2.2 The graded group algebra

Let $\mathbb {Z}_p[\![ U^1_D ]\!]=\varprojlim _{i\geq 1}\mathbb {Z}_p[U_D^1/U_D^i]$ be the Iwasawa algebra of $U^1_D$ over $\mathbb {Z}_p$. It is a pseudo-compact local $\mathbb {Z}_p$-algebra. For $\nu \geq 0$, let $J_{\nu }$ denote the smallest closed $\mathbb {Z}_p$-submodule of $\mathbb {Z}_p[\![ U^1_D]\!]$ which contains all elements of the form $p^{\ell } (h_1 - 1)\cdots (h_s - 1)$ with $\ell, s \geq 0$, $h_1,\ldots, h_s \in U^1_D$ and

\begin{align*} \ell + \omega(h_1) + \cdots +\omega(h_s) \geq \nu. \end{align*}

Let $J_{\nu +} : = \bigcup _{\nu ' > \nu } J_{\nu '}$. Let

\[ \operatorname{{\mathrm{gr}}}_J \mathbb{Z}_p[\![ U^1_D ]\! ] := \bigoplus_{\nu\geq 0} J_{\nu}/J_{\nu+}, \]

which is an associative graded algebra over $\operatorname {{\mathrm {gr}}} \mathbb {Z}_p := \bigoplus _{i\geq 0} p^i \mathbb {Z}_p/p^{i+1} \mathbb {Z}_p$. It naturally has a graded Lie algebra structure.

The homomorphism of abelian groups $\mathcal {L}_{\nu } : \operatorname {{\mathrm {gr}}}_{\nu } U^1_D \to J_{\nu } / J_{\nu +}$, $g (U^1_D)_{\nu + } \mapsto (g-1) + J_{\nu +}$ extends to a homomorphism of graded $\mathbb {F}_p[\varepsilon ]$-Lie algebras $\mathcal {L} : \operatorname {{\mathrm {gr}}} U^1_D \to \operatorname {{\mathrm {gr}}} \mathbb {Z}_p[\![ U^1_D ]\! ]$, where the $\mathbb {F}_p[\varepsilon ]$-algebra structure on $\operatorname {{\mathrm {gr}}}\mathbb {Z}_p[\![U_D^1]\!]$ is given through the isomorphism $\mathbb {F}_p[\varepsilon ] \xrightarrow {\sim } \operatorname {{\mathrm {gr}}} \mathbb {Z}_p$, $\varepsilon \mapsto p+ p^2 \mathbb {Z}_p \in \operatorname {{\mathrm {gr}}}^1 \mathbb {Z}_p$. Let $U_{\mathbb {F}_p[\varepsilon ]}(\operatorname {{\mathrm {gr}}} U^1_D)$ be the universal enveloping algebra of $\operatorname {{\mathrm {gr}}} U^1_D$ over $\mathbb {F}_p[\varepsilon ]$. By the universal property of $U_{\mathbb {F}_p[\varepsilon ]}(\operatorname {{\mathrm {gr}}} U_D^1)$, we have a homomorphism of associative $\operatorname {{\mathrm {gr}}} \mathbb {Z}_p$-algebras

(2.4)\begin{equation} \widetilde{\mathcal{L}} : U_{\mathbb{F}_p[\varepsilon]}(\operatorname{{\mathrm{gr}}} U^1_D) \to \operatorname{{\mathrm{gr}}}_J \mathbb{Z}_p[\![ U^1_D ]\!]. \end{equation}

By [Reference SchneiderSchn11, Theorem 28.3], $\widetilde {\mathcal {L}}$ is an isomorphism.

In practice, we will consider the Iwasawa algebra associated to the quotient group $U_D^1/Z_D^1$. Let $\mathbb {Z}_p[\![U_D^1/Z_D^1]\!]$ (respectively, $\mathbb {F}_p[\![U_D^1/Z_D^1]\!]$) be the Iwasawa algebra of $U_D^1/Z_D^1$ over $\mathbb {Z}_p$ (over $\mathbb {F}_p$). We have $\mathbb {F}_p[\![U_D^1/Z_D^1]\!]=\mathbb {Z}_p[\![U_D^1/Z_D^1]\!]\otimes _{\mathbb {Z}_p}\mathbb {F}_p$. The filtration $\{J_{\nu }, \nu \geq 0\}$ induces a filtration on $\mathbb {Z}_p[\![U_D^1/Z_D^1]\!]$ and on $\mathbb {F}_p[\![U_D^1/Z_D^1]\!]$. On the other hand, letting $\frak {m}_{D}$ denote the maximal ideal of $\mathbb {F}_p[\![ U^1_D/Z^1_D ]\!]$, we may consider the $\mathfrak {m}_{D}$-adic filtration on $\mathbb {F}_p[\![ U^1_D/Z^1_D ]\!]$. The following result shows that these two filtrations coincide up to rescaling indices.

One checks that $J_{\nu }\neq J_{\nu +}$ exactly when $\nu = {i}/{2}$ for some $i\geq 0$. Thus, by Lemma 2.4 the graded algebra

(2.5)\begin{equation} \operatorname{{\mathrm{gr}}}_{\mathfrak{m}_D} \mathbb{F}_p[\![ U^1_D /Z^1_D ]\! ] := \bigoplus_{i\geq 0} \frak{m}_{D}^i / \frak{m}_{D}^{i+1} \end{equation}

is identical to $\bigoplus _{\nu \geq 0}\overline {J}_{\nu }/\overline {J}_{\nu +}$.

Proposition 2.5 There is an isomorphism of graded $\mathbb {F}_p$-algebras

\[ \operatorname{{\mathrm{gr}}}_{\mathfrak{m}_D} \mathbb{F}_p[\![ U^1_D /Z^1_D ]\!] \cong U_{\mathbb{F}_p}(\frak{g}_{\mathbb{F}_{q}}). \]

Let $\mathbb {F}$ be a finite extension of $\mathbb {F}_p$ such that $\mathbb {F}_q$ embeds into $\mathbb {F}$. Let $\mathcal {J}$ denote the set of embeddings $\mathbb {F}_{q} \hookrightarrow \mathbb {F}$ and fix $\sigma _0\in \mathcal {J}$. We label the embeddings $\sigma _j = \sigma _0 \circ \varphi ^j$, so that $\mathcal {J}$ is identified with $\{0,\ldots, f-1\}$. Let $\frak {g}_j: = \mathbb {F} \otimes _{\mathbb {F}_{q}, \sigma _j} \frak {g}_{\mathbb {F}_{q}}$. We then have $\mathbb {F}\otimes _{\mathbb {F}_p}\frak {g}_{\mathbb {F}_{q}} = \bigoplus _{j=0}^{f-1} \frak {g}_j$. Let $e_j,f_j,h_j \in \frak {g}_j$ denote $1\otimes e$, $1\otimes f$, $1\otimes h \in \mathbb {F} \otimes _{\mathbb {F}_{q}, \sigma _j} \frak {g}_{\mathbb {F}_{q}}$.

We again denote by $\frak {m}_{D}$ the maximal ideal of $\mathbb {F}[\![ U^1_D/Z^1_D ]\!]=\mathbb {F}\otimes _{\mathbb {F}_p}\mathbb {F}_p[\![U_D^1/Z_D^1]\!]$. Then Proposition 2.5 implies that

(2.6)\begin{equation} \operatorname{{\mathrm{gr}}}_{\mathfrak{m}_D} \mathbb{F}[\![ U^1_D /Z^1_D ]\!] =\mathbb{F}\otimes_{\mathbb{F}_p} (\operatorname{{\mathrm{gr}}}_{\mathfrak{m}_D} \mathbb{F}_p[\![ U^1_D /Z^1_D ]\! ] ) \cong U_{\mathbb{F}}(\mathbb{F}\otimes_{\mathbb{F}_p} \frak{g}_{\mathbb{F}_q})\cong \bigotimes_{j=0}^{f-1} U_{\mathbb{F}} (\frak{g}_j). \end{equation}

In particular, we have $\operatorname {{\mathrm {gr}}}_{\mathfrak {m}_D}^1 \mathbb {F}[\![ U^1_D /Z^1_D ]\!]=\bigoplus _{j=0}^{f-1}(\mathbb {F} e_j\oplus \mathbb {F} f_j)$.

Theorem 2.6 is not enough for the application to Gelfand–Kirillov dimension, namely Corollary 2.12 below. We shall find eigenbases of $\mathbb {F}\otimes _{\mathbb {F}_p}\frak {g}_{\mathbb {F}_q}$ for the $\mathbb {F}_{q^2}^{\times }$-action in the next subsection.

2.3 Gelfand–Kirillov dimension

We regard $\mathbb {F}_{q^2}^{\times }$ as a subgroup of $\mathcal {O}_{L'}^{\times }$ via the Teichmüller lifting map, and then as a subgroup of $\mathcal {O}_D^{\times }$ via the fixed embedding $L'\hookrightarrow D$. It normalizes $U_D^1$, thus acts on $\overline {\operatorname {{\mathrm {gr}}} U_D^1}$ and on $\frak {g}_{\mathbb {F}_{q}}$. In practice, we need a basis of $\mathbb {F}\otimes _{\mathbb {F}_p}\frak {g}_{\mathbb {F}_q}$ consisting of eigenvectors for the action of $\mathbb {F}_{q^2}^{\times }$. Note that $e_j$ and $f_j$ are only eigenvectors for the action of $\mathbb {F}_q^{\times }$, but not for $\mathbb {F}_{q^2}^{\times }$.

