2.1 The Montréal functor and the trianguline correspondence

Following [Reference Colmez, Dospinescu and Paškūnas9, Reference Paškūnas21], we recall some basic facts about the p-adic and mod p local Langlands correspondence. As before, we fix a finite extension E of $\mathbb {Q}_p$ with a ring of integers $\mathcal {O}$ , a uniformizer $\varpi $ , and a residue field $\kappa $ .

Fix a continuous character $\zeta : Z \rightarrow \mathcal {O}^{\times }$ of the center Z of $\mathrm {G}$ . We denote by $\mathrm {Mod}^{\mathrm {sm}}_{\mathrm {G}}(\mathcal {O})$ the category of smooth $\mathcal {O}$ -torsion $\mathrm {G}$ -modules. Within this, let $\mathrm {Mod}^{\mathrm {sm}}_{\mathrm {G},\zeta }(\mathcal {O})$ be the full subcategory consisting of representations on which Z acts via $\zeta $ . Furthermore, we define $\mathrm {Mod}^{\mathrm {fin}}_{\mathrm {G}, \zeta }(\mathcal {O})$ and $\mathrm {Mod}^{\mathrm {lfin}}_{\mathrm {G}, \zeta }(\mathcal {O})$ as the full subcategories of representations that are, respectively, of finite length and locally of finite length as $\mathcal {O}[\mathrm {G}]$ -modules. A detailed study of these categories can be found in [Reference Emerton13, Section 2]. In his Montréal lectures, Colmez introduced an exact covariant $\mathcal {O}$ -linear functor $\mathbf {V}: \mathrm {Mod}^{\mathrm {fin}}_{\mathrm {G}, \zeta }(\mathcal {O})\rightarrow \mathrm {Mod}_{\mathrm {G}_{\mathbb {Q}_p}}(\mathcal {O})$ , where $\mathrm {Mod}_{\mathrm {G}_{\mathbb {Q}_p}}(\mathcal {O})$ is the category of $\mathcal {O}$ -modules with a continuous $\mathrm {G}_{\mathbb {Q}_p}$ -action. Let $\operatorname {\mathrm {Ban}}_{\mathrm {G}}^{\mathrm {adm}}(E)$ be the category of unitary admissible E-Banach space representations of $\mathrm {G}$ and let $\operatorname {\mathrm {Ban}}_{\mathrm {G}, \zeta }^{\mathrm {adm.fl}}(E)$ be the full subcategory consisting of finite length objects with central character $\zeta $ . For any $\Pi \in \operatorname {\mathrm {Ban}}_{\mathrm {G}}^{\mathrm {adm}}(E)$ , one may choose an open bounded $\mathrm {G}$ -invariant lattice $\Theta $ in $\Pi $ . Then $\Theta /\varpi ^n \Theta $ is a smooth admissible representation of $\mathrm {G}$ for all $n\ge 1$ , implying it is locally finite. This allows one to define $\mathbf {V}(\Pi ):=E\otimes \underset {\longleftarrow }{\lim }\, \mathbf {V}(\Theta /\varpi ^n \Theta )$ . Due to the commensurability of all open bounded lattices in $\Pi $ , the definition is independent of the specific lattice $\Theta $ chosen. The functor $\mathbf {V}$ satisfies the following properties:

1. (1) It is exact and covariant from $\operatorname {\mathrm {Ban}}_{\mathrm {G}, \zeta }^{\mathrm {adm.fl}}(E)$ to finite-dimensional continuous E-representations of $\mathrm {G}_{\mathbb {Q}_p}$ .

2. (2) The functor $\Pi \mapsto \mathbf {V}(\Pi )$ induces a bijection between the isomorphism classes of absolutely irreducible non-ordinary $\Pi \in \operatorname {\mathrm {Ban}}_{\mathrm {G}}^{\mathrm {adm}}(E) $ and two-dimensional absolutely irreducible continuous E-representations of $\mathrm {G}_{\mathbb {Q}_p}.$

3. (3) If $\psi : \mathbb {Q}_p^{\times }\rightarrow \mathcal {O}^{\times }$ is a continuous character, then we may also consider it as a continuous character $\psi : \mathrm {G}_{\mathbb {Q}_p}\rightarrow \mathcal {O}^{\times }$ via the local Artin map, and for all $\Pi \in \operatorname {\mathrm {Ban}}_{\mathrm {G}}^{\mathrm {adm}}(E)$ , we have the isomorphism:

   $$ \begin{align*}\mathbf{V}(\Pi\otimes \psi\circ \det)\cong \mathbf{V}(\Pi)\otimes \psi.\end{align*} $$

