} v^{\mu_{f-1-j}} \end{equation}
for some lower triangular matrix 
$U^{(j)}\in \mathrm {GL}_2(k_E(\!(v)\!))$. (Recall the isomorphism 
$\mathcal {O}_{\mathcal {E}, L} \otimes _{\mathcal {O}_L} k_E \cong k_E(\!(v)\!)$). Furthermore, 
$U^{(j)}$ is unipotent (respectively, the product of 
$\big (\begin {smallmatrix} \alpha & \\ & \alpha ' \end {smallmatrix}\big )$ and a unipotent matrix) if 
$j\neq 0$ (respectively, if 
$j = 0$), where 
$\alpha ' = -1$ if 
${\bar {\rho }}$ is irreducible.
Proof. We use Fontaine–Laffaille theory as in [Reference BreuilBre14, Appendix A]. Note that 
${\bar {\rho }}$ is Fontaine–Laffaille by the genericity condition. Suppose that 
${\bar {\rho }}$ is reducible. By twisting, we assume that 
$\mu _{2, j} = 0$ for all 
$j$. Then there is a Fontaine–Laffaille module 
$M = \oplus _{j=0}^{f-1} M^{(j)}$ with 
$M^{(j)} = k_E e^{(j)} \oplus k_E f^{(j)}$ such that
\[ \mathrm{Fil}^0 M^{(j)} = M^{(j)},\quad \mathrm{Fil}^1 M^{(j)} = \mathrm{Fil}^{\mu_{1, f-j}} M^{(j)} = k_E f^{(j)},\quad \mathrm{Fil}^{\mu_{1, f-j}+1} M^{(j)} = 0,\\[-1.8pc] \]
\begin{align*} \varphi(e^{(j)}) &= e^{(j+1)}, \\ \varphi_{\mu_{1, f-j}}(f^{(j)}) &= f^{(j+1)}+x_{j-1} e^{(j+1)},\quad \textrm{for } j\neq 1, \\ \varphi(e^{(1)}) &= \alpha' e^{(2)}, \\ \varphi_{\mu_{1, f-1}}(f^{(1)}) &= \alpha f^{(2)}+\alpha' x_0 e^{(2)}, \end{align*}
for some 
$x_j \in k_E$ and such that 
${\bar {\rho }} \cong \mathrm {Hom}_{\mathrm {Fil}^.,\varphi _.}(M,A_{\mathrm {cris}} \otimes _{\mathbf {Z}_p} \mathbf {F}_p)$ (see, e.g., [Reference BreuilBre14, (16)]).
Let 
$\mathcal {M}_{k_E}$ be defined as in (9) with
\[ U^{(j)} = \begin{pmatrix} 1 & \\ x_j & 1 \\ \end{pmatrix} \]
for 
$j\neq 0$ and
\[ U^{(0)} = \begin{pmatrix} \alpha & \\ & \alpha' \\ \end{pmatrix} \begin{pmatrix} 1 & \\ x_0 & 1 \\ \end{pmatrix}. \]
Let 
$\mathfrak {M}_{k_E}$ be the 
$\mathcal {O}_L[\![v]\!]\otimes _{\mathbf {Z}_p} k_E$-submodule of 
$\mathcal {M}_{k_E}$ generated by 
$(\gamma ^{(j)})_j$. Note that 
$\varphi$ maps 
$\mathfrak {M}_{k_E}$ to itself. Then a calculation (cf. [Reference Emerton, Gee and SavittEGS15, § 7.4] with 
$J = \emptyset$) shows that 
$\Theta _{p-1}(\mathfrak {M}_{k_E}) \cong \mathcal {F}_{p-1}(M)$, where the functors 
$\Theta _{p-1}$ and 
$\mathcal {F}_{p-1}$ are introduced in [Reference Emerton, Gee and SavittEGS15, Appendix A]. The result now follows from [Reference Emerton, Gee and SavittEGS15, Propositions A.3.2 and A.3.3]. The case 
${\bar {\rho }}$ is irreducible is proved similarly.
3.3 Modular Serre weights
In this section, we recall some of the various descriptions of the set of modular Serre weights of a Galois representation. More specifically, we fix a generic Galois representation 
${\bar {\rho }}: G_L \rightarrow \mathrm {GL}_2(k_E)$ with 
$V({\bar {\rho }}^{\mathrm {ss}}) \cong R_{s_{\bar {\rho }}}(\mu )$. In [Reference Buzzard, Diamond and JarvisBDJ10], a set of Serre weights is defined in terms of the restriction 
${\bar {\rho }}|_{I_L}$ of 
${\bar {\rho }}$ to the inertia group. We denote this set 
$W({\bar {\rho }})$, though it is denoted 
$\mathcal {D}({\bar {\rho }})$ in [Reference BreuilBre14] (see [Reference BreuilBre14, Proposition A.3]).
We need two perspectives on weights, one formulated in terms of combinatorial objects called ‘formal weights’, and another based on the map 
$\mathfrak {t}_\mu$ from § 2.4. The first perspective allows explicit computations and ready access to the results of [Reference Breuil and PaškūnasBP12, Reference BreuilBre11], whereas the second is better suited to the study of deformations of Kisin modules. Of course, they are ultimately equivalent, and we give an explicit dictionary in Proposition 3.5.
Formal weights. The genericity assumption implies that 
${\bar {\rho }}$ can be twisted by a character 
$\chi$ so that its restriction to inertia has one of the following two forms
\begin{gather*} {\bar{\rho}} |_{I_{\mathbf{Q}_p}} \cong \begin{pmatrix} \omega_f^{\sum_{j=0}^{f-1}(r_j+1)p^j} & * \\ 0 & 1 \end{pmatrix} \text{ in the reducible case,}\\ {\bar{\rho}}|_{I_{\mathbf{Q}_p}} \cong \begin{pmatrix} \omega_{2f}^{\sum_{j=0}^{f-1}(r_j+1)p^j} & 0 \\ 0 & \omega_{2f}^{q\sum_{j=0}^{f-1}(r_j+1)p^j} \end{pmatrix} \text{ in the irreducible case}. \end{gather*}
In the notation of the previous section, we have 
$r_j+1 = \mu _{1, j}-\mu _{2, j} = \langle \alpha ^{(j)}, \mu \rangle$, so that 
$1 < r_j < p-4$ for all values of 
$j$.
Remark 3.4 These conditions are slightly more restrictive than those in [Reference BreuilBre14]. These constraints come from our techniques for the study of deformation rings, and we have not tried to optimize our bounds.
A formal weight is an 
$f$-tuple of linear polynomials 
$\lambda = (\lambda _0(x_0), \ldots , \lambda _{f-1}(x_{f-1}))$ with coefficients in 
$\mathbf {Z}$ such that the leading coefficients are either 
$1$ or 
$-1$. Note that formal weights form a group under composition. There is a homomorphism from the group of formal weights to 
$W$ sending 
$\lambda$ to 
$w_\lambda \in W$ where 
$w_{\lambda ,j}$ is trivial if and only if the leading coefficient of 
$\lambda _j$ is 
$1$. (In fact, the group of formal weights is naturally isomorphic to the extended affine Weyl group of 
$\mathbf {G}^{\mathrm {der}}$, and the above homomorphism is the natural map to 
$W$, which is canonically isomorphic to the Weyl group of 
$\mathbf {G}^{\mathrm {der}}$.) For a formal weight 
$\lambda$, we define the linear polynomial in 
$f$-variables
\begin{equation} e(\lambda) =\begin{cases} \dfrac{1}{2} \bigg(\displaystyle \sum_{i=0}^{f-1} p^i(x_i - \lambda_i(x_i)) \bigg) & \mbox{if } w_{\lambda,f-1} = \mathrm{id}, \\[-4pt] \dfrac{1}{2} \bigg(p^f - 1 + \displaystyle\sum_{i=0}^{f-1} p^i(x_i - \lambda_i(x_i))\bigg) & \mbox{if } w_{\lambda,f-1} \neq \mathrm{id}. \end{cases} \end{equation}
Although 
$e(\lambda )$ a priori has coefficients in 
$\frac {1}{2}\mathbf {Z}$, for all formal weights that we consider, the coefficients in fact lie in 
$\mathbf {Z}$.
When 
${\bar {\rho }}$ is semisimple, its weights are parametrized by subsets of 
$\{0, \ldots , f-1 \}$ as follows. To a subset 
$J$, we associate the formal weight 
$\lambda _J = (\lambda _0(x_0), \ldots , \lambda _{f-1}(x_{f-1}))$ of 
${\bar {\rho }}$, where 
$\lambda _j$ is uniquely determined by whether 
$j-1, j$ are contained in 
$J$ or not, according to the following table. (We remark that this is the same normalization as [Reference BreuilBre11], and differs from that in [Reference Breuil and PaškūnasBP12, § 11]. Recall that 
$\operatorname {ch}_J$ is the characteristic function of 
$J$.)
Note that the image of 
$\lambda _J$ in 
$W$ is 
$F(w_J)$, which is defined in § 2.4: this is because the image of 
$\lambda _J$ is nontrivial precisely on 
$J$, 
$w_J$ is nontrivial on 
$\delta _\mathrm {red}(J)$, and 
$F$ is a right shift on the Weyl group. Moreover, the map from subsets of 
$\{0,\ldots ,f-1\}$ to 
$W$ taking 
$J$ to 
$F(w_J)$ is a bijection.
Then, the modular weights of 
${\bar {\rho }}$ are precisely the Serre weights of the form
\begin{equation} \sigma(\lambda, {\bar{\rho}}) = (\lambda_0(r_0), \ldots, \lambda_{f-1}(r_{f-1})) \otimes \mathrm{det}^{e(\lambda)(r_0, \ldots, r_{f-1})}\chi \end{equation}
as 
$\lambda$ runs through the formal weights of 
${\bar {\rho }}$. In this formula, we have followed our convention of writing 
$\det$ for the 
$k_E^\times$-valued character 
$\det _E = \sigma _0 \circ \det$ of 
$\mathrm {GL}_2(k_L)$.
To describe the weights of a nonsplit reducible 
${\bar {\rho }}$, let 
$J_{\bar {\rho }}$ be the set 
$\{f-1-j\mid U^{(j)} \textrm { is diagonal}\}$. (This coincides with the definition in [Reference BreuilBre14, (17)], see the proof of Proposition 3.3.) We define 
$W({\bar {\rho }}) \subset W({\bar {\rho }}^{\mathrm {ss}})$ to be the set of weights corresponding to subsets 
$J \subseteq \delta _{\mathrm {red}}(J_{{\bar {\rho }}})$. This is compatible with [Reference BreuilBre14]: indeed, the definition there is that the formal weight 
$\lambda$ defines an element of 
$W({\bar {\rho }})$ if and only if 
$\lambda _j \in \{p-2-x_j, p-3-x_j \}$ implies that a certain parameter 
$\mu _{f-j}$ of [Reference BreuilBre14] (unrelated to our use of the notation 
$\mu$) is 
$0$, and by [Reference BreuilBre14, (18)] we have 
$\delta _{\mathrm {red}}(J_{{\bar {\rho }}}) = \{f-j: \mu _j = 0 \}$.
Extension graph. Recall that 
$\omega _j$ is the fundamental weight of 
$\Lambda _W$ which is nonzero in embedding 
$j$, and that for a subset 
$J \subset J_{\bar {\rho }}$ we write 
$\omega _J = \sum _{j\in J} \omega _j$ (this differs slightly from the notation in [Reference Le, Morra and SchraenLMS20, § 2]). Our genericity condition on 
${\bar {\rho }}$ implies that 
$s_{\bar {\rho }} \omega _J \in \Lambda ^\mu _W$ for all 
$J$. Let 
$\sigma _J$ be the Serre weight 
$F(\mathfrak {t}_\mu (s_{\bar {\rho }}\omega _J))$.
Proposition 3.5 The set 
$W({\bar {\rho }})$ is equal to 
$\{\sigma _J\mid J\subset J_{\bar {\rho }}\}$. The formal weight corresponding to 
$\sigma _J$ is 
$\lambda _{\delta _\mathrm {red}(J)}$ (both in the reducible and irreducible case).
Proof. The case where 
${\bar {\rho }}$ is semisimple follows from [Reference Le, Morra and SchraenLMS20, Proposition 2.10]: we briefly recall the argument. As 
$J$ varies, the Serre weights 
$F(w_{J}t_{s_{\bar {\rho }}\omega '_J-p\pi ^{-1}\omega '_J}(\mu ) - \eta )$ are the ‘obvious weights’ of 
${\bar {\rho }}$, where by definition 
$\omega '_J \in X(\mathbf {T})$ is a lift of 
$\omega _J \in \Lambda _W$ and 
$\widetilde {w}_J = w_{J}t_{-\pi ^{-1}(\omega '_J)} \in ~\Omega$.
As 
$w_J t_{-\pi ^{-1}\omega '_J}t_{\pi ^{-1}s_{\bar {\rho }} \omega '_J} \in W_a$ (the translation part is in 
$\Lambda _R$) we find that
\[ \mathfrak{t}_\mu(s_{\bar{\rho}} \omega_J) = w_J t_{-\pi^{{-}1}\omega'_J}\cdot (\mu- \eta + s_{\bar{\rho}} \omega'_J) = w_Jt_{s_{\bar{\rho}}\omega'_J - p \pi^{{-}1}\omega'_J}(\mu) - \eta. \]
Hence 
$\sigma _J = F(\mathfrak {t}_\mu (s_{\bar {\rho }} \omega _J))$ is an obvious weight, and the claim follows because all modular weights of 
${\bar {\rho }}$ are obvious weights (when 
$n = 2$).
Now we make explicit the correspondence with formal weights, which make it clear that the statement of the proposition extends to nonsplit 
${\bar {\rho }}$. Assume that 
${\bar {\rho }}$ is split reducible, so that 
$s_{{\bar {\rho }}} = 1$, and that the character 
$\chi$ is trivial, so that
\[ {\bar{\rho}} |_{I_{\mathbf{Q}_p}} \cong \begin{pmatrix} \omega_f^{\sum_{j=0}^{f-1}(r_j+1)p^j} & * \\ 0 & 1 \end{pmatrix}. \]
Then the components of 
$\mu$ are 
$a_j = r_j+1$ and 
$b_j = 0$ for all 
$j$.
Recall that 
$\pi ^{-1}$ is a left shift on 
$X(\mathbf {T})$. Then we have the following correspondence.
Equivalently, the following table computes the components of the Serre weight 
$\sigma _J$.
A computation with 
$e(\lambda )$, as in Lemma 5.21 to follow, then shows that 
$\sigma _J$ is the weight attached to the formal weight corresponding to 
$\delta _{\mathrm {red}}(J)$. The remaining reducible cases follow by twisting 
${\bar {\rho }}$ and observing that 
$J \subseteq J_{\bar {\rho }}$ if and only if 
$\delta _\mathrm {red}(J) \subseteq \delta _\mathrm {red}(J_{\bar {\rho }})$. To do the irreducible case, it suffices to observe that because 
$\omega '_J - s_{\bar {\rho }} \omega '_J \in \Lambda _R$, we have
\[ \mathfrak{t}_\mu(s_{\bar{\rho}}\omega_J) = \widetilde{w}_J \cdot(\mu - \eta + s_{\bar{\rho}} \omega'_J). \]3.4 Orientations and crystalline Frobenius eigenvalues
In this section, 
$n$ is again an arbitrary natural number. Using Kisin's theory, we describe Frobenius eigenvalues of crystalline Dieudonné modules arising from Kisin modules. We also introduce the notion of an orientation which plays a key role in calculating Galois deformation rings.
Proposition 3.6 Let 
$\tau$ be a principal series tame inertial type for 
$W_L$. Suppose that 
$\mathfrak {M}$ is a Kisin module over 
$\mathcal {O}_E$ with descent datum of type 
$\tau$, such that
\[ \mathbf{V}^*_{\mathrm{dd}}(\mathfrak{M}\otimes_{\mathcal{O}_L[\![u]\!]} \mathcal{O}_{\mathcal{E},K}) \]
is the restriction to 
$G_{L_\infty }$ of a potentially crystalline representation 
$V$ of 
$G_L$. Then the 
$\varphi$-module with semilinear 
$\Delta$-action 
$D_{\mathrm {pst}}(V^\vee )$ is isomorphic to 
$(\mathfrak {M}/u)\otimes _{\mathcal {O}_E} E$ where 
$(-)^\vee$ denotes the contragredient representation.
Proof. This follows from [Reference KisinKis06, Proposition 2.1.5] by adding tame descent datum. Namely, this proposition shows that 
$(\mathfrak {M}/u)\otimes _{\mathcal {O}_E} E$ and 
$D_{\mathrm {pst}}(V^\vee |_{G_K})$ are canonically isomorphic as 
$\varphi$-modules. As the 
$\Delta$-action is compatible with the identification 
$V|_{G_{K_\infty }} = \mathbf {V}^*(\mathfrak {M}\otimes _{\mathcal {O}_L[\![u]\!]} \mathcal {O}_{\mathcal {E},K})$, the isomorphism between 
$(\mathfrak {M}/u)\otimes _{\mathcal {O}_E} E$ and 
$D_{\mathrm {pst}}(V^\vee |_{G_K})$ respects the induced 
$\Delta$-actions.
Corollary 3.7 Suppose that 
$\tau$ is a principal series tame inertial type as in § 3.1 with the characters 
$\eta _k$ distinct. Let 
$\mathfrak {M}$ be a Kisin module over 
$\mathcal {O}_E$ with descent datum of type 
$\tau$, 
$\beta$ an eigenbasis, and let 
$C^{(j)} = (c^{(j)}_{ik})_{ik}$ be the matrix defined in § 3.1. Then the 
$\varphi$-eigenvalue on the 
$\eta _k$-isotypic part of 
$(\mathfrak {M}/u)\otimes _{\mathcal {O}_E} E$ is given by
\[ \prod_j c_{kk}^{(j)} \pmod{u}. \]Definition 3.8 Let 
$(a_{k,j})_{k,j}$ be a tuple of integers for 
$1\leq k\leq n$ and 
$0\leq j\leq f-1$. An orientation of 
$(a_{k,j})_{k,j}$ is a tuple 
$(s_j)_j \in S_n^f$ such that
\[ a_{s_j(1),f-1-j} \geq a_{s_j(2),f-1-j} \geq \cdots \geq a_{s_j(n),f-1-j} \]
and 
$a_{s_j(1),f-1-j} - a_{s_j(n),f-1-j} < p$.
These inequalities, in particular, imply that if 
$(s_j)$ is an orientation of the tuple 
$(a_{k,j})_{k,j}$, then
\[ \mathbf{a}^{(j)}_{s_j(1)} \geq \mathbf{a}^{(j)}_{s_j(2)} \geq \cdots \geq \mathbf{a}^{(j)}_{s_j(n)}. \]
Given a principal series tame inertial type 
$\tau$ as above, there is a choice of the tuple 
$(a_{k,j})_{k,j}$ giving 
$\tau$ as in § 3.1 which has an orientation (though this choice is never unique).
We now fix a tuple of integers 
$(a_{k,j})_{k,j}$ for 
$1\leq k\leq n$ and 
$0\leq j\leq f-1$ for which there exists an orientation 
$(s_j)_j$, which we also fix. Let 
$\tau$ be the principal series tame inertial type defined in terms of 
$(a_{k,j})_{k,j}$ as in § 3.1, let 
$R$ be an 
$\mathcal {O}_E$-algebra, and let 
$\mathfrak {M}$ be a Kisin module over 
$R$ with descent datum of type 
$\tau$. Let 
$^\varphi \mathfrak {M}^{(j)}_k$ be the 
$R[\![v]\!]$-submodule of 
$\varphi ^*(\mathfrak {M}^{(j)})$ on which 
$\Delta$ acts by 
$\eta _k$. Note that 
$^\varphi \mathfrak {M}^{(j)}_k$ is typically strictly larger than 
$\varphi ^*(\mathfrak {M}^{(j)}_k)$.
If 
$\beta = (f^{(j)}_k)_{k,j}$ is an eigenbasis for 
$\mathfrak {M}$, then by the proof of [Reference Le, Le Hung, Levin and MorraLLLM18, Lemma 2.9]
\[ \beta^{(j)}_{s_j(n)} := \Big(u^{\mathbf{a}_{s_j(k)}^{(j)}-\mathbf{a}_{s_j(n)}^{(j)}} f^{(j)}_{s_j(k)}\Big)_k \]
is a basis for 
$\mathfrak {M}^{(j)}_{s_j(n)}$ for each 
$j$ and
\[ {}^\varphi \beta^{(j-1)}_{s_j(n)} := \Big(u^{\mathbf{a}_{s_j(k)}^{(j)}-\mathbf{a}_{s_j(n)}^{(j)}} \otimes f^{(j-1)}_{s_j(k)}\Big)_k \]
is a basis for 
$^\varphi \mathfrak {M}^{(j-1)}_{s_j(n)}$. For each 
$j$ let 
$A^{(j)}\in \mathrm {Mat}_n(R[\![v]\!])$ be the matrix such that
\[ \phi^{(j)}_{\mathfrak{M}}\big(^\varphi\beta^{(j)}_{s_{j+1}(n)}\big) = \beta^{(j+1)}_{s_{j+1}(n)}A^{(j)}. \]Then [Reference Le, Le Hung, Levin and MorraLLLM18, Proposition 2.13] states that
\[ C^{(j)} = \mathrm{ad}_{s_{j+1}} {\mathrm{Diag}}(u^{\mathbf{a}_1}, \ldots u^{\mathbf{a}_n}) (A^{(j)}) := \mathrm{ad} \Big( s_{j+1}{\mathrm{Diag}}\Big(u^{\mathbf{a}_{s_{j+1}(1)}^{(j+1)}}, \ldots, u^{\mathbf{a}_{s_{j+1}(n)}^{(j+1)}}\Big)\Big)(A^{(j)}), \]
where 
${\mathrm {Diag}}(u^{\mathbf {a}_1^{(j+1)}},\ldots ,u^{\mathbf {a}_n^{(j+1)}})$ denotes the diagonal matrix with 
$k$th diagonal entry 
$u^{\mathbf {a}_k^{(j+1)}}$ and 
$s_{j+1}$ is identified with the permutation matrix with 
$(s_{j+1}(k),k)$-entry equal to 
$1$ for all 
$k$. Corollary 3.7 then implies the following.
Corollary 3.9 Let 
$\tau$, 
$\mathfrak {M}$, and 
$\beta$ be as in Corollary 3.7. Let 
$(s_j)_j$ be an orientation for a tuple 
$(a_{k,j})_{k,j}$ giving rise to 
$\tau$ as in § 3.1. If we let 
$A^{(j)}=(x^{(j)}_{ik})_{ik}$ be as before, then the Frobenius eigenvalue on the 
$\eta _k$-isotypic part of 
$(\mathfrak {M}/u)\otimes _{\mathcal {O}_E} E$ is given by
\[ \prod_j x^{(j)}_{s_{j+1}^{{-}1}(k)s_{j+1}^{{-}1}(k)} \pmod{v}. \]3.4.1 Orientations and base change
Recall the notation 
$\tau$, 
$s_\tau$, 
$d$, 
$L'$, and 
$\tau '$ from § 3.1.1. Then there exists a tuple 
$(a_{k,j})_{k,j}$ giving 
$\tau '$ as in § 3.1 which admits an orientation and such that 
$a_{k,j} = a_{s_\tau ^{-1}(k),j+f}$ for all 
$j$ and 
$k$. We can and do choose an orientation 
$s' \in W'$ (the Weyl group of 
$\mathrm {Res}_{L'/\mathbf {Q}_p}\mathrm {GL}_2$) such that
\begin{equation} (s'_{j+f})^{{-}1} = s_\tau (s'_j)^{{-}1} s_\tau^{{-}1} \end{equation}
for all 
$j$. If 
$(\mathfrak {M}_R,\iota )$ is a Kisin module with descent data 
$\tau$ and 
$\beta$ is an eigenbasis for 
$(\mathfrak {M}_R,\iota )$, then we can define 
$A^{(j)}$ with respect to 
$s'$ as before. Then we have that 
$A^{(j)} = A^{(j+f)}$ for all 
$j$.
3.4.2 Orientations and étale $\varphi$-modules
Let 
$\tau$ be a tame inertial type and 
$\mathfrak {M}_R$ (or 
$(\mathfrak {M}_R,\iota )$) be a Kisin module over 
$R$ with descent datum of type 
$\tau$. Suppose that 
$\sigma (\tau )$ is 
$R_w(\mu )$ for 
$(w,\mu ) \in W\times X(T)$ where 
$\mu$ is dominant and 
$p$-restricted. There exists an element 
$s^*\in W$ such that 
$(s^*)^{-1}w F(s^*) = (s_\tau ,\mathrm {id},\ldots ,\mathrm {id})$ for some 
$s_\tau \in S_n$. By our discussion of Deligne–Lusztig induction in § 2.3.1 (compare also [Reference HerzigHer09, Lemma 4.2]), 
$R_w(\mu )$ is isomorphic to
\[ R_{(s_\tau,\mathrm{id},\ldots,\mathrm{id})}((s^*)^{{-}1} (\mu)). \]
Let 
$L'$ be the unramified extension of 
$L$ of degree equal to the order 
$d$ of 
$s_\tau$. Let 
$\tau '$ be the base change to 
$L'$, so that it is a principal series tame inertial type for 
$\mathrm {GL}_n(L')$.
Let 
$a_{k,j}$ be 
$((s^*)^{-1} \mu )_{k,j}$ for 
$1\leq k \leq n$ and 
$0\leq j\leq f-1$. Then define 
$a_{k,j}$ for 
$1\leq k \leq n$ and 
$0\leq j\leq df-1$ so that 
$a_{k,j+f} = a_{s_\tau ^{-1}(k),j}$ for all 
$k$ and 
$j$. Define 
$s\in W$ so that 
$s_j=(s^*_{f-1-j})^{-1}$ for 
$0\leq j \leq f-1$. There is a unique 
$s'\in W'$ satisfying (13) for all 
$j$ such that 
$s_j = s'_j$ for 
$0 \leq j \leq f-1$. Then 
$s'$ is an orientation for 
$(a_{k,j})_{k,j}$.
Proposition 3.10 Let 
$(\mathfrak {M}_R,\iota )$ be a Kisin module over 
$R$ with descent datum of type 
$\tau$. Let 
$\beta$ be an eigenbasis for 
$(\mathfrak {M}_R,\iota )$, and define 
$A^{(j)}$ with respect to the orientation 
$s'$ as before. Recall the definitions of 
$K'$ and 
$\Delta '$ from before. Then there is a basis for 
$(\mathfrak {M}_R\otimes _{\mathcal {O}_L[\![u]\!]}\mathcal {O}_{\mathcal {E},K'})^{\Delta '}$ such that
\[ B^{(j)} = A^{(j)} s_j^{{-}1}v^{\mu_j}. \]Proof. This follows from the proof of [Reference LeLe19, Proposition 3.4].
3.5 Deformation rings
For a local mod 
$p$ Galois representation 
${\bar {\rho }}: G_L \rightarrow \mathrm {GL}_n(k_E)$, let 
$R_{{\bar {\rho }}}$ be the framed unrestricted 
$\mathcal {O}_E$-deformation ring for 
${\bar {\rho }}$. If 
$\psi : G_L \rightarrow \mathcal {O}_E^\times$ is a lift of 
$\det {\bar {\rho }}\overline {\varepsilon }^{-1}$, then let 
$R^\psi _{\bar {\rho }}$ be the quotient of 
$R_{\bar {\rho }}$ parametrizing lifts of 
${\bar {\rho }}$ with determinant 
$\psi \varepsilon$. For an inertial type 
$\tau$, let 
$R_{{\bar {\rho }}}^{\tau }$ be the quotient of 
$R_{{\bar {\rho }}}$ corresponding to potentially crystalline lifts of 
${\bar {\rho }}$ of Galois type 
$\tau$ and parallel Hodge–Tate weights 
$(0,1)$.
From now on, we set 
$n = 2$. As in § 3.2, suppose that 
${\bar {\rho }}:G_L\rightarrow \mathrm {GL}_2(k_E)$ satisfies 
$V({\bar {\rho }}^{\mathrm {ss}}|_I) = R_{s_{\bar {\rho }}}(\mu )$.
3.5.1 Deformations of Kisin modules with tame descent data
We describe Galois deformation rings when 
$\sigma _{\mathrm {alg}}$ is trivial and 
$\sigma _{\mathrm {sm}} = \sigma (\tau )$ is a generic tame type, using the Kisin modules introduced in § 3.1. Let 
$\mathcal {M}_{k_E}$ be an element of 
$\Phi \textrm {-}\mathrm {Mod}^{\textrm {\unicode{x00E9}}}(k_E)$ with basis 
$\gamma ^{(j)}$ such that 
$B^{(j)}$ is given by (9) so that 
$\mathbf {V}^*(\mathcal {M}_{k_E}) \cong {\bar {\rho }}|_{G_{L_\infty }}$.
Construction of tame types. We now describe tame types whose reductions contain modular weights of a given 
${\bar {\rho }}$.
Proposition 3.11 Let 
${\bar {\rho }}$ be a generic Galois representation (possibly not semisimple). Suppose that 
$w\in W$, 
$\nu \in \Lambda _R$ are such that 
$-\eta +\nu +s_{\bar {\rho }} w\omega _J\in \Lambda _W^\mu$ for all subsets 
$J \subset \mathbf {Z}/f$, and 
$V$ is the tame type 
$R_{s_{\bar {\rho }} w}(\mu - \eta +\nu )$ for 
$\mathrm {GL}_2(L)$. Then 
$\mathrm {JH}(\bar {V}) \cap W({\bar {\rho }})$ is nonempty if and only if 
$-\eta +\nu = -s_{\bar {\rho }} w' \eta$ for some 
$w'\in W$ such that:
•
$(w_j,w'_j)\neq (s,\mathrm {id})$ for all $j\in \mathbf {Z}/f$; and• if
$(w_j,w'_j) = (\mathrm {id},s)$, then $j\in J_{\bar {\rho }}$.