Choose an embedding $\mathbb {F}_{q^2}\hookrightarrow \mathbb {F}$ which extends the fixed embedding $\sigma _0:\mathbb {F}_q\hookrightarrow \mathbb {F}$; we again denote it by $\sigma _0$ and let $\sigma _j=\sigma _0\circ \varphi ^j$ for $0\leq j\leq 2f-1$.

For $0\leq j\leq 2f-1$, define the following elements in $\mathbb {F}[\![U_D^1/Z_D^1]\!]$:

\[ Y_j:=\sum_{\lambda\in\mathbb{F}_{q^2}^{\times}}\sigma_j(\lambda)^{-1}(1+\varpi_D[\lambda]), \]

where the term $1+\varpi _D[\lambda ]$ is considered as an element in the group $U_D^1/Z_D^1$. Since $\sum _{\lambda \in \mathbb {F}_{q^2}^{\times }}\sigma _{j}(\lambda )^{-1}=0$, we have $Y_j\in \mathfrak {m}_{D}$. If $\mu \in \mathbb {F}_{q^2}^{\times }$, then one checks that

(2.7)\begin{equation} \mu\cdot Y_j:=[\mu]Y_j[\mu]^{-1}=\alpha_j(\mu)Y_j, \end{equation}

where $\alpha _j:\mathcal {O}_D^{\times }\rightarrow \mathbb {F}^{\times }$ denotes the character defined by

(2.8)\begin{equation} \alpha_j(x):=\sigma_j(\overline{x})^{q-1}. \end{equation}

Note that $\alpha _{j+f}=\alpha _j^q=\alpha _j^{-1}$.

For $0\leq j\leq 2f-1$, let $y_j:= Y_j+ \frak {m}_D^2 \in \operatorname {{\mathrm {gr}}}^1_{\mathfrak {m}_D}\mathbb {F}[\![U_D^1/Z_D^1]\!]$.

Lemma 2.8 For $g\in U_D^i/(U_D^i\cap Z_D^1)$ and $h\in U_D^j/(U_D^j\cap Z_D^1)$, we have

\[ gh-1\equiv (g-1)+(h-1) \mod \mathfrak{m}_D^{i+j}. \]

For $t\in \mathbb {F}_{q^2}^{\times }$, write

(2.9)\begin{equation} g_{t}:=1+p [t]\in U_D^1/Z_D^1. \end{equation}

Note that $\omega (g_t)=1$, so $g_t-1\in \mathfrak {m}_D^2$ by Lemma 2.4. Let $u_t$ denote the image of $g_t-1$ in $\operatorname {{\mathrm {gr}}}^2_{\mathfrak {m}_D}\mathbb {F}[\![U_D^1/Z_D^1]\!]$.

Proof. A direct computation shows

\[ Y_iY_{j}=\sum_{\lambda,\mu\in\mathbb{F}_{q^2}^{\times}}\sigma_i(\lambda)^{-1} \sigma_{j}(\mu)^{-1} (1+\varpi_D[\lambda]+\varpi_D[\mu]+p[\lambda^{q}\mu]). \]

We may write (in $U_D^1$)

\[ 1+\varpi_D[\lambda]+\varpi_D[\mu]+p[\lambda^q\mu]=(1+\varpi_D[\lambda]+ \varpi_D[\mu])(1+p[\lambda^q\mu]+x) \]

with $x\in \varpi _D^3\mathcal {O}_D$ and note that $(1+p[\lambda ^q\mu ]+x)-1$ has the same image as $(1+p[\lambda ^q\mu ])-1$ in $\operatorname {{\mathrm {gr}}}^2_{\mathfrak {m}_D}\mathbb {F}[\![U_D^1/Z_D^1]\!]$ by Lemma 2.8. Using Lemma 2.8 again, we have

\[ (1+\varpi_D[\lambda]+\varpi_D[\mu]+p[\lambda^q\mu])-1\equiv (h_{\lambda,\mu}-1)+(g_{\lambda^q\mu}-1) \mod \mathfrak{m}_D^3, \]

where $h_{\lambda,\mu }:=1+\varpi _D[\lambda ]+\varpi _D[\mu ]$ and $g_{\lambda ^q\mu }$ is defined by (2.9). Similarly, we have

\begin{align*} Y_jY_i\equiv\sum_{\lambda,\mu\in\mathbb{F}_{q^2}^{\times}}\sigma_i(\lambda)^{-1}\sigma_{j}(\mu)^{-1} ((h_{\lambda,\mu}-1)+(g_{\lambda\mu^q}-1))\mod \mathfrak{m}_D^3 \end{align*}

and so

\[ [Y_i,Y_j]\equiv \sum_{\lambda,\mu\in\mathbb{F}_{q^2}^{\times}} \sigma_i(\lambda)^{-1}\sigma_j(\mu)^{-1}((g_{\lambda^q\mu}-1)-(g_{\lambda\mu^q}-1))\mod \mathfrak{m}_D^3. \]

Taking the image in $\operatorname {{\mathrm {gr}}}^2_{\mathfrak {m}_D}\mathbb {F}[\![U_D^1/Z_D^1]\!]$ and noting that $\sigma _i(\lambda )=\sigma _{i-f}(\lambda ^q)$, we obtain

\[ [y_i,y_{j}]=\sum_{\lambda,\mu\in\mathbb{F}_{q^2}^{\times}} \frac{\sigma_j(\lambda^q)}{\sigma_{i-f}(\lambda^q)} \sigma_{j}(\lambda^q\mu)^{-1}(u_{\lambda^q\mu}-u_{\lambda\mu^q}). \]

The map

\[ \mathbb{F}_{q^2}^{\times}\times \mathbb{F}_{q^2}^{\times}\rightarrow \mathbb{F}_{q^2}^{\times},\quad (\lambda,\mu)\mapsto \lambda^q\mu \]

is surjective and each fibre is bijective to $\mathbb {F}_{q^2}^{\times }$ (by projecting to the second component), thus

\[ [y_i,y_{j}]=\sum_{t\in\mathbb{F}_{q^2}^{\times}} \bigg(\sum_{\lambda\in\mathbb{F}_{q^2}^{\times}} \frac{\sigma_j(\lambda^q)}{\sigma_{i-f}(\lambda^q)}\bigg)\cdot \sigma_j(t)^{-1}(u_{t}-u_{t^q}). \]

If $i-j\neq f$, then $\sum _{\lambda \in \mathbb {F}_{q^2}^{\times }} ({\sigma _j(\lambda ^q)}/{\sigma _{i-f}(\lambda ^q)})=0$ and so $[y_i,y_{j}]=0$, proving part (i). If $i-j=f$, then the last sum equals to $-1$, and so

\[ [y_{j+f},y_j]=-\sum_{t\in\mathbb{F}_{q^2}^{\times}}\sigma_j(t)^{-1}(u_t-u_{t^q}). \]

To prove part (ii), one could argue as in Lemma 2.7, but this needs to make explicit the $h_j$. Nonetheless, we can conclude by the following observation: since $y_i$ lies in $\bigoplus _{0\leq j\leq f-1}(\mathbb {F} e_j\oplus \mathbb {F} f_j)$, $[y_i,y_j]$ lies in the subspace spanned by $h_j=[e_j,f_j]$ (recall $[e_i,e_j]=[e_i,f_j] = [f_i,f_j]=0$ whenever $i\neq j$), and vice versa by part (i) and Lemma 2.7(ii).

To make the notation more transparent, we write $z_i:=y_{i+f}$ for $0\leq i\leq f-1$. Lemma 2.7 and Proposition 2.9 imply that the Lie algebra $\mathbb {F}\otimes _{\mathbb {F}_p}\frak {g}_{\mathbb {F}_q}$ has another basis over $\mathbb {F}$ given by $\{y_j,z_j,h_j'; 0\leq j\leq f-1\}$, with $y_j$ and $z_j$ in degree $1$, $h_j'$ in degree $2$ and satisfying the relations

\[ h_j'=[y_j,z_j],\quad [y_i,z_j]=0 \quad \mathrm{if}\ i\neq j,\quad [y_i,y_j]=[z_i,z_j]=[y_i,h_j']=[z_i,h_j']=0. \]

Let $I_D$ be the left ideal of $\operatorname {{\mathrm {gr}}}_{\mathfrak {m}_D} \mathbb {F}[\![ U^1_D /Z^1_D ]\!]$ generated by the degree-two elements $y_jz_j$ and $h'_j$ for all $0\leq j\leq f-1$. The ideal $I_D$ is, in fact, a two-sided ideal of $\operatorname {{\mathrm {gr}}}_{\mathfrak {m}_D} \mathbb {F}[\![ U^1_D /Z^1_D ]\!]$; it is also the left ideal generated by $(y_j z_j, h_j;\ 0\leq j \leq f-1)$ by Proposition 2.9(ii).

