1.1 Extensions

In this section we recall the computation of certain $\operatorname {Ext}^{1}$-groups of (locally analytic) generalized Steinberg representations due to Ding (see [Reference DingDin19]). We fix a finite extension ${E}$ of $ {\mathbb {Q}}_p$. If $V$ and $W$ are admissible locally $ {\mathbb {Q}}_p$-analytic ${E}$-representations of $G_ {\mathfrak {p}}$, we write $\operatorname {Ext}^{1}_{\operatorname {an}}(V,W)$ for the group of locally analytic extensions of $V$ by $W$.

Given an algebraic subgroup $H\subseteq G_{F_ {\mathfrak {p}}}$, we denote the group of $F_ {\mathfrak {p}}$-valued points of $H$ also by $H$. We fix a Borel subgroup $B$ of the split group $G_{F_ {\mathfrak {p}}}$ and a maximal split torus $T\subseteq B$ and denote by $\Delta$ the associated basis of simple roots. For a subset $I\subseteq \Delta$ we let $P_I\supseteq B$ be the corresponding parabolic containing $B$.

Suppose $M$ is a smooth representation of $P_I$ over a ring $R$; we define its smooth induction to $G_ {\mathfrak {p}}$ as

\[ i_{P_I}^{\infty}(M)=\{f\colon G_{\mathfrak{p}}\to M \mbox{ locally constant}\mid f(pg)=p.f(g)\ \forall p\in P_I,\ g\in G_{\mathfrak{p}} \}. \]

The generalized $R$-valued (smooth) Steinberg representation associated with $I\subseteq \Delta$ is given by the quotient

\[ v^{\infty}_{P_I}(R)= i^{\infty}_{P_I}(R) \bigg/ \sum_{I\subset J\subset \Delta, I\neq J} i^{\infty}_{P_J}(R). \]

Likewise, if $V$ is a locally $ {\mathbb {Q}}_ {\mathfrak {p}}$-analytic ${E}$-representation of $P_I$, we define its locally analytic induction to $G_ {\mathfrak {p}}$ as the space of functions

\[ {\mathbb{I}}_{P_I}^{\operatorname{an}}(V)=\{f\colon G_{\mathfrak{p}}\to \tau \mbox{ locally } {\mathbb{Q}}_p\mbox{-analytic}\mid f(pg)=p.f(g)\ \forall p\in P_I,\ g\in G_{\mathfrak{p}} \}. \]

We define the locally analytic generalized Steinberg representation with respect to $I$ as the quotient

\[ {\mathbb{V}}^{\operatorname{an}}_{P_I}({E})={\mathbb{I}}_{P_I}^{\operatorname{an}}({E})\bigg/\sum_{I\subset J\subset \Delta, I\neq J} {\mathbb{I}}^{\operatorname{an}}_{P_J}({E}). \]

We put $\operatorname {St}_{G_ {\mathfrak {p}}}^{\infty }(R)=v^{\infty }_B(R)$ and $\operatorname {St}_{G_ {\mathfrak {p}}}^{\operatorname {an}}({E})= {\mathbb {V}}^{\operatorname {an}}_B({E})$. Similarly, replacing locally analyticity with continuity we define the continuous Steinberg representation $\operatorname {St}_{G_ {\mathfrak {p}}}^{\operatorname {ct}}({E})$ and, more generally, $ {\mathbb {V}}^{\operatorname {ct}}_{P_I}({E})$. It is easy to see that $ {\mathbb {V}}^{\operatorname {ct}}_{P_I}({E})$ is the universal unitary completion of both $v^{\infty }_{P_I}({E})$ and $ {\mathbb {V}}^{\operatorname {an}}_{P_I}({E})$.