4. (4) The Montréal functor maps ordinary Banach space representations of $\mathrm {G}$ to one-dimensional representations of $\mathrm {G}_{\mathbb {Q}_p}$ . More precisely, as a functor from $\mathrm {Mod}^{\mathrm {fin}}_{\mathrm {G}, \zeta }(\mathcal {O})$ , $\mathbf {V}$ satisfies the following:

   $$ \begin{align*}\mathbf{V}(\delta \circ \det)=0,\,\, \mathbf{V}(St \otimes \delta \circ \det)=\varepsilon \delta, \,\, \mathbf{V}(\mathrm{Ind}_{P}^{\mathrm{G}}{(\chi_1\otimes \chi_2\varepsilon^{-1})})\cong \chi_2,\end{align*} $$
   where P is the Borel subgroup of upper-triangular matrices in $\mathrm {G}$ , $\mathrm {Ind}_{P}^{\mathrm {G}}$ denotes the (smooth) parabolic induction, $\chi _1$ , $\chi _2$ , and $\delta $ are continuous characters of $\mathbb {Q}_p^\times $ , and $St$ is the Steinberg representation.

Let $H \subset \mathrm {G}$ be a compact open subgroup, and let $\mathcal {O}[[H]]$ represent its completed group algebra. We define $\mathrm {Mod}^{\mathrm {pro \, aug}}_{\mathrm {G}}(\mathcal {O})$ as the category of profinite, linearly topological $\mathcal {O}[[H]]$ -modules M equipped with a continuous $\mathrm {G}$ -action, such that the action of $\mathcal {O}[H]$ induced by $\mathcal {O}[[H]]$ aligns with the restriction of the $\mathrm {G}$ -action. Morphisms in this category are continuous $\mathrm {G}$ -equivariant homomorphisms of topological $\mathcal {O}[[H]]$ -modules. This definition is well-defined regardless of the choice of H, as any two compact open subgroups of $\mathrm {G}$ are commensurable. As noted in [Reference Emerton13, Equation 2.2.8], taking Pontryagin dual yields an anti-equivalence between $\mathrm {Mod}^{\mathrm {sm}}_{\mathrm {G}}(\mathcal {O})$ and $\mathrm {Mod}^{\mathrm {pro \, aug}}_{\mathrm {G}}(\mathcal {O})$ . The Pontryagin dual of an $\mathcal {O}$ -module M is defined as

$$ \begin{align*}M^{\vee}:=\mathrm{Hom}^{cont}_{\mathcal{O}}(M, E/\mathcal{O}),\end{align*} $$

where $E/\mathcal {O}$ carries the discrete topology and $M^{\vee }$ the compact-open topology. There exists a canonical isomorphism $M^{\vee \vee }\cong M$ .

Let $\mathfrak {C}(\mathcal {O})$ be the full subcategory of $\mathrm {Mod}^{\mathrm {pro \, aug}}_{\mathrm {G}}(\mathcal {O})$ whose objects consist of all M isomorphic to $\pi ^{\vee }$ for some $\pi \in \mathrm {Mod}^{\mathrm {lfin}}_{\mathrm {G}, \zeta }(\mathcal {O})$ . This category $\mathfrak {C}(\mathcal {O})$ is abelian and has exact projective limits and projective envelopes. Let $\mathrm {Rep}_{\mathrm {G}_{\mathbb {Q}_p}}(\mathcal {O})$ denote the category of continuous $\mathrm {G}_{\mathbb {Q}_p}$ -representations on compact $\mathcal {O}$ -modules. Paškūnas [Reference Paškūnas21] introduced a functor $\check {\mathbf {V}}: \mathfrak {C}(\mathcal {O})\rightarrow \mathrm {Rep}_{\mathrm {G}_{\mathbb {Q}_p}}(\mathcal {O})$ defined as follows: for an object $M \in \mathfrak {C}(\mathcal {O})$ of finite length, set $\check {\mathbf {V}}(M):=\mathbf {V}(M^{\vee })^{\vee }(\varepsilon \zeta )$ , where $\vee $ denotes the Pontryagin dual and $\zeta $ is treated as a character of $\mathrm {G}_{\mathbb {Q}_p}$ via local class field theory. More generally, by writing $M\cong \underset {\longleftarrow }{\lim }\, M_i$ as a limit over finite-length quotients in $\mathfrak {C}(\mathcal {O})$ , one defines $\check {\mathbf {V}}(M):=\underset {\longleftarrow }{\lim }\, \check {\mathbf {V}}(M_i)$ . By construction, this functor is also covariant and exact. Under this normalization of $\check {\mathbf {V}}$ , the following relations hold:

(2.1) $$ \begin{align} \check{\mathbf{V}}(\pi^{\vee})\cong \mathbf{V}(\pi), \quad \check{\mathbf{V}}((\mathrm{Ind}_{P}^{\mathrm{G}}{\chi_1\otimes \chi_2 \omega^{-1}})^{\vee})= \chi_1, \quad \check{\mathbf{V}}((St \otimes \eta\circ \det)^{\vee})=\eta, \end{align} $$

where $\pi $ represents a supersingular representation and $\eta $ is a continuous character of  $\mathbb {Q}_p^\times $ .