Let $\chi : \mathcal {O}_D^{\times } \to \mathbb {F}^{\times }$ be a smooth character. Let $\operatorname {{\mathrm {Proj}}}_{\mathbb {F}[\![\mathcal {O}_{D}^{\times } / Z^1_D ]\!]} \chi$ denote the projective envelope of $\chi$ in the category of $\mathbb {F}[\![\mathcal {O}_{D}^{\times } / Z^1_D]\!]$-modules. For $n\geq 1$, let

(2.10)\begin{equation} W_{\chi_,n}: = (\operatorname{{\mathrm{Proj}}}_{\mathbb{F}[\![\mathcal{O}_{D}^{\times} / Z^1_D]\!]} \chi) / \frak{m}_{D}^n. \end{equation}

It is clear that $W_{\chi,n}\cong \chi \otimes W_{\mathbf {1},n}$, where $\mathbf {1}$ denotes the trivial character. The module $W_{\chi,3}$ is of particular importance to us.

Corollary 2.11 The module $W_{\mathbf {1},3}$ has the following graded structure:

\begin{gather*} \operatorname{{\mathrm{gr}}}^0 W_{\mathbf{1},3} = \mathbb{F},\quad \operatorname{{\mathrm{gr}}}^1W_{\mathbf{1},3} = \bigoplus_{i=0}^{f-1}\mathbb{F}\alpha_i \oplus \mathbb{F}\alpha_i^{-1},\\ \operatorname{{\mathrm{gr}}}^2 W_{\mathbf{1},3} = \mathbb{F}^{2f}\oplus \bigoplus_{0\leq i\leq j\leq f-1}\mathbb{F}\alpha_i\alpha_j\oplus \bigoplus_{0\leq i\leq j\leq f-1}\mathbb{F}\alpha_{i}^{-1}\alpha_{j}^{-1}\oplus \bigoplus_{0\leq i\neq j\leq f-1} \mathbb{F}\alpha_i\alpha_j^{-1}, \end{gather*}

where $\alpha _j:\mathcal {O}_D^{\times }\rightarrow \mathbb {F}^{\times }$ is the character defined in (2.8).

We have the following criterion which allows us to control the Gelfand–Kirillov dimension of an admissible smooth $\mathbb {F}$-representation of $\mathcal {O}_D^{\times }/Z_D^1$. It is an analogue of [Reference Breuil, Herzig, Hu, Morra and SchraenBHHMS23, Corollary 5.3.5]. Let $\overline {W}_{\chi,3}$ denote the quotient of $W_{\chi,3}$ by the sum of characters which occur in $\operatorname {{\mathrm {gr}}}^2W_{\chi,3}$ and non-isomorphic to $\chi$. For example, if $L=\mathbb {Q}_p$, then $\dim _{\mathbb {F}}\overline {W}_{\chi,3}=5$ and has a socle filtration as follows (with $\alpha =\alpha _0$):

\[ (\chi\oplus\chi)\ \textbf{---}\ (\chi\alpha\oplus\chi\alpha^{-1})\ \textbf{---}\ \chi. \]

Corollary 2.12 Let $\pi$ be an admissible smooth representation of $\mathcal {O}_D^{\times } / Z_D^1$ over $\mathbb {F}$. Assume for each character $\chi$ such that $\operatorname {{\mathrm {Hom}}}_{\mathcal {O}_D^{\times }}(\chi, \pi ) \neq 0$, the natural injection

(2.11)\begin{equation} \operatorname{{\mathrm{Hom}}}_{\mathcal{O}_D^{\times}}(\chi , \pi) \hookrightarrow \operatorname{{\mathrm{Hom}}}_{\mathcal{O}_D^{\times}}( \overline{W}_{\chi,3} , \pi) \end{equation}

is an isomorphism. Then $\dim _{\mathcal {O}_{D}^{\times }} (\pi ) \leq f$, where $\dim _{\mathcal {O}_{D}^{\times }}(\pi )$ is the Gelfand–Kirillov dimension of $\pi$ over $\mathcal {O}_{D}^{\times }$.

2.4 $\operatorname {{\mathrm {Ext}}}^i$ groups when $L=\mathbb {Q}_p$

We assume $L=\mathbb {Q}_p$ with $p\geq 5$. We write $\alpha =\alpha _0$.

Proposition 2.13 Let $\psi,\chi :\mathcal {O}_D^{\times }\rightarrow \mathbb {F}^{\times }$ be two smooth characters. Then $\operatorname {{\mathrm {Ext}}}^1_{\mathcal {O}^{\times }_D/Z_D^1}(\psi,\chi )$ is nonzero if and only if $\psi =\chi \alpha$ or $\psi =\chi \alpha ^{-1}$. Moreover,

\[ \dim_{\mathbb{F}}\operatorname{{\mathrm{Ext}}}^1_{\mathcal{O}_D^{\times}/Z_D^1}(\chi\alpha,\chi)=\dim_{\mathbb{F}} \operatorname{{\mathrm{Ext}}}^1_{\mathcal{O}_D^{\times}/Z_D^1}(\chi\alpha^{-1},\chi)=1. \]

Proof. First, we have isomorphisms

\[ \operatorname{{\mathrm{Ext}}}^i_{\mathcal{O}^{\times}_D/Z_D^1}(\tau_1,\tau_2)\cong\operatorname{{\mathrm{Ext}}}^i_{\mathcal{O}_D^{\times}/Z_D^1} (\mathbf{1},\tau_1^{\vee}\otimes\tau_2)\cong H^i(\mathcal{O}_D^{\times}/Z_D^1,\tau_1^{\vee}\otimes\tau_2) \cong H^i(U_D^1/Z_D^1,\tau_1^{\vee}\otimes\tau_2)^{\mathbb{F}_{p^2}^{\times}}. \]

Second, since $U_D^1/Z_D^1$ is a Poincaré group of dimension three (cf. [Reference SerreSer02, § 4.5]), Poincaré duality induces an isomorphism

\[ H^i(U_D^1/Z_D^1,\tau)\cong H^{3-i}(U_D^1/Z_D^1,\tau^{\vee})^{\vee} \]

for $0\leq i\leq 3$ and any finite-dimensional representation $\tau$. The result easily follows.

3.1 Preliminaries

Denote by $U(\mathbb {Z}_p)$ (respectively, $B(\mathbb {Z}_p)$) the (upper) unipotent (respectively, Borel) subgroup of $K$. Note that $H$ normalizes $U(\mathbb {Z}_p)$.

Proof. (i) Let $W_0:=W^{U(\mathbb {Z}_p)}\cong \chi$. We first prove that $(W/W_0)^{U(\mathbb {Z}_p)}$ is one-dimensional and isomorphic to $\chi \alpha ^{-1}$ as an $H$-representation. Since $\dim _{\mathbb {F}}W\geq 2$ by assumption, $W/W_0$ is nonzero, hence $(W/W_0)^{U(\mathbb {Z}_p)}$ is also nonzero because $U(\mathbb {Z}_p)$ is a pro-$p$-group. On the other hand, we have an $H$-equivariant injection

\[ 0\rightarrow (W/W_0)^{U(\mathbb{Z}_p)}\rightarrow H^1(U(\mathbb{Z}_p),W_0), \]

which is actually an isomorphism because $H^1(U(\mathbb {Z}_p),\chi )\cong \chi \alpha ^{-1}$ is one-dimensional (see, e.g., [Reference PaškūnasPaš10, Lemma 5.5]). This proves the claim.

Any element $w\in \mathrm {Sym}^1\mathbb {F}^2\otimes W$ can be written as $Y\otimes w_0+X\otimes w_1$ for (unique) $w_0,w_1\in W$. Let $g=\big (\begin{smallmatrix} {1} & {t}\\ {0} & {1}\end{smallmatrix}\big ) \in U(\mathbb {Z}_p)$. Then

\[ gw=(\bar{t}X+Y)\otimes gw_0+X\otimes gw_1 =Y\otimes gw_0+X\otimes(\bar{t}\cdot gw_0+gw_1). \]

Hence, $w$ is fixed by $U(\mathbb {Z}_p)$ if and only if

\[ \left\{\begin{array}{@{}ll} gw_0=w_0,\\ gw_1=w_1-\bar{t}gw_0. \end{array}\right. \]

We have two cases.

1. (a) If $w_0=0$, then the above condition becomes $gw_1=w_1$, i.e. $w_1\in W_0$.

2. (b) If $w_0\neq 0$, then $w_0\in W_0$ and $w_1\in (W/W_0)^{U(\mathbb {Z}_p)}$. Moreover, $(W/W_0)^{U(\mathbb {Z}_p)}$ is one-dimensional as seen above and the condition $gw_1=w_1-\bar {t}w_0$ determines uniquely $w_1$ (whenever $w_0\neq 0$ is fixed).

The result easily follows.