Let $i\in \Delta$ be a simple root and $\lambda \in \operatorname {Hom}^{\operatorname {ct}}(B,{E})$ a continuous homomorphism. Note that $\lambda$ is automatically locally analytic and is trivial on the unipotent radical of $B$. Thus, it can be identified with a character on $T$. We write $\tau _\lambda$ for the two-dimensional representation of $B$ given by

(1.1)\begin{equation} \tau_\lambda(b)=\begin{pmatrix} 1 & \lambda(b)\\ 0 & 1\end{pmatrix}. \end{equation}

By the exactness of the parabolic induction functor for locally analytic extensions (see [Reference KohlhaaseKoh11, Proposition 5.1 and Remark 5.4]) we have a short exact sequence of the form

\[ 0\longrightarrow {\mathbb{I}}_{B}^{\operatorname{an}}({E})\longrightarrow {\mathbb{I}}_{B}^{\operatorname{an}}(\tau_\lambda)\longrightarrow {\mathbb{I}}_{B}^{\operatorname{an}}({E})\longrightarrow 0. \]

We write

(1.2)\begin{equation} \widehat{\mathcal{E}}^{\operatorname{an}}(\lambda)\in \operatorname{Ext}^{1}_{\operatorname{an}}({\mathbb{I}}^{\operatorname{an}}_{B}({E}),{\mathbb{I}}_{B}^{\operatorname{an}}({E})) \end{equation}

for the associated extension class. Further, we define the class

(1.3)\begin{equation} \widetilde{\mathcal{E}}^{\operatorname{an}}_{i}(\lambda)\in \operatorname{Ext}^{1}_{\operatorname{an}}(i^{\infty}_{P_i}({E}),{\mathbb{I}}_{B}^{\operatorname{an}}({E})) \end{equation}

as the pullback of this extension along $i^{\infty }_{P_i}({E})\to {\mathbb {I}}^{\operatorname {an}}_{B}({E})$. Finally, taking pushforward along $ {\mathbb {I}}_{B}^{\operatorname {an}}({E})\to \operatorname {St}_{G_ {\mathfrak {p}}}^{\operatorname {an}}({E})$ yields the extension class

(1.4)\begin{equation} \mathcal{E}^{\operatorname{an}}_{i}(\lambda)\in \operatorname{Ext}^{1}_{\operatorname{an}}(i^{\infty}_{P_i}({E}),\operatorname{St}_{G_{\mathfrak{p}}}^{\operatorname{an}}). \end{equation}

By an easy calculation we see that the map

\[ \operatorname{Hom}^{\operatorname{ct}}(B,{E})\longrightarrow \operatorname{Ext}^{1}_{\operatorname{an}}(i^{\infty}_{P_i}({E}),\operatorname{St}_{G_{\mathfrak{p}}}^{\operatorname{an}}({E})),\quad \lambda\longmapsto \mathcal{E}^{\operatorname{an}}_{i}(\lambda) \]

defines a homomorphism. The inclusion $B\hookrightarrow P_i$ induces an injection

\[ \operatorname{Hom}^{\operatorname{ct}}(P_i,{E})\longrightarrow \operatorname{Hom}^{\operatorname{ct}}(B,{E}). \]

The quotient can be identified with the space $\operatorname {Hom}^{\operatorname {ct}}(F_ {\mathfrak {p}}^{\ast },{E})$ via the map

(1.5)\begin{equation} \operatorname{Hom}^{\operatorname{ct}}(F_{\mathfrak{p}}^{\ast},{E})\longrightarrow \operatorname{Hom}^{\operatorname{ct}}(B,{E})/\operatorname{Hom}^{\operatorname{ct}}(P_i,{E}),\quad \lambda\longmapsto \lambda\circ i. \end{equation}

Alternatively, let $i^{\vee }$ denote the coroot associated with $i$. Then, the kernel of the map