Consider $\Pi \in \operatorname {\mathrm {Ban}}_{\mathrm {G}, \zeta }^{\mathrm {adm}}(E)$ with an open, bounded $\mathrm {G}$ -invariant lattice $\Theta $ . Its Schikhof dual is defined as $ \Theta ^d:=\mathrm {Hom}_{\mathcal {O}}(\Theta , \mathcal {O})$ , endowed with the topology of pointwise convergence. It is the unit ball for the dual representation $\Pi ^*$ . The relationship between the Schikhof and Pontryagin duals is given by

$$ \begin{align*}\Theta^d \otimes_{\mathcal{O}} \mathcal{O}/\varpi^n \mathcal{O}\cong (\Theta/\varpi^n \Theta)^{\vee}. \end{align*} $$

According to [Reference Paškūnas21, Lemma 4.11], $\Theta ^d$ belongs to $\mathfrak {C}(\mathcal {O})$ . As $\Theta ^d$ is $\mathcal {O}$ -torsion free and $\check {\mathbf {V}}$ preserves exactness and covariance, $\check {\mathbf {V}}(\Theta ^d)$ remains $\mathcal {O}$ -torsion free. In this context, we define $\check {\mathbf {V}}(\Pi )$ as $\check {\mathbf {V}}(\Theta ^d)\otimes _{\mathcal {O}} E$ . The commensurability of different open lattices in $\Pi $ ensures that $\check {\mathbf {V}}(\Pi )$ is independent of the choice of $\Theta $ .

The surjectivity of the functor $\mathbf {V}$ is established in [Reference Colmez6] for odd p (and in [Reference Colmez, Dospinescu and Paškūnas10] for $p=2$ ) via the construction of a representation $\Pi (V)$ of $\mathrm {G}$ satisfying $\mathbf { V}(\Pi (V))=V$ for any two-dimensional representation V of $\mathrm {G}_{\mathbb {Q}_p}$ . The construction goes through Fontaine’s equivalence [Reference Fontaine16] relating representations of $\mathrm {G}_{\mathbb {Q}_p}$ and $(\varphi ,\Gamma )$ -modules, similar to the construction of $\Pi \mapsto \mathbf {V}(\Pi )$ . The resulting p-adic local Langlands correspondence $V \mapsto \Pi (V)$ exhibits several notable features:

• • compatibility with reduction modulo p;

• • consistent behavior within families;

• • compatibility with the classical local Langlands correspondence.

An explicit construction of $\Pi (V)$ for trianguline V is given in [Reference Emerton12, Section 6]. We recall it here. Note that Emerton’s description in the Banach space setting is the same as the characteristic zero translation of Colmez’s mod p results [Reference Colmez6, Théorème 0.11] via the extension of the Montréal functor to the Banach space setting described above. Throughout this construction, the superscript $\mathcal {C}^0$ indicates that the induction is taken in the category of continuous functions.

1. (1) (Absolutely reducible) Let $V \cong \begin {pmatrix} \eta _1 & 0 \\ 0 & \eta _2 \\ \end {pmatrix} \otimes \eta $ with $ \eta _1, \eta _2$ continuous integral characters and $\eta : \mathrm {G}_{\mathbb {Q}_p} \rightarrow E^{\times }$ be a continuous character. In this case,

   • • If $\eta _1 \eta _2^{-1}\neq \varepsilon ^{\pm 1}$ , then

     $$ \begin{align*} \Pi(V) \cong \mathrm{Ind}^{\mathrm{G}}_{P}(\eta_1 \otimes \eta_2 \varepsilon^{-1})^{\mathcal{C}^0} \otimes \eta \bigoplus \mathrm{Ind}^{\mathrm{G}}_ {P}(\eta_2 \otimes \eta_1 \varepsilon^{-1})^{\mathcal{C}^0} \otimes \eta. \end{align*} $$

   • • If $\eta _1 \eta _2^{-1}= \varepsilon ,$ then

     $$\begin{align*}\Pi(V) \cong \eta_1 \circ \det \otimes B(2, \infty) \oplus \mathrm{Ind}^{\mathrm{G}}_ {P} (\eta_1 \varepsilon^{-1} \otimes \eta_1 \varepsilon)^{\mathcal{C}^0} \otimes \eta, \end{align*}$$
     where $B(2, \infty )$ denotes the universal unitary completion of the smooth induction $\mathrm {Ind}_{P}^{\mathrm {G}}(|\,|_{p}^{-1}\otimes |\,|_{p})^{sm}. $