(ii) This follows from part (i) via the fact that $(W_{U(\mathbb {Z}_p)})^{\vee }\cong (W^{\vee })^{U(\mathbb {Z}_p)}$, and similarly for

\[ \mathrm{Sym}^1\mathbb{F}^2\otimes W. \]

Corollary 3.2 Let $V=\operatorname {{\mathrm {Ind}}}_{I}^{K}\chi$ for some smooth character $\chi :I\rightarrow \mathbb {F}^{\times }$. Then

\begin{gather*} (\mathrm{Sym}^1\mathbb{F}^2\otimes V)^{U(\mathbb{Z}_p)}\cong \chi^s\chi_{1,0}\oplus\chi^s\chi_{1,0}^s\oplus \chi\chi_{1,0},\\ (\mathrm{Sym}^1\mathbb{F}^2\otimes V)_{U(\mathbb{Z}_p)}\cong \chi^s\chi_{1,0}\oplus\chi^s\chi_{1,0}^s\oplus \chi\chi_{1,0}^s. \end{gather*}

Consider the following situation: $V_1, V_2$ are two irreducible locally algebraic representations of $K$, and $L_i\subset V_i$ is a lattice for $i=1,2$. Assume that we are given an $\mathbb {F}[K]$-module $W$, together with $K$-equivariant morphisms $r_i:L_i\rightarrow W$. Let $L$ be the fibered product of $r_1$ and $r_2$, namely

(3.1)\begin{equation} 0\rightarrow L\rightarrow L_1\oplus L_2 {\buildrel {r_1 - r_2} \over \longrightarrow} W. \end{equation}

Then $L$ is a lattice in $V_1\oplus V_2$. We also call $L$ the gluing lattice of $L_1$ and $L_2$ along $W$. We remark that, if either $r_1$ or $r_2$ is surjective, then so is $r_1-r_2$.

Proof. (i) We have the following commutative diagram.

By the snake lemma, it induces a short exact sequence $0\rightarrow \operatorname {{\mathrm {Ker}}}(r_1)\rightarrow L\rightarrow L_2\rightarrow 0$. We obtain the result by taking mod $p$ reduction (as $L_2$ is $\mathcal {O}$-flat).

(ii) It is clear from part (i) that $\operatorname {{\mathrm {Ker}}}(r_L)=\operatorname {{\mathrm {Ker}}}(r_1)+pL$, so we have a short exact sequence

\[ 0\rightarrow \operatorname{{\mathrm{Ker}}}(r_1)\cap pL\rightarrow \operatorname{{\mathrm{Ker}}}(r_1)\oplus pL\rightarrow \operatorname{{\mathrm{Ker}}}(r_L)\rightarrow0. \]

Taking mod $p$ reduction and noting that $\operatorname {{\mathrm {Ker}}}(r_1)\cap pL=p\operatorname {{\mathrm {Ker}}}(r_1)$ by part (i), we obtain an exact sequence

\[ 0\rightarrow p\operatorname{{\mathrm{Ker}}}(r_1)/p^2\operatorname{{\mathrm{Ker}}}(r_1)\rightarrow \operatorname{{\mathrm{Ker}}}(r_1)/p\operatorname{{\mathrm{Ker}}}(r_1)\oplus pL/p^2L\rightarrow \operatorname{{\mathrm{Ker}}}(r_L)/p\operatorname{{\mathrm{Ker}}}(r_L)\rightarrow0. \]

But the map $p\operatorname {{\mathrm {Ker}}}(r_1)/p^2\operatorname {{\mathrm {Ker}}}(r_1)\rightarrow \operatorname {{\mathrm {Ker}}}(r_1)/p\operatorname {{\mathrm {Ker}}}(r_1)$ is identically zero, so the result follows from part (i).

Proof. We need to show that the natural map

\[ \operatorname{{\mathrm{Hom}}}_{\mathcal{O}[K]}(L_2,\sigma)\rightarrow\operatorname{{\mathrm{Hom}}}_{\mathcal{O}[K]}(L,\sigma) \]

is an isomorphism for any Serre weight $\sigma$. By applying $\operatorname {{\mathrm {Hom}}}_{\mathcal {O}[K]}(-,\sigma )$ to (3.1) we obtain a long exact sequence

\begin{align*} 0\rightarrow \operatorname{{\mathrm{Hom}}}(W,\sigma)\rightarrow\operatorname{{\mathrm{Hom}}}(L_1,\sigma)\oplus \operatorname{{\mathrm{Hom}}}(L_2,\sigma) &\rightarrow \operatorname{{\mathrm{Hom}}}(L,\sigma)\\ &\rightarrow \operatorname{{\mathrm{Ext}}}^1(W,\sigma)\rightarrow \operatorname{{\mathrm{Ext}}}^1(L_1,\sigma)\oplus \operatorname{{\mathrm{Ext}}}^1(L_2,\sigma). \end{align*}

By assumption (a), the surjection $r_1:L_1\twoheadrightarrow W$ induces an isomorphism $\operatorname {{\mathrm {Hom}}}(W,\sigma )\xrightarrow {\sim } \operatorname {{\mathrm {Hom}}}(L_1,\sigma )$. To conclude we need to show that the morphism

\[ \operatorname{{\mathrm{Ext}}}^1(W,\sigma)\rightarrow \operatorname{{\mathrm{Ext}}}^1(L_1,\sigma)\oplus \operatorname{{\mathrm{Ext}}}^1(L_2,\sigma) \]

is injective. For this, it is enough to prove that either $\operatorname {{\mathrm {Hom}}}(\operatorname {{\mathrm {Ker}}}(r_1),\sigma )$ or $\operatorname {{\mathrm {Hom}}}(\operatorname {{\mathrm {Ker}}}(r_2),\sigma )$ vanishes, which is a consequence of assumption (b).

Finally, we record a result which will be used later on.

3.2 Lattices in tame types

We consider the following representations of $\Gamma$ over $E$, and view them as smooth representations of $K$ via the projection $K \twoheadrightarrow \Gamma$.

1. • Let $\chi _1,\chi _2: \mathbb {F}_{p}^{\times } \to E^{\times }$ be two characters. Let $I(\chi _1,\chi _2)$ denote the principal series representation $\operatorname {{\mathrm {Ind}}}_{B(\mathbb {F}_p)}^{\Gamma }\chi _1\otimes \chi _2$, where $B(\mathbb {F}_p)$ is the (upper) Borel subgroup of $\Gamma$. It is well-known that $I(\chi _1,\chi _2)$ is irreducible if $\chi _1 \neq \chi _2$. If $\chi _1 =\chi _2 =\chi$, then

   \[ I(\chi, \chi) \cong (\chi \circ\det) \oplus\ ({\rm sp} \otimes\chi\circ\det), \]
   where ${\rm sp}$ denotes the Steinberg representation.

2. • Let $\psi :\mathbb {F}_{p^2}^{\times } \to E^{\times }$ be a character which does not factor through the norm map $\mathbb {F}_{p^2}^{\times } \to \mathbb {F}_{p}^{\times }$. This is equivalent to requiring $\psi \neq \psi ^p$. There is an irreducible $(p-1)$-dimensional representation $\Theta (\psi )$ characterized by the isomorphism $\Theta (\psi )\otimes {\rm sp} \cong \operatorname {{\mathrm {Ind}}}_{\mathbb {F}_{p^2}^{\times }}^{\mathrm {GL}_2(\mathbb {F}_p)} \psi$, where $\mathbb {F}_{p^2}^{\times } \hookrightarrow \mathrm {GL}_2(\mathbb {F}_p)$ is a fixed group embedding. For two such characters $\psi,\psi '$, $\Theta (\psi ) \cong \Theta (\psi ')$ if and only if $\psi '\in \{\psi, \psi ^p\}$.

The Jordan–Hölder factors of the reduction mod $p$ of any lattice in the above representations are determined in [Reference DiamondDia07]. We recall the results in the next proposition.

Let $x: \mathbb {F}_p \hookrightarrow \mathbb {F}$ denote the natural embedding and $[x]: \mathbb {F}_p \to \mathcal {O}$ be the Teichmüller lift of $x$ which will be viewed as a multiplicative character of $\mathbb {F}_p^{\times }$. Let $\xi : \mathbb {F}_{p^2} \hookrightarrow \mathbb {F}$ be an embedding extending $x$. Let $\xi ':= \xi ^p$ and $\zeta : = \xi \xi '$. Let $[\xi ] : \mathbb {F}_{p^2} \to \mathcal {O}$ be the Teichmüller lift of $\xi$ which will be viewed as a multiplicative character of $\mathbb {F}_{p^2}^{\times }$. We have $[x]^{p-1} = \mathbf {1}$ and $[\xi ]^{p+1} = [x]$.

We recall Lemma 4.1.1 of [Reference Emerton, Gee and SavittEGS15] on the lattices of finite-dimensional irreducible residually multiplicity-free $E$-representations of $K$.

3.3 Lattices in $ \underline {\mathrm {Sym}}^1E^2 \otimes \Theta (\psi )$

Let $\mathrm {pr}:\mathbb {Q}_p^{\times }\rightarrow 1+p\mathbb {Z}_p$ denote the projection sending $p$ to $1$. As $p>2$, we can define the square root on $1+p\mathbb {Z}_p$ by the usual binomial formula. Define

(3.2)\begin{equation} \underline{\mathrm{Sym}}^1E^2:=\mathrm{Sym}^{1}E^2\otimes (\mathrm{pr}\circ\det)^{-1/2}. \end{equation}

The reason to introduce the twist is to make the central character of $\underline {\mathrm {Sym}}^1E^2$ to be trivial on $Z_1$. Note that the mod $p$ reduction of $\underline {\mathrm {Sym}}^1\mathcal {O}^2:=\mathrm {Sym}^{1}\mathcal {O}^2\otimes (\mathrm {pr}\circ \det )^{-1/2}$ still gives $\mathrm {Sym}^1\mathbb {F}^2$.