(1.6)\begin{equation} \operatorname{Hom}^{\operatorname{ct}}(B,{E})\longrightarrow \operatorname{Hom}^{\operatorname{ct}}(F_{\mathfrak{p}}^{\ast},{E}),\quad \lambda\longmapsto \lambda_{i}=\lambda\circ i^{\vee} \end{equation}

is equal to $\operatorname {Hom}^{\operatorname {ct}}(P_i,{E})$ and, hence, the map induces an isomorphism

\[ \operatorname{Hom}^{\operatorname{ct}}(B,{E})/\operatorname{Hom}^{\operatorname{ct}}(P_i,{E})\longrightarrow \operatorname{Hom}^{\operatorname{ct}}(F_{\mathfrak{p}}^{\ast},{E}), \]

which is the inverse of the isomorphism above up to multiplication by two.

1.2 Flawless lattices

We recall the notion of flawless smooth representations over a ring $R$.

The following is our main example.

1.3 Cohomology of $ {\mathfrak {p}}$-arithmetic groups

Let $A$ be a $ {\mathbb {Q}}_p$-affinoid algebra in the sense of Tate. Given a compact, open subgroup $K^{{\mathfrak {p}}}\subseteq G( {\mathbb {A}} {{}^{ {\mathfrak {p}},\infty }})$, an $A[G_ {\mathfrak {p}}]$-module $V$ and an $A[G(F)]$-module $W$, which is free and of finite rank over $A$, we define $ {\mathcal {C}}_A(K^{ {\mathfrak {p}}},V;W)$ as the space of all $A$-linear maps $\Phi \colon G( {\mathbb {A}} {{}^{ {\mathfrak {p}},\infty }})/K^{{\mathfrak {p}}} \times V\to W$. The $A$-module $ {\mathcal {C}}_{E}(K^{{\mathfrak {p}}},V;W)$ carries a natural $G(F)$-action given by

\[ (\gamma.\Phi)(g,v)=\gamma.(\Phi(\gamma^{-1}g,\gamma^{-1}.v)). \]

Suppose $V$ is a topological $A$-module equipped with a continuous $A$-linear $G_ {\mathfrak {p}}$-action we put $ {\mathcal {C}}^{\operatorname {ct}}_A(K^{{\mathfrak {p}}},V;W)=C(G( {\mathbb {A}} {{}^{ {\mathfrak {p}},\infty }})/K^{ {\mathfrak {p}}},\operatorname {Hom}_{A}^{\operatorname {ct}}(V,W))$. Here $W$ is endowed with its canonical topology as a finitely generated free $A$-module.

Let ${E}$ be a finite extension of $ {\mathbb {Q}}_p$ with ring of integers $\mathcal {O}_{E}$. Suppose that $V$ is a smooth ${E}$-representation of $G_ {\mathfrak {p}}$ that admits a flawless $G_ {\mathfrak {p}}$-stable $\mathcal {O}_{E}$-lattice $M$. As $M$ is finitely generated as an $\mathcal {O}_{E}[G_ {\mathfrak {p}}]$-module, the completion of $V$ with respect to $M$ is the universal unitary completion $V^{\operatorname {un}}$ of $V$. The following automatic continuity statement holds (see [Reference GehrmannGeh21, Proposition 3.11]).

Proposition 1.4 Suppose that $V$ is a smooth ${E}$-representation of $G_ {\mathfrak {p}}$ that admits a flawless $G_ {\mathfrak {p}}$-stable $\mathcal {O}_{E}$-lattice $M$. Then the canonical map

\[ \operatorname{H}^{d}(G(F),{\mathcal{C}}^{\operatorname{ct}}_{E}(K^{{\mathfrak {p}}},V^{\operatorname{un}};{E}(\epsilon)))\longrightarrow \operatorname{H}^{d}(G(F),{\mathcal{C}}_{E}(K^{{\mathfrak {p}}},V;{E}(\epsilon))) \]

is an isomorphism for every character $\epsilon \colon \pi _0(G_\infty )\to \{\pm 1\}$, every compact, open subgroup $K^{{\mathfrak {p}}}\subseteq G( {\mathbb {A}} {{}^{ {\mathfrak {p}},\infty }})$ and every degree $d\geqslant 0$.