In this case,
\[ W({\bar{\rho}})\cap \mathrm{JH}(\bar{V})=\{F(\mathfrak{t}_\mu(s_{\bar{\rho}}\omega_J)): J_1(V) \subset J \subset J_2(V)^\textrm{c} \cap J_{\bar{\rho}}\} \]where
\[ J_1(V) = \{j\in \mathbf{Z}/f: w_j = \mathrm{id} \ \textrm{and}\ w'_j \neq \mathrm{id}\},\quad J_2(V) = \{j\in \mathbf{Z}/f: w_j = \mathrm{id} \ \textrm{and}\ w'_j = \mathrm{id}\}, \]
and 
$J_2(V)^\textrm {c}$ denotes the complement of 
$J_2(V)$ in 
$\mathbf {Z}/f$.
Proof. By Propositions 2.15 and 3.5,
\[ \mathrm{JH}(\bar{V}) = \{F(\mathfrak{t}_\mu(s_{\bar{\rho}} w\omega_{J'} - \eta + \nu): J'\subset \mathbf{Z}/f\} \]and
\[ W({\bar{\rho}})=\{F(\mathfrak{t}_\mu(s_{\bar{\rho}}\omega_J)): J \subset J_{\bar{\rho}}\}. \]
If these sets intersect, then because 
$\mathfrak {t}_\mu$ is injective on 
$\Lambda ^\mu _W$ we must have that 
$J^\textrm {c} = J'$, and 
$-\eta +\nu = -s_{\bar {\rho }} w' \eta$ for some 
$w'\in W$.
To compute the intersection we need to classify equalities in 
$\Lambda _W$ of the form
\[ w\omega_{J^\textrm{c}} - w'\eta = \omega_J \]
for fixed 
$w, w'$ and varying 
$J, J^\textrm {c}$. If there exists 
$j$ such that 
$(w_j, w_j') = (s, \mathrm {id})$, the left-hand side is negative at 
$j$ (the coefficient of 
$\omega _j$ is negative) but the right-hand side at 
$j$ is nonnegative, so there cannot be any such equalities. Similarly, if 
$(w_j, w_j') = (\mathrm {id}, s)$, then the left-hand side is positive at 
$j$, so we need 
$j \in J$, hence 
$j \in J_{\bar {\rho }}$.
Thus, assume the two conditions in the proposition. If 
$(w_j,w'_j) = (\mathrm {id},s)$, then 
$(w\omega _{J^\textrm {c}}-w'\eta )_j$ is nonzero and so 
$j\in J$. If 
$(w_j,w'_j) = (\mathrm {id},\mathrm {id})$, then 
$(w\omega _{J^\textrm {c}}-w'\eta )_j$ cannot be 
$\omega _j$ and so 
$j\notin J$. This implies the inclusion
\[ W({\bar{\rho}})\cap \mathrm{JH}(\bar{V})\subset \{F(\mathfrak{t}_\mu(s_{\bar{\rho}}\omega_J)): J_1(V) \subset J \subset J_2(V)^\textrm{c} \cap J_{\bar{\rho}}\}. \]The reverse inclusion is checked similarly, and this completes the proof of the proposition.
Remark 3.12 As the types 
$R_{s_{\bar {\rho }} w}(\mu -s_{\bar {\rho }} w'\eta )$ have reductions with distinct sets of Jordan–Hölder factors, they are pairwise nonisomorphic. If 
${\bar {\rho }}$ is semisimple, then 
$J_{\bar {\rho }} = \mathbf {Z}/f$ so that there are 
$3^f$ tame types satisfying the conditions of Proposition 3.11. It is known that for 
${\bar {\rho }}$ generic and semisimple, there are 
$3^f$ tame types 
$V$ such that 
$\mathrm {JH}(\bar {V}) \cap W({\bar {\rho }})$ is nonempty. We conclude that all such tame types appear in Proposition 3.11, though we will not need this in what follows.
Proposition 3.11 gives another perspective on the parametrization of 
$I_1$-invariants in 
$D_0({\bar {\rho }})$ in [Reference Breuil and PaškūnasBP12, Corollary 14.10] as follows. If 
$\chi \in (\operatorname {soc}_K D_0({\bar {\rho }}))^{I_1}$, we will write 
$J(\chi )$ for the subset such that 
$\sigma _{J(\chi )} \cong \sigma (\chi )$.
Lemma 3.13 Let 
${\bar {\rho }}$ be a generic, possibly not semisimple, Galois representation. Let 
$\chi$ be a character appearing in 
$D_0({\bar {\rho }})^{I_1}$. Then there exist 
$s^*, w, w' \in W$ such that
\[ \theta(\chi)[1/p] \cong R_{s_{\bar{\rho}} w}(\mu - s_{\bar{\rho}} w' \eta),\quad s^*F(s^*)^{{-}1} = s_{\bar{\rho}} w,\quad \text{and}\quad \chi = (s^*)^{{-}1}(\mu-s_{\bar{\rho}} w'\eta)|_H. \]
Furthermore, if 
$\chi$ appears on 
$(\operatorname {soc}_K D_0({\bar {\rho }}))^{I_1}$ and 
$i\in \mathbf {Z}/f$, then 
$i\in J(\chi )$ if and only if 
$s^*_{i-1} \ne \mathrm {id}$.
Proof. A character 
$\chi : H \to \mathcal {O}_E^\times$ appears in 
$D_0({\bar {\rho }})^{I_1}$ if and only if 
$\bar {\theta }(\chi )\cap W({\bar {\rho }})$ is nonempty. By considerations of central characters, 
$\theta (\chi )[1/p]$ must then be isomorphic to 
$R_{s_{\bar {\rho }} w} (\mu -\eta +\nu )$ as in Proposition 3.11. Note that 
$-\eta +\nu +s_{\bar {\rho }} w\omega _J\in \Lambda _W^\mu$ for all subsets 
$J \subset \mathbf {Z}/f$ by the genericity assumption on 
${\bar {\rho }}$, which guarantees that 
$\chi$ appearing in [Reference Breuil and PaškūnasBP12, Corollary 14.10] is generic as well. Then Proposition 3.11 implies that there exist 
$w,w'\in W$ such that 
$\theta (\chi )[1/p]$ is isomorphic to 
$R_{s_{\bar {\rho }} w} (\mu -s_{\bar {\rho }} w'\eta )$. By our discussion of Deligne–Lusztig theory in § 2.3.1, there exists 
$s^* \in W$ such that
\[ s^*F(s^*)^{{-}1} = s_{\bar{\rho}} w \quad \text{and}\quad \chi = (s^*)^{{-}1}(\mu-s_{\bar{\rho}} w'\eta)|_H. \]
Now assume 
$\chi$ appears in 
$(\operatorname {soc}_K D_0({\bar {\rho }}))^{I_1}$. The weight 
$(s^*)^{-1}(\mu -s_{\bar {\rho }} w'\eta )$ need not be 
$p$-restricted, but there exists 
$\omega ' \in X(\mathbf {T})$ lifting some 
$-\omega _{J'} \in \Lambda _W$ such that 
$\sigma (\chi ) \cong \sigma _{J(\chi )}$ is isomorphic to 
$F((s^*)^{-1}(\mu -s_{\bar {\rho }} w'\eta )+\omega '-p\pi ^{-1}(\omega '))$. The image of 
$\omega '_i$ in 
$\Lambda _W/\Lambda _R$ is nonzero if and only if 
$s^*_{i-1} \neq \mathrm {id}$. Writing 
$\widetilde {w}$ for 
$t_{-\pi ^{-1}(\omega ')} (s^*)^{-1}$ and using the definition of 
$\sigma _{J(\chi )}$, we have that
\[ F(\widetilde{w}_{J(\chi)}\cdot(\mu-\eta+s_{\bar{\rho}}\omega'_{J(\chi)})) \cong \sigma_{J(\chi)} \cong F(\widetilde{w} \cdot (\mu-\eta-s_{\bar{\rho}} w'\eta + s^*\omega'+s^*\eta)) \]
for some choice of 
$\omega '_{J(\chi )}$ lifting 
$\omega _{J(\chi )}$. We conclude that 
$\mathfrak {t}_\mu (s_{\bar {\rho }} \omega '_{J(\chi )}) = \mathfrak {t}_\mu (s^*\eta -s_{\bar {\rho }} w'\eta +s^*\omega ')$ so that the image of 
$\omega '_{J(\chi )}$ and 
$\omega '$ in 
$\Lambda _W/\Lambda _R$ coincide (because 
$\mathfrak {t}_\mu$ is injective and 
$s^*\eta - s_{\bar {\rho }} w' \eta$ has image in 
$\Lambda _R$). It follows that 
$i \in J(\chi )$ if and only if 
$s^*_{i-1} \ne \mathrm {id}$.
Computation of deformation rings. Now let 
$w$ and 
$w'$ be as in Proposition 3.11, and let 
$\tau$ be the tame type with 
$\sigma (\tau ) \cong R_{s_{\bar {\rho }} w}(\mu -s_{\bar {\rho }} w' \eta )$. As before, let 
$d$ be the order of 
$s_\tau$. (Thus, 
$d = 1$ if 
$\tau$ is principal series, and 
$d = 2$ if 
$\tau$ is cuspidal.) For each 
$j \in \mathbf {Z} /df$, set
\begin{align*} A^{(j)} &= B^{(j)} v^{-(\mu_{f-1-j} - s_{{\bar{\rho}},f-1-j} w'_{f-1-j}\eta_{f-1-j})}s_{{\bar{\rho}},f-1-j}w_{f-1-j}\\ &\overset{\text{(9)}}= U^{(j)} v^{w'_{f-1-j}\eta_{f-1-j}}w_{f-1-j}, \end{align*}
where 
$j$ is interpreted on the right-hand sides as an element of 
$\mathbf {Z} /f$ via the natural projection. The matrices 
$A^{(j)}$ are of the form
\begin{equation} \begin{pmatrix} v & \\ x_jv & 1 \\ \end{pmatrix}, \begin{pmatrix} 1 & \\ & v \\ \end{pmatrix}, \textrm{ or } \begin{pmatrix} & 1 \\ v & x_j \\ \end{pmatrix} \end{equation}
when 
$j\neq 0$ and some matrix
\[ \begin{pmatrix} \alpha & \\ & \alpha' \\ \end{pmatrix} \]
multiplied by a matrix in (14) when 
$j=0$.
We define 
$(\overline {\mathfrak {M}}^\tau ,\iota )$ to be the Kisin module over 
$k_E$ with descent datum of type 
$\tau$ given by the matrices 
$A^{(j)}$ for a fixed eigenbasis. Then 
$\mathbf {V}_{\mathrm {dd}}^*(\overline {\mathfrak {M}}^\tau \otimes _{\mathcal {O}_L[\![u]\!]} \mathcal {O}_{\mathcal {E},K'})$ is isomorphic to 
${\bar {\rho }}$ by § 3.2 and Proposition 3.10. We are going to write down an explicit deformation of 
$\overline {\mathfrak {M}}^\tau$ to a complete Noetherian local 
$\mathcal {O}_E$-algebra 
$R^\tau _{\overline {\mathfrak {M}}^\tau }$. This will admit a formally smooth extension that will be simultaneously a formally smooth extension of the Galois deformation ring 
$R^\tau _{{\bar {\rho }}}$. Owing to this fact, we will be able to carry through most of our computations explicitly on 
$R^\tau _{\overline {\mathfrak {M}}^\tau }$.
Write 
$s$ for the nontrivial element of 
$W(\mathrm {GL}_2) = S_2$. Let 
$R^\tau _{\overline {\mathfrak {M}}^\tau }$ be the ring
\[ \mathcal{O}_E[\![(X_j,Y_j)_{j=0}^{f-1},X_\alpha,X_{\alpha'}]\!]/I(\tau), \]
where 
$I(\tau )= (f_j(\tau ))_{j=0}^{f-1}$ and
\[ f_j(\tau) =\begin{cases} Y_j &\mbox{if } (w_{f-1-j}, w'_{f-1-j}) = (\mathrm{id}, \mathrm{id}), \\ X_j &\mbox{if } (w_{f-1-j}, w'_{f-1-j}) = (\mathrm{id}, s),\\ Y_j &\mbox{if } (w_{f-1-j}, w'_{f-1-j}) = (s, s) \textrm{ and } x_j \neq 0, \\ X_jY_j-p &\mbox{if } (w_{f-1-j}, w'_{f-1-j}) = (s, s) \textrm{ and } x_j = 0. \end{cases} \]
Let 
$\mathfrak {M}^\tau$ be the Kisin module over 
$R^\tau _{\overline {\mathfrak {M}}^\tau }$ with descent datum of type 
$\tau$ given by the matrices 
$A^{\tau ,(j)}$, with respect to a fixed eigenbasis, which are defined to be
\begin{equation} \begin{cases} \begin{pmatrix} v+p & \\ (X_j+[x_j])v & 1 \end{pmatrix} &\mbox{if } A^{(j)} = \begin{pmatrix} v & \\ x_jv & 1 \end{pmatrix}, \\ \begin{pmatrix} 1 & -Y_j \\ & v+p \end{pmatrix} &\mbox{if } A^{(j)} = \begin{pmatrix} 1 & \\ & v \end{pmatrix}, \\ \begin{pmatrix} -Y_j & 1 \\ v & X_j \end{pmatrix} &\mbox{if } A^{(j)} = \begin{pmatrix} & 1 \\ v & \end{pmatrix}, \\ \begin{pmatrix} -p(X_j+[x_j])^{{-}1} & 1 \\ v & X_j+[x_j] \end{pmatrix} &\mbox{if } A^{(j)} = \begin{pmatrix} & 1 \\ v & x_j \end{pmatrix} \textrm{ and } x_j \neq 0, \end{cases} \end{equation}
when 
$j\neq 0$, and the product of the diagonal matrix
\[ \begin{pmatrix} X_\alpha +[\alpha] & \\ & X_{\alpha'}+[\alpha'] \\ \end{pmatrix} \]
with the appropriate matrix in (15) when 
$j=0$.
Let 
$R^{\tau ,\square }_{\overline {\mathfrak {M}}^\tau }$ represent the moduli problem which maps a local complete Noetherian 
$\mathcal {O}_E$-algebra 
$R$ to the set of pairs 
$(f,b)$ where 
$f: R^\tau _{\overline {\mathfrak {M}}^\tau } \rightarrow R$ is an 
$\mathcal {O}_E$-algebra homomorphism and 
$b$ is an 
$R$-basis for
\[ \mathbf{V}^*_{\mathrm{dd}}(\mathfrak{M}^\tau \otimes_{ R^\tau_{\overline{\mathfrak{M}}^\tau}, f} R \otimes_{\mathcal{O}_L[\![u]\!]} \mathcal{O}_{\mathcal{E},K'}). \]Then the natural forgetful map
\[ \mathrm{Spf} R^{\tau,\square}_{\overline{\mathfrak{M}}^\tau} \rightarrow \mathrm{Spf} R^\tau_{\overline{\mathfrak{M}}^\tau} \]
is formally smooth. We also denote by 
$I(\tau )$ the ideal 
$I(\tau )R^{\tau ,\square }_{\overline {\mathfrak {M}}^\tau } \subset R^{\tau ,\square }_{\overline {\mathfrak {M}}^\tau }$.
We now define a map 
$\kappa ^\tau : \mathrm {Spf} R^{\tau ,\square }_{\overline {\mathfrak {M}}^\tau } \rightarrow \mathrm {Spf} R^\tau _{\bar {\rho }}$. The natural map 
$\mathrm {Spf} R^\tau _{\bar {\rho }} \rightarrow \mathrm {Spf} R^\square _{{\bar {\rho }}|_{G_{L_\infty }}}$ induced by restriction is a closed embedding by [Reference Le, Le Hung, Levin and MorraLLLM18, Proposition 3.12]. The map
\begin{align*} \mathrm{Spf} R^{\tau,\square}_{\overline{\mathfrak{M}}^\tau} &\rightarrow \mathrm{Spf} R^\square_{{\bar{\rho}}|_{G_{L_\infty}}} \\ (f,b) &\mapsto (\mathbf{V}^*_{\mathrm{dd}}(\mathfrak{M}^\tau \otimes_{ R^\tau_{\overline{\mathfrak{M}}^\tau}, f} R \otimes_{\mathcal{O}_L[\![u]\!]} \mathcal{O}_{\mathcal{E},K'}),b) \end{align*}
factors through 
$\mathrm {Spf} R^\tau _{\bar {\rho }}$ by the proof of [Reference Caraiani and LevinCL18, Corollary 5.20]. We define this factoring to be 
$\kappa ^\tau$.
Theorem 3.14 The map 
$\kappa ^\tau$ is formally smooth.
Proof. This follows from [Reference Le, Le Hung, Levin and MorraLLLM18, § 5.2 and 6] (cf. [Reference LeLe19, Proposition 3.4] for more details). One first shows that the induced map 
$\mathrm {Spf} R^{\tau ,\square }_{\overline {\mathfrak {M}}^\tau } \rightarrow \mathrm {Spf} R^\square _{\bar {\rho }}$ is the composition of a formally smooth map and a closed immersion (see, e.g., the proof of [Reference LeLe19, Theorem 3.6]). Next, one shows that the scheme-theoretic image is 
$\mathrm {Spf} R^\tau _{\bar {\rho }}$. As 
$R^{\tau ,\square }_{\overline {\mathfrak {M}}^\tau }$ and 
$R^\tau _{\bar {\rho }}$ are both 
$\mathcal {O}_E$-flat with trivial nilradicals, it suffices to show that the map
\[ \operatorname{Spec} R^{\tau,\square}_{\overline{\mathfrak{M}}^\tau}[p^{{-}1}] \rightarrow \operatorname{Spec} R^\tau_{\bar{\rho}}[p^{{-}1}] \]
is surjective on 
$\bar {E}$-points. By [Reference KisinKis09b, Proposition 2.4.8], it suffices to show that any Kisin module over 
$\mathcal {O}_{\bar {E}}$ with descent data 
$\tau$ is isomorphic to a specialization of 
$\mathfrak {M}^\tau$. This follows from the proof of [Reference LeLe19, Proposition 3.4] (there is one new case, namely 
$A^{(j)} = \big (\begin {smallmatrix} 1 & \\ & v\end {smallmatrix}\big )$, which is easily handled).
3.5.2 Comparing deformation rings
Suppose that 
$\tau$ and 
$\tau _*$ are tame types such that 
$\sigma (\tau ) = R_{s_{\bar {\rho }} w}(\mu -s_{\bar {\rho }} w' \eta )$ and 
$\sigma (\tau _*) = R_{s_{\bar {\rho }} w_*}(\mu -s_{\bar {\rho }} w'_* \eta )$ are as in Proposition 3.11 so that, in particular, 
$(w_j,w'_j)$ and 
$(w_{*,j},w'_{*,j})$ are not equal to 
$(s,\mathrm {id})$ for all 
$j\in \mathbf {Z}/f$. Suppose furthermore that
\begin{equation} \text{for all}\ j \in \mathbf{Z}/f,\ (w_j,w'_j)\neq (w_{*,j},w'_{*,j})\ \text{implies that}\ w_j \neq \mathrm{id}. \end{equation}Lemma 3.15 Condition (16) is equivalent to the fact that
\[ W({\bar{\rho}}) \cap \mathrm{JH}(\bar \tau_*) \subseteq W({\bar{\rho}}) \cap \mathrm{JH}(\bar \tau). \]Proof. This is an easy consequence of Proposition 3.11.
By Lemma 3.15 and [Reference Emerton, Gee and SavittEGS15, Theorem 7.1.1], there is a closed embedding
\[ \operatorname{Spec} R^{\tau_*}_{\bar{\rho}} / p \to \operatorname{Spec} R^\tau_{\bar{\rho}} / p. \]On the other hand, the definition of our family of Kisin modules implies that
\[ I(\tau)+p\mathcal{O}_E[\![X_j, Y_j, X_\alpha, X_{\alpha'}]\!] \subset I(\tau_*)+p\mathcal{O}_E[\![X_j, Y_j, X_\alpha, X_{\alpha'}]\!]. \]This induces a surjection
\begin{equation} R^\tau_{\overline{\mathfrak{M}}^\tau}/p \rightarrow R^{\tau_*}_{\overline{\mathfrak{M}}^{\tau_*}}/p. \end{equation}Proposition 3.16 The following diagram commutes.
Proof. It suffices to show that the map (17) commutes with the maps to 
$\mathrm {Spf} R_{{\bar {\rho }}|_{G_{L_\infty }}}\!.$ This would follow from an isomorphism
\[ (\mathfrak{M}^\tau/p \otimes_{R^\tau_{\overline{\mathfrak{M}}^\tau}/p} R^{\tau_*}_{\overline{\mathfrak{M}}^{\tau_*}}/p \otimes_{\mathcal{O}_{L'}[\![u]\!]} \mathcal{O}_{\mathcal{E},K'})^{\Delta'} \cong (\mathfrak{M}^{\tau_*}/p \otimes_{\mathcal{O}_{L_*'}[\![u]\!]} \mathcal{O}_{\mathcal{E},K_*'})^{\Delta_*'}. \]
For 
$j\in \mathbf {Z}/f$, let 
$A^{\tau ,(j)}$ and 
$A^{\tau _*,(j)}$ be the matrices used to define 
$\mathfrak {M}^\tau$ and 
$\mathfrak {M}^{\tau _*}$, respectively, as before. For 
$j\in \mathbf {Z}/f$, let 
$B^{\tau ,(j)}$ and 
$B^{\tau _*,(j)}$ be the matrices for 
$\mathfrak {M}^\tau \otimes _{\mathcal {O}_{L'}[\![u]\!]} \mathcal {O}_{\mathcal {E},K'}$ and 
$\mathfrak {M}^{\tau _*} \otimes _{\mathcal {O}_{L'}[\![u]\!]} \mathcal {O}_{\mathcal {E},K'_*}$, respectively, defined as in Proposition 3.10. Then, by Proposition 3.10, it suffices to show that
\begin{equation} B^{\tau,(j)} \equiv B^{\tau_*,(j)} \pmod{I(\tau_*)+(p)}. \end{equation}
If 
$(w_{f-1-j},w'_{f-1-j}) = (w_{*,{f-1-j}},w'_{*,{f-1-j}})$, then it is easy to see that 
$A^{\tau ,(j)}=A^{\tau _*,(j)}$ and 
$B^{\tau ,(j)} = B^{\tau _*,(j)}$.
Now suppose that 
$(w_{f-1-j},w'_{f-1-j}) \neq (w_{*,{f-1-j}},w'_{*,{f-1-j}})$. We check the (18) in cases. If 
$x_j \neq 0$, then
\begin{align*} B^{\tau,(j)} &= \begin{pmatrix} v^{\mu_{1,f-1-j}} & -p(X_j+[x_j])^{{-}1}v^{\mu_{2,f-1-j}-1} \\ (X_j+[x_j])v^{\mu_{1,f-1-j}} & v^{\mu_{2,f-1-j}} \\ \end{pmatrix} ,\\ B^{\tau_*,(j)} &= \begin{pmatrix} (v+p)v^{\mu_{1,f-1-j}-1} & \\ (X_j+[x_j])v^{\mu_{1,f-1-j}} & v^{\mu_{2,f-1-j}} \\ \end{pmatrix}. \end{align*}
If 
$x_j = 0$, then
\[ B^{\tau,(j)} = \begin{pmatrix} v^{\mu_{1,f-1-j}} & -Y_jv^{\mu_{2,f-1-j}-1} \\ X_jv^{\mu_{1,f-1-j}} & v^{\mu_{2,f-1-j}} \\ \end{pmatrix} \]
and 
$B^{\tau _*,(j)}$ is either
\[ \begin{pmatrix} (v+p)v^{\mu_{1,f-1-j}-1} & \\ X_jv^{\mu_{1,f-1-j}} & v^{\mu_{2,f-1-j}} \\ \end{pmatrix} \quad \textrm{or} \quad \begin{pmatrix} v^{\mu_{1,f-1-j}} & -Y_jv^{\mu_{1,f-1-j}} \\ & (v+p)v^{\mu_{2,f-1-j}-1} \\ \end{pmatrix}. \]Then (18) is easy to check and the proposition follows.
3.5.3 Ideals and Serre weights
Recall the definition of the set 
$J_{\bar {\rho }}$ from § 3.3. In addition, recall the definition of 
$\sigma _J \in W({\bar {\rho }})$ for a subset 
$J$ of 
$J_{\bar {\rho }}$. Let 
$I(\sigma _J)$ be the ideal generated by 
$(\varpi _E,\{f_j(J)\}_j)$ in either 
$R^{\tau }_{\overline {\mathfrak {M}}^\tau }$ or 
$R^{\tau ,\square }_{\overline {\mathfrak {M}}^\tau }$ where
\[ f_j(J) =\begin{cases} X_j &\mbox{if } f-1-j \in J\\ Y_j &\mbox{if } f-1-j \notin J. \end{cases} \]Proposition 3.17 Suppose that 
$\sigma \in W({\bar {\rho }})$ and 
$\tau$ is a tame type such that 
$\sigma (\tau )$ is isomorphic to 
$R_{s_{\bar {\rho }} w}(\mu - s_{\bar {\rho }} w'\eta )$ with 
$w$ and 
$w' \in W$ as in Proposition 3.11. Then 
$I(\tau ) \subset I(\sigma )$ if and only if 
$\sigma \in \mathrm {JH}(\overline {\sigma (\tau )})$.
Proof. Suppose that 
$\sigma (\tau )$ is as before. Then 
$\sigma _J \in \mathrm {JH}(\overline {\sigma (\tau )})$ is equivalent to the following statement by Proposition 2.15. For all 
$j\in \mathbf {Z}/f$:
•
$f-1-j \in J$ implies that $(w_{f-1-j},w'_{f-1-j}) \neq (\mathrm {id},\mathrm {id})$; and•
$f-1-j \notin J$ implies that $(w_{f-1-j},w'_{f-1-j}) \neq (\mathrm {id},s)$.
(Compare with the proof of Proposition 3.11.) Now suppose that 
$\sigma = \sigma _J$ so that, in particular, 
$f-1-j\in J$ implies that 
$f-1-j \in J_{\bar {\rho }}$. Then this is equivalent to the fact that for all 
$j\in \mathbf {Z}/f$, 
$f_j(\tau ) \subset I(\sigma _J)$.
3.6 Local–global compatibility in families
We first recall the local Langlands correspondence for 
$\mathrm {GL}_n(L)$ from the introduction of [Reference Harris and TaylorHT01]. Fix an isomorphism 
$\iota : \bar {\mathbf {Q}}_p \rightarrow \mathbf {C}$. Then we can define a local Langlands correspondence 
$\mathrm {rec}_p$ for 
$\mathrm {GL}_n(L)$ over 
$\bar {\mathbf {Q}}_p$ by requiring that 
$\iota \circ \mathrm {rec}_p = \mathrm {rec} \circ \iota$ where 
$\mathrm {rec}$ defined in [Reference Harris and TaylorHT01] gives a bijection from isomorphism classes of smooth irreducible representations of 
$\mathrm {GL}_n(L)$ over 
$\mathbf {C}$ to Frobenius semisimple Weil–Deligne representations of the Weil group 
$W_L$ over 
$\mathbf {C}$. (More specifically, 
$\mathrm {rec}$ is the correspondence over 
$\mathbf {C}$ that preserves 
$L$-functions and 
$\epsilon$-factors.) We define 
$r_p(\pi )$ to be 
$\mathrm {rec}_p(\pi \otimes |\det |^{(n-1)/2})$. Then 
$r_p$ does not depend on the choice of 
$\iota$. (The definition of 
$r_p$ differs from that in [Reference Caraiani, Emerton, Gee, Geraghty, Paškūnas and ShinCEG+16] by a twist.)
Let 
$\sigma$ be the smooth 
$\sigma (\tau )$. Let 
$\mathcal {H}(\sigma )$ denote the Hecke algebra
\[ \mathrm{End}_{\mathrm{GL}_2(L)}\big(\mathrm{ind}_{\mathrm{GL}_2(\mathcal{O}_L)}^{\mathrm{GL}_2(L)} \sigma\big) \]
for 
$\sigma$. We write
\[ \eta: \mathcal{H}(\sigma) \rightarrow R^{\tau}_{\bar{\rho}}[p^{{-}1}] \]
for the map from [Reference Caraiani, Emerton, Gee, Geraghty, Paškūnas and ShinCEG+16, Theorem 4.1], but normalized using 
$r_p$ defined previously.