Let $\psi : \mathbb {F}_{p^2}^{\times } \to E^{\times }$ be a character with $\psi \neq \psi ^p$. Write $\psi = [\xi ]^{a+1 + (p+1)b}$ with $0\leq a \leq p-1$ and $0\leq b\leq p-2$. By Proposition 3.6, $\overline {\Theta (\psi )}^{\rm ss}$ is multiplicity free and has two (respectively, one) Jordan–Hölder factors if $1 \leq a \leq p-2$ (respectively, if $a\in \{0,p-1\}$).

Assume first $1\leq a\leq p-2$. By Propositions 3.6 and 3.7, there are two lattices $T,T'$ in $\Theta (\psi )$ such that

(3.3)$$\begin{gather} 0\rightarrow \sigma_{p-2-a, a+b+1} \rightarrow T/pT\rightarrow \sigma_{a-1, b+1}\rightarrow 0 , \end{gather}$$

(3.4)$$\begin{gather}0\rightarrow \sigma_{a-1, b+1} \rightarrow T'/pT'\rightarrow \sigma_{p-2-a, a+b+1}\rightarrow 0, \end{gather}$$

where both extensions are nonsplit. Note that $T/pT$ and $T'/pT'$ are $\Gamma$-representations as $\Theta (\psi )$ itself is. Moreover, if we fix $T$ and normalize $T'$ (by a scalar) so that $T'\subset T$ and $T'\nsubseteq pT$, then by [Reference Emerton, Gee and SavittEGS15, Proposition 5.2.3(1)] we have

(3.5)\begin{equation} pT\subset T'\subset T. \end{equation}

Proof. (i) We only give the proof in the case of $T/pT$. Using (3.3) we obtain an exact sequence

\[ 0\rightarrow (\sigma_{p-2-a,a+b+1})^{I_1} \to (T/pT)^{I_1} \to (\sigma_{a-1 ,b+1})^{I_1}. \]

Assume for a contradiction that $(T / pT)^{I_1}$ is two-dimensional. Then we would obtain

\[ (T/pT)^{I_1}\cong \chi_{p-2-a,a+b+1}\oplus \chi_{a-1,b+1}, \]

and consequently an $I$-equivariant injection $\chi _{a-1,b+1}\hookrightarrow T/pT$. By Frobenius reciprocity, we would get a nonzero $K$-equivariant map

\[ \operatorname{{\mathrm{Ind}}}_{I}^{K} \chi_{a-1,b+1}\rightarrow T/pT. \]

By comparing the Jordan–Hölder factors, this map cannot be injective and must have image isomorphic to $\sigma _{a-1,b+1}$ (see [Reference Breuil and PaškūnasBP12, Lemma 2.3]). This gives a contradiction because the sequence (3.3) is nonsplit.

(ii) This is proved in a similar way as part (i). Alternatively, it can be deduced from part (i) by taking dual.

Recall that $E$ is unramified over $\mathbb {Q}_p$. Consider $\underline {\mathrm {Sym}}^1\mathcal {O}^2:=\mathcal {O} Y\oplus \mathcal {O} X$, the standard lattice in $\underline {\mathrm {Sym}}^1E^2$ and set

(3.6)\begin{gather} L:= \underline{\mathrm{Sym}}^1\mathcal{O}^2\otimes_{\mathcal{O}} T , \end{gather}

(3.7)\begin{gather}L' := \underline{\mathrm{Sym}}^1\mathcal{O}^2\otimes_{\mathcal{O}} T'. \end{gather}

Then we haveFootnote ²

\begin{align*} L / p L &\cong \mathrm{Sym}^1\mathbb{F}^2\otimes T/pT,\\ L' / p L' &\cong \mathrm{Sym}^1\mathbb{F}^2\otimes T'/p T', \end{align*}

and (3.5) implies $p L \subset L' \subset L$.

Proof. It suffices to prove part (i). Recall the following facts (see [Reference Breuil and PaškūnasBP12, Lemma 3.8])

\begin{align*} \mathrm{Sym}^1\mathbb{F}^2\otimes \sigma_{a-1, b+1} &\cong \sigma_{a, b+1} \oplus \sigma_{a-2, b+2}, \\ \mathrm{Sym}^1\mathbb{F}^2\otimes\sigma_{p-2-a,a+b+1} &\cong \sigma_{p-1-a,a+b+1}\oplus\sigma_{p-3-a,a+b+2}, \end{align*}

with the convention $\sigma _{-1,b+1} = \sigma _{-1,b+2}= 0$. Using (3.3) this implies that $L/pL\cong \mathrm {Sym}^1\mathbb {F}^2\otimes T/pT$ has a two-step filtration as claimed in (3.8), and is multiplicity free. By Lemma 3.10, the filtration gives exactly the socle (and cosocle) filtration. This also completes the proof if $a\in \{1,p-2\}$.

Assume $2\leq a \leq p-3$ in the rest of the proof. For a Serre weight $\sigma$, denote by ${\rm Inj}_{\Gamma }(\sigma )$ the injective envelope of $\sigma$ in the category of $\mathbb {F}[\Gamma ]$-modules; we remark that $\mathrm {Inj}_{\Gamma }(\sigma )$ is also projective. Let $W_1$ denote the image of the unique (up to scalar) nonzero map ${\rm Inj}_{\Gamma }(\sigma _{a-2,b+2}) \to L/ p L$. Since $\sigma _{p-3-a, a+b+2}$ is not a Jordan–Hölder factor of ${\rm Inj}_{\Gamma }(\sigma _{a-2,b+2})$ (see [Reference Breuil and PaškūnasBP12, Lemma 3.2]), $W_1$ does not admit $\sigma _{p-3-a,a+b+2}$ as a subquotient. Since $\mathrm {cosoc}(W_1)\cong \sigma _{a-2,b+2}$ by construction, we deduce from (3.8) that

\[ W_1\cong (\sigma_{p - 1-a, a+b+1} \ \textbf{---}\ \sigma_{a-2, b+2}), \]

i.e. the nonsplit extension $E_2$ occurs in $L/pL$. Consequently, the cokernel of the inclusion $W_1\hookrightarrow L/ p L$, denoted by $W_2$, has $\{\sigma _{a,b+1},\sigma _{p-3-a,a+b+2}\}$ as the set of Jordan–Hölder factors, hence is isomorphic to the nonsplit extension $E_1=(\sigma _{p- 3-a, a+b+2} \ \textbf {---}\ \sigma _{a , b+1})$ because $\sigma _{p-3-a,a+b+2}$ does not occur in the cosocle of $W_2$ by (3.8).

We are left to show that $L / pL$ is a nonsplit extension of $W_2$ by $W_1$ (this implies that $E_3$ occurs in $L/pL$). Assume for a contradiction that $L / pL \cong W_1 \oplus W_2$. Let $V$ denote the principal series $\operatorname {{\mathrm {Ind}}}_{I}^{K}\chi _{a+1,b}^s$ which is isomorphic to the (unique) nonsplit extension $(\sigma _{a+1,b}\ \textbf {---}\ \sigma _{p-2-a,a+b+1})$. By [Reference Breuil and PaškūnasBP12, § 3], there exists a short exact sequence

\[ 0 \to T / pT \to {\rm Inj}_{\Gamma} (\sigma_{p - 2 - a, a+b +1}) \to V \to 0, \]

which induces a short exact sequence

\[ 0 \to L/pL \to \mathrm{Sym}^1 \mathbb{F}^2 \otimes {\rm Inj}_{\Gamma} (\sigma_{p - 2 - a, a+b +1}) \to \mathrm{Sym}^1 \mathbb{F}^2 \otimes V \to 0. \]

By Lemma 3.12 below, if $2 \leq a \leq p-4$, then

\[ \mathrm{Sym}^1 \mathbb{F}^2 \otimes {\rm Inj}_{\Gamma} (\sigma_{p - 2 - a, a+b +1}) = {\rm Inj}_{\Gamma} (\sigma_{p -1-a, a+b+1}) \oplus {\rm Inj}_{\Gamma} (\sigma_{p-3-a,a+b+2}). \]

Comparing the socles, it is clear that $W_2\cap {\rm Inj}_{\Gamma } (\sigma _{p -1-a, a+b+1})\,{=}\,0$, thus $W_2\!\hookrightarrow\! {\rm Inj}_{\Gamma } (\sigma _{p-3-a,a+b+2})$. Moreover, we have

\[ L/pL \cap \mathrm{Inj}_{\Gamma}\sigma_{p-3-a,a+b+2}=W_2, \]

which induces a (nonzero) morphism

\[ {\rm Inj}_{\Gamma} (\sigma_{p-3-a,a+b+2}) / W_2 \rightarrow\mathrm{Sym}^1 \mathbb{F}^2 \otimes V. \]

However, by [Reference Breuil and PaškūnasBP12, § 3] we have

\[ {\rm Inj}_{\Gamma} (\sigma_{p-3-a,a+b+2}) / W_2 \cong \operatorname{{\mathrm{Ind}}}_{B(\mathbb{F}_p)}^{\Gamma}\chi_{p-3-a,a+b+2}, \]

so by Frobenius reciprocity we obtain a nonzero $I$-equivariant morphism

\[ \chi_{p-3-a,a+b+2}\rightarrow \mathrm{Sym}^1\mathbb{F}^2\otimes V. \]

But this contradicts Corollary 3.2, by which $(\mathrm {Sym}^{1}\mathbb {F}^2\otimes V)^{I_1}\cong \chi _{a+2,b}\oplus \chi _{a,b+1}\oplus \chi _{p-1-a,a+b+1}$.