This proposition combined with Theorem 1.3 implies the following.

Corollary 1.5 The canonical map

\[ \operatorname{H}^{d}(G(F),{\mathcal{C}}^{\operatorname{ct}}_{E}(K^{{\mathfrak {p}}},\operatorname{St}_{G_{\mathfrak{p}}}^{\operatorname{ct}}({E});{E}(\epsilon))) \longrightarrow \operatorname{H}^{d}(G(F),{\mathcal{C}}_{E}(K^{{\mathfrak {p}}},\operatorname{St}_{G_{\mathfrak{p}}}^{\infty}({E});{E}(\epsilon))) \]

is an isomorphism.

For a compact, open subset $K_ {\mathfrak {p}}\subseteq G_ {\mathfrak {p}}$ and a character $\epsilon \colon \pi _0(G_\infty ) \to \{\pm 1\}$ we put

\[ \operatorname{H}^{d}(X_{K^{{\mathfrak {p}}}\times K_{\mathfrak{p}}},W)^{\epsilon}=\operatorname{H}^{d}(G(F),{\mathcal{C}}_{E}(K^{{\mathfrak {p}}},{E}[G_{\mathfrak{p}}/K_{\mathfrak{p}}];W(\epsilon))). \]

If the level $K^{{\mathfrak {p}}}\times K_ {\mathfrak {p}}$ is neat, this group is naturally isomorphic to (the epsilon component of) the singular cohomology with coefficients in $W$ of the locally symmetric space of level $K^{{\mathfrak {p}}}\times K_ {\mathfrak {p}}$ associated with $G$. More generally, let $V$ be a topological $A$-module, which is locally convex as a $ {\mathbb {Q}}_p$-vector space, with a continuous $A$-linear $K_ {\mathfrak {p}}$-action. We consider $\operatorname {c-ind}_{K_ {\mathfrak {p}}}^{G_ {\mathfrak {p}}}V$ with the locally convex inductive limit topology. This is a continuous $A$-module with a continuous $G_p$-action. We put

\[ \operatorname{H}^{d}(X_{K^{{\mathfrak {p}}}\times K_{\mathfrak{p}}},\operatorname{Hom}_{A}^{\operatorname{ct}}(V,W))^{\epsilon}=\operatorname{H}^{d}(G(F),{\mathcal{C}}_{A}^{\operatorname{ct}}(K^{{\mathfrak {p}}},\operatorname{c-ind}_{K_{\mathfrak{p}}}^{G_{\mathfrak{p}}}V;W(\epsilon))). \]

Again, this space can be identified with the cohomology of the corresponding locally symmetric space with values in the sheaf associated with $\operatorname {Hom}_{A}^{\operatorname {ct}}(V,A)$.

1.4 Automorphic $\mathcal {L}$-invariants

For the remainder of the article we fix a finite extension ${E}$ of $ {\mathbb {Q}}_p$ that contains $ {\mathbb {Q}}_\pi$ and a compact, open subgroup $K^{{\mathfrak {p}}}=\prod _{v\nmid {\mathfrak {p}} \infty }K_v\subseteq G( {\mathbb {A}} {{}^{ {\mathfrak {p}},\infty }})$ such that $(\pi {{}^{ {\mathfrak {p}},\infty }})^{K^{{\mathfrak {p}}}}\neq 0$. We may assume that $K_v$ is hyperspecial for every finite place $v$ such that $\pi _v$ is spherical.