We now specialize to the case when 
$\sigma$ is 
$\theta (\chi )_E$, the principal series tame type of 
$\mathrm {GL}_2(L)$ over 
$E$ given by induction of an 
$\mathcal {O}_E$-character 
$\chi = \chi _1\otimes \chi _2$ of the Iwahori subgroup 
$I$ to 
$\mathrm {GL}_2(\mathcal {O}_L)$. Suppose that 
$\tau$ is the principal series tame inertial type such that 
$\sigma (\tau ) \cong \theta (\chi )_E$. Recall that 
$U_p$ is the element of 
$\mathcal {H}(\theta (\chi )_E)$ associated to the 
$I$-double coset of 
$\big (\begin {smallmatrix} p & 0 \\ 0 & 1 \end {smallmatrix}\big )$. Suppose that 
$x\in \operatorname {Spec} R^\tau _{\bar {\rho }}(\bar {E})$ gives a Galois representation 
$\rho _x:G_L \rightarrow \mathrm {GL}_2(\bar {E})$. We have that 
$\mathrm {WD}(\rho _x)^{F\text {-ss}}$ is of the form 
$(\widetilde {\chi }_1 \oplus \widetilde {\chi }_2) \circ \mathrm {Art}_L^{-1}$ (and 
$N=0$) where 
$\widetilde {\chi }_j: L^\times \rightarrow \bar {E}^\times$ is an extension of 
$\chi _j$ for 
$j =1$ and 
$2$. Let 
$\pi _x$ be 
$r_p^{-1}(\mathrm {WD}(\rho _x)^{F\text {-ss}})$ so that 
$\pi _x$ is isomorphic to 
$\mathrm {Ind}_{B(L)}^{\mathrm {GL}_2(L)} \widetilde {\chi }_1 \otimes \widetilde {\chi }_2 |\cdot |^{-1}$. As in the proof of Lemma 2.14, 
$U_p$ acts on 
$\pi _x^{I,\chi _1}$ by 
$q\widetilde {\chi }_1(p)$. Theorem 4.1 of [Reference Caraiani, Emerton, Gee, Geraghty, Paškūnas and ShinCEG+16] implies that 
$\eta (U_p)$ at 
$x$ is 
$p^f$ times the 
$\mathrm {Art}_L(p)$-eigenvalue on the 
$\chi _1$-isotypic line.
Let 
$\eta '$ be the composition
\[ \mathcal{H}(\sigma) \overset{\eta}{\rightarrow} R^{\tau}_{\bar{\rho}}[p^{{-}1}] \overset{\kappa^\tau}{\rightarrow} R^{\tau,\square}_{\overline{\mathfrak{M}}^\tau}[p^{{-}1}]. \]Proposition 3.18 We have
\begin{equation} \eta'(U_p) = \prod_j p\Big(x^{(j)}_{s_{j+1}^{{-}1}(1)s_{j+1}^{{-}1}(1)}\Big)^{{-}1} \pmod{v}, \end{equation}
where 
$A^{\sigma , (j)}=(x^{(j)}_{ik})_{ik}$ gives 
$\mathfrak {M}^\tau$ as in § 3.5.1.
Proof. Let 
$\mathfrak {M}$ correspond to an 
$\bar {E}$-point of 
$\operatorname {Spec} R^{\tau ,\square }_{\overline {\mathfrak {M}}^\tau }[p^{-1}]$ and set
\[ \rho = \mathbf{V}^*_{\mathrm{dd}}(\mathfrak{M}\otimes_{\mathcal{O}_L[\![u]\!]} \mathcal{O}_{\mathcal{E},K}). \]
By Proposition 3.6 and Corollaries 3.7 and 3.9, 
$\varphi$ acts on the 
$\chi _1^{-1}$-isotypic line of 
$\mathfrak {M}/u \cong D_{\mathrm {pst}}(\rho ^\vee )$ by 
$\prod _j x^{(j)}_{s_{j+1}^{-1}(1)s_{j+1}^{-1}(1)} \pmod {v}$. By the definition of 
$\mathrm {WD}(-)$ in [Reference FontaineFon90], 
$\mathrm {Art}_L(p)$ acts on the 
$\chi _1^{-1}$-isotypic line in 
$\mathrm {WD}(\rho ^\vee )$ by 
$\prod _j x^{(j)}_{s_{j+1}^{-1}(1)s_{j+1}^{-1}(1)} \pmod {v}$. Then because 
$\mathrm {Art}_L(p)$ acts on the 
$\chi _1$-isotypic line in 
$\mathrm {WD}(\rho )$ by 
$\prod _j (x^{(j)}_{s_{j+1}^{-1}(1)s_{j+1}^{-1}(1)})^{-1} \pmod {v}$, 
$\eta '(U_p) = \prod _j p(x^{(j)}_{s_{j+1}^{-1}(1)s_{j+1}^{-1}(1)})^{-1} \pmod {v}$ for any 
$\bar {E}$-point of 
$\operatorname {Spec} R^{\tau ,\square }_{\overline {\mathfrak {M}}^\tau }[p^{-1}]$. The result now follows from the fact that 
$R^{\tau ,\square }_{\overline {\mathfrak {M}}^\tau }[p^{-1}]$ is a reduced Jacobson ring whose maximal ideals have residue field of finite dimension over 
$E$. (These properties are inherited from 
$R_{\bar {\rho }}^\tau [p^{-1}]$, for example.)
Proposition 3.19 With the notation given previously, 
$p(x^{(j)}_{s_{j+1}^{-1}(1)s_{j+1}^{-1}(1)})^{-1} \pmod {v}$ is one of
\begin{equation} 1,\ p,\ -X_j,\ Y_j,\ -( X_j+[x_j]),\quad \textrm{and}\quad p(X_j+[x_j])^{{-}1} \end{equation}
if 
$j \neq 0$ and the product of one of 
$(X_\alpha +[\alpha ])^{-1}$ and 
$(X_{\alpha '}+[\alpha '])^{-1}$ and one of (20) if 
$j=0$. In particular, 
$\eta '(U_p) \in R^{\tau ,\square }_{\overline {\mathfrak {M}}^\sigma }$.
Proof. This follows by inspection of (15).
Remark 3.20 Note that the last two possibilities 
$-(X_j+[x_j])$ and 
$p(X_j+[x_j])^{-1}$ can be removed if 
${\bar {\rho }}$ is semisimple.
3.6.1 Comparison with the central inertial type
Definition 3.21 If 
$w_0\in W$ is the longest element (nontrivial in all embeddings), then we define the central inertial type of 
${\bar {\rho }}$ to be the tame inertial type 
$\tau$ such that 
$\sigma (\tau ) = R_{s_{{\bar {\rho }}}w_0}(\mu -s_{{\bar {\rho }}}w_0\eta )$.
By Proposition 3.11, we have that 
$\mathrm {JH}(\overline {\sigma (\tau )})$ coincides with 
$W({\bar {\rho }}^{\mathrm {ss}})$, which contains 
$W({\bar {\rho }})$. Hence, whenever 
$\sigma = \sigma _\mathrm {sm} = \theta (\chi )_E$ is a principal series tame type as before, and 
$\tau (\chi )$ is the principal series tame inertial type such that 
$\sigma (\tau (\chi )) \cong \theta (\chi )_E$, the condition (16) holds and there exists a commutative diagram as in Proposition 3.16 (substituting 
$\tau (\chi )$ for 
$\tau _*$). We are going to define an element of 
$R^{\tau ,\square }_{\overline {\mathfrak {M}}^\tau }$ that specializes to a normalized version of 
$\eta '(U_p)$.
Definition 3.22 Using Proposition 3.19, let 
$\widetilde {U}'_p$ be the product 
$\prod _j \widetilde {x}^{(j)} \in R^{\tau ,\square }_{\overline {\mathfrak {M}}^\tau }$, where
\begin{equation} \widetilde{x}^{(j)} = 1, \text{ respectively, $1$}, Y_j, -X_j, (X_j+[x_j])^{{-}1},-(X_j+[x_j]) \end{equation}if
\[ x^{(j)}_{s_{j+1}^{{-}1}(1)s_{j+1}^{{-}1}(1)} = 1, \text{ respectively, $p$}, X_j, -Y_j, X_j+[x_j], -p(X_j+[x_j])^{{-}1} \]
if 
$j \neq 0$ and the product of the appropriate one of 
$(X_\alpha +[\alpha ])^{-1}$ and 
$(X_{\alpha '}+[\alpha '])^{-1}$ and one of (21) if 
$j=0$.
Again, 
$\widetilde {x}^{(j)}$ cannot be 
$(X_j+[x_j])^{-1}$ or 
$-(X_j+[x_j])$ if 
${\bar {\rho }}$ is semisimple. Note that 
$p$ does not divide 
$\widetilde {U}'_p$.
Proposition 3.23 Let 
$e(\chi )\in \mathbf {N}$ be such that 
$p^{-e(\chi )} \eta '(U_p)\in R^{\sigma ,\square }_{\overline {\mathfrak {M}}^\sigma }$, but is not divisible by 
$p$. Then the image of 
$\widetilde {U}'_p$ in 
$R^{\sigma ,\square }_{\overline {\mathfrak {M}}^\sigma }/p$ under the map (17) is the image of 
$p^{-e(\chi )} \eta '(U_p)$.
Proof. This is a direct consequence of Proposition 3.18.
3.6.2 Fixed determinant deformation rings
In applications, it is often necessary to use deformation rings parameterizing lifts with fixed determinant. Let 
${\bar {\rho }}$ be as in § 3.2.2. Let 
$\psi : G_L \rightarrow \mathcal {O}_E^\times$ be a character lifting 
$\det {\bar {\rho }} \overline {\varepsilon }^{-1}$. For an inertial type 
$\tau$, let 
$R_{\bar {\rho }}^{\psi ,\tau }$ be the quotient of 
$R_{\bar {\rho }}^\tau$ parameterizing lifts of 
${\bar {\rho }}$ of Galois type 
$\tau$, parallel Hodge–Tate weights 
$(0,1)$, and determinant 
$\psi \varepsilon$. This quotient is nonzero if and only if 
$\psi |_{I_L} \cong \det \tau$. If 
$\tau$ is a tame inertial type with 
$R^\tau _{\bar {\rho }}$ nonzero, we similarly define 
$R^{\psi ,\tau }_{\overline {\mathfrak {M}}^\tau }$ and 
$R^{\psi ,\tau ,\square }_{\overline {\mathfrak {M}}^\tau }$. We let 
$I^\psi (\sigma )$ denote the image of 
$I(\sigma )$ in 
$R^{\psi ,\tau }_{\overline {\mathfrak {M}}^\tau }$ and 
$R^{\psi ,\tau ,\square }_{\overline {\mathfrak {M}}^\tau }$. Let 
$\widetilde {U}^{\psi ,\prime }_p$ be the image of 
$\widetilde {U}_p'$ in 
$R^{\psi ,\tau ,\square }_{\overline {\mathfrak {M}}^\tau }$. Then the fixed determinant analogues of Propositions 3.16 and 3.23 hold compatibly with the natural quotient maps from deformation rings to fixed determinant deformation rings.
4. Diagrams, patching, and groupoids
4.1 Some generalities on diagrams
We define a 
$0$-diagram to be a triple 
$(D_0,D_1,r)$ where 
$D_0$ is a 
$L^\times \mathrm {GL}_2(\mathcal {O}_L)$-representation with central character, 
$D_1$ is an 
$N(I)$-representation (where 
$N(I)$ is the normalizer in 
$\mathrm {GL}_2(L)$ of 
$I$), and 
$r: D_1|_{L^\times I} \hookrightarrow D_0|_{L^\times I}$ is an injection that induces an isomorphism 
$D_1 \cong D_0^{I_1}$ (we have weaker hypotheses on the central character than [Reference Breuil and PaškūnasBP12, § 9]). We consider only diagrams over 
$k_E$. We assume throughout that the 
$K_1$-action on 
$D_0$ is trivial.
Let 
${\bar {\rho }}: G_L \to \mathrm {GL}_2(k_E)$ be generic and let 
$D_0 = D_0({\bar {\rho }})$ be the 
$\mathrm {GL}_2(k_L)$-representation over 
$k_E$ defined in [Reference Breuil and PaškūnasBP12, § 13] (where it is defined over 
$\bar {\mathbf {F}}_p$, but descends to 
$k_L$ and, in particular, 
$k_E$ via 
$\sigma _0$). Let 
$D_1^\circ$ be 
$D_0^{I_1}$. Recall that the Jordan–Hölder factors of 
$D_0$ appear with multiplicity one by [Reference Breuil and PaškūnasBP12, Theorem 1.1(ii)] so that in particular the isotypic space 
$D_1^\circ [\chi ]$ has dimension at most one.
To give a diagram 
$(D_0,D_1,\mathrm {can})$ such that 
$D_1|_I = D_1^\circ$ and 
$\mathrm {can}$ is the inclusion 
$D_1^\circ \hookrightarrow D_0$ is equivalent to extending the action of 
$I$ on 
$D_1^\circ$ to an action of the normalizer 
$N(I)$ of 
$I$. Such an extension is equivalent to giving a map 
$\Pi : D_1^\circ \rightarrow D_1^\circ$ taking 
$D_1^\circ [\chi ]$ to 
$D_1^\circ [\chi ^s]$ for all 
$\chi$, and such that 
$\Pi ^2: D_1^\circ \rightarrow D_1^\circ$ acts by a nonzero scalar.
By definition, a morphism of diagrams consists of a pair 
$(\psi _0, \psi _1)$ of morphisms, equivariant for 
$L^\times \mathrm {GL}_2(\mathcal {O}_L)$ and 
$N(I)$, respectively, that commute with the inclusion 
$r$. It follows that every diagram is isomorphic to one for which 
$D_1 = D_1^\circ$ and 
$r = \mathrm {can}$. We introduce an invariant of a diagram that classifies it up to isomorphism and will be accessible to computation.
Definition 4.1 Let 
$R: D_0^{I_1} \rightarrow (\operatorname {soc}_K D_0)^{I_1}$ be the unique map so that if 
$D_0^{I_1}[\chi ]$ is nonzero, then 
$R|_{D_0^{I_1}[\chi ]}$ is nonzero and given by 
$S_{i(\chi )}$ for some 
$0 \leq i(\chi ) \leq q-1$, except when 
$\chi$ appears in 
$(\operatorname {soc}_K D_0)^{I_1}$, in which case 
$R|_{(\operatorname {soc}_K D_0)^{I_1}[\chi ]}$ is the identity. (The existence and uniqueness of 
$R$ follow from [Reference Breuil and PaškūnasBP12, Lemma 2.7], compare also with Lemma 2.10.) Given 
$\chi$, we write 
$R\chi$ for the character such that 
$R(D_0^{I_1}[\chi ]) = (\operatorname {soc}_K D_0)^{I_1}[R\chi ]$.
Remark 4.2 In the previous definition, the operator 
$S_{i(\chi )}$ is unique whenever 
$\chi$ is not contained in 
$(\operatorname {soc}_K D_0)^{I_1}$. This is because in most cases 
$R\chi$ has multiplicity one in the 
$\mathrm {GL}_2(\mathcal {O}_L)$-representation generated by 
$\chi$, and the only exception is when 
$R\chi = \chi$ or 
$R\chi = \chi ^s$. The first case is excluded because we are assuming 
$(\operatorname {soc}_K D_0)^{I_1}[\chi ] = 0$. In the second case, only 
$S_0$ preserves 
$I_1$-invariants. Thus, we can attach a uniquely determined integer 
$i(\chi )$ to any character 
$\chi \in D_0^{I_1}$ not appearing in 
$(\operatorname {soc}_K D_0)^{I_1}$. This integer has appeared before: relative to the tame principal series type 
$\mathrm {Ind}_I^K[\chi ]$, it is what we called 
$i(\chi , R\chi )$ in Lemma 2.10. To shorten notation, we continue writing 
$i(\chi )$ for this integer, and when 
$\chi = R\chi$ we write 
$S_{i(\chi )}$ for the identity map.
Definition 4.3 We write 
$\mathcal {G}$ for the groupoid with an object 
$\mathbf {x}_\xi$ for each character such that 
$(\operatorname {soc}_K D_0)^{I_1}[\xi ]$ is nonzero, and morphisms freely generated by 
$g_\chi : \mathbf {x}_{R\chi } \to \mathbf {x}_{R\chi ^s}$ for each nonzero 
$D_ 0^{I_1}[\chi ]$.
Let 
$D = (D_0,D_1,\mathrm {can})$ be a diagram such that 
$D_1|_I = D_1^\circ$. Then we can construct a 
$1$-dimensional representation (i.e. a character) 
$\chi _D$ of 
$\mathcal {G}$ as follows: the objects 
$\mathbf {x}_\xi$ go to 
$(\operatorname {soc}_K D_0)^{I_1}[\xi ]$ and the morphisms 
$g_\chi$ go to maps
\[ (\operatorname{soc}_K D_0)^{I_1}[R\chi] \rightarrow (\operatorname{soc}_K D_0)^{I_1}[R\chi^s] \]
defined by 
$g_\chi (R(v)) = R(\Pi v)$.
Proposition 4.4 The diagram 
$D$ can be recovered from 
$\chi _D$.
Proof. This is clear from the fact that 
$R: D_1^\circ [\chi ] \rightarrow D_1^\circ [R\chi ]$ is an isomorphism for all 
$\chi$, hence 
$\Pi$ is the only map that makes all the following diagrams commute.
From a representation of a groupoid 
$\mathcal {G}$, one obtains by restriction a representation of the group 
$\mathcal {G}_\mathbf {x}$ of automorphisms of any object 
$\mathbf {x}$ of 
$\mathcal {G}$. The following result is standard.
Proposition 4.5 Suppose that 
$X$ is a set of representatives for 
$\pi _0(\mathcal {G})$. The product of restriction functors
\[ \mathrm{Rep}_{\mathcal{G}} \rightarrow \prod_{\mathbf{x} \in X} \mathrm{Rep}_{\mathcal{G}_\mathbf{x}} \]is an equivalence of categories.
Proof. Consider the category 
$G$ whose object set is 
$X$ and 
$\mathrm {Hom}_G(\mathbf {x}, \mathbf {x}) = \mathcal {G}_{\mathbf {x}}$ for all 
$\mathbf {x} \in X$, while 
$\mathrm {Hom}_G(\mathbf {x}_1, \mathbf {x}_2)$ is empty if 
$\mathbf {x}_1 \ne \mathbf {x}_2$. The inclusion functor 
$G \to \mathcal {G}$ is fully faithful and essentially surjective, hence it is an equivalence of categories. The proposition follows because a representation of 
$\mathcal {G}$ is a functor from 
$\mathcal {G}$ to vector spaces.
Let 
$\chi _0, \ldots , \chi _{k-1}$ be characters such that 
$R\chi _i^s = R\chi _{i+1}$ for each 
$i$ (where we set 
$\chi _k$ to be 
$\chi _0$). Then there is a map
\[ g_{\chi_{k-1}} \circ \cdots \circ g_{\chi_0}: (\operatorname{soc}_K D_0)^{I_1}[R\chi_0] \rightarrow (\operatorname{soc}_K D_0)^{I_1}[R\chi_0]. \]
By Proposition 4.5, if one determines the constants giving all such maps, one determines the isomorphism class of 
$D$.
4.2 Diamond diagrams from patched modules
There is a special class of diagrams, called Diamond diagrams, attached to many suitable globalizations of 
${\bar {\rho }}$: for example, representations of 
$\mathrm {Gal}(\bar {F}/F)$, for 
$F/\mathbf {Q}$ a totally real field, that are modular for a totally definite (or split at a precisely one real place) quaternion algebra over 
$F$, and that recover 
${\bar {\rho }}$ by restricting to decomposition groups (see § 6.1). In §§ 4 and 5, we work in the more general context of 
$\mathcal {O}_E[\mathrm {GL}_2(L)]$-modules with an arithmetic action, in the sense of [Reference Gee and NewtonGN20, § 5.2] (see also [Reference Caraiani, Emerton, Gee, Geraghty, Paškūnas and ShinCEG+18, § 3.1]), that recover these examples through Taylor–Wiles–Kisin patching (see § 6.2). In this section, we recall the definition of 
$\mathcal {O}_E[\mathrm {GL}_2(L)]$-modules with an arithmetic action and state our first main theorem.
Let 
${\bar {\rho }}: G_L \rightarrow \mathrm {GL}_2(k_E)$ be a local Galois representation. Recall from § 3.5 the following notation. The ring 
$R_{{\bar {\rho }}}$ is the framed unrestricted 
$\mathcal {O}_E$-deformation ring for 
${\bar {\rho }}$. If 
$\psi : G_L \rightarrow \mathcal {O}_E^\times$ is a lift of 
$\det {\bar {\rho }}\overline {\varepsilon }^{-1}$, then 
$R^\psi _{\bar {\rho }}$ is the quotient of 
$R_{\bar {\rho }}$ parameterizing lifts of 
${\bar {\rho }}$ with determinant 
$\psi \varepsilon$. We assume throughout that 
$\psi$ factors through 
$W(k_E)^\times$. For an inertial type 
$\tau$, recall that 
$R_{{\bar {\rho }}}^{\tau }$ is the quotient of 
$R_{{\bar {\rho }}}$ corresponding to potentially crystalline lifts of 
${\bar {\rho }}$ of Galois type 
$\tau$ and Hodge–Tate weight 
$(0,1)$. Fix a natural number 
$h$ and let 
$R_\infty$ be 
$R_{{\bar {\rho }}}\widehat {\otimes }_{\mathcal {O}_E} \mathcal {O}_E[\![x_1,x_2,\ldots , x_h]\!]$ and 
$R_\infty ^\psi$ be 
$R^\psi _{{\bar {\rho }}}\widehat {\otimes }_{\mathcal {O}_E} \mathcal {O}_E[\![x_1,x_2,\ldots , x_h]\!]$. Let 
$R_\infty (\tau )$ be 
$R_{{\bar {\rho }}}^{\tau } \otimes _{R_{{\bar {\rho }}}} R_\infty$ and 
$R_\infty ^\psi (\tau )$ be 
$R_{{\bar {\rho }}}^{\psi ,\tau } \otimes _{R_{{\bar {\rho }}}^\psi } R_\infty ^\psi$.
We consider the case where the determinant is not fixed (the necessary modifications for the fixed determinant case are straightforward). By an 
$\mathcal {O}_E[\mathrm {GL}_2(L)]$-module 
$M_\infty$ with an arithmetic action of 
$R_\infty$ we mean a nonzero 
$\mathcal {O}_E$-module with commuting actions of 
$R_\infty$ and a right action of 
$\mathrm {GL}_2(L)$ satisfying the following properties.
(i) The module
$M_\infty$ is a finitely generated right $R_\infty [\![\mathrm {GL}_2(\mathcal {O}_L)]\!]$-module.(ii) The module
$M_\infty$ is projective in the category of pseudocompact right $\mathcal {O}_E[\![\mathrm {GL}_2(\mathcal {O}_L)]\!]$-modules.
For a finite 
$\mathcal {O}_E[\![\mathrm {GL}_2(\mathcal {O}_L)]\!]$-module 
$\sigma$, we let 
$M_\infty (\sigma )$ be
\[ M_\infty\otimes_{\mathcal{O}_E[\![\mathrm{GL}_2(\mathcal{O}_L)]\!]} \sigma. \](iii) If
$\theta$ is an $\mathcal {O}_E$-lattice in a locally algebraic type $\sigma (\tau )$, then $M_\infty (\theta )$ is maximal Cohen–Macaulay over $R_\infty (\tau )$.(iv) Moreover, the action of
$\mathcal {H}(\theta )$ on $M_\infty (\theta )[1/p]$ is given by the composite
with\[ \mathcal{H}(\theta) \overset{\eta}{\rightarrow} R_{{\bar{\rho}}}^{\tau}[1/p] \rightarrow R_\infty(\tau)[1/p] \]$\eta$ as in § 3.6.
In the fixed determinant case, we furthermore require the following.
(v) The center
$L^\times$ of $\mathrm {GL}_2(L)$ acts on $M_\infty$ through the fixed determinant character $\psi : L^\times \rightarrow \mathcal {O}_E^\times$.
Then, in particular, 
$M_\infty (-)$ is a (fixed determinant) patching functor in the sense of [Reference Emerton, Gee and SavittEGS15, Definition 6.1.3]. This gives access to results in [Reference Emerton, Gee and SavittEGS15, § 8-10]. For example, if 
$\sigma$ is a Serre weight, then 
$M_\infty (\sigma ) \neq 0$ if and only if 
$\sigma \in W({\bar {\rho }})$ by [Reference Emerton, Gee and SavittEGS15, Theorem 9.1.1]. We say further that an 
$\mathcal {O}_E[\mathrm {GL}_2(L)]$-module 
$M_\infty$ with an arithmetic action of 
$R_\infty$ is minimal if the following holds.
(vi) For
$\theta$ as previously, $\dim _{\bar {E}}M_\infty (\theta ) \otimes _{R_\infty (\tau )} \bar {E}$ is at most one for any $\bar {E}$-point of $R_\infty (\tau )$.
In this case, 
$M_\infty (-)$ is a minimal (fixed determinant) patching functor in the sense of [Reference Emerton, Gee and SavittEGS15, § 6.1].
Let us elaborate on the action of 
$\mathcal {H}(\theta )$ on 
$M_\infty (\theta )[1/p]$. In fact, 
$\mathcal {H}(\theta )$ acts on 
$M_\infty (\theta )$ by the identification of the latter with 
$\mathrm {Hom}_{\mathrm {GL}_2(\mathcal {O}_L)}(\theta , M_\infty ^\vee )^\vee$ in the following lemma and Frobenius reciprocity. Note that 
$M_\infty ^\vee$ inherits a natural left action of 
$\mathrm {GL}_2(L)$.
Lemma 4.6 Let 
$\theta$ be a finite 
$\mathcal {O}_E[\![\mathrm {GL}_2(\mathcal {O}_L)]\!]$-module. Then 
$M_\infty (\theta )$ is Pontrjagin dual to
\[ \mathrm{Hom}_{\mathrm{GL}_2(\mathcal{O}_L)}(\theta, M_\infty^\vee). \]Proof. This follows from [Reference Gee and NewtonGN20, Lemma B.3].
The following theorem, which essentially follows from [Reference Emerton, Gee and SavittEGS15, Theorem 10.1.1], will be fundamental to our method.
Theorem 4.7 Suppose that 
$M_\infty$ is a minimal 
$\mathcal {O}_E[\mathrm {GL}_2(L)]$-module with an arithmetic action of 
$R_\infty$. Let 
$\tau$ be a nonscalar tame inertial type such that 
$R_{\bar {\rho }}^\tau$ is nonzero. If 
$\sigma (\tau )^\circ$ is an 
$\mathcal {O}_E$-lattice in 
$\sigma (\tau )$ with irreducible cosocle, then 
$M_\infty (\sigma (\tau )^\circ )$ is free of rank one over 
$R_\infty (\tau )$. If 
$\psi$ factors through 
$W(k_E)^\times$, then the analogous fixed determinant result holds as well.
Proof. If 
$E$ is an unramified extension of 
$\mathbf {Q}_p$, then this follows from [Reference Emerton, Gee and SavittEGS15, Theorem 10.1.1]. We begin with the following lemma. Let 
$R_{\infty ,0}$ and 
$R_{\infty ,0}(\tau )$ be defined similarly to 
$R_\infty$ and 
$R_\infty (\tau )$, respectively, with the Witt vectors 
$W(k_E)$ in place of 
$\mathcal {O}_E$.
Lemma 4.8 Suppose that 
$M_\infty$ is an 
$\mathcal {O}_E[\mathrm {GL}_2(L)]$-module with an arithmetic action of 
$R_\infty$. Then the inclusion 
$W(k_E) \subset \mathcal {O}_E$ makes 
$M_\infty$ a 
$W(k_E)[\mathrm {GL}_2(L)]$-module with an arithmetic action of 
$R_{\infty ,0}$. If 
$\psi$ factors through 
$W(k_E)^\times$, then the analogous fixed determinant result holds as well.
Proof. This is straightforward with the exception of part (iii). Let 
$\mathfrak {m}_0\subset R_{\infty ,0}$ be the maximal ideal, and note that 
$R_{\infty }$ is finite free over 
$R_{\infty , 0}$. Then
\begin{align*} \mathrm{depth}_{R_{\infty,0}} M_\infty(\theta) &= \mathrm{depth}_{R_{\infty,0}} M_\infty(\theta)+\mathrm{depth}_{k_E} R_\infty/\mathfrak{m}_0 R_\infty \\ &= \mathrm{depth}_{R_\infty} M_\infty(\theta) \otimes_{R_{\infty,0}} R_\infty\\ &= \mathrm{depth}_{R_\infty} M_\infty(\theta)\\ &= \dim \mathrm{Supp}_{R_\infty} M_\infty(\theta)\\ &= \dim \mathrm{Supp}_{R_{\infty,0}} M_\infty(\theta), \end{align*}
where the first equality is true because 
$\dim k_E = 0$, the second follows from [Reference GrothendieckGro65, Proposition 6.3.1], and the third and fifth follow from the fact that 
$R_\infty$ is finite free over 
$R_{\infty ,0}$.
The only place where the proof of [Reference Emerton, Gee and SavittEGS15, Theorem 10.1.1] requires that 
$E$ is an unramified extension of 
$\mathbf {Q}_p$ is in the proof of [Reference Emerton, Gee and SavittEGS15, Lemma 10.1.12] (
$\mathcal {O}_E[\![X,Y]\!]/(XY-p)$ is a regular ring if and only if 
$E$ is unramified). However, using Lemma 4.8, the same proof shows that 
$M_\infty (\sigma (\tau )^\circ )$ is free over 
$R_{\infty ,0}(\tau )$. Now we deduce that
\[ M_\infty(\sigma(\tau)^\circ) \otimes_{R_{\infty, 0}(\tau)} R_\infty(\tau) \cong M_\infty(\sigma(\tau)^\circ) \otimes_{R_\infty(\tau)} (R_\infty(\tau) \otimes_{R_{\infty, 0}(\tau)} R_\infty(\tau)) \]
is 
$R_\infty (\tau )$-free. As 
$M_\infty (\sigma (\tau )^\circ )$ is a direct summand of the right-hand side (as 
$R_\infty (\tau )$-modules), it is finite projective over 
$R_\infty (\tau )$, hence finite free because 
$R_\infty (\tau )$ is a local ring. The rest of the proof of [Reference Emerton, Gee and SavittEGS15, Theorem 10.1.1], where key computations are done modulo 
$\varpi _E$, works without modification.