2. (2) (Reducible non-split, case I)Let $V \cong \begin {pmatrix} \eta _1 & \star \\ 0 & \eta _2 \\ \end {pmatrix} \otimes \eta $ with $ \eta _1, \eta _2 $ as above, $\star \neq 0$ and $\eta _1 \cdot \eta _2^{-1}\neq \varepsilon ^{\pm 1}$ , then $ \Pi (V)$ sits inside the following exact sequence in the category $\operatorname {\mathrm {Ban}}_{\mathrm {G}}^{\mathrm {adm}}(E)$ :

   $$ \begin{align*} 0 \rightarrow \pi_1 \otimes \eta \rightarrow \Pi(V) \rightarrow \pi_2 \otimes \eta \rightarrow 0, \end{align*} $$
   where $\pi _1:=\mathrm {Ind}^{\mathrm {G}}_{ P}(\eta _2 \otimes \eta _1)^{\mathcal {C}^0}$ and $\pi _2:=\mathrm {Ind}^{\mathrm {G}}_{ P}(\eta _1 \varepsilon \otimes \eta _2 \varepsilon ^{-1})^{\mathcal {C}^0}$ .

3. (3) (Reducible non-split, case II) Suppose $V \cong \begin {pmatrix} \eta & \star \\ 0 & \eta \varepsilon ^{-1}\\ \end {pmatrix} $ or $V \cong \begin {pmatrix} \eta \varepsilon ^{-1} & \star \\ 0 & \eta \\ \end {pmatrix}$ with $\eta $ as above and $\star \neq 0$ . Then the corresponding $\mathrm {G}$ -representation $\Pi (V)$ has a Jordan–Hölder filtration with Jordan–Hölder factors $\widehat {St}$ , $\underline {1}$ , and $\mathrm {Ind}_{P}^{\mathrm {G}} (\eta \varepsilon ^{-1} \otimes \eta \varepsilon )^{\mathcal {C}^0}$ , where $\widehat {St}$ denotes the universal unitary completion of the Steinberg representation $St$ .

2.2 Blocks and semisimple mod p LLC

In [Reference Paškūnas21], Paškūnas defined an equivalence relation on the set of (isomorphism classes of) irreducible objects of $\mathrm {Mod}^{\mathrm {lfin}}_{\mathrm {G},\zeta }(\mathcal {O})$ , where $\tau \sim \pi $ if and only if there exists a sequence of irreducible representations $\tau =\tau _0, \tau _1, \ldots , \tau _n=\pi $ such that for each i, one of the following holds: $1)\,\tau _i=\tau _{i+1}$ ; $2)\,\mathrm {Ext}^1_{\mathrm {G}}(\tau _i, \tau _{i+1})\neq 0$ ; and $3)\,\mathrm {Ext}^1_{\mathrm {G}}(\tau _{i+1}, \tau _i)\neq 0$ . An equivalence class is called a block. The category $\mathrm {Mod}^{\mathrm {lfin}}_{\mathrm {G}, \zeta }(\mathcal {O})$ is locally finite and, by general results of Gabriel [Reference Gabriel17], it decomposes as a product

$$ \begin{align*}\mathrm{Mod}^{\mathrm{lfin}}_{\mathrm{G}, \zeta}(\mathcal{O})\cong \prod_{\mathfrak{B}} \mathrm{Mod}^{\mathrm{lfin}}_{\mathrm{G}, \zeta}(\mathcal{O})^{\mathfrak{B}}\end{align*} $$

of indecomposable subcategories, where the product runs over the set of all blocks $\mathfrak {B}$ of $\mathrm {Mod}^{\mathrm {lfin}}_{\mathrm {G},\zeta }(\mathcal {O})$ . Here, $\mathrm {Mod}^{\mathrm {lfin}}_{\mathrm {G},\zeta }(\mathcal {O})^{\mathfrak {B}}$ denotes the full subcategory of $\mathrm {Mod}^{\mathrm {lfin}}_{\mathrm {G},\zeta }(\mathcal {O})$ consisting of those representations such that all their irreducible subquotients lie in $\mathfrak {B}$ . Dually, we obtain a decomposition:

$$ \begin{align*} \mathfrak{C}(\mathcal{O})\cong \prod_{\mathfrak{B}} \mathfrak{C}(\mathcal{O})^{\mathfrak{B}}, \end{align*} $$

where $M\in \mathfrak {C}(\mathcal {O})$ lies in $\mathfrak {C}(\mathcal {O})^{\mathfrak {B}}$ if and only if $M^{\vee }$ lies in $\mathrm {Mod}^{\mathrm {lfin}}_{\mathrm {G}, \zeta }(\mathcal {O})^{\mathfrak {B}}$ .