As $v_{P_I}^{\infty }({E})=v_I^{\infty }( {\mathbb {Z}})\otimes {E}$ we have a canonical isomorphism

\[ {\mathcal{C}}_{\mathbb{Z}}(K^{{\mathfrak {p}}},v^{\infty}_{P_I}({\mathbb{Z}});W)\cong {\mathcal{C}}_{{E}}(K^{{\mathfrak {p}}},v^{\infty}_{P_I}({E});W) \]

for any ${E}$-vector space $W$. Hence, we abbreviate this space by $ {\mathcal {C}}(K^{{\mathfrak {p}}},v^{\infty }_{P_I};W)$ (and similarly for $\operatorname {St}_{G_ {\mathfrak {p}}}^{\infty }$ in place of $v^{\infty }_{P_I}$).

Let $I_ {\mathfrak {p}}\subseteq G_ {\mathfrak {p}}$ be an Iwahori subgroup. By Frobenius reciprocity the choice of an Iwahori-fixed vector yields a $G_ {\mathfrak {p}}$-equivariant homomorphism

(1.7)\begin{equation} \operatorname{c-ind}_{I_{\mathfrak{p}}}^{G_{\mathfrak{p}}}{E}\longrightarrow \operatorname{St}_{G_{\mathfrak{p}}}^{\infty}, \end{equation}

which, in turn, induces a Hecke-equivariant map

(1.8)\begin{equation} \operatorname{ev}^{(d)}\colon \operatorname{H}^{d}(G(F),{\mathcal{C}}(K^{{\mathfrak {p}}},\operatorname{St}_{G_{\mathfrak{p}}}^{\infty};{E}(\epsilon)))\longrightarrow \operatorname{H}^{d}(X_{K^{{\mathfrak {p}}}\times I_{\mathfrak{p}}},{E})^{\epsilon}. \end{equation}

Let

\[ {\mathbb{T}}={\mathbb{T}}(K^{{\mathfrak {p}}}\times I_{\mathfrak{p}})_{E}=C_c(K^{{\mathfrak {p}}}\times I_{\mathfrak{p}}\backslash G({\mathbb{A}}^{\infty})/K^{{\mathfrak {p}}}\times I_{\mathfrak{p}},{E}) \]

be the ${E}$-valued Hecke algebra of level $K^{{\mathfrak {p}}}\times I_ {\mathfrak {p}}$. By abuse of notation, we denote the model of $\pi ^{\infty }$ over ${E}$ also by $\pi ^{\infty }$. If $V$ is a $\mathbb { {\mathbb {T}}}$-module, we put

\[ V[\pi]=\sum_f \operatorname{im}(f) \]

where we sum over all $\mathbb {T}$-homomorphisms $f\colon (\pi ^{\infty })^{K^{{\mathfrak {p}}}\times I_ {\mathfrak {p}}}\to V.$ Similarly, we define

\[ {\mathbb{T}}^{{\mathfrak{p}}}={\mathbb{T}}(K^{{\mathfrak {p}}})_{E}=C_c(K^{{\mathfrak {p}}}\backslash G({\mathbb{A}}{{}^{{\mathfrak{p}},\infty}})/K^{{\mathfrak {p}}},{E}) \]

to be the Hecke algebra away from $ {\mathfrak {p}}$ and, given a $ {\mathbb {T}}^{ {\mathfrak {p}}}$-module $V$, we put

\[ V[\pi^{{\mathfrak{p}}}]=\sum_f \operatorname{im}(f) \]

where we sum over all $ {\mathbb {T}}^{ {\mathfrak {p}}}$-homomorphisms $f\colon (\pi {{}^{ {\mathfrak {p}},\infty }})^{K^{{\mathfrak {p}}}}\to V.$

For the proof of the following proposition that crucially relies on the hypothesis (SMO) we refer to [Reference GehrmannGeh21, Proposition 3.6].