Let 
$\mathfrak {m}$ denote the maximal ideal of 
$R_\infty$.
Definition 4.9 We let 
$\pi ({\bar {\rho }}^\vee )$ be the 
$\mathrm {GL}_2(L)$-representation 
$M_\infty ^\vee [\mathfrak {m}]$ (where 
$(-)^\vee$ denotes the Pontrjagin dual). Let 
$D_0$ be 
$\pi ({\bar {\rho }}^\vee )^{K_1}$ and 
$D_1$ be 
$\pi ({\bar {\rho }}^\vee )^{I_1}$ and set 
$D = \mathcal {D}(\pi ({\bar {\rho }}^\vee )) = (D_0,D_1,\mathrm {can})$ for the inclusion 
$\mathrm {can}: D_1 \to D_0$.
Remark 4.10 We denote 
$M_\infty ^\vee [\mathfrak {m}]$ by 
$\pi ({\bar {\rho }}^\vee )$ and not 
$\pi ({\bar {\rho }})$ to emphasize that our conventions in § 6 are dual to those in [Reference BreuilBre14, § 9].
We now suppose that 
${\bar {\rho }}:G_L\rightarrow \mathrm {GL}_2(k_E)$ is generic as in § 3.2.2. Then 
$D_0$ is isomorphic to 
$D_0({\bar {\rho }})$ by (the proof of) [Reference LeLe19, Corollary 5.2].
Theorem 4.11 If 
${\bar {\rho }}$ is generic, then the diagram 
$D$ is independent of the choice of 
$M_\infty$ satisfying properties (i)–(vi).
The proof of Theorem 4.11 occupies the rest of § 4, and will proceed by determining the compositions 
$g_{\chi _{k-1}}\circ \cdots \circ g_{\chi _0}$ for every cycle of characters 
$\chi _0, \ldots , \chi _{k-1}$ such that 
$R\chi _i^s = R\chi _{i+1}$ for every 
$i\in \mathbf {Z}/k$.
Example 4.12 We describe the groupoid 
$\mathcal {G}$ when 
$L = \mathbf {Q}_{p^2}$ and 
${\bar {\rho }}$ is semisimple generic. There are always four weights in the socle, parametrized by the subsets 
$J \subseteq \{0, 1\}$ through a map 
$J \mapsto \sigma (\lambda _J, {\bar {\rho }})$ (see §§ 11 and 12). Hence, there are four objects in the groupoid 
$\mathcal {G}$, corresponding to the decomposition
\[ (\operatorname{soc}_K D_0({\bar{\rho}}))^{I_1} = \chi_1 \oplus \chi_2 \oplus \chi_3 \oplus \chi_4. \]
If 
${\bar {\rho }}$ is irreducible, by [Reference Breuil and PaškūnasBP12, Proposition 14.7] the representation 
$D_0({\bar {\rho }})^{I_1}$ has dimension eight, and the morphisms of 
$\mathcal {G}$ are generated by
\begin{align*} & g_{\chi_i} : (\operatorname{soc}_K D_0({\bar{\rho}}))^{I_1}[\chi_i] \rightarrow (\operatorname{soc}_K D_0({\bar{\rho}}))^{I_1}[\chi_{i+1}], \\ & g_{\chi_i^s} : (\operatorname{soc}_K D_0({\bar{\rho}}))^{I_1}[\chi_{i+1}] \rightarrow (\operatorname{soc}_K D_0({\bar{\rho}}))^{I_1}[\chi_i] \end{align*}
(with some ordering of the characters 
$\chi _i$). The diagram 
$D$ is determined by the central character and a single parameter in 
$k_E^\times$, corresponding to the composition 
$g_{\chi _3} \circ g_{\chi _2} \circ g_{\chi _1} \circ g_{\chi _0}$.
For a split reducible 
${\bar {\rho }}$, the dimension of 
$D_0({\bar {\rho }})^{I_1}$ is 
$10$, and hence there will be 
$10$ generating morphisms for 
$\mathcal {G}$. In this case, the diagram 
$D$ is determined by the central character and five parameters in 
$k_E^\times$ (although really one of these is determined by the central character).
4.3 Lifting $g_\chi$
The first step in the proof of Theorem 4.11 is to construct specific lifts of the maps 
$g_\chi$ (or, rather, their duals) to patched modules in characteristic zero, via composition of saturated inclusions and 
$U_p$ operators. In this section, we fix a character 
$\chi : I/I_1 \to \mathcal {O}_E^\times$ that appears in 
$D_0({\bar {\rho }})^{I_1}$. If 
$\xi$ is another character, recall from Definition 2.12 the definitions of 
$\theta (\chi )$ and 
$\theta (\chi )^\xi$ if 
$\sigma (\xi )$ is a Jordan–Hölder factor of 
$\bar {\theta }(\chi )$.
Lemma 4.13 The vector 
$S_0\varphi ^{\chi ^s}$ generates 
$\theta (\chi ^s)^\chi$ over 
$\mathcal {O}_E[\mathrm {GL}_2(k_L)]$. If 
$\chi \not = R\chi ^s$, then the vector 
$S_{i(\chi ^s)} \varphi ^{\chi ^s}$ generates 
$\theta (\chi ^s)^{R\chi ^s}$ over 
$\mathcal {O}_E[\mathrm {GL}_2(k_L)]$. (See Remark 4.2 for the definition of 
$i(\chi ^s)$.)
Proof. This follows from Lemma 2.13, because by Remark 4.2 we have 
$i(\chi ^s) = i(\chi ^s, R\chi ^s)$.
We are going to need an isomorphism
\[ \Pi: M_\infty(\theta(\chi^s)) \to M_\infty(\theta(\chi)) \]arising as follows. By Lemma 4.6, it is equivalent to construct an isomorphism
\[ \mathrm{Hom}_{\mathcal{O}_E[\mathrm{GL}_2(\mathcal{O}_L)]}(\theta(\chi), M_\infty^\vee) \to \mathrm{Hom}_{\mathcal{O}_E[\mathrm{GL}_2(\mathcal{O}_L)]}(\theta(\chi^s), M_\infty^\vee). \]
However, by Frobenius reciprocity, these 
$\mathrm {Hom}$-spaces are isomorphic to 
$(M_\infty ^\vee )^{I_1}[\chi ]$ and 
$(M_\infty ^\vee )^{I_1}[\chi ^s]$, respectively. Hence, multiplication by the matrix 
$\Pi$ induces an isomorphism
\[ (M_\infty^\vee)^{I_1}[\chi]\rightarrow (M_\infty^\vee)^{I_1}[\chi^s]. \]We summarize this construction in the following lemma, for ease of reference.
Lemma 4.14 Multiplication by 
$\Pi$ on 
$M_\infty ^\vee$ induces an isomorphism
\[ \Pi: M_\infty(\theta(\chi^s)) \to M_\infty(\theta(\chi)). \] We are also going to need the following related construction. As in Lemma 4.13, there is an isomorphism 
$\theta (\chi ) \rightarrow \theta (\chi ^s)^{\chi }$ sending 
$\varphi ^{\chi }$ to 
$S_0 \varphi ^{\chi ^s}$. Precomposing with the isomorphism from Lemma 4.14, we get an isomorphism 
$M_\infty (\theta (\chi ^s)^{\chi ^s}) \rightarrow M_\infty (\theta (\chi ^s)^{\chi })$, which we also denote by 
$\Pi$. Now consider a sequence
\[ \theta(\chi^s)^{R\chi} \to \theta(\chi^s)^{\chi} \to \theta(\chi^s)^{R\chi^s} \to \theta(\chi^s)^{\chi^s} = \theta(\chi^s) \]of saturated inclusions, as defined in § 2.2. In this context we have the following commutative diagram, where the unlabelled arrows are induced by the saturated inclusions above.
Lemma 4.15 In the diagram (22) the diagonal arrows are multiplication by the image through 
$\eta$ of the 
$U_p$-element of 
$\mathcal {H}(\theta (\chi )[1/p])$, and the vertical arrows are isomorphisms.
Proof. The inclusions are dual to restrictions, hence the short diagonal arrow is dual to
\begin{align*} \mathrm{Hom}_{\mathcal{O}_E[\mathrm{GL}_2(k_L)]}(\theta(\chi^s)^\chi, M_\infty^\vee) &\to \mathrm{Hom}_{\mathcal{O}_E[\mathrm{GL}_2(k_L)]}(\theta(\chi^s)^\chi, M_\infty^\vee)\\ f &\mapsto f^+|_{\theta(\chi^s)^\chi}, \end{align*}
where 
$f^+: \theta (\chi ^s) \to M_\infty ^\vee$ is characterized by the equation 
$f^+(\varphi ^{\chi ^s}) = \Pi f(S_0 \varphi ^{\chi ^s})$. Then 
$f^+|_{\theta (\chi ^s)^\chi }$ is characterized by the equation
\[ f^+(S_0 \varphi^{\chi^s}) = S_0\Pi f(S_0 \varphi^{\chi^s}) = U_p f(S_0\varphi^{\chi^s}), \]
(recall from the proof of Lemma 2.14 that 
$S_0 \Pi = U_p$). As 
$f(S_0 \varphi ^{\chi ^s})$ is an 
$I/I_1$-eigenvector with eigencharacter 
$\chi$, 
$U_p$ is an element in 
$\mathcal {H}(\theta (\chi )[1/p])$. The result now follows from [Reference Caraiani, Emerton, Gee, Geraghty, Paškūnas and ShinCEG+16, Lemma 4.17(2)]. The long diagonal arrow is the restriction of this arrow (recall that 
$\sigma \mapsto M_\infty (\sigma )$ is an exact functor).
For the second claim, we only need to consider the unlabelled vertical arrows in (22) by Lemma 4.14. These arrows are injective because 
$\sigma \mapsto M_\infty (\sigma )$ is an exact functor. By Nakayama's lemma, it suffices to show that 
$M_\infty (\theta (\chi ^s)^{R\chi ^s})/\mathfrak {m} \rightarrow M_\infty (\theta (\chi ^s))/\mathfrak {m}$ is surjective (and similarly for 
$\chi$). Dualizing and using that 
$M_\infty ^\vee [\mathfrak {m}]^{K(1)} \cong D_0({\bar {\rho }})$, it suffices to show that 
$\mathrm {Hom}_K(\theta (\chi ^s)/\theta (\chi ^s)^{R\chi ^s},D_0({\bar {\rho }}))$ is zero. Any map in 
$\mathrm {Hom}_K(\theta (\chi ^s),D_0({\bar {\rho }}))$ factors through 
$D_{0,\sigma (R\chi ^s)}({\bar {\rho }})$ in the notation of [Reference Breuil and PaškūnasBP12, § 14]. The image of a nonzero such map would then contain 
$\sigma (R\chi ^s)$. As 
$\theta (\chi ^s)/\theta (\chi ^s)^{R\chi ^s}$ does not contain 
$\sigma (R\chi ^s)$ as a Jordan–Hölder factor by construction, we are done. Alternatively, one could show that 
$\mathrm {JH}(\theta (\chi ^s)/\theta (\chi ^s)^{R\chi ^s}) \cap W({\bar {\rho }}) = \emptyset$, which immediately implies that 
$M_\infty (\theta (\chi ^s)/\theta (\chi ^s)^{R\chi ^s})$ is zero.
We let 
$U_p(\chi )$ be 
$\eta (U_p)$ where 
$U_p$ is the Hecke operator in 
$\mathcal {H}(\theta (\chi )[1/p])$. We denote by 
$h_\chi : M_\infty (\theta (\chi ^s)^{R\chi ^s}) \to M_\infty (\theta (\chi ^s)^{R\chi })$ the composition of the three vertical arrows (the last one inverted). Its mod 
$\mathfrak {m}$ reduction is closely related to 
$g_\chi$. To see this, we deduce from Lemma 4.13 that there exist surjections
\begin{align} \mathrm{pr}_{\chi^s}: \theta(\chi^s)^{R\chi^s} &\to \sigma(R\chi^s) \nonumber\\ S_{i(\chi^s)} \varphi^{\chi^s} &\mapsto \varphi^{R\chi^s}, \\ \mathrm{pr}_{\chi}: \theta(\chi^s)^{R\chi} &\to \sigma(R\chi) \nonumber \end{align}
\begin{align} S_{i(\chi)}S_0 \varphi^{\chi^s} &\mapsto \varphi^{R\chi}, \end{align}where in the second case we use the isomorphism
\begin{align*} \theta(\chi^s)^\chi &\to \theta(\chi) \\ S_0\varphi^{\chi^s} &\mapsto \varphi^\chi. \end{align*}
(Recall that, by definition 
$\sigma (\xi )$ is a quotient of 
$\theta (\xi )$ with generator 
$\varphi ^{\xi }$. Although the existence of the surjections is automatic from the fact that, for example, 
$\sigma (R\chi ) \cong \operatorname {cosoc} \theta (\chi ^s)^{R\chi }$, we characterize them using, for example, the generator 
$S_{i(\chi )}S_0\varphi ^{\chi ^s}$ of 
$\theta (\chi ^s)^{R\chi }$; see Lemma 4.13.) In the special case that 
$\chi = R\chi ^s$, so that 
$S_{i(\chi ^s)} = S_0$ and 
$S_{i(\chi )} = \mathrm {id}$, we have that 
$\mathrm {pr}_{\chi ^s} = \mathrm {pr}_{\chi }: S_0\varphi ^{\chi ^s} \mapsto \varphi ^\chi$.
Proposition 4.16 There is a commutative diagram
where the unlabelled arrows are induced from saturated inclusions, the maps labelled 
$\mathrm {pr}_\chi$ and 
$\mathrm {pr}_{\chi ^s}$ are induced from these surjections, and all vertical arrows are isomorphisms. Moreover, the map 
$\bar {h}_\chi$ is dual to 
$g_\chi$.
Proof. The top vertical arrows are isomorphisms by Lemma 4.15. By exactness of 
$M_\infty (-)$, the bottom vertical arrows are surjective. As 
$M_\infty (\theta (\chi ^s)^{R\chi ^s})/\mathfrak {m}$ and 
$M_{\infty }(\theta (\chi ^s)^{R\chi })/\mathfrak {m}$ are one-dimensional by Theorem 4.7 and 
$M_{\infty }(\sigma (R\chi ^s))/\mathfrak {m}$ and 
$M_{\infty }(\sigma (R\chi ))/\mathfrak {m}$ are nonzero, the bottom vertical arrows are isomorphisms. We let 
$\bar {h}_\chi$ be the unique map which makes the diagram commute.
Let 
$f$ be an element in 
$\mathrm {Hom}_{k_E[\mathrm {GL}_2(k_L)]}(\sigma (R\chi ), M_\infty ^\vee [\mathfrak {m}])$. Let
\[ f_\mathrm{infl}\in\mathrm{Hom}_{k_E[\mathrm{GL}_2(k_L)]}(\theta(\chi^s)^{R\chi}, M_\infty^\vee[\mathfrak{m}])\quad \textrm{and}\quad \widetilde{f}\in\mathrm{Hom}_{k_E[\mathrm{GL}_2(k_L)]}(\theta(\chi^s)^\chi, M_\infty^\vee[\mathfrak{m}]) \]be the elements obtained by the vertical isomorphisms. Then
\begin{align*} g_\chi(f(\varphi^{R\chi})) &= g_\chi(\widetilde{f}(S_{i(\chi)} S_0 \varphi^{\chi^s})) \\ &= g_\chi(S_{i(\chi)} \widetilde{f}( S_0 \varphi^{\chi^s})) \\ &= g_\chi(R \widetilde{f}( S_0 \varphi^{\chi^s})) \\ &= R\Pi (\widetilde{f}( S_0 \varphi^{\chi^s})) \\ &= S_{i(\chi^s)} h_\chi^\vee(f_\mathrm{infl})(\varphi^{\chi^s}) \\ &= \bar{h}_\chi^\vee(f)(\varphi^{\chi^s}). \end{align*} We deduce the following special case of Theorem 4.11, which is also a consequence of the results of [Reference Breuil and DiamondBD14]. Accordingly, in what follows we often make the assumption that 
$\chi \not = R\chi ^s$.
Lemma 4.17 Assume 
$\chi = R\chi ^s$. Then 
$g_\chi$ does not depend on 
$M_\infty$.
Proof. The discussion has shown that the dual 
$h_\chi$ to 
$g_\chi$ can be identified with the reduction of 
$U_p(\chi )$.
4.4 Lifting $U_p(\chi )$
We translate the calculation for the principal series type 
$\theta (\chi ^s) = \sigma (\tau (\chi ^s))$ (this defines the tame inertial type 
$\tau (\chi ^s)$) into one for the central type 
$\theta = \sigma (\tau )$ (recall that 
$\tau$ is the central inertial type). Again, 
$\chi : I \to \mathcal {O}_E^\times$ is a character appearing in 
$D_0({\bar {\rho }})^{I_1}$ with 
$\chi \not = R\chi ^s$. As a composition of saturated inclusions need not be saturated, from the top line of (22) we see that there is a unique 
$e(\chi )\in \mathbf {N}$ such that the following diagram commutes, where the unlabelled horizontal map is induced from a saturated inclusion.
Proposition 4.18 There exists a unique quotient 
$Q(\chi ^s)^{R\chi ^s}$ of 
$\theta (\chi ^s)^{R\chi ^s}/\varpi _E$ which is isomorphic to a quotient of 
$\theta ^{R\chi ^s}/\varpi _E$ and whose Jordan–Hölder factors are exactly 
$\mathrm {JH}(\bar {\theta }(\chi ^s)) \cap \mathrm {JH}(\bar {\theta })$. Similarly, there exists a unique quotient 
$Q(\chi ^s)^{R \chi }$ of 
$\theta (\chi ^s)^{R\chi }/\varpi _E$ which is isomorphic to a quotient of 
$\theta ^{R\chi }/\varpi _E$ and whose Jordan–Hölder factors are exactly 
$\mathrm {JH}(\bar {\theta }(\chi ^s)) \cap \mathrm {JH}(\bar {\theta })$.
Proof. The duals 
$(\theta (\chi ^s)^{R\chi ^s}/\varpi _E)^\vee$ and 
$(\theta ^{R\chi ^s}/\varpi _E)^\vee$ are representations with the same socle, which is isomorphic to 
$\sigma (R\chi ^s)^\vee$, is generic and appears with multiplicity one as a Jordan–Hölder factor. By [Reference Breuil and PaškūnasBP12, Proposition 3.6], these representations embed in a twist of what they denote 
$V_{\textbf {2p-2-r}}$, where 
$\mathbf {r}$ is such that 
$\sigma (R\chi ^s)^\vee \cong (r_0, r_1, \ldots , r_{f-1}) \otimes \eta$ for some character 
$\eta$. As 
$\sigma (R\chi ^s)$ is a modular weight for the generic representation 
${\bar {\rho }}$, all the 
$r_i$ satisfy the constraint 
$0 \leq r_i \leq p-2$ (and, indeed, even stronger ones). Then the proposition follows from [Reference Breuil and PaškūnasBP12, Theorem 4.7].
Consider the following diagram, where 
$\pi$ denotes the natural projection maps and 
$\iota$ denotes a saturated inclusion.
Then 
$\pi \circ \iota$ factors through the top map. Let 
$\iota _Q$ be the unique map that makes the diagram commute.
Note that the quotient maps 
$\theta (\chi ^s)^{R\chi }/\varpi _E \to Q(\chi ^s)^{R\chi }$ and 
$\theta (\chi ^s)^{R\chi ^s}/\varpi _E \to Q(\chi ^s)^{R\chi ^s}$ induce isomorphisms on 
$M_{\infty }$ (indeed, their kernels have no Jordan–Hölder factors in 
$W({\bar {\rho }})$ because 
$W({\bar {\rho }}) \subseteq \mathrm {JH}(\bar {\theta })$). Moreover, the surjections 
$\mathrm {pr}_{\chi ^s}$ and 
$\mathrm {pr}_\chi$ factor through these quotients. Passing to quotients, (26) becomes the following diagram.
Fix a surjection 
$\alpha : \theta ^{R\chi ^s} \to Q(\chi ^s)^{R\chi ^s}$. Consider the following diagram.
The map 
$\alpha \circ \iota$ factors through 
$\iota _Q$. We call the unique top map that makes the diagram commute 
$\alpha$ as well.
We have the following diagram, where 
$\widetilde {h}_\chi$ is any lift that makes the diagram commute.
Note that 
$\widetilde {h}_\chi$ exists because 
$M_\infty (\theta ^{R\chi ^s})$ is free of rank one over 
$R_\infty (\tau )$. In fact, 
$\widetilde {h}_\chi$ is necessarily an isomorphism. It is surjective by Nakayama's lemma because 
$h_\chi$ is surjective. This implies that it is injective because source and target are free of rank one over 
$R_\infty (\tau )$ by Theorem 4.7. This free of rank one fact also implies that the composition of the left column is given by multiplication by some element 
$\widetilde {U}_p(\chi ) \in R_\infty (\tau )$. Then the following commutative diagram is a lift of (27).
Now recall that because
\[ \mathrm{JH}(\bar{\theta}(\chi^s)) \cap W({\bar{\rho}}) \subseteq \mathrm{JH}(\bar{\theta}) \cap W({\bar{\rho}}) \]
by construction, by [Reference Emerton, Gee and SavittEGS15, Theorem 7.2.1] there exists an ideal 
$I(\chi )$ of 
$R_\infty (\tau )/\varpi _E$ such that the special fiber 
$R_\infty (\tau (\chi ^s)) / \varpi _E$ is the quotient 
$R_\infty (\tau )/(\varpi _E, I(\chi ))$.
Proposition 4.19 If we let 
$1+\mathfrak {m}_{R_\infty (\tau )}$ denote the 
$1$-units of 
$R_\infty (\tau )$, then the equivalence class
\[ \widetilde{U}_p(\chi) (1+\mathfrak{m}_{R_\infty(\tau)}) \in R_\infty(\tau)/(1+\mathfrak{m}_{R_\infty(\tau)}) \](the orbit under multiplication) is uniquely characterized by the following two properties:
(i) we have the equality
(30)\begin{equation} M_\infty(\iota)(M_\infty(\theta^{R\chi})) = \widetilde{U}_p(\chi) M_\infty(\theta^{R\chi^s}); \end{equation}(ii) the image of
$\widetilde {U}_p(\chi )$ in $R_\infty (\tau (\chi ^s))/\varpi _E$ is equal to that of $p^{-e(\chi )}U_p(\chi )$.
Moreover, 
$\widetilde {U}_p(\chi ) (1+\mathfrak {m}_{R_\infty (\tau )})$ is independent of the choice of 
$M_\infty$ (once the integer 
$h$ is fixed).
Proof. Note that 
$\widetilde {U}_p(\chi )$ satisfies both of these conditions. Indeed,
\[ \widetilde{U}_p(\chi) \circ \iota = \iota \circ \widetilde{U}_p(\chi) = (\iota \circ \widetilde{h}_\chi) \circ \iota \]whereas
\begin{align*} h_\chi\circ \iota_Q &\equiv \widetilde{U}_p(\chi) \pmod{\varpi_E+I(\tau(\chi))} \\ &\equiv p^{{-}e(\chi)}U_p(\chi) \pmod{\varpi_E} \end{align*}by (28).
The condition (30) uniquely characterizes 
$\widetilde {U}_p(\chi )$ up to multiplication by 
$R_\infty (\tau )^\times$. The comparison with 
$U_p(\chi )$ characterizes it up to 
$1+\mathfrak {m}_{R_\infty (\tau )}$. To prove the last part, it suffices to show that these two properties are independent of 
$M_\infty$. This is clear for the second property. We now consider the first property. We begin by noting that 
$R_\infty (\tau )$ is normal (
$R_1$ and 
$S_2$) because it is 
$\mathcal {O}_E$-flat and the special fiber is reduced and so 
$R_0$ and 
$S_1$. Then the principal ideal giving 
$M_\infty (\iota )(M_\infty (\theta ^{R\chi }))$ in 
$M_\infty (\theta ^{R\chi ^s})$ is determined by its corresponding Weil divisor, which is
\begin{equation} \sum_{\sigma \in \mathrm{JH}(\bar{\theta})} m(\sigma) \bar{X}^\psi(\sigma), \end{equation}
where 
$m(\sigma )$ denotes the multiplicity of 
$\sigma$ as a Jordan–Hölder factor of 
$\mathrm {coker}\, \iota$ and 
$\bar {X}^\psi (\sigma )$ is as in [Reference Emerton, Gee and SavittEGS15, § 6]. We conclude by noting that (31) does not depend on the choice of 
$M_\infty$.
4.5 Proof of Theorem 4.11
Now fix a cycle 
$\chi _0, \ldots , \chi _{k-1}$ so that 
$R\chi _i^s = R\chi _{i+1}$ and 
$\chi _i \not = R\chi _i^s$ for all 
$i$ (otherwise we apply Lemma 4.17). We prove that 
$\bar {h}_{\chi _0} \cdots \bar {h}_{\chi _k-1}$ is independent of 
$M_{\infty }$, which will imply Theorem 4.11 by Propositions 4.5 and 4.16. The last preliminary step is to iterate the constructions for 
$\chi _i$.
Fix a surjection 
$\alpha _0: \theta ^{R \chi ^{s}_0} \to Q(\chi ^s_0)^{R\chi ^s_0}$. We have seen that there exists a map 
$\widetilde {h}_{\chi _0}$, well-defined up to 1-units, such that the following diagram commutes.
Now define a surjection 
$\alpha _1: \theta ^{R\chi _1^s} \to Q(\chi _1^s)^{R\chi _1^s}$ by requiring that 
$\alpha _1$ makes the following diagram commute.
Then define inductively maps 
$\alpha _i: \theta ^{R\chi _i^s} \to Q(\chi _i^s)^{R\chi _i^s}$. These are maps between representations of 
$\mathrm {GL}_2(k_L)$, defined independently of 
$M_\infty$.
Remark 4.20 It is usually not the case that 
$\alpha _k = \alpha _0$ for a cycle of length 
$k$, and determining the scalar by which they differ is actually the main representation-theoretic input to the computation of Breuil's constants. It is for this reason that we have taken some care in the definition of 
$\alpha _i$.
We are now in a position to complete the proof of Theorem 4.11. Consider the amalgamation of diagrams (29) using (32) along the cycle 
$\chi _0, \ldots , \chi _{k-1}$, which we draw next for 
$k = 3$.
Here, the unlabelled horizontal arrows are induced by saturated inclusions. The diagonal arrows are 
$\widetilde {U}_p$-operators. The rectangles commute by definition of the 
$\alpha _i$ together with (27) and (28). By construction, there exists a positive integer 
$\nu = \nu (\chi _0, \ldots , \chi _{k-1})$ such that the composition
\[ \theta^{R\chi^s_0} \to \theta^{R\chi^s_1} \to \cdots \to \theta^{R\chi^s_{k-1}} \to \theta^{R\chi^s_0} \]
of saturated inclusions is multiplication by 
$p^\nu$. From the diagram, we see that
\begin{equation} p^\nu \widetilde{h}_{\chi_{1}}\cdots\widetilde{h}_{\chi_{k-1}}\widetilde{h}_{\chi_{0}} = \widetilde{U}_p(\chi_{0})\cdots\widetilde{U}_p(\chi_{k-1}). \end{equation}
If 
$c(\chi _0, \ldots , \chi _{k-1}) \in k_E^\times$ is such that 
$\alpha _k = c(\chi _0, \ldots , \chi _{k-1}) \alpha _0$, then 
$c(\chi _0, \ldots , \chi _{k-1})$ is independent of 
$M_\infty$ (it is even independent of the choice of surjection 
$\alpha _0$) and we have
\begin{equation} c(\chi_0, \ldots, \chi_{k-1}) \bar{h}_{\chi_1}\cdots\bar{h}_{\chi_k}\bar{h}_{\chi_0} = p^{-\nu}\widetilde{U}_p(\chi_{0})\cdots\widetilde{U}_p(\chi_{k-1}) \end{equation}
in 
$R_{\infty }(\tau )^\times /1+\mathfrak {m}_{R_\infty (\tau )}$, and the theorem follows because the 
$\widetilde {U}_p(\chi _i)$ are well-defined and independent of 
$M_\infty$ up to multiplication by 
$1+\mathfrak {m}_{R_\infty (\tau )}$ by Proposition 4.19.
5. Breuil's constants
We now proceed to the computation of the constant 
$g_{\chi _{k-1}}\cdots g_{\chi _0} \in k_E^\times$ in the cases considered by Breuil in [Reference BreuilBre11]. By Proposition 4.16 and (34), this amounts to a computation of two units
\[ c(\chi_0, \ldots, \chi_{k-1}) \in k_E^\times \quad \text{and}\quad p^{-\nu(\chi_0, \ldots, \chi_{k-1})} \widetilde{U}_p(\chi_0)\cdots \widetilde{U}_p(\chi_{k-1}) \in R_\infty(\tau)^\times{/}(1+\mathfrak{m}_{R_\infty(\tau)}) \cong k_E^\times. \]
Although Breuil's cycles are only defined for semisimple mod 
$p$ representations, we make some of our calculations on Galois deformation rings in more generality. Until Lemma 5.5, the only assumptions on 
${\bar {\rho }}$ are our running assumptions of genericity from § 3.2.2.