For each fixed block $\mathrm {Mod}^{\mathrm {lfin}}_{\mathrm {G}, \zeta }(\mathcal {O})^{\mathfrak {B}}$ , there is a finite extension $E'$ of E with ring of integers $\mathcal {O}'$ , such that $\mathrm {Mod}^{\mathrm {lfin}}_{\mathrm {G}, \zeta }(\mathcal {O})^{\mathfrak {B}}\otimes _{\mathcal {O}} \mathcal {O}'$ decomposes into a finite product of indecomposable subcategories, each of which remains indecomposable after further extension of scalars. Such absolutely indecomposable blocks have been classified in [Reference Paškūnas23], which are described as follows:

1. (1) $\mathfrak {B}=\{\pi \}$ , with $\pi $ supersingular,

2. (2) $\mathfrak {B}= \{\mathrm {Ind}_P^{\mathrm {G}} \chi _1 \otimes \chi _2 \omega ^{-1}, \mathrm {Ind}_P^{\mathrm {G}} \chi _2 \otimes \chi _1 \omega ^{-1} \}$ , with $\chi _1 \chi _2 ^{-1} \notin \{1, \omega ^{\pm 1}\}$ ,

3. (3) $\mathfrak {B}=\{ \mathrm {Ind}_P^{\mathrm {G}} \chi \otimes \chi \omega ^{-1}\}$ , with $p \geq 3$ ,

4. (4) $\mathfrak {B}=\{1, St\} \otimes \chi \circ \det $ , with $p=2$ ,

5. (5) $\mathfrak {B}=\{1, St, \mathrm {Ind}_P^{\mathrm {G}} \omega \otimes \omega ^{-1}\} \otimes \chi \circ \det $ , with $p\geq 5$ ,

6. (6) $\mathfrak {B}=\{1, St, \omega \circ \det , St \otimes \omega \circ \det \} \otimes \chi \circ \det $ , with $p=3$ ,

where $\chi _1$ , $\chi _2,$ and $\chi $ are smooth characters of $\mathbb {Q}_p^\times $ and $\omega $ is the mod p cyclotomic character given by $x \mapsto x|x|(\mathrm{mod}\ p)$ .

To each block above, one may attach a semi-simple two-dimensional $\kappa $ -representation $\overline {\rho _{\mathfrak {B}}}$ of $\mathrm {G}_{\mathbb {Q}_p}$ using the semi-simple mod p correspondence of Breuil [Reference Breuil2, Reference Breuil3]: in case (1), $\overline {\rho _{\mathfrak {B}}}$ is absolutely irreducible, and Colmez’s functor $\mathbf {V}$ maps $\pi $ to $\overline {\rho _{\mathfrak {B}}}$ , in case (2), $\overline {\rho _{\mathfrak {B}}}=\chi _1\oplus \chi _2$ , in cases (3) and (4), $\overline {\rho _{\mathfrak {B}}}= \chi \oplus \chi $ , and in cases (5) and (6), $\overline {\rho _{\mathfrak {B}}}=\chi \oplus \chi \omega $ , where we consider characters of $\mathrm {G}_{\mathbb {Q}_p}$ as characters of $\mathbb {Q}_p^{\times }$ via local class field theory, normalized so that uniformizers correspond to geometric Frobenii. We note that the determinant of $\overline {\rho _{\mathfrak {B}}}$ is equal to $\zeta \varepsilon $ modulo $\varpi $ , where $\varepsilon $ is the p-adic cyclotomic character.

3.1 Two-dimensional pseudocharacters

We begin by reviewing the definition and some properties of two-dimensional pseudocharacters. Since these facts are standard, we quote much of this and the following paragraph directly from [Reference Paškūnas21, Appendix A]. Let $\mathcal {G}$ denote a profinite group, and let $(A, \mathfrak {m}_A)$ be a local artinian $\mathcal {O}$ -algebra. Throughout, we assume that $p>2$ . A continuous map $T: \mathcal {G} \rightarrow A$ is defined as a two-dimensional A-valued pseudocharacter if it adheres to the following conditions:

1. (1) $T(1)=2$ ;

2. (2) $T(gh)=T(hg)$ for every $g,h \in \mathcal {G}$ ;

3. (3) the relation $T(g)T(h)T(k)-T(g) T(hk)- T(h) T(gk)- T(k) T(gh)+T(ghk)+T(gk h)=0$ holds for all $g,h, k \in \mathcal {G}$ .

It is a well-known result that for any continuous representation $\rho : \mathcal {G} \rightarrow {\mathrm {GL}}_2(A)$ , the associated trace function $\mathrm {tr} \rho $ constitutes a two-dimensional pseudocharacter. Given a two-dimensional pseudocharacter $T: \mathcal {G}\rightarrow A$ , one can define a continuous group homomorphism $\mathcal {G} \rightarrow A^{\times }$ via the formula $D(g):=\frac {T(g)^2-T(g^2)}{2}$ (see [Reference Chenevier5, Proof of Proposition 7.23]). As shown in [Reference Chenevier5, Lemma 7.7 and Proposition 7.23], the mapping ${T\mapsto (T, D)}$ establishes a bijection between the set of two-dimensional pseudocharacters and the set of pairs $(T, D)$ , where $D:\mathcal {G} \rightarrow A^{\times }$ is a continuous group homomorphism and $T:\mathcal {G}\rightarrow A$ is a continuous function which satisfies: $T(1)=2$ , $T(gh)=T(hg)$ , and the identity $D(g)T(g^{-1}h)-T(g)T(h)+T(gh)=0$ for all $g, h\in \mathcal {G}$ .