The case $a = p-3$ is a little subtle. By Lemma 3.12 below we have

\[ \mathrm{Sym}^1 \mathbb{F}^2 \otimes {\rm Inj}_{\Gamma} (\sigma_{1, b -1}) = {\rm Inj}_{\Gamma} (\sigma_{2, b- 1}) \oplus {\rm Inj}_{\Gamma} (\sigma_{0,b}) \oplus \sigma_{p-1,b}. \]

Comparing the socles, one checks that $W_2$ embeds into ${\rm Inj}_{\Gamma } (\sigma _{0,b})$ and actually

\[ L/pL\cap \mathrm{Inj}_{\Gamma}(\sigma_{0,b})=W_2. \]

Hence, we obtain a nonzero morphism from ${\rm Inj}_{\Gamma } (\sigma _{0,b})/ W_2 \cong \sigma _{0,b}$ to $\mathrm {Sym}^1\mathbb {F}^2 \otimes V$. On the other hand, $\sigma _{p-1,b}$ also occurs in $\mathrm {Sym}^1\mathbb {F}^2\otimes V$ and, in fact, is a direct summand because $\sigma _{p-1,b}$ is an injective $\mathbb {F}[\Gamma ]$-module. Thus, there exists an embedding

\[ \sigma_{0,b}\oplus\sigma_{p-1,b}\hookrightarrow \mathrm{Sym}^1\mathbb{F}^2\otimes V. \]

However, by Corollary 3.2 we have $(\mathrm {Sym}^{1}\mathbb {F}^2\otimes V)^{I_1}\cong \chi _{p-1,b}\oplus \chi _{p-3,b+1}\oplus \chi _{2,b-1}$, in which the character $\chi _{p-1,b}$ ($=\chi _{0,b}$) occurs only once. This gives a contradiction and finishes the proof.

Recall the following facts (see [Reference Breuil and PaškūnasBP12, § 3]): ${\rm Inj}_{\Gamma } (\sigma _{a , b})$ is of dimension $2p$ if $1\leq a \leq p-2$; ${\rm Inj}_{\Gamma }(\sigma _{p-1,b}) \cong \sigma _{p-1 , b}$ is of dimension $p$; ${\rm Inj}_{\Gamma } (\sigma _{0,b}) \cong (\sigma _{0,b} \ \textbf {---}\ \sigma _{p-3, b+1} \ \textbf {---}\ \sigma _{0,b})$ is of dimension $p$.

Finally, we treat the case $a\in \{0,p-1\}$.

Proposition 3.13 Assume $a\in \{0,p-1\}$. There are two lattices (unique up to homothety) $L, L'$ of $\underline {\mathrm {Sym}}^1 E^2\otimes \Theta (\psi )$ such that $pL \subset L' \subset L$ and

\begin{align*} L/pL &\cong \sigma_{p-1, b+1} \oplus \sigma_{p-3, b+2},\\ L'/pL' &\cong (\sigma_{p-1, b+1} \ \textbf{---}\ \sigma_{p-3, b+2}). \end{align*}

The lattice $pL$ is then identified with the kernel of the natural projection $L' \twoheadrightarrow \sigma _{p-3,b+2}$. Moreover, $(L'/pL')_{K_1} \cong \sigma _{p-3, b+2}$.

Proof. Let $T$ be any lattice in $\Theta (\psi )$ and $L: = \underline {\mathrm {Sym}}^1 \mathcal {O}^2 \otimes T$. Then $T/pT \cong \sigma _{p-2, b+1}$ by Proposition 3.6, and consequently $L/pL \cong \sigma _{p-1, b+1} \oplus \sigma _{p-3,b+2}$ by [Reference Breuil and PaškūnasBP12, Lemma 3.8]. Let $L'$ be the kernel of the composition $L \to L/pL \overset {p_1}{\twoheadrightarrow } \sigma _{p-1,b+1}$. Then $pL \subsetneq L'\subsetneq L$. Moreover, we have a short exact sequence

(3.9)\begin{equation} 0\to pL/pL' \to L'/pL' \to \sigma_{p-3,b+2} \to 0. \end{equation}

We claim that (3.9) induces an isomorphism $(L'/pL')_{K_1} \xrightarrow {\sim } \sigma _{p-3,b+2};$ this will imply that $L'/pL'$ is a nonsplit extension of $\sigma _{p-3,b+2}$ by $\sigma _{p-1,b+1}$.

The proof of Proposition 3.1 shows that there exist $\overline {w}_0,\overline {w}_1\in T/pT$ such that $X\otimes \overline {w}_0$ and $Y\otimes \overline {w}_0+X\otimes \overline {w}_1$ span $(L/pL)^{I_1}$. Comparing the $H$-action, we must have

\[ (\sigma_{p-1,b+1})^{I_1}=\mathbb{F} (X\otimes \overline{w}_0),\quad (\sigma_{p-3,b+2})^{I_1}=\mathbb{F} (Y\otimes \overline{w}_0+X\otimes \overline{w}_1). \]

Let $w_0,w_1 \in T$ be a lift of $\overline {w}_0,\overline {w}_1$, respectively. From the definition of $L'$ we see that $Y\otimes w_0+X\otimes w_1\in L'$. As $K_1$ acts trivially on $T$, we have

\[ \bigg(\bigg(\begin{matrix} {1} & {p}\\ {0} & {1}\end{matrix}\bigg) - 1\bigg) (Y\otimes w_0 + X \otimes w_1) = (pX )\otimes w_0 \in pL. \]

Since $(pX )\otimes \overline {w}_0$ generates $pL/pL'$, the claim follows.

The uniqueness of $L'$ (up to homothety) follows from Proposition 3.7. Since $pL$ is identified with the kernel of the natural projection $L' \twoheadrightarrow \sigma _{p-3,b+2}$, the uniqueness of $L$ follows.

3.3.1 Sublattices in $L$

In this subsection we specify some sublattices in $L$ in the case $1 \leq a \leq p-2$. Recall that $\sigma _{-1, b+1} = \sigma _{-1, b+2} = 0$ by our convention.

Let $L_1 : = \operatorname {{\mathrm {Ker}}}( L \twoheadrightarrow L/p L \twoheadrightarrow \sigma _{a-2, b+2} )$. It is clear that $pL\subset L_1\subset L$.

Proposition 3.14 The following nonsplit extensions

\begin{align*} & (\sigma_{p-3-a, a+b +2} \ \textbf{---}\ \sigma_{a, b+1}),\\ & (\sigma_{p-1-a, a+b+1} \ \textbf{---}\ \sigma_{a, b+1}),\\ & (\sigma_{a-2, b+2} \ \textbf{---}\ \sigma_{p-1-a, a+b+1}) \end{align*}

occur in $L_1/ p L_1$ as subquotients. Consequently, $L_1/pL_1$ has a cosocle filtration

\[ \sigma_{a-2, b+2} \ \textbf{---}\ ( \sigma_{p-1-a, a+b+1} \oplus \sigma_{p-3-a, a+b+2} ) \ \textbf{---}\ \sigma_{a, b+1} \]

and $L_1$ is the unique (up to homothety) lattice in $\underline {\mathrm {Sym}}^1 E^2 \otimes \Theta (\psi )$ whose reduction has cosocle $\sigma _{a, b+1}$. Moreover, we have

\[ (L_1/ p L_1)_{K_1} = ( \sigma_{p-1-a, a+b+1} \oplus \sigma_{p-3-a, a+b+2} ) \ \textbf{---}\ \sigma_{a, b+1}. \]

Proof. By construction, we have $pL\subset L_1$ and $L_1 / pL = \operatorname {{\mathrm {Ker}}} (L/pL \to \sigma _{a-2, b+2})$. By Proposition 3.11(i), the cosocle filtration of $L_1/pL$ is

\[ (\sigma_{p-1-a,a+b+1}\oplus\sigma_{p-3-a,a+b+2}) \ \textbf{---}\ \sigma_{a,b+1}, \]

thus the nonsplit extensions $(\sigma _{p-3-a, a+b +2} \ \textbf {---}\ \sigma _{a, b+1})$ and $(\sigma _{p-1-a, a+b+1} \ \textbf {---}\ \sigma _{a, b+1})$ occur in $L_1/pL$, hence also in $L_1/pL_1$.

We need to show that the nonsplit extension $(\sigma _{a-2, b+2} \ \textbf {---}\ \sigma _{p-1-a, a+b+1})$ also occurs in $L_1/pL_1$. For this we note that $pL_1 \subset L' \subset L_1$, where $L'$ is defined in (3.7). Consequently, $L' / pL_1$ is a subrepresentation of $L_1 / pL_1$, and it is easy to see that

\[ \operatorname{{\mathrm{JH}}}(L'/pL_1)=\{\sigma_{a-2,b+2},\sigma_{p-1-a,a+b+1},\sigma_{p-3-a,a+b+2}\}. \]

As a quotient of $L' / pL'$, $L' / pL_1$ admits the nonsplit extension $(\sigma _{a-2, b+2} \ \textbf {---}\ \sigma _{p-1-a, a+b+1})$ as a subquotient, see Proposition 3.11(ii). The structure of $(L_1 / pL_1)_{K_1}$ and other statements easily follow.