5.1 The product $\widetilde {U}_p(\chi _0)\cdots \widetilde {U}_p(\chi _{k-1})$
In order to compute this product, we must relate the patching constructions given previously with deformation ring calculations in § 3. This involves a translation of the results of [Reference Emerton, Gee and SavittEGS15, §§ 7–8] into the notation of § 3. Rather than make this translation explicit, we indicate how the methods of [Reference Emerton, Gee and SavittEGS15, §§ 7–8] can be used to prove the results that we need. By construction of the 
$\widetilde {U}_p$, our final result will only be well-defined up to 1-units in the deformation ring, but this will suffice for our purposes.
Notation. Recall from § 3.6.1 that 
$\tau$ is the central inertial type for 
${\bar {\rho }}$. If 
$\tau _*$ is a tame inertial type, then the map 
$\kappa ^{\tau _*}$ from § 3.5.1 induces an injective, formally smooth map
\begin{equation} R^{\tau_*}_{\bar{\rho}} \rightarrow R^{\tau_*,\square}_{\overline{\mathfrak{M}}^{\tau_*}}. \end{equation}
Recall also that we have attached a weight 
$\sigma _J$ to each subset 
$J \subset \mathbf {Z}/f$, and that all modular weights of 
${\bar {\rho }}$ have this form. If 
$\chi \in (\operatorname {soc}_K D_0({\bar {\rho }}))^{I_1}$, we write 
$J(\chi )$ for the subset such that 
$\sigma _{J(\chi )}^{I_1} \cong \chi$.
Lemma 5.1 Suppose that 
$\sigma \in W({\bar {\rho }})$ and recall from § 3.5.3 the ideal 
$I(\sigma )$. Then the scheme-theoretic support of
\[ M_\infty(\sigma)\otimes_{R^\tau_{\bar{\rho}}} R^{\tau,\square}_{\overline{\mathfrak{M}}^\tau} \]
is 
$\operatorname {Spec} R_\infty \otimes _{R^\tau _{\bar {\rho }}} (R^{\tau ,\square }_{\overline {\mathfrak {M}}^\tau }/I(\sigma ))$.
Proof. There exists a type 
$\tau _*$ such that 
$\mathrm {JH}(\bar {\sigma }(\tau _*)) \cap W({\bar {\rho }}) = \{\sigma \}$. By Proposition 3.11, if 
$\sigma \cong F(\mathfrak {t}_\mu (s_{\bar {\rho }}\omega _J))$ one could take 
$w_{*,j} = \mathrm {id}$ for all 
$j$ and 
$w'_{*,j}$ to be 
$\mathrm {id}$ if and only if 
$j\notin J$ in the notation of § 3.5.2. Then 
$I(\tau _*)+(\varpi _E)$ coincides with 
$I(\sigma )$. Let 
$\theta _*^\sigma$ be the lattice in 
$\sigma (\tau _*)$ with cosocle 
$\sigma$. Then the surjection 
$\theta _*^\sigma \rightarrow \sigma$ induces an isomorphism
\[ M_\infty(\bar{\theta}_*^\sigma) \rightarrow M_\infty(\sigma). \]
The scheme-theoretic support of 
$M_\infty (\bar {\theta }_*^\sigma )$ is 
$\operatorname {Spec} R^{\tau _*}_{\bar {\rho }}/\varpi _E$, and 
$R_{\bar {\rho }}^{\tau _*}/\varpi _E$ is a quotient of 
$R_{\bar {\rho }}^{\tau }/\varpi _E$ because 
$\tau$ is central for 
${\bar {\rho }}$. As (35) is flat, it suffices to show that
\[ R_\infty(\tau^*)/\varpi_E \otimes_{R^\tau_{\bar{\rho}}} R^{\tau,\square}_{\overline{\mathfrak{M}}^\tau} = R_\infty\otimes_{R^\tau_{\bar{\rho}}} (R^{\tau,\square}_{\overline{\mathfrak{M}}^\tau}/I(\tau_*)+(\varpi_E)). \]This follows from Proposition 3.16.
Lemma 5.2 Let 
$J_1 \subset J_{\bar {\rho }}$ be such that 
$j\in J_{\bar {\rho }} \setminus J_1$. Let 
$J_2$ be 
$J_1 \cup \{j\}$. Suppose that 
$\tau _*$ is a tame inertial type such that 
$\mathrm {JH}(\bar {\sigma }(\tau _*))$ contains both 
$\sigma _1 = \sigma _{J_1}$ and 
$\sigma _2 = \sigma _{J_2}$. Let 
$\theta _*^2 \subset \theta _*^1$ be a saturated inclusion of 
$\mathcal {O}_E$-lattices in 
$\sigma (\tau _*)$ with cosocles isomorphic to 
$\sigma _2$ and 
$\sigma _1$, respectively. Then
\begin{equation} M_\infty(\theta_*^2) \otimes_{R^{\tau_*}_{\bar{\rho}}} R^{\tau_*,\square}_{\overline{\mathfrak{M}}^{\tau_*}} = Y_{f-1-j}M_\infty(\theta_*^1) \otimes_{R^{\tau_*}_{\bar{\rho}}} R^{\tau_*,\square}_{\overline{\mathfrak{M}}^{\tau_*}}. \end{equation}Proof. One checks from Lemma 5.1 that 
$Y_{f-j-2}$ annihilates the quotient
\[ (M_\infty(\theta_*^1)/M_\infty(\theta_*^2)) \otimes_{R^{\tau_*}_{\bar{\rho}}} R^{\tau_*,\square}_{\overline{\mathfrak{M}}^{\tau_*}} \]
(we can assume without loss of generality that 
$E/\mathbf {Q}_p$ is an unramified extension by Lemma 4.8). The inclusion 
$\supset$ in (36) then follows. By [Reference Emerton, Gee and SavittEGS15, Theorem 5.2.4], we see that 
$p\tau _*^1\subset \tau _*^2$ is also a saturated inclusion. Arguing as before we see that
\begin{equation} pM_\infty(\theta_*^1) \otimes_{R^{\tau_*}_{\bar{\rho}}} R^{\tau_*,\square}_{\overline{\mathfrak{M}}^{\tau_*}} \supset X_{f-1-j}M_\infty(\theta_*^2) \otimes_{R^{\tau_*}_{\bar{\rho}}} R^{\tau_*,\square}_{\overline{\mathfrak{M}}^{\tau_*}}. \end{equation}
Combining (37) with the earlier inclusion and the equation 
$X_{f-1-j}Y_{f-1-j} = p$ yields
\[ pM_\infty(\theta_*^1) \otimes_{R^{\tau_*}_{\bar{\rho}}} R^{\tau_*,\square}_{\overline{\mathfrak{M}}^{\tau_*}} \supset X_{f-1-j}M_\infty(\theta_*^2) \otimes_{R^{\tau_*}_{\bar{\rho}}} R^{\tau_*,\square}_{\overline{\mathfrak{M}}^{\tau_*}} \supset pM_\infty(\theta_*^1) \otimes_{R^\tau_{\bar{\rho}}} R^{\tau_*,\square}_{\overline{\mathfrak{M}}^{\tau_*}}. \]This forces that all inclusions above are, in fact, equalities.
Lemma 5.2 and an induction argument yields the following.
Proposition 5.3 Suppose that 
$\tau _*$ is a tame type such that 
$\mathrm {JH}(\bar {\tau }_*)$ contains both 
$\sigma _1 = \sigma _{J_1}$ and 
$\sigma _2 = \sigma _{J_2}$. Let 
$\theta _*^2 \subset \theta _*^1$ be a saturated inclusion of 
$\mathcal {O}_E$-lattices in 
$\sigma (\tau _*)$ with cosocles isomorphic to 
$\sigma _2$ and 
$\sigma _1$, respectively. Let
\begin{equation} f_j(J_1,J_2) = \begin{cases} X_j &\mbox{if } f-1-j\in J_1 \setminus J_2\\ Y_j &\mbox{if } f-1-j\in J_2 \setminus J_1\\ 1 &\mbox{otherwise.} \end{cases} \end{equation}Then
\[ M_\infty(\theta_*^2) \otimes_{R^{\tau_*}_{\bar{\rho}}} R^{\tau_*,\square}_{\overline{\mathfrak{M}}^{\tau_*}} = \bigg(\prod_j f_j(J_1,J_2)\bigg) M_\infty(\theta_*^1) \otimes_{R^{\tau_*}_{\bar{\rho}}} R^{\tau_*,\square}_{\overline{\mathfrak{M}}^{\tau_*}}. \] We are now ready to compare our calculations on deformation rings of Galois representations and Kisin modules. Fix a character 
$\chi$ appearing in 
$D_0({\bar {\rho }})^{I_1}$. Let 
$\widetilde {U}'_p(\chi )$ be the element of 
$R^{\tau ,\square }_{\overline {\mathfrak {M}}^\tau }$ defined in § 3.6.1, with the type 
$\theta (\chi )[p^{-1}]$ playing the role of 
$\sigma$. We begin by relating 
$\widetilde {U}'_p(\chi )$ to 
$\widetilde {U}_p(\chi )$.
Lemma 5.4 We have
\[ \kappa^\tau(\widetilde{U}_p(\chi)) R^{\tau,\square}_{\overline{\mathfrak{M}}^\tau} = \widetilde{U}'_p(\chi) R^{\tau,\square}_{\overline{\mathfrak{M}}^\tau}. \]Proof. Let 
$\theta ^2 \subset \theta ^1$ be a saturated inclusion of 
$\mathcal {O}_E$-lattices in 
$\sigma (\tau )$ with cosocles isomorphic to 
$\sigma (R\chi )$ and 
$\sigma (R\chi ^s)$, respectively. By the first property of 
$\widetilde {U}_p(\chi )$ in Proposition 4.19, it suffices to show that
\begin{equation} M_\infty(\theta^2) \otimes_{R^\tau_{\bar{\rho}}} R^{\tau,\square}_{\overline{\mathfrak{M}}^\tau} = \widetilde{U}'_p(\chi) \Big(M_\infty(\theta^1) \otimes_{R^\tau_{\bar{\rho}}} R^{\tau,\square}_{\overline{\mathfrak{M}}^\tau}\Big). \end{equation} Let 
$\theta (\chi )^2 \subset \theta (\chi )^1$ be a saturated inclusion of lattices in 
$\sigma (\tau (\chi ))$ with cosocles isomorphic to 
$\sigma (R\chi )$ and 
$\sigma (R\chi ^s)$, respectively. By the definition of 
$U_p(\chi )$, we have
\[ M_\infty(\theta(\chi)^2) \otimes_{R^{\tau(\chi)}_{\bar{\rho}}} R^{{\tau(\chi)},\square}_{\overline{\mathfrak{M}}^{\tau(\chi)}} = p^{{-}e(\chi)}\kappa^{\tau(\chi)}(U_p(\chi)) \Big(M_\infty(\theta(\chi)^1) \otimes_{R^{\tau(\chi)}_{\bar{\rho}}} R^{{\tau(\chi)},\square}_{\overline{\mathfrak{M}}^{\tau(\chi)}}\Big). \]By Proposition 5.3, this implies that
\begin{equation} p^{{-}e(\chi)}\kappa^{\tau(\chi)}(U_p(\chi)) \equiv \prod_j f_j(J_1,J_2) \mod \Big(R^{\tau(\chi),\square}_{\overline{\mathfrak{M}}^{\tau(\chi)}}\Big)^\times \end{equation}
with 
$J_1$ and 
$J_2$ corresponding to 
$\sigma (R\chi ^s)$ and 
$\sigma (R\chi )$, respectively. By definition, 
$\widetilde {U}_p'(\chi )$ is congruent modulo 
$(R^{\tau ,\square }_{\overline {\mathfrak {M}}^\tau })^\times$ to a product 
$\prod _j g_j$ for certain factors 
$g_j \in \{X_j, Y_j, 1 \}$ (note that 
$X_\alpha + [\alpha ], X_{\alpha '} + [\alpha ']$ are units). Furthermore, Proposition 3.23 and (40) imply that the image of 
$\prod _j g_j$ in 
$R_{\bar{\mathfrak{M}}^{\tau (\chi )}}^{\tau (\chi ), \square }/p$ is
\[ \prod_j f_j(J_1, J_2) \in R_{\bar{\mathfrak{M}}^{\tau(\chi)}}^{\tau(\chi), \square}/p. \]Then necessarily we must have
\[ g_j \equiv f_j(J_1, J_2) \mod \Big(R^{\tau,\square}_{\overline{\mathfrak{M}}^\tau}\Big)^\times \]
for all 
$j$. However, then (39) follows from Proposition 5.3 because the formula (38) is independent of the tame type.
Lemma 5.5 Let 
$\chi$ be a character appearing in 
$D_0({\bar {\rho }})^{I_1}$. Then 
$\kappa ^\tau (\widetilde {U}_p(\chi ))$ and 
$\widetilde {U}'_p(\chi )$ coincide up to multiplication by a 
$1$-unit of 
$R^{\tau ,\square }_{\overline {\mathfrak {M}}^\tau }$.
We now make some extra assumptions.
(i) We assume that
${\bar {\rho }}$ is semisimple and we keep this assumption until the end of § 5.(ii) We fix a cycle
$\chi _0, \ldots , \chi _{k-1}$ of characters appearing in $(\operatorname {soc}_K D_0({\bar {\rho }}))^{I_1}$, such that $R\chi _i^s = \chi _{i+1}$ for all $i \in \mathbf {Z}/k$: this is precisely the situation of [Reference BreuilBre14, Question 9.5] (though our genericity assumptions are slightly less general).(iii) By [Reference BreuilBre11, Lemme 6.1], we can assume that
$R\chi _i^s \not = \chi _i^s$ for all $i$ (i.e. $\sigma (\chi _i^s) \notin W({\bar {\rho }})$), and by Lemma 4.17 we can assume that $R\chi _i^s \not = \chi _i$ for all $i$.
Lemma 5.6 Assumption (iii) implies that each 
$H$-character 
$\chi _i$ has multiplicity one in the central type 
$\theta _E = \sigma (\tau )$.
Proof. If 
$\theta _E$ is cuspidal, this is true because every 
$H$-character has multiplicity one in 
$\theta _E$. If 
$\theta _E$ is principal series, then the only exceptions correspond to the two characters coming from 
$I_1$-invariants. However, if 
$\chi _i$ is one of these characters, then 
$\sigma (\chi _i^s)$ is in 
$\mathrm {JH}(\bar {\sigma }(\tau )) = W({\bar {\rho }})$ so that 
$R\chi _i^s = \chi _i^s$, which contradicts assumption (iii).
Proposition 5.7 Let 
$J\subset \mathbf {Z}/f$ be 
$J(\chi _i)$ for some 
$i\in \mathbf {Z}/k$, so that 
$\chi _i \cong \sigma _J^{I_1}$. Let 
$J_0 = \delta _\mathrm {red} J$, so that 
$\chi _i$ corresponds to the formal weight 
$\lambda _{J_0}$. Let 
$\delta$ be 
$\delta _\mathrm {irr}$ or 
$\delta _\mathrm {red}$ according to whether 
${\bar {\rho }}$ is irreducible or reducible. Then the product
\begin{equation} \prod_{i=0}^{k-1} \widetilde{U}_p(\chi_i) \end{equation}
is the product of a 
$1$-unit of 
$R^\tau _{\bar {\rho }}$, the scalar
\[ ({-}p)^{({1}/{2})k|J_0\Delta \delta(J_0)|}, \]and the scalar
\[ \begin{cases} [\alpha]^{-{k|J_0^\textrm{c}|}/{f}}[\alpha']^{- {k|J_0|}/{f}} &\mbox{if } {\bar{\rho}} \textrm{ is reducible}\\ [-\alpha]^{- {k}/{2}} &\mbox{if } {\bar{\rho}} \textrm{ is irreducible} \end{cases} \]
where 
$\alpha$ and 
$\alpha '$ are as in § 3.2.2.
Remark 5.8 For any 
$J \subset \{0,\ldots ,f-1\}$, 
$|J_0 \Delta \delta (J_0)| = |\delta (J_0) \Delta \delta ^2(J_0)|$ by Lemma 1.5. Hence, the formulas in Proposition 5.7 do not depend on the choice of 
$i \in \mathbf {Z}/k$ in the definition of 
$J$.
Proof. By Lemma 5.5, it suffices to show that
\begin{equation} \prod_{i=0}^{k-1} \widetilde{U}'_p(\chi_i) \end{equation}
is of the same form up to multiplying by a 
$1$-unit of 
$R^{\tau ,\square }_{\overline {\mathfrak {M}}^\tau }$. Recall that for each 
$i$, 
$\widetilde {U}'_p(\chi _i)$ has a factorization
\[ \prod_{j= 0}^{f-1} \widetilde{x}^{(j)}_i, \]
where 
$\widetilde {x}^{(j)}_i$ is 
$1$, 
$Y_{j}$, or 
$-X_{j}$ (multiplied by one of 
$(X_\alpha +[\alpha ])^{-1}$ and 
$(X_{\alpha '}+[\alpha '])^{-1}$ if 
$j=0$). Let 
$\widetilde {y}^{(j)}_i$ be 
$\widetilde {x}^{(j)}_i$ without the modification at 
$j=0$.
Fix 
$0 \leq i \leq k-1$ and let 
$J$ be 
$J(\chi _i)$. We claim that
\begin{equation} \big|\{\widetilde{y}^{(j)}_i\neq 1\mid 0\leq j \leq f-1\} \big|=|J_0\Delta \delta(J_0)|. \end{equation} By Lemma 3.13, there are permutations 
$s^*, w, w' \in W$ such that
\[ \theta(\chi)[1/p] \cong R_{s_{\bar{\rho}} w}(\mu - s_{\bar{\rho}} w' \eta),\quad s^*F(s^*)^{{-}1} = s_{\bar{\rho}} w \quad \text{and}\quad \chi = (s^*)^{{-}1}(\mu-s_{\bar{\rho}} w'\eta)|_H, \]
and, furthermore, 
$j \in J_0$ if and only if 
$s^*_j \ne \mathrm {id}$. (Recall that the formal weight of 
$\sigma _{J(\chi _i)}$ corresponds in all cases to 
$\delta _\mathrm {red}(J(\chi _i))$, hence 
$\chi _j \in \{p-2-r_j, p-3-r_j\}$ if and only if 
$j \in J_0$.) From § 3.5.1, we see that 
$\widetilde {y}^{(f-1-j)}_i = 1$ if and only if 
$w_j = \mathrm {id}$. However, because 
$w_j = s_{{\bar {\rho }},j}^{-1} s^*_j (s^*_{j-1})^{-1}$, one finds that 
$\widetilde {y}_i^{(f-1-j)} \neq 1$ if and only if 
$j \in J_0\triangle \delta ^{-1}(J_0)$. This establishes (43).
By Remark 5.8, we conclude that
\begin{equation} \big|\{\widetilde{y}^{(j)}_i\neq 1\mid 0\leq i\leq k-1,\ 0\leq j \leq f-1\} \big|=k|J_0\Delta \delta(J_0)|. \end{equation} As (41) is a power of 
$p$ up to a unit by (33), the same is true for (42). This implies that 
$Y_j$ and 
$-X_j$ appear among 
$\{\widetilde {y}^{(j)}_i\}_{i,j}$ with equal frequency. We conclude from (44) that
\begin{equation} \prod_{i=0}^{k-1}\prod_{j=0}^{f-1} y^{(j)}_i = ({-}p)^{({1}/{2})k|J_0\Delta \delta(J_0)|}. \end{equation} For each 
$i$, 
$\widetilde {x}^{(0)}_i$ contains a factor of either 
$X_\alpha +[\alpha ]$ or 
$X_{\alpha '} + [\alpha ']$. Moreover, we see from Corollary 3.9, § 3.4.2, and (15) that these two possibilities correspond precisely to 
$f-2 \notin J(\chi _i)$ and 
$f-2 \in J(\chi _i)$, respectively. Thus, we have
\[ \prod_{i=0}^{k-1} \frac{\widetilde{x}^{(0)}_i}{\widetilde{y}^{(0)}_i} = \begin{cases} (X_\alpha+[\alpha])^{- {k|J_0^\textrm{c}|}/{f}}(X_{\alpha'}+[\alpha'])^{- {k|J_0|}/{f}} &\mbox{if } {\bar{\rho}} \textrm{ is reducible}\\ (X_\alpha+[\alpha])^{- {k}/{2}}(X_{\alpha'}+[\alpha'])^{- {k}/{2}} &\mbox{if } {\bar{\rho}} \textrm{ is irreducible} \end{cases} \]
(by the beginning of § 5.3, 
$J_0^\textrm {c}$ is always in the 
$\delta$-orbit of 
$J_0$ when 
${\bar {\rho }}$ is irreducible). In conjunction with (45) and recalling that 
$\alpha ' = -1$ if 
${\bar {\rho }}$ is irreducible, this finishes the proof.
Remark 5.9 In the rest of this section, we work with the parametrization by formal weights instead of the parametrization by the extension graph. This is the reason for working with 
$J_0$ in Proposition 5.7.
5.2 The constant $c(\chi _0, \ldots , \chi _{k-1})$
Recall our running assumptions from § 4.5 that 
$R\chi _i = \chi _i$, that 
$R\chi _i^s = \chi _{i+1}$, and that 
$R\chi _i^s \not = \chi _i^s, \chi _i$ for each 
$i\in \mathbf {Z}/k$. We actually compute the constant 
$c(\chi _{k-1}^s, \chi _{k-2}^s, \ldots , \chi _0^s)$, which is enough because of the following lemma.
Lemma 5.10 The equality 
$c(\chi _0, \ldots , \chi _{k-1}) = c(\chi _{k-1}^s, \ldots , \chi _{0}^s)^{-1}$ holds.
Proof. Informally, this is going around the cycle in the opposite direction. More precisely, we have defined 
$c(\chi _0, \ldots , \chi _{k-1})$ starting from a map 
$\alpha _0: \theta ^{R \chi _0^s} \to Q(\chi _0^s)^{R\chi _0^s}$, constructing 
$\alpha _i$ inductively, and putting 
$\alpha _k = c(\chi _0, \ldots , \chi _{k-1})\alpha _0$. Now we have to start from some 
$\theta ^{R\chi _{k-1}} \to Q(\chi _{k-1})^{R\chi _{k-1}}$ and we know that 
$R \chi _{k-1} = R \chi _{k-2}^s$, so we choose to start from 
$\alpha _{k-2}$. The next map to be constructed will be some 
$\theta ^{R\chi _{k-2} }\to Q(\chi _{k-2})^{R\chi _{k-2}}$, and we know that 
$R\chi _{k-2} = R\chi _{k-3}^s$ and 
$\alpha _{k-3}$ makes the pertinent diagram commute. Iterating, we deduce that 
$\alpha _0 = c(\chi _{k-1}^s, \ldots , \chi _{0}^s)\alpha _k$, and the lemma follows.
We begin by reducing our goal to certain explicit computations on the central type 
$\theta$, that can be performed via the methods of § 2. Write 
$\psi _i = \chi _{k-1-i}^s$, and fix a surjection 
$\alpha _0: \theta ^{\psi _0^s} \to Q(\psi _0^s)^{\psi _0^s}$ (observe that 
$R\psi _0^s = R\chi _{k-1} = \chi _{k-1} = \psi _0^s$), and construct the 
$\alpha _i$ as in § 4.5. This relabels the 
$\alpha _i$ for our new cycle 
$\psi _0, \ldots , \psi _{k-1}$, and they will not change until the end of the paper.
By construction, there exists a unique 
$\nu (\psi _i^s) \in \mathbf {Z}_{\geq 0}$ such that the composition
\begin{equation} \theta(\psi_i^s)^{R\psi_i} \to \theta(\psi_i^s)^{\psi_i} \to \theta(\psi_i^s) \end{equation}
of saturated inclusions is 
$p^{\nu (\psi _i^s)}$ times the saturated inclusion. By Lemma 4.13 together with the isomorphism
\begin{align*} \theta(\psi_i) &\to \theta(\psi_i^s)^{\psi_i} \\ \varphi^{\psi_i} &\mapsto S_0 \varphi^{\psi_i^s} \end{align*}
it follows that 
$p^{-\nu (\psi _i^s)}S_{i(\psi _i)}S_0 \varphi ^{\psi _i^s}$ generates the image of the saturated inclusion 
$\theta (\psi _i^s)^{R\psi _i} \to \theta (\psi _i^s)$ (see Definition 4.1 for the definition of 
$S_{i(\psi _i)}$).
Lemma 5.11 There exists a unique 
$0 < i^+(\psi _i^s) < q-1$ and constant 
$c(\psi _i^s) \in W(k_L) \setminus \{0\} \subset W(k_E)$ such that
\[ S_{i(\psi_i)} S_0 \varphi^{\psi_i^s} = c(\psi_i^s) S^+_{i^+(\psi_i^s)}\varphi^{\psi_i^s} \]
in the integral induction 
$\theta (\psi _i^s) = \mathrm {Ind}_I^K(\psi _i^s)$. The equality 
$i^+(\psi _i^s) = q-1-i(\psi _i)$ holds.
Proof. This follows from Lemma 2.11 as the character 
$R\psi _i$ has multiplicity one in 
$\theta (\psi _i^s)$. Indeed, otherwise 
$R\chi _{k-1-i}^s = \chi _{k-1-i}$ or 
$R \chi _{k-1-i}^s = \chi _{k-1-i}^s$, which we have assumed not to hold. The equality follows from Lemma 2.7.
Definition 5.12 We write 
$\widetilde {c}(\psi _i^s)$ for the leading Teichmüller coefficient in the 
$p$-adic expansion of 
$c(\psi _i^s)$.
Next, we show that the reductions mod 
$p$ of these constants 
$\widetilde {c}(\psi _i^s)$ are closely related to the constant 
$c(\chi _{k-1}^s, \ldots , \chi _0^s)$ we are trying to compute.
Lemma 5.13 Let 
$\iota : \theta (\psi _i^s)^{R\psi _i} \to \theta (\psi _i^s)$ be a saturated inclusion. Then the vector 
$S_{i^+(\psi _i^s)}^+\varphi ^{\psi _i^s}$ is contained in and generates 
$\theta (\psi _i^s)^{R\psi _i}$ over 
$\mathcal {O}_E[\mathrm {GL}_2(k_L)]$.
Proof. Recall that 
$R\psi _i$ has multiplicity one in 
$\theta (\psi _i^s)[1/p]$. The result then follows from Lemma 2.13.
Proposition 5.14 Recall the maps 
$\mathrm {pr}$ from (32). Assume 
$x \in \theta ^{\psi _{i}^s}$ is such that
\[ \mathrm{pr}_{\psi_i^s}\alpha_{i}(x) = \varphi^{\psi_{i}^s} \in \sigma(\psi_{i}^s). \]
Then 
$S^+_{i^+(\psi _i^s)}(x)$ is contained in the image of the saturated inclusion 
$\theta ^{\psi _{i-1}^s} \to \theta ^{\psi _i^s}$, and
\[ \mathrm{pr}_{\psi_{i-1}^s}\alpha_{i-1} \big(\widetilde{c}(\psi_i^s)S^+_{i^+(\psi_i^s)}(x) \big) = \varphi^{\psi_{i-1}^s} \in \sigma(\psi_{i-1}^s). \]Proof. Taking definitions into account, this follows from the previous discussion and inspection of the following commutative diagram.
We give the details as a sequence of lemmas.
Lemma 5.15 The equality 
$\mathrm {pr}_{\psi _i} \pi (\widetilde {c}(\psi _i^s)S_{i^+(\psi _i^s)}^+\varphi ^{\psi _i^s}) = \varphi ^{\psi _{i-1}^s}$ holds in 
$\sigma (\psi _{i-1}^s)$.
Proof. By definition,
\[ \mathrm{pr}_{\psi_i}\pi(p^{-\nu(\psi_i^s)}S_{i(\psi_i^s)}S_0\varphi^{\psi_i^s}) = \varphi^{\psi_{i-1}^s}, \]
and the claim follows from Lemmas 5.11 and 5.13. (The map 
$\mathrm {pr}_{\psi _i}$ in (24) is defined in terms of a nonsaturated inclusion, so we need to divide by 
$p^{-\nu (\psi _i^s)}$.)
Lemma 5.16 The equality 
$\alpha _i(x) = \pi (\varphi ^{\psi _i^s})$ holds in 
$Q(\psi _i^s)^{\psi _i^s}$.
Proof. By [Reference Breuil and PaškūnasBP12, Lemma 2.7(i)], the only quotient of 
$\theta (\psi _i^s)/\varpi _E$ in which 
$\psi _i^s$ has multiplicity two is 
$\theta (\psi _i^s)/\varpi _E$ itself. As 
$\psi _i$ does not appear in 
$\operatorname {soc}_K D_0({\bar {\rho }})$, the type 
$\theta (\psi _i^s)$ is not the central type 
$\theta$, and so 
$\psi _i^s$ has multiplicity one in 
$Q(\psi _i^s)^{\psi _i^s}$. It follows that 
$\alpha _i(x) = \pi (\varphi ^{\psi _i^s})$, since they have the same image 
$\varphi ^{\psi _i^s}$ in 
$\sigma (\psi _i^s)$.
Lemma 5.17 The vector 
$S_{i^+(\psi _i^s)}^+(x)$ is an element of 
$\theta ^{\psi _{i-1}^s}$.
Proof. By assumption, the character 
$R\psi _i = \psi _{i-1}^s$ has multiplicity one in 
$\theta [1/p]$. The result then follows from Lemma 2.13.