Consider a continuous representation $\rho :\mathcal {G}\rightarrow {\mathrm {GL}}_2(\kappa )$ . Let $D_{\mathrm {tr} \rho }^{\mathrm {ps}}$ denote the functor from local artinian augmented $\mathcal {O}$ -algebras with residue field $\kappa $ to the category of sets, where $D_{\mathrm {tr} \rho }^{\mathrm {ps}}(A)$ consists of all two-dimensional A-valued pseudocharacters T lifting $\mathrm {tr}\rho $ , that is, satisfying the condition $T\equiv \mathrm {tr} \rho\ \pmod {\mathfrak {m}_A}$ . Provided that $\mathrm {Hom}^{cont}(\mathcal {H}, \mathbb {F}_p)$ is a finite-dimensional $\mathbb {F}_p$ -vector space for every open subgroup $\mathcal {H}$ of $\mathcal {G}$ , this functor $D_{\mathrm {tr} \rho }^{\mathrm {ps}}$ is pro-represented by a complete local Noetherian $\mathcal {O}$ -algebra. Notably, this finiteness criterion is met when $\mathcal {G}$ is the absolute Galois group of a local field.

We usually work with a modified version where the determinant is fixed. Let ${\psi : \mathcal {G}\rightarrow \mathcal {O}^{\times }}$ be a continuous character that lifts $\det \rho $ . We define $D_{\mathrm {tr} \rho }^{\mathrm {ps}, \psi }$ as the subfunctor of $D_{\mathrm {tr} \rho }^{\mathrm {ps}}$ , where $T\in D_{\mathrm {tr} \rho }^{\mathrm {ps}, \psi }(A)$ if and only if $\frac {T(g)^2-T(g^2)}{2}$ coincides with (the image of) $\psi (g)$ for all $g\in \mathcal {G}$ . We refer to $D_{\mathrm {tr} \rho }^{\mathrm {ps}, \psi }$ as a deformation problem with a fixed determinant. If $R_{\mathrm {tr} \rho }^{\mathrm {ps}}$ pro-represents $D_{\mathrm {tr} \rho }^{\mathrm {ps}}$ , it follows that $D_{\mathrm {tr} \rho }^{\mathrm {ps}, \psi }$ is pro-represented by a quotient of this ring, which we denote by $R_{\mathrm {tr} \rho }^{\mathrm {ps}, \psi }$ .

Recall that for the identity functor $I_{\mathcal {C}}$ of any category $\mathcal {C}$ , the natural transformations $\eta : I_{\mathcal {C}} \rightarrow I_{\mathcal {C}}$ form a commutative monoid, called the center of the category $\mathcal {C}$ . The following theorem characterizes the center of the category $\mathrm {Mod}^{\mathrm {lfin}}_{\mathrm {G}, \psi }(\mathcal {O})^{\mathfrak {B}}$ , as established in [Reference Paškūnas22, Theorem 1.3] (refer also to [Reference Paškūnas21]).

3.2 Block decomposition in the Banach setting

In the context of Banach space representation, for a block $\mathfrak {B}$ , let $\operatorname {\mathrm {Ban}}^{\mathrm {adm}}_{\mathrm {G}, \psi }(E)^{\mathfrak {B}}$ be the full subcategory of $\operatorname {\mathrm {Ban}}^{\mathrm {adm}}_{\mathrm {G}, \psi }(E)$ consisting of those $\Pi $ , such that for some (equivalently any) open bounded $\mathrm {G}$ -invariant lattice $\Theta $ in $\Pi $ , all the irreducible subquotients of $\Theta \otimes _{\mathcal {O}} \kappa $ , which is an object of $\mathrm {Mod}^{\mathrm {lfin}}_{\mathrm {G}, \psi }(\mathcal {O})$ , lie in $\mathfrak {B}$ . It is shown in [Reference Paškūnas21, Proposition 5.36] that $\operatorname {\mathrm {Ban}}^{\mathrm {adm}}_{\mathrm {G}, \psi }(E)$ decomposes into a direct sum of subcategories:

(3.1) $$ \begin{align} \operatorname{\mathrm{Ban}}^{\mathrm{adm}}_{\mathrm{G}, \psi}(E)\cong \bigoplus_{\mathfrak{B}} \operatorname{\mathrm{Ban}}^{\mathrm{adm}}_{\mathrm{G}, \psi}(E)^{\mathfrak{B}}, \end{align} $$

where the sum runs over the set of all blocks $\mathfrak {B}$ of $\mathrm {Mod}^{\mathrm {lfin}}_{\mathrm {G},\psi }(\mathcal {O})$ .