Let $L_1':=\operatorname {{\mathrm {Ker}}}(L'\twoheadrightarrow L'/pL'\twoheadrightarrow \sigma _{p-3-a,a+b+2})$, where $L'$ is defined in (3.7). Then $pL'\subset L_1'\subset L'$. Alternatively, $L_1'$ is characterized by the following exact sequence:

(3.10)\begin{equation} 0\rightarrow pL\rightarrow L_1'\rightarrow \sigma_{p-1-a,a+b+1}\rightarrow0. \end{equation}

In a similar way to Proposition 3.14, we have the following result.

Proposition 3.15 The nonsplit extensions

\begin{align*} & (\sigma_{p-3-a, a+b +2} \ \textbf{---}\ \sigma_{a, b+1}), \\ & (\sigma_{a, b+1} \ \textbf{---}\ \sigma_{p-1-a, a+b+1} ), \\ & (\sigma_{a-2, b+2} \ \textbf{---}\ \sigma_{p-1-a, a+b+1}) \end{align*}

occur in $L'_1/ p L'_1$. Consequently, $L'_1/pL'_1$ has a cosocle filtration

\[ \sigma_{p-3-a, a+b+2} \ \textbf{---}\ (\sigma_{a-2, b+2}\oplus \sigma_{a, b+1}) \ \textbf{---}\ \sigma_{p-1-a, a+b+1}, \]

and $L_1'$ is the unique (up to homothety) lattice in $\underline {\mathrm {Sym}}^1 E^2 \otimes \Theta (\psi )$ whose reduction has cosocle $\sigma _{p-1-a, a+b+1}$. Moreover, we have

\[ (L'_1/ p L'_1)_{K_1} = (\sigma_{a-2, b+2}\oplus \sigma_{a, b+1} ) \ \textbf{---}\ \sigma_{p-1-a, a+b+1}. \]

Let $L_2 : = \operatorname {{\mathrm {Ker}}}( L \twoheadrightarrow L/p L \twoheadrightarrow (\sigma _{p-3-a, a+b+2} \ \textbf {---}\ \sigma _{a, b+1}) )$. Then $pL\subset L_2\subset L$ and there is a short exact sequence

(3.11)\begin{equation} 0\rightarrow pL\rightarrow L_2\rightarrow (\sigma_{p-1-a,a+b+1}\ \textbf{---}\ \sigma_{a-2,b+2})\rightarrow0. \end{equation}

Proposition 3.16 Assume $2\leq a \leq p-2$. Then the nonsplit extensions

\begin{align*} & (\sigma_{p-3-a, a+b +2} \ \textbf{---}\ \sigma_{a, b+1}),\\ & (\sigma_{a,b+1} \ \textbf{---}\ \sigma_{p-1-a, a+b+1} ),\\ & (\sigma_{p-1-a, a+b+1} \ \textbf{---}\ \sigma_{a-2, b+2}) \end{align*}

occur in $L_2/ p L_2$. Consequently, $L_2/pL_2$ has a cosocle filtration

\[ \sigma_{p-3-a, a+b +2} \ \textbf{---}\ \sigma_{a,b+1} \ \textbf{---}\ \sigma_{p-1-a, a+b+1} \ \textbf{---}\ \sigma_{a-2, b+2}, \]

and $L_2$ is the unique (up to homothety) lattice in $\underline {\mathrm {Sym}}^1 E^2 \otimes \Theta (\psi )$ whose reduction has cosocle $\sigma _{a-2, b+2}$. Moreover, we have

\[ (L_2 / pL_2)_{K_1} = (\sigma_{p-1-a, a+b+1} \ \textbf{---}\ \sigma_{a-2, b+2}). \]

Proof. Since $L_2 / pL_2$ is an extension of $L_2 / pL$ by $pL / pL_2$, it suffices to show that the nonsplit extension $(\sigma _{a, b+1} \ \textbf {---}\ \sigma _{p-1-a, a+b+1})$ occurs in $L_2 / pL_2$.

It follows from (3.10) and (3.11) that $pL\subset L'_1\subset L_2$ and there is a short exact sequence

\[ 0\rightarrow L_1'\rightarrow L_2\rightarrow \sigma_{a-2,b+2}\rightarrow0. \]

This implies that

\[ L_1'/pL_2\cong (L_1'/pL_1')/(\sigma_{a-2,b+2}), \]

and the nonsplit extension $(\sigma _{a, b+1} \ \textbf {---}\ \sigma _{p-1-a, a+b+1})$ occurs in $L'_1 / pL_2$ by Proposition 3.15. Since $L_1'/pL_2$ embeds in $L_2/pL_2$, this nonsplit extension also occurs in $L_2/pL_2$.

The sublattices $L_1$ and $L_2$ of $L$ satisfy the following property.

Proposition 3.17 Assume $3\leq a\leq p-2$. Fix $i\in \{1,2\}$. Then for every $x\in p L$, there exist $r\in \mathbb {N}$, $k_1,\ldots, k_r \in K_1$, $y_1,\ldots, y_r \in L_i$, such that

\[ x= (k_1 -1) y_1 + \cdots + (k_r - 1) y_r. \]

The proof of Proposition 3.17 requires a technique introduced in [Reference Breuil, Herzig, Hu, Morra and SchraenBHHMS23, § 7], so we first recall some notation. Let $\frak {sl}_{2,\mathbb {F}_p}$ be the Lie algebra consisting of trace-zero $2\times 2$ matrices with coefficients in $\mathbb {F}_p$. It is a three-dimensional vector space over $\mathbb {F}_p$ with a basis

\[ e = \bigg(\begin{array}{cc} 0 & 1 \\ 0 & 0\\ \end{array}\bigg),\quad f = \bigg(\begin{array}{cc} 0 & 0 \\ 1 & 0\\ \end{array}\bigg),\quad h = \bigg(\begin{array}{cc} 1 & 0 \\ 0 & -1\\ \end{array}\bigg) \]

subject to the Lie bracket relations

\[ [e,f] = h,\quad [h,e] = 2e,\quad [h,f] = -2f. \]

Let $(V,\rho )$ be a continuous finite-dimensional representation of $K/Z_1$ over $E$. Assume that $V^{\circ }$ is a $K$-stable $\mathcal {O}$-lattice in $V$ such that $K_1$ acts trivially on $V^{\circ } / p V^{\circ }$. Breuil et al. [Reference Breuil, Herzig, Hu, Morra and SchraenBHHMS23, § 7.1] defined a Lie algebra action of $\frak {sl}_{2,\mathbb {F}_p}$ on $V^{\circ }/pV^{\circ }$, which induces an $\mathbb {F}$-linear map

\[ \beta_{V^{\circ}} : \frak{sl}_{2,\mathbb{F}_p} \otimes_{\mathbb{F}_p} (V^{\circ}/pV^{\circ} )\to V^{\circ}/pV^{\circ} \]

sending $x \otimes v$ to $p^{-1} (\rho (\exp (p \widetilde {x})) \widetilde {v} - \widetilde {v}) \pmod {p V^{\circ }}$, where $\widetilde {x} \in \frak {sl}_{2,\mathbb {Z}_p}$ is a trace-zero $2\times 2$ matrix with coefficients in $\mathbb {Z}_p$ lifting $x\in \frak {sl}_{2,\mathbb {F}_p}$, and $\widetilde {v} \in V^{\circ }$ is a lift of $v \in V^{\circ }/pV^{\circ }$. The definition does not depend on the choice of the lifts. Moreover, letting $K$ act on $\frak {sl}_{2,\mathbb {F}_p}$ by conjugation $k\cdot x:=\overline {k}x\overline {k}^{-1}$ (for $k\in K$ and $k\mapsto \overline {k}\in \mathrm {GL}_2(\mathbb {F}_p)$), $\beta _{V^{\circ }}$ is $K$-equivariant. Indeed,

\begin{align*} \beta_{V^{\circ}}(k(x\otimes v))=\beta_{V^{\circ}} (\overline{k}x\overline{k}^{-1}\otimes kv) &= p^{-1} (\rho(\exp(pk\widetilde{x}k^{-1}))k\widetilde{v}-k\widetilde{v}) \pmod{p V^{\circ}}\\ &=p^{-1}(\rho(k\exp(p\widetilde{x}))\widetilde{v}-k\widetilde{v})\pmod{p V^{\circ}}\\ &=k\beta_{V^{\circ}}(x\otimes v). \end{align*}

In the special case $V=\underline {\mathrm {Sym}}^1E^2$ and $V^{\circ }=\underline {\mathrm {Sym}}^1\mathcal {O}^2$, we easily check that the map $\beta _{\underline {\mathrm {Sym}}^1\mathcal {O}^2}:\frak {sl}_{2,\mathbb {F}_p}\otimes \mathrm {Sym}^1\mathbb {F}^2\rightarrow \mathrm {Sym}^2\mathbb {F}^2$ is given by

(3.12)\begin{equation} \beta_{\underline{\mathrm{Sym}}^1\mathcal{O}^2}(\alpha\otimes \ell(X,Y))=\ell(a_{11}X+a_{21}Y,a_{12}X-a_{11}Y) \end{equation}

for $\alpha = \big (\begin{smallmatrix} a_{11} & a_{12} \\ a_{21} & -a_{11} \end{smallmatrix}\big ) \in \frak {sl}_{2,\mathbb {F}_p}$ and $\ell (X,Y) \in \mathrm {Sym}^1 \mathbb {F}^2$.