Now we conclude the proof of the proposition. Lemmas 5.16 and 5.17 imply that
\begin{equation} \iota\alpha_iS_{i^+(\psi_i^s)}^+(x) = S_{i^+(\psi_i^s)}^+ \alpha_i(x) = S_{i^+(\psi_i^s)}^+\pi (\varphi^{\psi_i^s}) = \pi S_{i^+(\psi_i^s)}^+(\varphi^{\psi_i^s}) \end{equation}
in 
$Q(\psi _{i}^s)^{\psi _i^s}$. By Lemma 5.13 we deduce an equality
\[ \iota\alpha_iS_{i^+(\psi_i^s)}^+(x) = \iota \pi S_{i^+(\psi_i^s)}^+(\varphi^{\psi_i^s}). \]
Now 
$\iota$ need not be injective, but we know that it is nonzero and that the vector 
$\pi S_{i^+(\psi _i^s)}^+(\varphi ^{\psi _i^s})$ generates 
$Q(\psi _i^s)^{R\psi _i}$, so we deduce that
\begin{equation} \alpha_iS_{i^+(\psi_i^s)}^+(x) = \pi S_{i^+(\psi_i^s)}^+(\varphi^{\psi_i^s}) \in Q(\psi_i^s)^{R\psi_i}. \end{equation}We conclude that
\begin{align*} \hspace{8pc}\mathrm{pr}_{\psi_{i-1}^s} \alpha_{i-1}(\widetilde{c}(\psi_i^s)S^+_{i^+(\psi_i^s)}(x)) &= \mathrm{pr}_{\psi_i} \alpha_i(\widetilde{c}(\psi_i^s) S_{i^+(\psi_i^s)}^+(x)) \\ &\overset{\text{(48)}}= \mathrm{pr}_{\psi_i} \pi (\widetilde{c}(\psi_i^s) S_{i^+(\psi_i^s)}^+(\varphi^{\psi_i^s})) \\ &= \varphi^{\psi_{i-1}^s} \text{ by Lemma {5.15}}. \hspace{8.3pc}\Box \end{align*} By Proposition 5.14, we deduce that if 
$x \in \theta ^{\psi _{k}^s}$, and 
$\alpha _k(x)$ maps to 
$\varphi ^{\psi _{k}^s} \in \sigma (\psi _{k}^s)$, then
\[ \alpha_0 \big(\widetilde{c}(\psi_1^s)S^+_{i^+(\psi_1^s)}\cdots \widetilde{c}(\psi_{k}^s)S^+_{i^+(\psi_{k}^s)}(x) \big) \]
maps to 
$\varphi ^{\psi _0^s} = \varphi ^{\psi _{k}^s}$. The composition 
$\theta ^{\psi _0^s} \to \theta ^{\psi _1^s} \to \cdots \to \theta ^{\psi _k^s}$ of saturated inclusions is multiplication by some 
$p^\nu$, and we deduce that
\[ p^\nu[c(\psi_0, \ldots, \psi_{k-1})]\cdot x = \widetilde{c}(\psi_1^s)S^+_{i^+(\psi_1^s)}\cdots \widetilde{c}(\psi_{k}^s)S^+_{i^+(\psi_{k}^s)}(x). \]
In the next sections, we compute the product 
$\prod _{i = 1}^{k} \widetilde {c}(\psi _i^s)$ and the composition 
$S_{i^+(\psi _1^s)}^+\cdots S_{i^+(\psi _{k}^s)}^+$, using the methods of § 2.
5.3 The irreducible case
Here we compute the constants of the previous section when 
${\bar {\rho }}$ is irreducible. Recall the bijections 
$J \mapsto \lambda _J$ and 
$J \mapsto \sigma (\lambda _J, {\bar {\rho }})$, from subsets 
$J \subseteq \{0, \ldots , f-1 \}$ to formal weights of 
${\bar {\rho }}$ or modular Serre weights of 
${\bar {\rho }}$, respectively (they are defined in (11) and (12), respectively). The integers 
$r_i$ are defined through
\[ {\bar{\rho}}|_{I_{\mathbf{Q}_p}} \cong \begin{pmatrix} \omega_{2f}^{\sum_{j=0}^{f-1} (r_j+1)p^j} & 0 \\ 0 & \omega_{2f}^{q\sum_{j=0}^{f-1}(r_j+1)p^j} \end{pmatrix} \]
up to twist. In this section, we write 
$\delta$ for 
$\delta _{\mathrm {irr}}$. If 
$\lambda = \lambda _J$, then we define 
$\delta \lambda = \lambda _{\delta J}$, and similarly, if 
$\chi = \sigma (\lambda _J, {\bar {\rho }})^{I_1}$, then we define 
$\delta \chi = \sigma (\lambda _{\delta J}, {\bar {\rho }})^{I_1}$. By the proof of [Reference BreuilBre11, Proposition 5.1], and our assumptions on the characters 
$\chi _i$, we have 
$\chi _{i+1} = \delta \chi _i$ for all 
$i$.
We remark that 
$k$, the number of characters in our cycle, is an even number when 
${\bar {\rho }}$ is irreducible: this is because 
$\delta ^f(J) = J^\textrm {c}$ (the complement of 
$J$). For future reference, we record the following lemma.
Lemma 5.18 The number 
$|J \triangle \delta J|$ is odd.
Proof. It is enough to prove that 
$|J \triangle \delta _{\mathrm {red}}(J)|$ is even, because 
$\delta _{\mathrm {red}}(J)$ differs from 
$\delta _\mathrm {irr}(J)$ precisely at 
$f-1$. We have 
$j \in J \triangle \delta _\mathrm {red} J$ if and only if 
$\operatorname {ch}_J(j, j+1) \in \{(1, 0), (0, 1)\}$.
Define an involution 
$*: J \triangle \delta _\mathrm {red} J \to J \triangle \delta _\mathrm {red} J$ as follows: if 
$\operatorname {ch}_J(j, j+1) = (1, 0)$, then 
$*j$ is the first number following 
$j$ such that 
$\operatorname {ch}_J(*j, *j+1) = (0, 1)$. If 
$\operatorname {ch}_J(j, j+1) = (0, 1)$, then 
$*j$ is the first number preceding 
$j$ such that 
$\operatorname {ch}_J(*j, *j+1) = (1, 0)$. The lemma follows because 
$*$ has no fixed points.
Computation of 
$i^+(\psi _{k-1-i}^s)$. By definition, 
$i^+(\psi _{k-1-i}^s)$ is the only number in the interval 
$[1, q-2]$ such that 
$S_{i^+(\psi _{k-1-i}^s)}^+$ sends the character 
$\chi _i$ to 
$\chi _{i+1}$. As such, we also write it as 
$i^+(\chi _i)$. The modular weight of 
${\bar {\rho }}$ corresponding to 
$\chi _i$ can be written as 
$\sigma (\lambda _J, {\bar {\rho }}) = (\lambda _0(r_0), \ldots , \lambda _{f-1}(r_{f-1}) )\otimes \det ^{e(\lambda )(r)}$ for some 
$J$, and so
\[ \chi_i: \begin{pmatrix} a & 0 \\ 0 & d \end{pmatrix} \mapsto a^{\lambda(r)} (ad)^{e(\lambda)(r)} \]
where we have written 
$\lambda (r) = \sum _{i=0}^{f-1} \lambda _i(r_i)p^i$. Our goal in this paragraph is to prove the following proposition.
Proposition 5.19 The 
$j$th 
$p$-adic digit of 
$i^+(\chi _i)$ is
\[ i^+(\chi_i)_j =\begin{cases} (\delta\lambda)_j(r_j) &\text{when~${\bar{\rho}}$ is irreducible},\\ p-1 &\text{when~${\bar{\rho}}$ is reducible}. \end{cases} \]Remark 5.20 Some of the computations in this section have appeared in [Reference HuHu16] for reducible 
${\bar {\rho }}$ (the combinatorics is similar). To see the compatibility between Proposition 5.19 and [Reference HuHu16, Lemme 5.4(ii)], note that the set there denoted 
$I(\lambda )$ coincides with 
$\delta _\mathrm {red} J_\lambda$ in our notation. As we also need the (slightly different) irreducible case, we have decided to give a full argument.
We need to recall [Reference Breuil and PaškūnasBP12, Lemma 2.2]. The Jordan–Hölder factors of 
$\mathrm {Ind}_I^{\mathrm {GL}_2(k_L)}(\chi _i^s)$ are in bijection with subsets of 
$\{ 0, \ldots , f-1 \}$. We first define a bijection between the set of subsets of 
$\{ 0, \ldots , f-1 \}$ and a set of formal weights 
$\mathcal {P}(x_i)$, which we denote 
$\mu \mapsto J(\mu )$. Writing 
$\mu$ as an 
$f$-tuple 
$(\mu _i)$, it is characterized by
\[ \mu_i = \begin{cases} p-1-x_i & \text{ if } \operatorname{ch}_{J(\mu)}(i-1, i) = (1, 1),\\ x_i - 1 & \text{ if } \operatorname{ch}_{J(\mu)}(i-1, i) = (1, 0),\\ p-2-x_i & \text{ if } \operatorname{ch}_{J(\mu)}(i-1, i) = (0, 1),\\ x_i & \text{ if } \operatorname{ch}_{J(\mu)}(i-1, i) = (0, 0). \end{cases} \]
Note that 
$j \in J(\mu )$ if and only if 
$\mu _j \in \{p-2-x_j, p-1-x_j\}$. The Jordan–Hölder factor of 
$\mathrm {Ind}_I^{\mathrm {GL}_2(k_L)}(\chi _i^s)$ corresponding to 
$J(\mu )$ is 
$\tau = \mu (\lambda (r)) \otimes \det ^{e(\mu )(\lambda (r))}\eta$, where 
$e(\mu )$ is as defined in (10).
The following lemma is implicit in [Reference Breuil and PaškūnasBP12], but because it is fundamental to our method we give some details.
Lemma 5.21 Let 
$t \in [1, q-2]$ be an integer and define 
$\chi : \big (\begin {smallmatrix} a & * \\ 0 & d \end {smallmatrix}\big ) \to a^t\eta (ad)$. Let 
$\mu \in \mathcal {P}(x_i)$, corresponding to the constituent 
$\tau = \mu (t) \otimes \det ^{e(\mu )(t)}\eta$ of 
$\mathrm {Ind}_I^{\mathrm {GL}_2(k_L)}\chi ^s$. (We write 
$\mu (t)$ for the evaluation of the formal weight 
$\mu$ at the 
$p$-adic digits of 
$t$.) Assume that 
$\tau$ is neither the socle nor the cosocle of the induction. Let 
$\varphi$ be a generator of the 
$\chi ^s$-eigenspace for 
$I/I_1$ in 
$\mathrm {Ind}_I^{\mathrm {GL}_2(k_L)}\chi ^s$.
Then 
$S_{\sum _{j \in J(\mu )}p^j(p-1-\mu _j(t_j))} \varphi$ generates 
$\tau ^{I_1}$ in every quotient of 
$\mathrm {Ind}_I^{\mathrm {GL}_2(k_L)}\chi ^s$ where 
$\tau$ is a subrepresentation.
Proof. Compare [Reference Breuil and PaškūnasBP12, Lemma 2.7]. By our assumption on 
$\tau$, the character 
$\tau ^{I_1}$ has multiplicity one, hence it is enough to prove that the given vector has the same 
$H$-eigencharacter as
\[ \tau^{I_1} = a^{\mu(t)}(ad)^{e(\mu)(t)}\eta(ad) = a^t (a^{{-}1}d)^{e(\mu)(t)}\eta(ad). \]
An application of Lemma 2.7 implies that the 
$H$-eigencharacter of 
$S_{\sum _{j \in J(\mu )}p^j(p-1-\mu _j(t_j))} \varphi$ is
\[ a^{t}\eta(ad)(a^{{-}1}d)^{\sum_{j \in J(\mu)}p^j(p-1-\mu_j(t_j))}, \]
so we need to prove that 
$e(\mu )(t) \equiv \sum _{j \in J(\mu )}p^j(p-1-\mu _j(t_j)) \mod q-1$. We actually prove this as an equality in 
$\mathbf {Z}$.
Write 
$e(\mu )_i = 2x_i-p+1, 1, 2x_i-p+2, 0$ according to whether 
$\operatorname {ch}_{J(\mu )}(i-1, i) = (1, 1)$, 
$(1, 0)$, 
$(0, 1)$, 
$(0, 0)$, respectively. By definition, we have
\begin{equation} e(\mu) = \begin{cases} \displaystyle\frac{1}{2} \bigg(\sum_{i=0}^{f-1} e(\mu)_i p^i\bigg) &\text{ if } \mu_{f-1} \in \{x_{f-1}, x_{f-1}-1\}\\ \displaystyle\frac{1}{2} \bigg(p^f - 1 + \sum_{i=0}^{f-1} e(\mu)_i p^i\bigg) &\text{ otherwise}. \end{cases} \end{equation} We are going to rearrange this sum so that the summands 
$-p$ do not appear. Define an integer
\begin{equation} d(\mu)_i =\begin{cases} e(\mu)_i + p - 1 &\text{ if } \operatorname{ch}_{J(\mu)}(i-1, i) = (1, 1), \\ e(\mu)_i - 1 &\text{ if } \operatorname{ch}_{J(\mu)}(i-1, i) = (1, 0), \\ e(\mu)_i +p &\text{ if } \operatorname{ch}_{J(\mu)}(i-1, i) = (0, 1), \\ e(\mu)_i &\text{ if } \operatorname{ch}_{J(\mu)}(i-1, i) = (0, 0), \end{cases} =\begin{cases} 2x_i & \text{if } \operatorname{ch}_{J(\mu)}(i-1, i) = (1, 1), \\ 0 & \text{if } \operatorname{ch}_{J(\mu)}(i-1, i) = (1, 0), \\ 2x_i + 2 & \text{if } \operatorname{ch}_{J(\mu)}(i-1, i) = (0, 1), \\ 0 & \text{if } \operatorname{ch}_{J(\mu)}(i-1, i) = (0, 0). \end{cases} \end{equation}Then
\begin{equation} \sum_i d(\mu)_ip^i = \begin{cases} \displaystyle\sum_i e(\mu)_ip^i & \text{if } f-1 \notin J(\mu),\\ p^f - 1 + \sum_i e(\mu)_ip^i & \text{if } f-1 \in J(\mu). \end{cases} \end{equation}
(It may be helpful to visualize this process as ‘carrying’ a coefficient 
$e(\mu )_i$ to the left, when it is equal to 
$1$.)
As 
$f-1 \notin J(\mu )$ is equivalent to 
$\mu _{f-1} \in \{x_{f-1}-1, x_{f-1}\}$, we deduce from (49), (50), and (51) that
\[ e(\mu) = \frac{1}{2} \bigg(\sum_{i-1 \not \in J(\mu), i \in J(\mu)}(2x_i+2)p^i + \sum_{i-1, i \in J(\mu)} 2 x_i p^i \bigg) \]
and because 
$x_i+1 = p-1-(p-2-x_i)$ and 
$x_i = p-1-(p-1-x_i)$, the claim follows.
Lemma 5.22 Let 
$\lambda$ be a formal weight for 
${\bar {\rho }}$ and let 
$\mu \in \mathcal {P}(x_i)$ be such that 
$\delta (\lambda ) = \mu \circ \lambda$ (that is, its 
$i$th component is the composition 
$\mu _i \circ \lambda _i$). Then
\[ \mathrm{det}^{e(\mu\circ\lambda)(r)} = \mathrm{det}^{e(\mu)(\lambda(r))}\mathrm{det}^{e(\lambda)(r)}. \]Proof. Working in the group of formal weights under composition, we see that the expression for 
$e(\mu )$ contains a 
$p^f-1$ if and only if exactly one of 
$e(\delta (\lambda ))$ and 
$e(\lambda )$ contains a 
$p^f-1$. The rest follows from the identity 
$r_i - (\mu \circ \lambda )_i(r_i) = r_i - \lambda _i(r_i) + \lambda _i(r_i) - \mu _i(\lambda _i(r_i))$.
Lemma 5.23 Let 
$\chi _i = \sigma (\lambda _J, {\bar {\rho }})^{I_1}$, as before. Assume 
$\delta (\lambda ) = \mu \circ \lambda$ with 
$\mu \in \mathcal {P}(x_i)$. Then
\[ i^+(\chi_i) = q-1-\sum_{j \in J(\mu)} p^j(p-1-(\delta\lambda)_j(r_j)). \]Proof. By our assumptions, 
$\chi _{i+1}$ is the 
$I/I_1$-character appearing in the weight
\[ ((\delta \lambda)_0(r_0), \ldots, (\delta\lambda)_{f-1}(r_{f-1})) \otimes \mathrm{det}^{e(\delta \lambda)(r)}, \]
where, by definition, 
$\delta \lambda$ is the formal weight corresponding to the set 
$\delta J$. We claim that the hypotheses of Lemma 5.21 hold: then it follows from Lemma 5.22 and Lemma 5.21 that if 
$\delta (\lambda ) = \mu \circ \lambda$, then 
$S_{\sum _{j \in J(\mu )} p^j(p-1-(\delta \lambda )_j(r_j))}$ sends 
$\chi _i^s$ to 
$\chi _{i+1}$. But then 
$i^+(\chi _i) = q-1-\sum _{j \in J(\mu )} p^j(p-1-(\delta \lambda )_j(r_j))$ by Lemma 2.7.
To check the assumptions of Lemma 5.21, observe that 
$\chi _{i+1}$ is the eigencharacter of
\[ \operatorname{soc}_K \operatorname{im}(\bar{\theta}(\chi_i^s) \to D_0({\bar{\rho}})). \]
If 
$\bar {\theta }(\chi _i^s)$ embeds in 
$D_0({\bar {\rho }})$, then 
$R\chi _i^s = (\operatorname {soc}_K \bar {\theta }(\chi _i^s))^{I_1} = \chi _i$, which contradicts our assumptions. Similarly, if 
$\bar {\theta }(\chi _i^s) \to D_0({\bar {\rho }})$ factors through 
$\operatorname {cosoc}_K \bar {\theta }(\chi _i^s)$, then 
$R \chi _i^s = \chi _i^s$, another contradiction.
To conclude the proof of Proposition 5.19, it is enough to prove the following.
Proposition 5.24 Let 
$\lambda$ be a formal weight of 
${\bar {\rho }}$ and define 
$\mu \in \mathcal {P}(x_i)$ by 
$\delta (\lambda ) = \mu \circ \lambda$. Then 
$J(\mu ) = J \triangle \delta J$.
Proof. The group of formal weights maps onto 
$W$, the Weyl group of 
$\mathrm {Res}_{\mathbf {Q}_{p^f}/\mathbf {Q}_p} \mathrm {GL}_2$, according to the sign of the leading coefficients. By construction, 
$j \in J(\mu )$ if and only if 
$\mu _j$ has negative leading coefficient, but this happens if and only if 
$\lambda _j, (\delta \lambda )_j$ have opposite leading coefficients. By construction, this is equivalent to 
$j \in J \triangle \delta J$.
Computation of 
$S_{i^+(\psi _1^s)}^+\cdots S_{i^+(\psi _k^s)}^+(x)$. By construction, Proposition 2.5 applies to the product 
$S_{i^+(\psi _1^s)}^+\cdots S_{i^+(\psi _k^s)}^+$. Let
\[ \beta = \beta(i^+(\psi_1^s), \ldots, i^+(\psi_k^s)) \in W(k_L) \]
be as in Proposition 2.5, and choose 
$x \in \theta ^{\psi _k^s}$ as in Proposition 5.14, where 
$\theta [1/p]$ denotes the central type for 
${\bar {\rho }}$. Then Lemma 2.9 implies that
\begin{equation} S_{i^+(\psi_1^s)}^+\cdots S_{i^+(\psi_k^s)}^+(x) = \beta x \in \theta^{\psi_k^s}. \end{equation}Proposition 5.25 We have 
$v_p(\beta ) = \frac {1}{2} k |J \triangle \delta J|$ and the leading term of 
$\beta$ is congruent to
\[ ({-}1)^{(1/2) k |J \triangle \delta J| + ({kh}/{2f})(1+\sum r_i)} \]
modulo 
$p$, where the invariant 
$h$ is defined in [Reference BreuilBre11, Lemme 6.2(ii)] (we do not need the explicit definition).
Remark 5.26 The equality 
$v_p(\beta ) = \frac {1}{2} k |J \triangle \delta J|$ follows from Proposition 5.7 and the fact that 
$v_p(\beta )=v_p(\prod _i \widetilde {U}_p(\chi _i))$. Here, we give a different proof of this fact.
Proof. By Proposition 2.5 we know that the leading term 
$c(\beta )$ of 
$\beta$ is the Teichmüller lift of
\[ ({-}1)^{v_p(\beta) + (k-2)(f-1)} \prod_{\substack{0 \leq t \leq k-1 \\ 0 \leq j \leq f-1}} i^+(\psi_t^s)_j! \]and the valuation is
\begin{equation} v_p(\beta) = \frac{1}{p-1} \bigg(\sum_{\substack{0 \leq t \leq k-1 \\ 0 \leq j \leq f-1}} p - 1 - i^+(\psi_t^s)_j \bigg). \end{equation} We begin to compute 
$c(\beta )$. By Proposition 5.19 and the proof of [Reference BreuilBre11, Lemme 6.2(ii)], we have
\[ \prod_{\substack{0 \leq t \leq k-1 \\ 0 \leq j \leq f-1}}(p-1- i^+(\psi_t^s)_j)! = ({-}1)^{({kh}/{2f})(1+\sum r_i)}. \]
Hence, 
$c(\beta )$ is also congruent modulo 
$p$ to
\begin{equation} ({-}1)^{v_p(\beta) + (k-2)(f-1) + kf+\sum_{j, t}(p-1- i^+(\psi_t^s)_j)+ ({kh}/{2f})(1+\sum r_i)}, \end{equation}
using that 
$x! (p-1-x)! \equiv (-1)^{x+1} \pmod p$ for 
$0\leq x\leq p-1$. Hence, because 
$k$ is even and 
$p-1$ divides 
$\sum _{j, t}(p-1- i^+(\psi _t^s)_j)$, we see that (54) also equals
\[ ({-}1)^{v_p(\beta)+ ({kh}/{2f})(1+\sum r_i)}. \]
Now we consider 
$v_p(\beta )$. There is some cancellation in pairs in the sum (53). If the character 
$\chi _{k-1}$ corresponds to 
$J \subset \{0, \ldots , f-1\}$, then we can apply Proposition 5.19 and rewrite (53) as
\begin{align} \frac{1}{p-1} \bigg(\sum_{j \in J \triangle \delta J} p-1-(\delta\lambda)_j(r_j) + \sum_{j \in \delta J \triangle \delta^2 J} p-1-(\delta^2 \lambda)_j(r_j) + \cdots + \sum_{j \in \delta^{k-1} J \triangle J} p-1-\lambda_j(r_j)\bigg). \end{align}
Recall that the complement 
$J^\textrm {c}$ is contained in the 
$\delta$-orbit of 
$J$: more precisely, it is equal to 
$\delta ^{k/2}(J)$, by the definition of 
$\delta = \delta _\mathrm {irr}$. By Lemma 5.27 to follow, we find that
\[ p-1-\delta\lambda_j(r_j) + p-1-(\delta^{k/2+1}\lambda)_j(r_j) = p-1. \]
As the integer 
$|J \triangle \delta J|$ is constant throughout the 
$\delta$-orbit of 
$J$, Equation (55) equals 
$\frac {1}{2} k |J \triangle \delta J|$.
Lemma 5.27 If 
$j \in J \triangle \delta J$, then 
$\lambda _{\delta J, j} + \lambda _{\delta (J^\textrm {c}), j} = p-1$.
Proof. By Lemma 1.5, it suffices to prove that if 
$j \in \delta ^{-1}J \triangle J$, then 
$\lambda _{J, j} + \lambda _{J^\textrm {c}, j} = p-1$. This is immediate from the definition of 
$J \mapsto \lambda _J$, see (11).
Computation of 
$\prod _{i=1}^{k} \widetilde {c}(\psi _i^s)$. Recall that the constants 
$c(\psi _i^s)$ are defined by 
$S_{i(\psi _i)}S_0 \varphi ^{\psi _i^s} = c(\psi _i^s) S^+_{i^+(\psi _i^s)} \varphi ^{\psi _i^s}$, that 
$i(\psi _i) = q-1 - i^+(\psi _i^s) \not \in \{0, q-1\}$, and that 
$\widetilde {c}(\psi _i^s)$ is the leading term of 
$c(\psi _i^s)$. In this section, we compute the product 
$\prod _{i=1}^{k} \widetilde {c}(\psi _i^s)$ by the method of [Reference Breuil and DiamondBD14], and we prove that it equals 
$1$.
Fix an index 
$i$. The character 
$\psi _i^s = \chi _{k-1-i}$ appears in the socle of 
$D_0({\bar {\rho }})$ and so it is the 
$I/I_1$-eigencharacter of a modular weight of 
${\bar {\rho }}$. We write 
$\lambda$ for the corresponding formal weight, and 
$J$ for the corresponding subset of 
$\{0, \ldots , f-1\}$. We therefore have the formula 
$\psi _i^s: \big (\begin {smallmatrix} a & 0 \\ 0 & d \end {smallmatrix}\big ) \mapsto [a]^{\lambda (r)}[ad]^{e(\lambda )(r)}$.
Lemma 5.28 We have an equality
\[ c(\psi_i^s) = \mathbf{J}(i(\psi_i), q-1-\lambda(r)). \]Proof. Our genericity assumptions imply that 
$i(\psi _i), \lambda (r) \not \in \{0, q-1\}$. Thus, we have 
$\mathbf {J}(i(\psi _i), q-1-\lambda (r)) = \mathbf {J}_0(i(\psi _i), q-1-\lambda (r))$, and applying Lemma 2.8 we find
\[ S_{i(\psi_i)}S_0 \varphi^{\psi_i^s} = [({-}1)^{e(\lambda)(r)}] \mathbf{J}_0(i(\psi_i), q-1-\lambda(r)) S_{i(\psi_i)- \lambda(r)}\varphi^{\psi_i^s}. \]
The assumptions of the lemma are satisfied because 
$i(\psi _i) \not \in \{0, q-1\}$, and if 
$q-1 | i(\psi _i) - \lambda (r)$, then 
$R\psi _i = \psi _i$, which contradicts our assumption 
$R\chi _{k-1-i}^s \ne \chi _{k-1-i}^s$. (Recall that, by Definition 4.1, the operator 
$S_{i(\psi _i)}$ applied to the 
$\psi _i$-eigenvector 
$S_0\varphi ^{\psi _i^s}$ yields an 
$R\psi _i$-eigenvector. On the other hand, 
$S_{\lambda (r)}$ applied to a 
$\psi _i$-eigenvector would yield a 
$\psi _i^s\alpha ^{-\lambda (r)} = \psi _i$-eigenvector, by Lemma 2.7.) Hence, we have an equality
\[ [({-}1)^{e(\lambda)(r)}] \mathbf{J}_0(i(\psi_i), q-1-\lambda(r)) S_{i(\psi_i)- \lambda(r)}\varphi^{\psi_i^s} = c(\psi_i^s) S_{i^+(\psi_i^s)}^+ \varphi^{\psi_i^s}. \]
To conclude, we are going to apply an auxiliary operator 
$S_z$ to both sides of this equality. The number 
$z$ can be chosen arbitrarily, except that we require that Lemmas 2.8 and 2.4 can be applied to deduce that
\begin{gather*} S_z S_{i(\psi_i)- \lambda(r)}\varphi^{\psi_i^s} = [({-}1)^{e(\lambda)(r)}] \mathbf{J}_0(z,q-1-i(\psi_i)) S_{z - i(\psi_i)} \varphi^{\psi_i^s},\\ S_z S_{i^+(\psi_i^s)}^+ \varphi^{\psi_i^s} = \mathbf{J}_0(z, i^+(\psi_i^s)) S_{z+i^+(\psi_i^s)} \varphi^{\psi_i^s}. \end{gather*}
This rules out at most four choices of 
$z$, which is fine because 
$p \geq 5$. As 
$i(\psi _i) = q-1-i^+(\psi _i^s)$ and these Jacobi sums do not vanish, we deduce that
\[ \mathbf{J}_0(i(\psi_i), q-1- \lambda(r)) S_{z+i^+(\psi_i^s)} \varphi^{\psi_i^s} = c(\psi_i^s) S_{z+i^+(\psi_i^s)} \varphi^{\psi_i^s}. \]
The lemma follows because 
$S_{z+i^+(\psi _i^s)}\varphi ^{\psi _i^s} \not = 0$ (all the 
$S_i$-operators are nonzero on 
$\varphi ^{\psi _i^s}$, which generates the cosocle of the lattice 
$\mathrm {Ind}_I^K(\psi _i^s)$ where these computations are taking place).
Lemma 5.29 Define 
$\epsilon _j = 1$ if 
$j \not \in J \triangle \delta J \text { and } j-1 \in J \triangle \delta J$, and 
$\epsilon _j = 0$ otherwise. Then 
$v_p(c(\psi _i^s)) = f - |J \triangle \delta J|$, and the leading term of 
$c(\psi _i^s)$ is
\[ ({-}1)^{f-1+v_p(c(\psi_i^s))} \prod_{j \in J \triangle \delta J} \frac{(p-1-\lambda_j(r_j))!(p-1-(\delta \lambda)_j(r_j))!}{0!} \prod_{j \not \in J \triangle \delta J}\frac{(p-1-\lambda_j(r_j))!}{(p-1-\lambda_j(r_j) + \epsilon_j)!}. \]Proof. In this proof we shorten notation to 
$\lambda _j = \lambda _j(r_j)$. For 
$j \in [0,f-1]$, let 
$\Sigma _j \in [0,p-1]$ and 
$Q \in \{0,1\}$ be such that
\[ \sum_{j \in \{0, \ldots, f-1\}} p^j(p-1-\lambda_j) + \sum_{j \in J \triangle \delta J} p^j(p-1-(\delta\lambda)_j) = \sum_{j \in \{ 0, \ldots, f-1\}} \Sigma_j p^j + Q(q-1). \]
Then 
$\Sigma _j$ is equal to
\[ \Sigma_j = \begin{cases} 0 & \text{if } j \in J \triangle \delta J,\\ p-1-\lambda_j + \epsilon_j & \text{if } j \notin J \triangle \delta J. \end{cases} \]This follows immediately (by ‘carrying’) from the fact that
\[ (p-1-\lambda_j) + (p-1-(\delta \lambda)_j) = \begin{cases} p & \text{if } \operatorname{ch}_{J \triangle \delta J}(j, j-1) = (1, 0),\\ p-1 & \text{if } \operatorname{ch}_{J \triangle \delta J}(j, j-1) = (1, 1), \end{cases} \]as can be checked directly from (11) (or see the Appendix).