Proposition 1.6 The map $\operatorname {ev}^{(d)}$ induces an isomorphism

\[ \operatorname{H}^{d}(G(F),{\mathcal{C}}(K^{{\mathfrak {p}}},\operatorname{St}_{G_{\mathfrak{p}}}^{\infty};{E}(\epsilon)))[\pi^{{\mathfrak{p}}}]\xrightarrow{\cong} \operatorname{H}^{d}(X_{K^{{\mathfrak {p}}}\times I_{\mathfrak{p}}},{E})^{\epsilon}[\pi]. \]

on isotypic components. There exists an integer $m_\pi \geqslant 0$ such that

\[ \dim_{E} \operatorname{H}^{d}(X_{K^{{\mathfrak {p}}}\times I_{\mathfrak{p}}},{E})^{\epsilon}[\pi]= m_\pi\cdot \dim(\pi^{\infty})^{K^{{\mathfrak {p}}}\times I_{\mathfrak{p}}} \cdot\binom{\delta}{d-q}\ \forall d\geqslant 0 \]

for each sign character $\epsilon$.

By Corollary 1.5, we have a canonical isomorphism

\[ \operatorname{H}^{d}(G(F),{\mathcal{C}}(K^{{\mathfrak {p}}},\operatorname{St}_{G_{\mathfrak{p}}}^{\infty};{E}(\epsilon))) \cong \operatorname{H}^{d}(G(F),{\mathcal{C}}_{E}^{\operatorname{ct}}(K^{{\mathfrak {p}}},\operatorname{St}_{G_{\mathfrak{p}}}^{\operatorname{ct}};{E}(\epsilon))). \]

Composing with the homomorphism coming from dualizing the continuous inclusion $\operatorname {St}_{G_ {\mathfrak {p}}}^{\operatorname {an}}\hookrightarrow \operatorname {St}_{G_ {\mathfrak {p}}}^{\operatorname {ct}}$ yields the map

\[ \operatorname{H}^{d}(G(F),{\mathcal{C}}(K^{{\mathfrak {p}}},\operatorname{St}_{G_{\mathfrak{p}}}^{\infty};{E}(\epsilon)))\longrightarrow \operatorname{H}^{d}(G(F),{\mathcal{C}}_{E}^{\operatorname{ct}}(K^{{\mathfrak {p}}},\operatorname{St}_{G_{\mathfrak{p}}}^{\operatorname{an}};{E}(\epsilon))) \]

in cohomology. Thus, for every $i\in \Delta$ we get a well-defined cup-product pairing

\begin{align*} &\operatorname{H}^{d}(G(F),{\mathcal{C}}(K^{{\mathfrak {p}}},\operatorname{St}_{G_{\mathfrak{p}}}^{\infty};{E}(\epsilon))) \times \operatorname{Ext}^{1}_{\operatorname{an}}(v_{P_i}^{\infty}({E}),\operatorname{St}_{G_{\mathfrak{p}}}^{\operatorname{an}}({E}))\\ &\quad \longrightarrow \operatorname{H}^{d+1}(G(F),{\mathcal{C}}(K^{{\mathfrak {p}}},v^{\infty}_{P_i};{E}(\epsilon))), \end{align*}

which commutes with the action of the Hecke algebra $ {\mathbb {T}}^{ {\mathfrak {p}}}$. By Theorem 1.1 we have a canonical isomorphism $\operatorname {Hom}^{\operatorname {ct}}(F_ {\mathfrak {p}}^{\ast },{E})\cong \operatorname {Ext}^{1}_{\operatorname {an}}(v^{\infty }_{P_i}({E}),\operatorname {St}_{G_ {\mathfrak {p}}}^{\operatorname {an}}({E}))$. Hence, taking cup product with the extension $\mathcal {E}^{\operatorname {an}}_{i}(\lambda \circ i)$ associated with a homomorphism $\lambda \in \operatorname {Hom}^{\operatorname {ct}}(F_ {\mathfrak {p}}^{\ast },{E})$ in (1.4) yields a map

\[ c^{(d)}_{i}(\lambda)^{\epsilon}\colon \operatorname{H}^{d}(G(F),{\mathcal{C}}(K^{{\mathfrak {p}}},\operatorname{St}_{G_{\mathfrak{p}}}^{\infty};{E}(\epsilon)))[\pi^{{\mathfrak{p}}}]\longrightarrow \operatorname{H}^{d+1}(G(F),{\mathcal{C}}(K^{{\mathfrak {p}}},v^{\infty}_{P_i};{E}(\epsilon)))[\pi^{{\mathfrak{p}}}] \]

on $\pi {{}^{ {\mathfrak {p}},\infty }}$-isotypic parts.