To understand the finer structure of these blocks, we first make explicit the action of the generic fiber of the relevant pseudodeformation ring. For any $\Pi \in \operatorname {\mathrm {Ban}}^{\mathrm {adm}}_{\mathrm {G}, \psi }(E)^{\mathfrak {B}}$ equipped with an open bounded $\mathrm {G}$ -invariant lattice $\Theta $ , the quotients $\Theta /\varpi ^n$ are objects of $\mathrm {Mod}^{\mathrm {lfin}}_{\mathrm {G}, \psi }(\mathcal {O})^{\mathfrak {B}}$ for all $n \geq 1$ . Theorem 3.1 gives a natural action of $R_{\mathrm {tr}\overline {\rho _{\mathfrak {B}}}}^{\mathrm {ps}, \psi }$ on $\Theta /\varpi ^n$ . Upon taking the inverse limit and inverting p, we obtain a natural homomorphism $R_{\mathrm {tr}\overline {\rho _{\mathfrak {B}}}}^{\mathrm {ps}, \psi }[1/p]\rightarrow \mathrm {End}_{\mathrm {G}}^{\mathrm {cont}}(\Pi )$ .

This action allows us to decompose the finite length objects. The Chinese remainder theorem, see [Reference Paškūnas21, Theorem 4.36], yields an equivalence of categories:

(3.2) $$ \begin{align} \operatorname{\mathrm{Ban}}^{\mathrm{adm}.\mathrm{fl}}_{\mathrm{G}, \psi}(E)^{\mathfrak{B}}\cong \bigoplus_{\mathfrak{n} \in \mathrm{maxSpec} R_{\mathrm{tr}\overline{\rho_{\mathfrak{B}}}}^{\mathrm{ps}, \psi}[1/p]} \operatorname{\mathrm{Ban}}^{\mathrm{adm}.\mathrm{fl}}_{\mathrm{G}, \psi}(E)^{\mathfrak{B}}_{\mathfrak{n}}, \end{align} $$

where, for a maximal ideal $\mathfrak {n}$ of $R_{\mathrm {tr}\overline {\rho _{\mathfrak {B}}}}^{\mathrm {ps}, \psi }[1/p]$ , $\operatorname {\mathrm {Ban}}^{\mathrm {adm}.\mathrm {fl}}_{\mathrm {G}, \psi }(E)^{\mathfrak {B}}_{\mathfrak {n}}$ consists of those finite length representations that are killed by a power of $\mathfrak {n}.$

3.3 Proof of main theorem 1.1

Consider an absolutely irreducible representation $\Pi \in \operatorname {\mathrm {Ban}}^{\mathrm {adm}}_{\mathrm {G}, \psi }(E)^{\mathfrak {B}}$ . As established in the previous section, we have a natural homomorphism $x: R_{\mathrm {tr}\overline {\rho _{\mathfrak {B}}}}^{\mathrm {ps}, \psi }[1/p] \rightarrow \mathrm {End}_{\mathrm {G}}^{\mathrm {cont}}(\Pi )$ . Because $\Pi $ is absolutely irreducible, Schur’s Lemma [Reference Paškūnas21, Corollary 4.42] implies the isomorphism $\mathrm {End}_{\mathrm {G}}^{\mathrm {cont}}(\Pi ) \cong E$ .

Finally, assuming the block $\mathfrak {B}$ corresponds to types (1) or (2) as defined in the preceding section, [Reference Paškūnas22, Corollary 2.25] provides the subsequent commutative diagram:

(3.3)

where $t^{\mathrm {univ}}$ is the universal pseudocharacter of $\mathrm {G}_{\mathbb {Q}_p}$ . Let $\mathfrak {n}$ be the maximal ideal of $ R_{\mathrm {tr}\overline {\rho _{\mathfrak {B}}}}^{\mathrm {ps}, \psi }[1/p]$ corresponding to $\Pi $ in the above diagram. Then, by the construction of the specialization map x, we have $\mathfrak {n} \subseteq \mathrm {Ann}_{R_{\mathrm {tr}\overline {\rho _{\mathfrak {B}}}}^{\mathrm {ps}, \psi }[1/p]}(\Pi ).$ Therefore, $\Pi $ is an element of $\operatorname {\mathrm {Ban}}^{\mathrm {adm}.\mathrm {fl}}_{\mathrm {G}, \psi }(E)^{\mathfrak {B}}_{\mathfrak {n}}$ .