By Lemma 5.28, Proposition 5.19 and Theorem 2.1, it follows that
\begin{align*} v_p(c(\psi_i^s)) &= \frac 1 {p-1} \bigg(\sum_{j \in J \triangle \delta J} p-1-(p-1-\lambda_j+p-1-\delta\lambda_j - \Sigma_j)\\ &\quad + \sum_{j \not \in J \triangle \delta J} p-1-(p-1-\lambda_j + 0 -\Sigma_j)\bigg)\\ &= \frac{1}{p-1} \bigg(\sum_{j \in J \triangle \delta J} p-1-(p-1-\lambda_j + p-1-\delta\lambda_j) + \sum_{j \not \in J \triangle \delta J} p-1+\epsilon_j\bigg) \\ &= f-|J \triangle \delta J| + \frac{1}{p-1} \bigg(\sum_{j \in J \triangle \delta J, j-1 \not \in J \triangle \delta J} -1 + \sum_{j \not \in J \triangle \delta J, j-1 \in J \triangle \delta J} 1\bigg) = f-|J \triangle \delta J| \end{align*}and the leading term
\[ ({-}1)^{f-1+v_p(c(\psi_i^s))} \prod_{j \in J \triangle \delta J} \frac{(p-1-\lambda_j)!(p-1-\delta\lambda_j)!}{\Sigma_j!} \prod_{j \not \in J \triangle \delta J} \frac{(p-1-\lambda_j)!}{\Sigma_j!} \]is as claimed.
Lemma 5.30 The leading term of 
$c(\psi _i^s)$ is the Teichmüller lift of
\[ ({-}1)^{f-1+v_p(c(\psi_i^s))+ |J \triangle \delta J|+ \sum_{j \in J \triangle \delta J}(p-1-\delta \lambda_j(r_j))} \frac{\prod_{j \in J \triangle \delta J, j-1 \not \in J \triangle \delta J}(p-1- \lambda_j(r_j))}{\prod_{j \not \in J \triangle \delta J, j-1 \in J \triangle \delta J}(p-\lambda_j(r_j))}. \]Proof. Again we shorten notation to 
$\lambda _j = \lambda _j(r_j)$. We simplify the two factors in the expression of Lemma 5.29. For the second factor, by definition of 
$\epsilon _j$ we have cancellation of numerator with denominator whenever 
$j, j-1 \not \in J \triangle \delta J$. In the numerator of the first factor, we apply the identity 
$x!(p-1-x)! \equiv (-1)^{x+1} \mod p$ together with
\[ p-1- \lambda_j = p-1 - (p-1-\delta\lambda_j) \quad \text{when } \operatorname{ch}_{J \triangle \delta J}(j, j-1) = (1, 1), \]to simplify the whole leading term to
\begin{align*} & ({-}1)^{f-1+v_p(c(\psi_i^s))+\sum_{j,\ j-1 \in J \triangle \delta J}(p-1-\delta\lambda_j+1)} \prod_{\operatorname{ch}_{J \triangle \delta J}(j ,j-1) = (1, 0)}(p-1-\lambda_j)!(p-1-\delta\lambda_j)! \\ &\quad \times \prod_{\operatorname{ch}_{J \triangle \delta J} (j, j-1) = (0, 1)} \frac{(p-1-\lambda_j)!}{(p-\lambda_j)!}. \end{align*}
To conclude, we simplify the second factor to 
$\prod _{\operatorname {ch}_{J \triangle \delta J}(j, j-1) = (0, 1)} 1/ ({p-\lambda _j})$, and we apply the equality
\[ p-1-\lambda_j = p - (p-1- \delta\lambda_j)\quad \text{when } \operatorname{ch}_{J \triangle \delta J}(j, j-1) = (1, 0) \]
together with 
$x!(p-1-x)! \equiv (-1)^{x+1} \mod p$ to simplify the first factor to
\[ ({-}1)^{\sum_{\operatorname{ch}_{J \triangle \delta J}(j, j-1) = (1, 0)} (p-1-\delta \lambda_j + 1)} \prod_{\operatorname{ch}_{J \triangle \delta J}(j, j-1) = (1, 0)}(p-1-\lambda_j). \]Proposition 5.31 The product 
$\prod _{i=1}^k c(\psi _i^s)$ has valuation 
$kf - k|J\triangle \delta J|$ and leading term equal to 
$1$.
Proof. By Lemma 5.29, 
$v_p(c(\psi _i^s))$ does not depend on 
$i$ and the product has valuation 
$kf - k|J \triangle \delta J|$. For the leading term, we have seen in the proof of Proposition 5.25 that
\[ \sum_{i=0}^{k-1}\sum_{j \in \delta^i J \triangle \delta^{i+1} J}(p-1-\delta^{i+1}\lambda_j(r_j)) \]
is divisible by 
$p-1$, hence it is even when 
$p$ is odd. It follows that the product of all the sign terms from Lemma 5.30 is a positive sign, because 
$k$ is even.
To conclude, we need to prove that
\[ \prod_{t=0}^{k-1}\frac{\prod_{j \in \delta^t J \triangle \delta^{t+1}J, j-1 \not \in \delta^t J \triangle \delta^{t+1}J}(p-1-(\delta^t\lambda)_j(r_j))}{\prod_{j \not \in \delta^t J \triangle \delta^{t+1}J, j-1 \in \delta^t J \triangle \delta^{t+1}J} (p-(\delta^t\lambda)_j(r_j))} = 1. \]
For 
$\mathfrak {j} \in \{J, \delta J \ldots , \delta ^{k-1} J\}$, let us define
\begin{align*} (j, \mathfrak{j}) &= p-1-\lambda_{\mathfrak{j}, j}(r_j)\quad \text{if } j \in \mathfrak{j} \triangle \delta \mathfrak{j},\enspace j-1 \not \in \mathfrak{j} \triangle \delta \mathfrak{j}\\ &= (p-\lambda_{\mathfrak{j}, j}(r_j))^{{-}1}\quad \text{if } j \not \in \mathfrak{j} \triangle \delta \mathfrak{j},\enspace j-1 \in \mathfrak{j} \triangle \delta \mathfrak{j}\\ &= 1 \quad \text{otherwise}. \end{align*}Then we need to prove
\begin{equation} \prod_{j=0}^{f-1}\prod_{\mathfrak{j} \in \{J, \ldots, \delta^{k-1}J\}}\quad (j, \mathfrak{j}) = 1, \end{equation}
and for this it will suffice to fix an index 
$j$ and prove that the inner product is equal to 
$1$.
Having fixed 
$j$, we claim that there are two commuting involutions on the set
\[ \mathcal{S}_j = \{ \mathfrak{j} \in \{J, \ldots, \delta^{k-1}J\} : \operatorname{ch}_{\mathfrak{j} \triangle \delta \mathfrak{j}}(j, j-1) = (1, 0) \text{ or } (0, 1)\}. \]
One is the complement 
$\mathfrak {j} \mapsto \mathfrak {j}^\textrm {c}$, because 
$\mathfrak {j} \triangle \delta \mathfrak {j} = \mathfrak {j}^\textrm {c} \triangle \delta \mathfrak {j}^\textrm {c}$. The other, denoted 
$\mathfrak {j} \mapsto *\mathfrak {j}$, is defined in the same way as in the proof of Lemma 5.18: for instance, if 
$\operatorname {ch}_{\mathfrak {j} \triangle \delta \mathfrak {j}}(j, j-1) = (1, 0)$, then 
$*\mathfrak {j}$ is the first element of 
$\mathcal {S}_j$ following 
$\mathfrak {j}$ such that 
$\operatorname {ch}_{*\mathfrak {j} \triangle \delta *\mathfrak {j}}(j, j-1) = (0, 1)$. This is well-defined because 
$\delta _{\mathrm {red}}(\mathfrak {j} \triangle \delta \mathfrak {j}) = \delta \mathfrak {j} \triangle \delta ^2 \mathfrak {j}$.
Decomposing 
$\mathcal {S}_j$ in orbits for the resulting action of 
$\mathbf {Z}/2 \times \mathbf {Z}/2$, it suffices to prove that, if 
$\mathfrak {j} \in \mathcal {S}_j$, then
\[ (j, \mathfrak{j})(j, \mathfrak{j}^\textrm{c})(j, *\mathfrak{j})(j, *\mathfrak{j}^\textrm{c}) = 1. \]
To do so, we assume without loss of generality that 
$j \in \mathfrak {j} \triangle \delta \mathfrak {j}$ (otherwise, we relabel 
$\mathfrak {j}$ to 
$*\mathfrak {j}$). Now it suffices to prove that
\[ \{p-1-\lambda_{\mathfrak{j}, j}(r_j),\ p-1-\lambda_{\mathfrak{j}^\textrm{c}, j}(r_j)\} = \{p-\lambda_{*\mathfrak{j}, j}(r_j),\ p-\lambda_{*\mathfrak{j}^\textrm{c}, j}(r_j)\}, \]or, equivalently, that
\[ \{\lambda_{\mathfrak{j}, j}, \lambda_{\mathfrak{j}^\textrm{c}, j}\} = \{\lambda_{*\mathfrak{j}, j}-1, \lambda_{*\mathfrak{j}^\textrm{c}, j}-1\}. \]
Knowing that 
$\operatorname {ch}_{\mathfrak {j} \triangle \delta \mathfrak {j}}(j, j-1) = (1, 0)$ does not determine 
$\lambda _{\mathfrak {j}, j}$ uniquely. However, because 
$\operatorname {ch}_\mathfrak {j} + \operatorname {ch}_{\mathfrak {j}^\textrm {c}} = 1$, it determines 
$\{\lambda _{\mathfrak {j}, j}, \lambda _{\mathfrak {j}^\textrm {c}, j}\}$. More precisely, we have
\[ \{\operatorname{ch}_\mathfrak{j}(j-1, j, j+1), \operatorname{ch}_{\mathfrak{j}^\textrm{c}}(j-1, j, j+1)\} = \begin{cases} \{(0, 0, 1), (1, 1, 0)\}& \text{if } 0 < j < f-1\\ \{(1, 0, 1), (0, 1, 0)\}& \text{if } j = 0\\ \{(1, 1, 1), (0, 0, 0)\}& \text{if } j = f-1, \end{cases} \]
and we deduce from the definition of 
$\mathfrak {j} \mapsto \lambda _\mathfrak {j}$ (see (11)) that
\[ \{\lambda_{\mathfrak{j}, j}, \lambda_{\mathfrak{j}^\textrm{c}, j}\} = \{p-3-x_j, x_j\} \text{ or } \{x_0-1, p-2-x_0\}. \]
An entirely analogous argument using 
$\operatorname {ch}_{*\mathfrak {j} \triangle \delta *\mathfrak {j}}(j, j-1) = \operatorname {ch}_{*\mathfrak {j}^\textrm {c} \triangle \delta *\mathfrak {j}^\textrm {c}}(j, j-1) = (0, 1)$ implies that
\[ \{\lambda_{*\mathfrak{j}, j}, \lambda_{*\mathfrak{j}^\textrm{c}, j}\} = \{x_j+1, p-2-x_j\} \text{ or } \{p-1-x_0, x_0\} \]and the claim follows.
5.4 The split reducible case
In this section, 
${\bar {\rho }}$ is a split reducible representation with tame inertial weights 
$r_i$ (defined as in § 3.2.2). The situation is similar to the irreducible case, with some combinatorial differences, and we will indicate how the computations of the previous sections can be modified to account for this case. Recall the following notational convention: for 
$J \subset \{0, \ldots , f-1\}$ and 
$x, y, z \in \{0, 1\}$, we write 
$\mathbf {r}(j, J) = (x, y, z)$ if 
$\operatorname {ch}_J(j-1, j, j+1) = (x, y, z)$.
The weights of 
${\bar {\rho }}$ are again in bijection with the subsets of 
$\{0, \ldots , f-1\}$. If 
$\chi _i = \sigma (\lambda _J, {\bar {\rho }})^{I_1}$ corresponds to 
$J$, the subset corresponding to 
$\chi _{i+1} = R\chi _i^s$ is 
$\delta _{\mathrm {red}}(J)$, by the proof of [Reference BreuilBre11, Proposition 5.1] (compare also [Reference Breuil and PaškūnasBP12, Lemma 15.2]). In this section, we write 
$\delta$ for 
$\delta _{\mathrm {red}}$.
Computation of 
$i^+(\psi _{k-1-i}^s)$. Assume 
$\chi _i = \sigma (\lambda , {\bar {\rho }})^{I_1}$ and 
$\lambda = \lambda _J$, so that 
$j \in J$ if and only if 
$\lambda _j \in \{p-2-x_j, p-3-x_j \}$, as in [Reference BreuilBre11] and § 3.3. The following proposition is proved in the same way as Proposition 5.19.
Proposition 5.32 Write 
$i^+(\psi _{k-1-i}^s) = i^+(\chi _i) = \sum _{j=1}^{f-1} i^+(\chi _i)_jp^j$. Then we have the equality
\begin{align*} i^+(\chi_i)_j= \begin{cases} \delta(\lambda)_j \quad \text{if } j \in J \triangle \delta J\\ p-1 \quad \text{if } j \not \in J \triangle \delta J. \end{cases}\end{align*} Computation of 
$S_{i^+(\psi _1^s)}^+\cdots S_{i^+(\psi _k^s)}^+(x)$. As in the previous section, there exists a nonzero
\[ \beta = \beta(i^+(\psi_1^s), \ldots, i^+(\psi_k^s)) \in~W(k_L) \]such that
\[ S_{i^+(\psi_1^s)}^+\cdots S_{i^+(\psi_k^s)}^+(x) = \beta x \]
in the central type of 
${\bar {\rho }}$ (where 
$x$ is as in Proposition 5.14). Here, however, 
$k$ need not be even.
Proposition 5.33 The valuation of 
$\beta$ is
\[ v_p(\beta) = \dfrac12 k |J \triangle \delta J|, \]and its leading term is the Teichmüller lift of
\[ ({-}1)^{v_p(\beta) + ({kh}/{f})\sum_{j=0}^{f-1}r_j}\cdot({-}1)^k. \]Remark 5.34 Again, the computation of 
$v_p(\beta )$ also follows from Proposition 5.7, and we do not need the explicit definition of the invariant 
$h$, which can be found in [Reference BreuilBre11, Lemme 6.2].
Proof. We apply Proposition 2.5 and the identity appearing in the proof of [Reference BreuilBre11, Lemme 6.2(i)] to deduce that the leading coefficient 
$c(\beta )$ is the Teichmüller lift of
\[ ({-}1)^{v_p(\beta) + ({kh}/{f})\sum_{j=0}^{f-1}r_j}\cdot({-}1)^k \]
where the factor 
$(-1)^k$ comes from the fact that 
$(k-2)(f-1) + kf$ has the same parity as 
$k$.
To compute the valuation, we consider the equality
\[ v_p(\beta) = \frac{1}{p-1} \bigg(\sum_{t=0}^{k-1} \sum_{j \in \delta^{t}J \triangle \delta^{t+1} J} p-1-(\delta^{t+1}\lambda)_j \bigg) \]
where 
$J$ corresponds to 
$\chi _{k-1}$ (of course, because we are summing over 
$t$, we could have taken any other 
$\chi _i$).
Fix an index 
$j$ and sum over 
$t$. Recall that 
$\delta ^t J \triangle \delta ^{t+1}J = \delta ^t(J \triangle \delta J)$, so that
\begin{equation} \frac{1}{p-1} \bigg(\sum_{t=0}^{k-1}\sum_{j \in \delta^{t}J \triangle \delta^{t+1} J} p-1-(\delta^{t+1}\lambda)_j(r_j)\bigg) = \frac{1}{p-1} \bigg(\sum_{t=0}^{k-1}\sum_{j+t \in J \triangle \delta J} p-1-(\delta\lambda)_{j+t}(r_j)\bigg). \end{equation}
(Recall that 
$\delta ^{t+1}(\lambda )_j$ is a polynomial in a variable 
$x_j$, and it depends on 
$\operatorname {ch}_J(j+t, j+t-1)$ in the same way as 
$(\delta \lambda )_{j+t}$, which is a polynomial in 
$x_{j+t}$. Here we are evaluating 
$x_{j+t}$ at 
$r_j$.)
As 
$\operatorname {ch}_J$ is periodic of period 
$k$, so is 
$\operatorname {ch}_{J \triangle \delta J}$, hence we can rewrite (57) as
\[ \frac k {f(p-1)} \bigg(\sum_{t=0}^{f-1} \sum_{j+t \in J \triangle \delta J} p-1- (\delta \lambda)_{j+t}(r_j)\bigg) = \frac k {f(p-1)} \bigg(\sum_{x \in J \triangle \delta J} p-1-(\delta \lambda)_x(r_j)\bigg). \]
By definition (11), we see that if 
$x\in J \Delta \delta J$, then
\[ p-1-(\delta\lambda)_x(r_j) = \begin{cases} p-2-r_j & \text{if } x\in J, \\ r_j+1 & \text{if } x \notin J. \end{cases} \]
In Lemma 5.18, we have constructed an involution 
$*$ on 
$J \triangle \delta J$, with the property that 
$x \in J$ if and only if 
$*x \notin J$. It follows immediately that
\[ \frac k {f(p-1)} \bigg(\sum_{x \in J \triangle \delta J} p-1-(\delta \lambda)_x(r_j)\bigg) = \frac k {f(p-1)} \frac{1}{2} |J \triangle \delta J|(p-1). \]
The claim follows since summing over indices 
$j$ multiplies this number by 
$f$.
Computation of 
$\prod _{i=1}^{k} \widetilde {c}(\psi _i^s)$. Fix an index 
$i$, and assume 
$\psi _i^s$ corresponds to a subset 
$J$. Here Lemmas 5.29 and 5.30 are still valid, and proved in the same way, hence
\[ v_p(c(\psi_i^s)) = f - |J \triangle \delta J| \]
and the leading coefficient 
$\widetilde {c}(\psi _i^s)$ is the Teichmüller lift of
\[ ({-}1)^{f-1+v_p(c(\psi_i^s))+\sum_{j \in J \triangle \delta J}(p-1-\delta \lambda_j(r_j)) + |J \triangle \delta J|} \frac{\prod_{j \in J \triangle \delta J, j-1 \not \in J \triangle \delta J}(p-1- \lambda_j(r_j))}{\prod_{j \not \in J \triangle \delta J, j-1 \in J \triangle \delta J}(p-\lambda_j(r_j))}. \]Proposition 5.35 We have the equality 
$\prod _{i=1}^{k} \widetilde {c}(\psi _i^s) = (-1)^{k(f-1)+kf}$.
Proof. Let 
$J$ correspond to 
$\psi _k^s$. In the product 
$\prod _{i=1}^{k} \widetilde {c}(\psi _i^s)$ we first notice a sign contribution of
\[ ({-}1)^{k(f-1)+ kv_p(c(\psi_i^s)) + k|J \triangle \delta J|} = ({-}1)^{k(f-1)+kf} \]
(it will cancel together with the factor of 
$(-1)^k$ from the previous section, because the exponent has the same parity as 
$k$). What remains is
\begin{equation} ({-}1)^{\sum_{t=0}^{k-1}\sum_{j \in \delta^t J \triangle \delta^{t+1} J}(p-1-\delta^{t+1} \lambda_j(r_j))} \prod_{t=0}^{k-1}\bigg(\frac{\prod_{j \in \delta^tJ \triangle \delta^{t+1} J, j-1 \not \in \delta^t J \triangle \delta^{t+1} J}(p-1- \delta^t\lambda_j(r_j))}{\prod_{j \not \in \delta^t J \triangle \delta^{t+1} J, j-1 \in \delta^t J \triangle \delta^{t+1} J}(p-\delta^t\lambda_j(r_j))}\bigg). \end{equation}
The sign is positive because the exponent is divisible by 
$p-1$. Note also that 
$j-1 \in \delta ^t J \Delta \delta ^{t+1}J$ if and only if 
$j \in \delta ^{t-1} J \Delta \delta ^t J$. Then (58) is equal to
\[ \prod_{j=0}^{f-1} \prod_{\mathfrak{j} \in \{J, \ldots, \delta^{k-1} J\}} \frac {\prod_{j\in \delta^{{-}1}\mathfrak{j} \Delta \delta \mathfrak{j},\ j \notin \delta^{{-}1}\mathfrak{j} \Delta \mathfrak{j}}p-1-\lambda_{\mathfrak{j}, j}(r_j)}{\prod_{j\in \delta^{{-}1}\mathfrak{j} \Delta \delta \mathfrak{j},\ j \in \delta^{{-}1}\mathfrak{j} \Delta \mathfrak{j}} p - \lambda_{\mathfrak{j}, j}(r_j)}. \] Fix an index 
$j$. We prove that the corresponding factor in the product above is equal to 
$1$. By (11) (see also Appendix), for any 
$\mathfrak {j} \in \{J, \ldots , \delta ^{k-1} J\}$ we see that
\[ p-1-\lambda_{\mathfrak{j}, j}(r_j) = \begin{cases} r_j+2 & \text{if } \mathbf{r}(j, \mathfrak{j}) = (1, 1, 0),\\ p-1-r_j & \text{if } \mathbf{r}(j, \mathfrak{j}) = (0, 0, 1), \end{cases} \]
\[ p-\lambda_{\mathfrak{j}, j}(r_j) = \begin{cases} r_j+2 & \text{if } \mathbf{r}(j, \mathfrak{j}) = (0, 1, 1),\\ p-1-r_j & \text{if } \mathbf{r}(j, \mathfrak{j}) = (1, 0, 0). \end{cases} \]
If 
$\mathbf {r}(j, \mathfrak {j}) = (1, 1, 0), (0, 0, 1), (0, 1, 1), (1, 0, 0)$ (equivalently, 
$j \in \delta ^{-1} \mathfrak {j} \triangle \delta \mathfrak {j}$), define
\[ f(\mathfrak{j},j)= \begin{cases} r_j+2 & \text{if } \operatorname{ch}_{\mathfrak{j}}(j-1, j) = (1, 1),\\ (p-1-r_j)^{{-}1} & \text{if } \operatorname{ch}_{\mathfrak{j}}(j-1, j) = (1, 0),\\ (r_j+2)^{{-}1} & \text{if } \operatorname{ch}_{\mathfrak{j}}(j-1, j) = (0, 1),\\ p-1-r_j & \text{if } \operatorname{ch}_{\mathfrak{j}}(j-1, j) = (0, 0). \end{cases} \]
Otherwise, let 
$f(\mathfrak {j},j)$ be 
$1$.
Then we have to prove that
\[ \prod_{\mathfrak{j} \in \{J, \ldots, \delta^{k-1} J\}} f(\mathfrak{j},j) = 1. \]
To do so, it suffices to prove that the number of 
$\mathfrak {j}$ such that 
$\mathbf {r}(j, \mathfrak {j}) = (1, 1, 0)$ (respectively, (0, 0, 1)) is the same as the number of 
$\mathfrak {j}$ such that 
$\mathbf {r}(j, \mathfrak {j}) = (0, 1, 1)$ (respectively, (1, 0, 0)). For this, define
\[ \mathcal{R}_j = \{ \mathfrak{j} \in \{J, \ldots, \delta^{k-1} J\}: j \in \delta^{{-}1} \mathfrak{j} \Delta \delta \mathfrak{j} \}. \]
It is enough to construct an involution 
${\dagger} : \mathcal {R}_j \to \mathcal {R}_j$ such that 
$\mathbf {r}(\mathfrak {j},j) \mapsto \mathbf {r}({\dagger} \mathfrak {j},j)$ exchanges 
$(1, 1, 0)$ with 
$(0, 1, 1)$ and 
$(0, 0, 1)$ with 
$(1, 0, 0)$. Note that if 
$\mathbf {r}(j, \mathfrak {j}) = (1, 1, 0)$ and 
$\mathfrak {j} = \delta ^t J$, then 
$\mathbf {r}(j+t, J) = (1, 1, 0)$, so there exists some positive integer 
$r$ such that 
$\mathbf {r}(j + t + r, J) = (0, 1, 1)$. Hence it makes sense to define 
${\dagger} \mathfrak {j}$ as the first element of 
$\mathcal {R}_j$ following 
$\mathfrak {j}$ such that 
$\mathbf {r}(j, {\dagger} \mathfrak {j}) = (0, 1, 1)$. The other 
${\dagger} \mathfrak {j}$ are defined similarly, as in the proof of Lemma 5.18, and the proposition follows.
5.5 Computation of the constant
Here we put together all of the above and prove the following theorem, which by [Reference BreuilBre11, Théorème 6.4] implies Theorem 1.3, our second main result.
Theorem 5.36 Let 
${\bar {\rho }}: G_L \to \mathrm {GL}_2(k_E)$ be a semisimple Galois representation as in § 3.2.2. Let 
$\chi _0, \ldots , \chi _{k-1}$ be a cycle of characters on 
$D_0({\bar {\rho }})$ satisfying the conditions in § 5.1. Assume that 
$\det ({\bar {\rho }}) \circ \mathrm {Art}_L(p) = 1$. Then we have
\[ g_{\chi_{k-1}}\cdots g_{\chi_0} = \begin{cases} ({-}1)^{(1/2) k + ({kh}/{2f})(1+\sum_{j=0}^{f-1} r_j)} & \text{when ${\bar{\rho}}$ is irreducible,}\\ ({-}1)^{({kh}/{f})\sum_{j=0}^{f-1}r_j} \alpha^{(|J|-|J^\textrm{c}|)k/f} & \text{when~${\bar{\rho}}$ is reducible}. \end{cases} \]Proof. By Lemma 5.10 and the discussion after Proposition 5.14, 
$c(\chi _0, \ldots , \chi _{k-1})^{-1}$ is the product of 
$\prod _{i=0}^{k-1}\widetilde {c}(\psi _i^s)$ with the leading term of 
$\beta = \beta (\psi _1^s, \ldots , \psi _k^s)$ (the valuation of 
$\beta$ does not intervene because we are working with saturated inclusions). Combining Propositions 5.25, 5.31, 5.33, and 5.35, this product is equal to
\begin{gather*} ({-}1)^{(1/2) k |J \triangle \delta J| + ({kh}/{2f})(1+\sum_{j=0}^{f-1} r_j)} \text{ in the irreducible case,}\\ ({-}1)^{(1/2) k |J \triangle \delta J| + ({kh}/{f})\sum_{j=0}^{f-1}r_j} \text{ in the reducible case}. \end{gather*}
By Proposition 5.7, the factor 
$(-1)^{(1/2) k |J \triangle \delta J|}$ cancels in both cases, and there is an extra factor of
\begin{gather*} (-\alpha)^{-({1}/{2}) k} \text{ in the irreducible case,}\\ \alpha^{{-}k|J^\textrm{c}|/f}(\alpha')^{{-}k|J|/f} \text{ in the reducible case}. \end{gather*}
Recall that by our conventions the determinant of 
${\bar {\rho }}$ evaluated at 
$\mathrm {Art}_L(p)$ is 
$\alpha$ in the irreducible case and 
$\alpha \alpha '$ in the reducible case. As we are assuming 
$\det ({\bar {\rho }})\circ \mathrm {Art}_L(p) = 1$, this means that the factor contributes 
$(-1)^{(1/2) k}$, respectively 
$\alpha ^{(|J|-|J^\textrm {c}|)k/f}$, which is in accordance with [Reference BreuilBre11, Théorème 6.4], because we defined 
$r_i$ to be 
$\mu _{1, i} - \mu _{2, i}-1$ (see § 3.3) and 
$M_\infty [\mathfrak {m}]$ to be 
$\pi ({\bar {\rho }}^\vee )$ rather than 
$\pi ({\bar {\rho }})$ (see Definition 4.9 and Remark 6.1).
6. Global applications
We now give examples of 
$\mathrm {GL}_2(L)$-representations 
$\pi _{\mathrm {glob}}({\bar {\rho }}^\vee )$ from global contexts for which there exists an 
$\mathcal {O}_E[\mathrm {GL}_2(L)]$-module 
$M_\infty$ with an arithmetic action of 
$R_\infty ^\psi$ (for some character 
$\psi : G_L \rightarrow \mathcal {O}_E^\times$ and some natural number 
$h$) such that 
$\pi _{\mathrm {glob}}({\bar {\rho }}^\vee ) \cong M_\infty ^\vee [\mathfrak {m}]$. The construction of 
$M_\infty$ comes from the Taylor–Wiles–Kisin method as augmented in [Reference Caraiani, Emerton, Gee, Geraghty, Paškūnas and ShinCEG+16] and simplified in [Reference ScholzeSch18]. The 
$\mathrm {GL}_2(L)$-representations 
$\pi _{\mathrm {glob}}({\bar {\rho }}^\vee )$ we define appear in the cohomology of Shimura curves and in spaces of algebraic modular forms on definite quaternion algebras over totally real fields. One could also consider algebraic modular forms on definite unitary groups (see, e.g., [Reference Emerton, Gee and SavittEGS15, § 6.6]).