Definition 1.7 Given a character $\epsilon \colon \pi _0(G_\infty ) \to \{\pm 1\}$, an integer $d\in {\mathbb {Z}}$ with $0\leqslant d\leqslant \delta$ and a root $i\in \Delta$ we define

\[ \mathcal{L}_{i}^{(d)}(\pi,{\mathfrak{p}})^{\epsilon}\subseteq \operatorname{Hom}^{\operatorname{ct}}(F_{\mathfrak{p}}^{\ast},{E}) \]

as the kernel of the map $\lambda \mapsto c^{(q+d)}_{i}(\lambda )^{\epsilon }$.

Alternatively, we can consider cup products with extensions associated with homomorphisms $\lambda \colon B \to {E}$ and define the $\mathcal {L}$-invariant as a subspace of

\[ \operatorname{Hom}^{\operatorname{ct}}(B,{E})/\operatorname{Hom}^{\operatorname{ct}}(P_i,{E}). \]

This subspace is mapped to the $\mathcal {L}$-invariant defined above via the map (1.6) induced by the coroot $i^{\vee }$ associated with $i$.

Proposition 1.8 For all sign characters $\epsilon$, every degree $d\in [0,\delta ]\cap {\mathbb {Z}}$ and every root $i \in \Delta$ the $\mathcal {L}$-invariant

\[ \mathcal{L}_{i}^{(d)}(\pi,{\mathfrak{p}})^{\epsilon}\subseteq \operatorname{Hom}^{\operatorname{ct}}(F_{\mathfrak{p}}^{\ast},{E}) \]

is a subspace of codimension at least one, which does not contain the space of smooth homomorphisms.

Suppose $m_\pi =1$. Then, in the extremal cases $d=0$ and $d=\delta$ the codimension is exactly one.

Remark 1.9 Let $\log _p\colon {E}^{\ast }\to {E}$ denote the branch of the $p$-adic logarithm such that $\log _p(p)=0.$ Let us assume that ${E}$ contains the images of all embeddings $\sigma \colon F_ {\mathfrak {p}}\to \overline { {\mathbb {Q}}_p}$. We put $\log _{p,\sigma }=\log _p \circ \sigma \colon F_ {\mathfrak {p}}^{\ast }\to {E}$. The set

\[ \{{\rm ord}_{\mathfrak{p}}\} \cup \{{\rm log}_{p,\sigma}\mid \sigma\colon F_{\mathfrak{p}} \to {E} \} \]

is a basis of $\operatorname {Hom}^{\operatorname {ct}}(F_ {\mathfrak {p}}^{\ast },{E})$. Suppose $\mathcal {L}\subseteq \operatorname {Hom}^{\operatorname {ct}}(F_ {\mathfrak {p}}^{\ast },{E})$ is a subspace of codimension one that does not contain $\operatorname {ord}_ {\mathfrak {p}}$. Then for each embedding $\sigma$ there exists a unique element $\mathcal {L}^{\sigma }\in {E}$ such that $\log _{p,\sigma }- \mathcal {L}^{\sigma }\operatorname {ord}_ {\mathfrak {p}} \in \mathcal {L}$ and these elements clearly form a basis of $\mathcal {L}$.