Proof Since $\pi $ and $\Pi (V)$ are absolutely irreducible, their continuous duals $\pi ^*$ and $\Pi (V)^*$ are also absolutely irreducible. Therefore, by [Reference Colmez, Dospinescu and Paškūnas9, Theorem 1.4 and Corollary 1.6], they are of finite length. Now if $\pi ^*$ and $\Pi (V)^*$ are in $\operatorname {\mathrm {Ban}}_{\mathrm {G},\zeta }^{\mathrm {adm}}(E)^{\mathfrak {B}}$ for different $\mathfrak {B}$ , then by the block decomposition (3.1), we are done. Assume they lie in $\operatorname {\mathrm {Ban}}_{\mathrm {G},\zeta }^{\mathrm {adm}}(E)^{\mathfrak {B}}$ for the same block $\mathfrak {B}$ . By Lemma 2.1, we have $\check {\mathbf {V}}(\Pi (V)^*)=V$ , which is an absolutely irreducible representation of dimension $2$ . Set $r := \check {\mathbf {V}}(\pi ^*)$ , which is also an absolutely irreducible E-representation of $\mathrm {G}_{\mathbb {Q}_p}$ . Let $\mathfrak {n}$ be the maximal ideal of $R^{ps,\zeta \varepsilon }_{\mathrm {tr}\,\overline {\rho _{\mathfrak {B}}}}[1/p]$ corresponding to $\mathrm {tr}\,V$ , and let $\mathfrak {m}$ be the maximal ideal of $R^{ps,\zeta \varepsilon }_{\mathrm {tr}\,\overline {\rho _{\mathfrak {B}}}}[1/p]$ corresponding to $\mathrm {tr}\,r$ . Since $\overline {V}$ is generic, the above discussion implies $\Pi (V)^*\in \operatorname {\mathrm {Ban}}_{\mathrm {G},\zeta }^{\mathrm {adm}}(E)^{\mathfrak {B}}_{\mathfrak {n}}$ and ${\pi ^*\in \operatorname {\mathrm {Ban}}_{\mathrm {G},\zeta }^{\mathrm {adm}}(E)^{\mathfrak {B}}_{\mathfrak {m}}}$ . By the Brauer–Nesbitt theorem, the trace character is determined by the semisimplification, so $\mathrm {tr}\,V$ and $\mathrm {tr}\,r$ depend only on $V^{ss}\cong V$ and $r^{ss}\cong r$ , respectively. Thus, $\mathfrak {n}=\mathfrak {m}$ can occur only when $V \cong r.$

If $\pi $ is ordinary, then by the isomorphisms in (2.1), r is one-dimensional, so r can never be isomorphic to V. When $\pi $ is non-ordinary, the first isomorphism in (2.1) tells us $r = \mathbf {V}(\pi )$ . Therefore, if $\mathbf {V}(\pi )\neq V$ , then [Reference Paškūnas and Tung24, Corollary 6.8] yields $\mathrm {Ext}_{\mathrm {G},\zeta }^i(\pi ^*,\Pi (V)^*)=0$ for all $i\ge 0$ .

For the case $\mathbf {V}(\pi )=V=\mathbf {V}(\Pi (V))$ , we have the equality $\mathfrak {n}=\mathfrak {m}$ between the maximal ideals of $R^{ps,\zeta \varepsilon }_{\mathrm {tr}\,\overline {\rho _{\mathfrak {B}}}}[1/p]$ . Following the proof of [Reference Paškūnas22, Corollary 2.26], we get that $\operatorname {\mathrm {Ban}}^{\mathrm {adm}.\mathrm {fl}}_{\mathrm {G}, \zeta }(E)^{\mathfrak {B}}_{\mathfrak {n}}$ is anti-equivalent to the category of A-modules of finite length, where A denotes the completion of $R^{ps,\zeta \varepsilon }_{\mathrm {tr}\,\overline {\rho _{\mathfrak {B}}}}[1/p]$ at $\mathfrak {n}$ . Via this anti-equivalence, the objects $\pi ^*$ and $\Pi (V)^*$ are both identified with the residue field $\kappa (\mathfrak {n})$ of A. Consequently, we obtain the identification:

$$ \begin{align*} \mathrm{Ext}_{\mathrm{G},\zeta}^i(\pi^*,\Pi(V)^*) \cong \mathrm{Ext}_A^i(\kappa(\mathfrak{n}),\kappa(\mathfrak{n})). \end{align*} $$

Moreover, in the proof of [Reference Paškūnas22, Corollary 2.26], it is shown that A is a complete regular local ring of dimension $3$ . Using the Koszul complex, we compute $\mathrm {Ext}_A^i(\kappa (\mathfrak {n}),\kappa (\mathfrak {n}))$ , and by [Reference Weibel26, Exercise 4.5.6], this group is a $\binom {3}{i}$ -dimensional $\kappa (\mathfrak {n})$ -vector space. As the residue field $\kappa (\mathfrak {n})$ is identified with E via the specialization map x, we have the required dimension results.