6.1 Spaces of mod $p$ automorphic forms on some quaternion algebras
Let 
$p$ be an odd prime. Let 
$F$ be a totally real number field in which 
$p$ is unramified. Let 
$D_{/F}$ be a quaternion algebra which is unramified at all places dividing 
$p$ and at zero (the definite case) or one infinite place (the indefinite case). If 
$D_{/F}$ is indefinite (unramified at exactly one infinite place) and 
$K = \prod _w K_w \subset (D\otimes _F \mathbf {A}_F^\infty )^\times$ is an open compact subgroup, then there is a smooth projective curve 
$X_K$ defined over 
$F$ and we define 
$S(K,k_E)$ to be the cohomology group 
$H^1(X_K \times _F \bar {F}, k_E)$. If 
$D_{/F}$ is definite, then we let 
$S(K,k_E)$ be the space of 
$K$-invariant smooth functions
\[ f: D^\times\backslash (D\otimes_F \mathbf{A}_F^\infty)^\times \rightarrow k_E. \]
Let 
$\bar {r}: G_F \rightarrow \mathrm {GL}_2(k_E)$ be a Galois representation. Let 
$S$ be the union of the set of places in 
$F$ where 
$\bar {r}$ is ramified, the set of places where 
$D$ is ramified, and the set of places dividing 
$p$. As 
$p>3$, we can choose a place 
$w_1$ of 
$F$ such that 
$\bar {r}$ is unramified at 
$w_1$ and 
$\bar {r}(\mathrm {Frob}_{w_1})$ has distinct eigenvalues (where 
$\mathrm {Frob}_{w_1}$ denotes a geometric Frobenius element at 
$w_1$). Enlarging 
$E$ if necessary, we assume that these eigenvalues are in 
$k_E$. Let 
$\mathbf {T}^{S,\mathrm {univ}}$ be the commutative polynomial algebra over 
$\mathcal {O}_E$ generated by the formal variables 
$T_w$ and 
$S_w$ for each 
$w \notin S\cup \{w_1\}$ where 
$w_1$ is chosen as in [Reference Emerton, Gee and SavittEGS15, § 6.2]. Then 
$\mathbf {T}^{S,\mathrm {univ}}$ acts on 
$S(K,k_E)$ with 
$T_w$ and 
$S_w$ acting by the usual double coset action of
\[ \Bigg[\mathrm{GL}_2(\mathcal{O}_{F_w}) \begin{pmatrix} \varpi_w & \\ & 1 \\ \end{pmatrix}\mathrm{GL}_2(\mathcal{O}_{F_w})\Bigg] \]and
\[ \Bigg[\mathrm{GL}_2(\mathcal{O}_{F_w}) \begin{pmatrix} \varpi_w & \\ & \varpi_w \\ \end{pmatrix}\mathrm{GL}_2(\mathcal{O}_{F_w})\Bigg], \]
respectively. Define a map 
$\mathbf {T}^{S,\mathrm {univ}}\rightarrow k_E$ such that the image of 
$X^2 - (\mathbf {N} w)^{-1} T_w X + (\mathbf {N} w)^{-1} S_w$ in 
$k_E[X]$ is the characteristic polynomial of 
$\bar {r}(\mathrm {Frob}_w)$, where 
$\mathrm {Frob}_w$ is a geometric Frobenius element at 
$w$. Let the kernel of this map be 
$\mathfrak {m}_{\bar {r}}$.
Remark 6.1 Given a maximal ideal of 
$\mathbf {T}^{S,\mathrm {univ}}$, [Reference BreuilBre14, § 9] (respectively, [Reference Emerton, Gee and SavittEGS15, § 6.2]) defines a Galois representation which is dual to (respectively, a twist by 
$\varepsilon ^{-1}$ of) the representation 
$\bar {r}$ that we define. This is the reason for the presence of a dual in Definition 4.9 and the different normalization of the map 
$\eta$ in § 3.6.
For the rest of the section, suppose that:
(i)
$\bar {r}$ is modular, that is, that there exists a compact open subgroup $K\subset (D\otimes _F \mathbf {A}_F^\infty )^\times$ such that $S(K,k_E)_{\mathfrak {m}_{\bar {r}}}$ is nonzero;(ii)
$\bar {r}|_{G_{F(\zeta _p)}}$ is absolutely irreducible;(iii) if
$p=5$, then the image of $\bar {r}(G_{F(\zeta _p)})$ in $\mathrm {PGL}_2(k_E)$ is not isomorphic to $A_5$;(iv)
$\bar {r}|_{G_{F_w}}$ is generic (see § 3.2.2, though the weaker [Reference Emerton, Gee and SavittEGS15, Definition 2.1.1] suffices) for all places $w|p$; and(v)
$\bar {r}|_{G_{F_w}}$ is non-scalar at all finite places where $D$ ramifies.
Let 
$v|p$ be a place of 
$F$, and let 
$L$ be 
$F_v$. Let 
${\bar {\rho }}$ be the restriction 
$\bar {r}|_{G_{F_v}}$. We define a compact open subgroup 
$K^v = \prod _{w\neq v} K_w \subset (D\otimes _F \mathbf {A}_F^{v,\infty })^\times$ as in [Reference Emerton, Gee and SavittEGS15, § 6.5]. The subgroup 
$K_w$ is 
$(\mathcal {O}_D)_w^\times$ unless 
$\bar {r}|_{G_{F_w}}$ is ramified (which is always the case if 
$w$ divides 
$p$ by the genericity assumption), reducible, and 
$D$ is unramified at 
$w$. If these three conditions hold and moreover 
$w|p$ or 
$w = w_1$, then 
$K_w$ is the standard (upper triangular) Iwahori subgroup. If these three conditions hold, but 
$w\nmid p$, then 
$K_w$ is the subgroup of matrices which are upper triangular modulo 
$\varpi _w^{n_w}$ where 
$\varpi _w$ is a uniformizer of 
$\mathcal {O}_{F_w}$ and 
$n_w$ is defined in [Reference Emerton, Gee and SavittEGS15, § 6.5].
Following [Reference Emerton, Gee and SavittEGS15, § 6.5], we let 
$S$ be the set of places 
$w$ where 
$\bar {r}|_{G_{F_w}}$ is ramified, 
$D$ is ramified at 
$w$, or 
$w$ is 
$w_1$. For each 
$w\in S$ distinct from 
$v$, one defines a finite free 
$\mathcal {O}_E$-module 
$V_w$ with a smooth 
$K_w$-action denoted 
$L_w$ in [Reference Emerton, Gee and SavittEGS15, § 6.5]. Rather than define 
$V_w$, we only note that the 
$\mathcal {O}_E$-rank of 
$V_w$ is one unless 
$\bar {r}|_{G_{F_w}}$ is irreducible. For any compact open subgroup 
$K_v \subset \mathrm {GL}_2(L)$, let 
$V$ be the 
$\mathcal {O}_E[\![K^vK_v]\!]$-module 
$\bigotimes _{w\in S,\, w\neq v} V_w$ where 
$K_w$ acts trivially if 
$w \notin S$ or 
$w$ is 
$v$.
Fix a finite-order character 
$\psi : G_F \rightarrow W(k_E)^\times \subset \mathcal {O}_E^\times$ such that 
$\det \bar {r} = \overline {\varepsilon \psi }$. We extend the action of 
$K^vK_v$ to 
$K^vK_v (\mathbf {A}_F^\infty )^\times$ by letting 
$(\mathbf {A}_F^\infty )^\times$ act by 
$\psi \circ \mathrm {Art}_F$ (the compatibility of the actions here follows from the condition in the choice of 
$\psi$).
We let 
$S'$ be the set of places 
$w$ away from 
$v$ where 
$\bar {r}|_{G_{F_w}}$ is reducible and 
$w|p$, 
$D$ is ramified at 
$w$, or 
$w$ is 
$w_1$. For 
$w\in S'$, one also introduces Hecke operators 
$T_w$ and scalars 
$\beta _w \in k_E^\times$, in the same way as in [Reference Emerton, Gee and SavittEGS15, § 6.5]. Let 
$\mathbf {T}^{S,S',\mathrm {univ}}$ be the free polynomial algebra over 
$\mathbf {T}^{S,\mathrm {univ}}$ generated by the variables 
$(T_w)_{w\in S'}$. Let 
$\mathfrak {m}'_{\bar {r}} \subset \mathbf {T}^{S,S',\mathrm {univ}}$ be the maximal ideal generated by 
$\mathfrak {m}_{\bar {r}}$ and 
$(T_w - \beta _w)_{w\in S'}$.
Fix a compact open subgroup 
$K_v \subset \mathrm {GL}_2(L)$. In the definite case, we let 
$S(K^vK_v,V)$ be the space of functions
\[ f: D^\times\backslash (D\otimes_F \mathbf{A}_F^\infty)^\times \rightarrow V^\vee \]
such that 
$f(gu) = u^{-1}f(g)$ for all 
$g\in (D\otimes _F \mathbf {A}_F^\infty )^\times$ and 
$u\in K^vK_v (\mathbf {A}_F^\infty )^\times$. In the indefinite case, 
$V^\vee$ defines a local system 
$\mathcal {F}_{V^\vee }$ on 
$X_{K^vK_v}$, and we let 
$S(K^vK_v,V)$ be the cohomology group 
$H^1(X_{K^vK_v} \times _F \bar {F},\mathcal {F}_{V^\vee })$. In either case, we then let 
$S(K^v,V)$ be the limit
\[ \varinjlim_{K_v} S(K^vK_v,V). \]
Then 
$S(K^v,V)$ has natural commuting actions of 
$\mathbf {T}^{S,S',\mathrm {univ}}$ and 
$\mathrm {GL}_2(L)$. In the indefinite case, there is also an action of 
$G_F$ that commutes with the two aforementioned actions. We let 
$\pi _{\mathrm {glob}}({\bar {\rho }}^\vee )$ be 
$S(K^v,V)[\mathfrak {m}'_{\bar {r}}]$ in the definite case and 
$\mathrm {Hom}_{G_F}(\bar {r},S(K^v,V)[\mathfrak {m}'_{\bar {r}}])$ in the indefinite case. By the assumption that 
$\bar {r}$ is modular and the choices of 
$K^v$, 
$L$, and 
$(\beta _w)_{w\in S'}$, we know that 
$\pi _{\mathrm {glob}}({\bar {\rho }}^\vee )$ is nonzero. We emphasize that despite the suggestive notation, it is far from clear that 
$\pi _{\mathrm {glob}}({\bar {\rho }}^\vee )$ depends only on 
${\bar {\rho }}^\vee$.
Theorem 6.2 There exists a natural number 
$h$ and an 
$\mathcal {O}_E[\mathrm {GL}_2(L)]$-module 
$M_\infty$ with an arithmetic action of 
$R_\infty ^\psi$ such that 
$M_\infty ^\vee [\mathfrak {m}]$ is isomorphic to 
$\pi _{\mathrm {glob}}({\bar {\rho }}^\vee )$ (where 
$\psi$ denotes the restriction 
$\psi |_{G_{F_v}}$).
The proof of this theorem uses the Taylor–Wiles method described in the next section.
6.2 The Taylor–Wiles patching construction
6.2.1 Galois deformation rings
For each 
$w\in S$, let 
$\bar {r}_w$ denote 
$\bar {r}|_{G_{F_w}}$ and 
$\psi _w$ denote 
$\psi _{G_{F_w}}$. Let 
$R_{\bar {r}_w}^{\square ,\psi _w}$ be the framed deformation ring parametrizing liftings of 
$\bar {r}_w$ with determinant 
$\psi _w \varepsilon$. Let 
$R_S^\psi$ be 
$\widehat {\otimes }_{w\in S,\mathcal {O}_E} R_{\bar {r}_w}^{\square ,\psi _w}$.
Fix a finite set of places 
$Q$ disjoint from 
$S$. For any finite set 
$T$ of places in 
$F$, let 
$G_{F,T}$ denote the Galois group of the maximal extension of 
$F$ unramified outside of 
$T$ over 
$F$. Let 
$R_{F,S_Q}^\psi$ denote the deformation ring of 
$\bar {r}$ as a 
$G_{F,S\cup Q}$-representation with fixed determinant 
$\psi \varepsilon$. Let 
$r(Q): G_{F,S\cup Q} \rightarrow \mathrm {GL}_2(R_{S_Q}^{\psi })$ be a versal lifting of 
$\bar {r}$. Let 
$R_{F,S_Q}^{\square ,\psi }$ denote the complete 
$\mathcal {O}_E$-algebra which prorepresents the functor assigning to a local Artinian 
$\mathcal {O}_E$-algebra 
$A$ with residue field 
$k_E$ the set of equivalence classes of tuples 
$(r,\{\alpha _w\}_{w\in S})$ where 
$r$ is an 
$A$-lifting of 
$\bar {r}$ as a 
$G_{F,S\cup Q}$-representation and 
$\alpha _w \in \ker (\mathrm {GL}_2(A) \rightarrow \mathrm {GL}_2(k_E))$ for each 
$w\in S$.
6.2.2 Spaces of automorphic forms
Let 
$K' = \prod _w K'_w \subset (D\otimes _F \mathbf {A}_F^\infty )^\times$ be a compact open subgroup such that 
$K'_w = K_w$ for all 
$w \notin Q$ and 
$w\neq v$. Let 
$\mathbf {T}(K')$ denote the image of 
$\mathbf {T}^{S\cup Q,S',\mathrm {univ}}$ in 
$\mathrm {End}_{\mathcal {O}_E}(S(K',V))$. We denote the image of 
$\mathfrak {m}'_{\bar {r}} \cap \mathbf {T}^{S\cup Q,\mathrm {univ}}$ in 
$\mathbf {T}(K')$ by 
$\mathfrak {m}_Q$: it is a maximal ideal in 
$\mathbf {T}(K')$. There is a map 
$R_{F,S_Q}^\psi \rightarrow \mathbf {T}(K')_{\mathfrak {m}_Q}$ such that the image of the characteristic polynomial of 
$r(Q)(\mathrm {Frob}_w)$ is 
$X^2 - (\mathbf {N} w)^{-1}T_w X + (\mathbf {N} w)^{-1} S_w \in \mathbf {T}(K')_{\mathfrak {m}_Q}[X]$ for 
$w \notin S\cup Q \cup \{w_1\}$. Let 
$r(K'): G_{F,S\cup Q} \rightarrow \mathrm {GL}_2(\mathbf {T}(K')_{\mathfrak {m}_Q})$ be the Galois representation obtained from 
$r(Q)$.
6.2.3 Auxiliary primes
We can and do choose an integer 
$q \geq [F:\mathbf {Q}]$ and for each integer 
$N\geq 1$ a tuple 
$(Q_N,\{\bar {\psi }_w\}_{w\in Q_N})$ where:
•
$Q_N$ is a set of $q$ places of $F$ disjoint from $S$;• for each
$w\in Q_N$, $\mathbf {N} w \equiv 1 \pmod {p^N}$;• for each
$w\in Q_N$, $\bar {r}|_{G_{F_w}} \cong \bar {\psi }_w \oplus \bar {\psi }'_w$ for some $\bar {\psi }'_w \neq \bar {\psi }_w$; and• the ring
$R_{F,S_{Q_N}}^{\square ,\psi }$ can be topologically generated over $R_S^\psi$ by $q-[F:\mathbf {Q}]$ elements.
Fix a compact open subgroup 
$K_v \subset \mathrm {GL}_2(L)$. For each 
$N\geq 1$, let
\[ K(N)^v = \prod_{w\neq v} K(N)_w \subset (D\otimes_F \mathbf{A}_F^{v,\infty})^\times \]
be the compact open subgroup such that 
$K(N)_w = K_w$ for 
$w \notin Q_N$ and 
$w\neq v$ and 
$K(N)_w$ is the subgroup of matrices of 
$\mathrm {GL}_2(\mathcal {O}_{F_w})$ congruent to 
$\big (\begin {smallmatrix} * & * \\ 0 & 1 \end {smallmatrix}\big )$ modulo 
$\varpi _w$ if 
$w\in Q_N$.
Choose a lift 
$\varphi _w \in G_{F_w}$ of the geometric Frobenius element, and let 
$\varpi _w \in \mathcal {O}_{F_w}$ be the uniformizer such that 
$\mathrm {Art}_{F_w} \varpi _w = \varphi _w|_{F_w^{\mathrm {ab}}}$. For 
$w\in Q_N$, let 
$U_w$ be the Hecke operator corresponding to the double coset
\[ K(N)_w \begin{pmatrix} 1 & 0 \\ 0 & \varpi_w \end{pmatrix} K(N)_w. \]
Using that 
$\bar {\psi }_w(\varphi _w) \neq \bar {\psi }'_w(\varphi _w)$, the characteristic polynomial 
$P_w(X) \in \mathbf {T}(K(N)^vK_v)[X]$ of 
$r(K(N)^vK_v)(\varphi _w)$ factors as 
$(X-A_w)(X-B_w)$ by Hensel's lemma. Let 
$\mathbf {T}(K(N)^vK_v)'$ be the 
$\mathbf {T}(K(N)^vK_v)$-subalgebra of 
$\mathrm {End}_{\mathcal {O}_E}(S(K(N)^v K_v,V))$ generated by 
$U_w$ for 
$w\in Q_N$, and let 
$\mathfrak {m}_{Q_N}' \subset \mathbf {T}(K(N)^vK_v)'$ be the ideal generated by 
$\mathfrak {m}_{Q_N}$ and 
$U_w - A_w$. For each 
$w\in Q_N$, let 
$k_w^\times (p)$ denote the 
$p$-Sylow subgroup of 
$k_w^\times$. Let 
$\mathcal {O}_{E,N}$ denote the group algebra of 
$\prod _{w\in Q_N} k_w^\times (p)$ over 
$\mathcal {O}_E$ with augmentation ideal 
$\mathfrak {a}_N$. Note that the natural inclusion composed with multiplication by 
$\prod _{w\in Q_N} (U_w-B_w)$ gives an isomorphism
\[ S(K^vK_v,V)_{\mathfrak{m}} \cong S(K(N)^v K_v,V)_{\mathfrak{m}'_Q}[\mathfrak{a}_N]. \]
In the definite case, let 
$M(K_v,N)$ be
\[ S(K(N)^vK_v,V)_{\mathfrak{m}'_Q}^\vee \otimes_{R_{F,S_{Q_N}}^\psi} R_{F,S_{Q_N}}^{\square,\psi}. \]
In the indefinite case, let 
$M(K_v,N)$ be
\[ \mathrm{Hom}_{\mathbf{T}(K(N)^vK_v)_{\mathfrak{m}_{Q_N}}[G_{F,S\cup Q_N}]}(r(K(N)^vK_v),S(K(N)^vK_v,V)_{\mathfrak{m}'_Q})^\vee \otimes_{R_{F,S_{Q_N}}^\psi} R_{F,S_{Q_N}}^{\square,\psi}. \]6.2.4 Patching
For 
$w\in S$ with 
$w\neq v$, define 
$R_w^\textrm{min}$ as in [Reference Emerton, Gee and SavittEGS15, § 6.5]. Let 
$R^{\mathrm {loc}}$ be
\[ R_{{\bar{\rho}}}^{\square,\psi_v}\widehat{\otimes}_{\mathcal{O}_E}(\widehat{\otimes}_{w \in S,w\neq v,\mathcal{O}_E} R_w^{{\rm min}}). \]
Note that 
$R^{\mathrm {loc}}$ is formally smooth over 
$R_{{\bar {\rho }}}^{\square ,\psi _v}$.
Let 
$g$ be 
$q-[F:\mathbf {Q}]$. Let 
$R_\infty ^\psi$ be 
$R^{\mathrm {loc}}[\![t_1,\ldots ,t_g]\!]$, which is isomorphic to
\[ R_{{\bar{\rho}}}^{\square,\psi_v}\widehat{\otimes}_{\mathcal{O}_E} \mathcal{O}_E[\![x_1,\ldots,x_h]\!] \]
for some integer 
$h\geq 0$. We fix a versal lifting
\[ r(\varnothing): G_{F,S} \rightarrow \mathrm{GL}_2(R_{F,S}^\psi) \]
of 
$\bar {r}$ with determinant 
$\psi \varepsilon$ and versal liftings
\[ r(Q_N): G_{F,S} \rightarrow \mathrm{GL}_2(R_{F,S}^\psi) \]
of 
$\bar {r}$ with determinant 
$\psi \varepsilon$ compatible with 
$r(\varnothing )$. This gives an isomorphism
\[ R_{F,S}^{\square,\psi} \cong R_{F,S}^\psi\widehat{\otimes}_{\mathcal{O}_E}\mathcal{O}_E[\![z_1,\ldots,z_{4\#S-1}]\!] \]and compatible isomorphisms
\[ R_{F,S_{Q_N}}^{\square,\psi} \cong R_{F,S_{Q_N}}^\psi \widehat{\otimes}_{\mathcal{O}_E}\mathcal{O}_E[\![z_1,\ldots,z_{4\#S-1}]\!]. \]
For each 
$N\geq 1$, choose a surjection of 
$R^{\mathrm {loc}}$-algebras
\[ R_\infty^\psi \twoheadrightarrow R_{F,S_{Q_N}}^{\square,\psi} \]
and a surjection 
$\mathcal {O}_{E,\infty } := \mathcal {O}_E[\![y_1,\ldots ,y_q]\!] \twoheadrightarrow \mathcal {O}_{E,N}$ whose kernel is contained in the ideal generated by 
$((1+y_i)^{p^N}-1)_{i=1}^q$. Let 
$J \subset \mathcal {O}_{E,\infty }$ be an open ideal for the adic topology defined by the maximal ideal, and let 
$I_J \subset \mathbf {N}$ be the cofinite subset of elements 
$N$ such that 
$J$ contains the kernel of 
$\mathcal {O}_{E,\infty } \twoheadrightarrow \mathcal {O}_{E,N}$. If 
$N\in I_J$, let 
$M(K_v,J,N)$ be 
$\mathcal {O}_{E,\infty }/J \otimes _{\mathcal {O}_{E,\infty }} M(K_v,N)$.
Let 
$(\mathcal {O}_{E,\infty }/J)_{I_J}$ be the product 
$\prod _{i\in I_J} \mathcal {O}_{E,\infty }/J$. Fix a nonprincipal ultrafilter 
$\mathfrak {F}$ on the set 
$\mathbf {N} = \{N\geq 1\}$. This defines a point in 
$x \in \operatorname {Spec} (\mathcal {O}_{E,\infty }/J)_{I_J}$ (see [Reference Gee and NewtonGN20, Lemma 2.2.2]). Then let 
$M(K_v,J,\infty )$ be
\[ (\mathcal{O}_{E,\infty}/J)_{I_J,x} \otimes_{(\mathcal{O}_{E,\infty}/J)_{I_J}} \prod_{N\in I_J} M(K_v,J,N). \]
The product 
$\prod _{N\in I_J} M(K_v,J,N)$, and thus 
$M(K_v,J,\infty )$, has a diagonal 
$R_\infty ^\psi$-action. For an open subideal 
$J' \subset J$ and an open compact subgroup 
$K'_v \subset K_v$, there is a natural map 
$M(K'_v,J',\infty )\rightarrow M(K_v,J,\infty )$ (see [Reference Gee and NewtonGN20, Lemma 3.4.11(1)]). We let 
$M_\infty$ be the 
$R_\infty ^\psi$-module
\[ \varprojlim_{J,K_v} M(K_v,J,\infty). \]Proof of Theorem 6.2 There is a natural right 
$\mathcal {O}_E[\mathrm {GL}_2(L)]$-action (by duality) on 
$M_\infty$ coming from the maps 
$M(K'_v,J,\infty )\rightarrow M(K_v,J,\infty )$ for any inclusion 
$K'_v \subset K_v$ of compact open subgroups (see [Reference ScholzeSch18, § 9]). As the action of 
$L^\times$ on 
$M_\infty$ factors through 
$\psi _v$ on each 
$M(K_v,N)$, it does on 
$M_\infty$ as well. Let 
$S_\infty$ be the ring 
$\mathcal {O}_E[\![z_1,\ldots ,z_{4\# S-1},y_1,\ldots ,y_q]\!]$ with augmentation ideal 
$\mathfrak {a}$. Then 
$M(K_v,N)$ is an 
$S_\infty$-module for each 
$N\geq 1$ in an apparent way. As 
$M(K_v,N)/\mathfrak {a}$ is isomorphic to 
$S(K^vK_v,V)_{\mathfrak {m}}^\vee$ in the definite case and
\[ \mathrm{Hom}_{\mathbf{T}(K^vK_v)_{\mathfrak{m}'_{\bar{r}}}[G_{F,S}]}(r(K^vK_v),S(K^vK_v,V)_{\mathfrak{m}'_{\bar{r}}})^\vee \]
in the indefinite case, 
$M(K_v,N)$ is finitely generated over 
$S_\infty$ by (topological) Nakayama's lemma. We conclude that 
$M_\infty$ is a finitely generated 
$S_\infty [\![\mathrm {GL}_2(\mathcal {O}_L)]\!]$-module. The 
$S_\infty$-action factors through 
$R_\infty ^\psi$ so that 
$M_\infty$ is a finitely generated 
$R_\infty ^\psi [\![\mathrm {GL}_2(\mathcal {O}_L)]\!]$-module.
For proofs of statements in this section, see [Reference Caraiani, Emerton, Gee, Geraghty, Paškūnas and ShinCEG+16, Lemma 4.18]. The finitely generated 
$S_\infty [\![\mathrm {GL}_2(\mathcal {O}_L)]\!]$-module 
$M_\infty$ is in fact projective (see also [Reference Gee and NewtonGN20, Proposition 3.4.16]). If 
$\theta$ is a lattice in 
$\sigma (\tau )$, then 
$M_\infty (\theta )$ is supported in 
$R_\infty ^\psi (\tau )$. As 
$\dim S_\infty = \dim R_\infty ^\psi (\tau )$, 
$M_\infty (\theta )$ is maximal Cohen–Macaulay over 
$R_\infty ^\psi (\tau )$. The action of 
$\mathcal {H}(\theta )$ on 
$M_\infty (\theta )$ factors through the map 
$\eta$ by the proof of [Reference Caraiani, Emerton, Gee, Geraghty, Paškūnas and ShinCEG+16, Theorem 4.19]. Our different normalization of 
$\eta$ stems from our normalization of Galois representations (see Remark 6.1) For an 
$\mathcal {O}_E$-lattice 
$\theta \subset \sigma (\tau )$ in an inertial 
$K$-type, 
$M_\infty (\theta )[1/p]$ is locally free of rank at most one over 
$R_\infty (\tau )[1/p]$ by [Reference Breuil and DiamondBD14, Proposition 3.5.1].
Finally, in the definite case, we have that
\[ M_\infty^\vee[\mathfrak{m}] \cong (M_\infty/\mathfrak{a})^\vee[\mathfrak{m}] \cong S(K^v,V)_{\mathfrak{m}'_{\bar{r}}}[\mathfrak{m}'_{\bar{r}}] = \pi_{\mathrm{glob}}({\bar{\rho}}^\vee). \]In the indefinite case, we have that
\begin{align*} (M_\infty/\mathfrak{a})^\vee[\mathfrak{m}] &\cong \varinjlim_{K_v} \mathrm{Hom}_{\mathbf{T}(K^vK_v)_{\mathfrak{m}'_{\bar{r}}}[G_{F,S}]}(r(K^vK_v),S(K^vK_v,V)_{\mathfrak{m}'_{\bar{r}}})[\mathfrak{m}'_{\bar{r}}] \\ &\cong \mathrm{Hom}_{k_E[G_{F,S}]}(\bar{r},S(K^v,V)[\mathfrak{m}'_{\bar{r}}]) \\ &= \pi_{\mathrm{glob}}({\bar{\rho}}^\vee). \end{align*}Acknowledgements
The authors would like to thank A. Caraiani, M. Emerton, T. Gee, D. Geraghty, V. Paškūnas, and S.W. Shin for sharing a note which contains an inspiring reinterpretation of [Reference Breuil and DiamondBD14] using Taylor–Wiles patching. DL would like to thank C. Breuil for proposing [Reference BreuilBre14, Question 9.5] and encouraging him to work on it, as well as M. Emerton for related conversations. AD joins DL in thanking Breuil and Emerton for their interest and encouragement, and is grateful to T. Gee for mentioning the problem to him. The authors thank the referee for a close reading of and comments and corrections on earlier versions of this article. The authors would like to thank L. Berger, G. Dospinescu, P. Gille, W. Niziol, V. Pilloni, S. Rozensztajn, A. Thuillier, l'Université de Lyon, and l'ENS de Lyon for organizing and hosting the trimester on Algebraic Groups and Geometrization of the Langlands program where this collaboration began. AD was supported by the Engineering and Physical Sciences Research Council [EP/L015234/1], The EPSRC Centre for Doctoral Training in Geometry and Number Theory (The London School of Geometry and Number Theory), University College London, and Imperial College London. DL was supported by the National Science Foundation under agreement No. DMS-1703182 and an AMS-Simons travel grant.
Appendix. Explicit formulas for weight cycling
Although the computations in §§ 5.3 and 5.4 are ultimately based only on the definitions in (11), it might be helpful to have explicit formulas for the behaviour of 
$\delta \in \{\delta _\mathrm {red}, \delta _\mathrm {irr}\}$ on formal weights. We collect this information in the following tables.
Here, we assume 
$0 < j < f-1$ and 
$\delta \in \{\delta _\mathrm {red}, \delta _\mathrm {irr}\}$. The remaining cases are treated in what follows.