Proof of Proposition 3.17 We only give the proof for $L_2$ (which will be used in the proof of Proposition 3.19). Take $V=\underline {\mathrm {Sym}}^1E^2\otimes \Theta (\psi )$ and $V^{\circ }=L=\underline {\mathrm {Sym}}^1\mathcal {O}^2\otimes T$ in the above discussion. Since $K_1$ acts trivially on $T$, the map $\beta _{L}:\frak {sl}_{2,\mathbb {F}_p}\otimes \mathrm {Sym}^1\mathbb {F}^2\otimes T/pT\rightarrow \mathrm {Sym}^1\mathbb {F}^2\otimes T/pT$ is given by (cf. [Reference Breuil, Herzig, Hu, Morra and SchraenBHHMS23, Remark 7.1.3])

(3.13)\begin{equation} \beta_{L}=\beta_{\underline{\mathrm{Sym}}^1\mathcal{O}^2}\otimes \mathrm{Id}_{T/pT}. \end{equation}

Let $W_1=(\sigma _{p - 1-a, a+b+1} \ \textbf {---}\ \sigma _{a-2, b+2})$ be the subrepresentation of $L/ pL$ defined in the proof of Proposition 3.11. Then $L_2$ is exactly the preimage of $W_1$ in $L$ under the surjection $L \twoheadrightarrow L/pL$. Taking $W= W_1$ (and $V^\circ = L$) in [Reference Breuil, Herzig, Hu, Morra and SchraenBHHMS23, Lemma 7.1.4], we obtain the following commutative diagram in which both rows are exact.

To prove the proposition, it suffices to check that the dotted map $\varphi$, which is the composite

(3.14)\begin{equation} \varphi: \frak{sl}_{2, \mathbb{F}_p} \otimes W_1 \hookrightarrow \frak{sl}_{2,\mathbb{F}_p} \otimes L / pL {\buildrel {\beta_{L}} \over \longrightarrow} L/ pL, \end{equation}

is surjective, since this implies that the images of $pL$ and $p^2L$ in $(L_2)_{K_1}$ coincide. Recall that $T/pT$ fits into a short exact sequence

\[ 0\rightarrow \sigma_{p-2-a,a+b+1}\rightarrow T/pT\rightarrow \sigma_{a-1,b+1}\rightarrow0 \]

and $W_1\cap (\mathrm {Sym}^1\mathbb {F}^2\otimes \sigma _{p-2-a,a+b+1})=\sigma _{p-1-a,a+b+1}$ (see the proof of Proposition 3.11). Using (3.13), we see that $\varphi$ sends $\frak {sl}_{2,\mathbb {F}_p}\otimes \sigma _{p-1-a,a+b+1}$ to $\mathrm {Sym}^1\mathbb {F}^2\otimes \sigma _{p-2-a,a+b+1}$, so that $\varphi$ induces a $K$-equivariant map

\[ \overline{\varphi}: \frak{sl}_{2,\mathbb{F}_p}\otimes \sigma_{a-2,b+2}\hookrightarrow \frak{sl}_{2,\mathbb{F}_p}\otimes \mathrm{Sym}^1\mathbb{F}^2\otimes\sigma_{a-1,b+1}\rightarrow \mathrm{Sym}^1\mathbb{F}^2\otimes\sigma_{a-1,b+1}, \]

where the second map is given by $\beta _{\underline {\mathrm {Sym}}^1\mathcal {O}^2}\otimes \mathrm {Id}_{\sigma _{a-1,b+1}}$. By Proposition 3.11(i), $\mathrm {cosoc}(L/pL)=\mathrm {Sym}^1\mathbb {F}^2\otimes \sigma _{a-1,b+1}$. Hence, to prove $\varphi$ is surjective it suffices to prove $\overline {\varphi }$ is surjective.

We prove that $\overline {\varphi }$ is surjective by a direct computation (analogous to [Reference Breuil, Herzig, Hu, Morra and SchraenBHHMS23, Lemma 7.2.1]). Fix a nonzero element $v\in (\sigma _{a-1,b+1})^{U(\mathbb {Z}_p)}$. Then $(\sigma _{a-1,b+1})^{U(\mathbb {Z}_p)} = \mathbb {F} v$ and the group $H = \{\big (\begin{smallmatrix} {[a]} & {0}\\ {0} & {[d]}\end{smallmatrix}\big ),\ a,d\in \mathbb {F}_p^{\times }\}$ acts on $v$ by $\chi _{a-1,b+1}$. Recall that there is a natural action of $\frak {sl}_{2,\mathbb {F}_p}$ on $\sigma _{a-1,b+1}$ and the set $\{v,f(v),\ldots, f^{a-1}(v)\}$ forms an $\mathbb {F}$-basis of $\sigma _{a-1,b+1}$, see [Reference PaškūnasPaš04, § 4.2.1]. By construction, we have $e(v)=0$ and $h(v)=(a-1)v$. Consider the following nonzero elements of $\mathrm {Sym}^1\mathbb {F}^2\otimes \sigma _{a-1,b+1}$:

\[ l_1 : = X \otimes v,\quad l_2: = Y\otimes v - \frac{1}{a-1} X \otimes f(v). \]

It is easy to see that $H$ acts on $l_1$ (respectively, $l_2$) by $\chi _{ a, b +1}$ (respectively, $\chi _{ a - 2, b +2}$). Moreover, using the fact $ef(v)=fe(v)+h(v)=(a-1)v$, one checks that $e(l_1)=e(l_2)=0$, so that $l_1$ and $l_2$ are fixed by $U(\mathbb {Z}_p)$. Since $\mathrm {Sym}^1\mathbb {F}^2\otimes \sigma _{a-1,b+1}\cong \sigma _{a,b+1}\oplus \sigma _{a-2,b+2}$, we deduce that $l_1\in (\sigma _{a,b+1})^{U(\mathbb {Z}_p)}$ and $l_2\in (\sigma _{a-2,b+2})^{U(\mathbb {Z}_p)}$ under this decomposition. In particular, $\mathrm {Sym}^1\mathbb {F}^2\otimes \sigma _{a-1,b+1}$ is generated by $l_1$ and $l_2$ as a $K$-representation.

Since $\overline {\varphi }$ is $K$-equivariant, to finish the proof it suffices to prove that both $l_1$ and $l_2$ lie in the image of $\overline {\varphi }$. We let (recall $a\geq 3$)

\[ w_1 := e \otimes l_2,\quad w_2 : = -h\otimes l_2 - \frac{2}{a-2} e\otimes f(l_2) \]

be elements of $\frak {sl}_{2,\mathbb {F}_p} \otimes \sigma _{a-2,b+2}$ and claim that $\overline {\varphi }(w_i) = l_i$, $i=1,2$. Indeed, as $\beta _{\underline {\mathrm {Sym}}^1 \mathcal {O}^2}(e\otimes X)=0$ and $\beta _{\underline {\mathrm {Sym}}^1 \mathcal {O}^2}(e\otimes Y)=X$ by (3.12),

\begin{align*} \overline{\varphi} (w_1) &= \beta_L(e \otimes l_2) = \beta_L\bigg(e\otimes \bigg(Y\otimes v - \frac{1}{a-1} X \otimes f(v)\bigg)\bigg) \\ &= \beta_{\underline{\mathrm{Sym}}^1 \mathcal{O}^2} (e\otimes Y) \otimes v - \frac{1}{a-1} \beta_{\underline{\mathrm{Sym}}^1 \mathcal{O}^2} (e\otimes X) \otimes f(v) = X\otimes v = l_1. \end{align*}

Similarly, $\overline {\varphi }(w_2) = \beta _L(-h\otimes l_2 - ({2}/({a-2})) e\otimes f(l_2))$. We have

\begin{align*} f (l_2) &= f \bigg(Y\otimes v - \frac{1}{a-1} X \otimes f(v)\bigg)\\ &= Y \otimes f(v) - \frac{1}{a-1} (f(X) \otimes f(v) + X \otimes f^2(v))\\ &= \frac{a-2}{a-1}Y\otimes f(v) - \frac{1}{a-1}X \otimes f^2 (v), \end{align*}

hence

\begin{align*} \beta_L( e\otimes f (l_2)) &= \frac{a-2}{a-1}\beta_{\underline{\mathrm{Sym}}^1 \mathcal{O}^2} (e\otimes Y) \otimes f(v) - \frac{1}{a-1} \beta_{\underline{\mathrm{Sym}}^1 \mathcal{O}^2}(e\otimes X) \otimes f^2 (v) \\ &= \frac{a-2}{a-1} X\otimes f(v). \end{align*}

As $h(X)=X$ and $h(Y)=-Y$ by (3.12), we obtain

\begin{align*} \overline{\varphi} (w_2) &=- \beta_{L} \bigg(h \otimes \bigg(Y\otimes v - \frac{1}{a-1}X \otimes f(v)\bigg)\bigg) - \frac{2}{a-2}\cdot \frac{a-2}{a-1} X\otimes f(v) \\ &= Y \otimes v + \frac{1}{a-1}X \otimes f(v) - \frac{2}{a-1} X\otimes f(v) = l_2. \end{align*}

This proves the claim and finishes the proof of the proposition.