2.1 Linear algebra over the dual numbers

Let $R$ be a ring and $R[\varepsilon ]=R[X]/X^{2}$ the ring of dual numbers over $R$. Let $M$ be an $R[\varepsilon ]$-module. We define several maps on dual spaces associated with $M$. First, multiplication with $\varepsilon$ induces the map $\mu \colon M/\varepsilon M\to \varepsilon M\hookrightarrow M$. We denote by

\[ \mu_\varepsilon^{\ast}\colon \operatorname{Hom}_R(M,R)\longrightarrow \operatorname{Hom}_R(M/\varepsilon M,R) \]

its $R$-dual. Second, reducing modulo $\varepsilon$ yields the map

\[ \operatorname{red}\colon \operatorname{Hom}_{R[\varepsilon]}(M,R[\varepsilon])\longrightarrow \operatorname{Hom}_{R}(M/\varepsilon M,R). \]

Finally, the map $\operatorname {add}\colon R[\varepsilon ]\to R, a+b\varepsilon \mapsto a+b$ induces the map

\[ \operatorname{add}_\ast\colon\operatorname{Hom}_{R[\varepsilon]}(M,R[\varepsilon])\longrightarrow \operatorname{Hom}_{R}(M,R). \]

The following easy computation is left to the reader.

2.2 Infinitesimal deformations of principal series

Let $T\subseteq B\subset G_{F_ {\mathfrak {p}}}$ be the maximal torus respectively the Borel subgroup chosen in § 1.1. The map

\[ \tau\colon\operatorname{Hom}^{\operatorname{ct}}(B,{E})\longrightarrow\operatorname{Hom}^{\operatorname{ct}}(B,{E}[\varepsilon]^{\ast}),\quad \lambda \longmapsto \left[x\mapsto 1+\lambda(x)\epsilon\right] \]

defines an injective group homomorphism. Its image is the set of all continuous characters $\chi \colon B\to {E}[\varepsilon ]^{\ast }$ such that $\chi \equiv 1 \bmod \varepsilon$. The underlying ${E}$-representation of $\tau (\lambda )$ is the two-dimensional representation $\tau _\lambda$ defined in (1.1). Thus, we can view $ {\mathbb {I}}_{B}^{\operatorname {an}}(\tau _\lambda )= {\mathbb {I}}_{B}^{\operatorname {an}}(\tau (\lambda ))$ as an ${E}[\varepsilon ]$-representation of $G_ {\mathfrak {p}}$. Reducing modulo $\varepsilon$ induces the map

\[ \operatorname{red}_\lambda^{d,\epsilon}\colon \operatorname{H}^{d}(G(F),{\mathcal{C}}^{\operatorname{ct}}_{{E}[\varepsilon]}(K^{{\mathfrak {p}}},{\mathbb{I}}_{B}^{\operatorname{an}}(\tau_\lambda);{E}[\varepsilon](\epsilon)))\longrightarrow\operatorname{H}^{d}(G(F),{\mathcal{C}}^{\operatorname{ct}}_{E}(K^{{\mathfrak {p}}},{\mathbb{I}}_{B}^{\operatorname{an}}({E});{E}(\epsilon))) \]

in cohomology.

Let $i^{\vee }$ be the coroot associated with $i$. Given a character $\lambda \colon B\to {E}$ we put $\lambda _i=\lambda \circ i^{\vee }$.

Lemma 2.2 Let $\lambda \colon B\to {E}$ be a continuous character. If the isotypic component

\[ \operatorname{H}{}^{q+d}(G(F),{\mathcal{C}}{}^{\operatorname{ct}}_{E}(K^{{\mathfrak {p}}},{\mathbb{I}}_{B}^{\operatorname{an}}({E});{E}(\epsilon)))[\pi^{{\mathfrak{p}}}] \]

is contained in the image of $\operatorname {red}_\lambda ^{q+d,\epsilon }$, then the homomorphism $\lambda _i$ belongs to $\mathcal {L}_{i}^{(d)}(\pi, {\mathfrak {p}})^{\epsilon }.$

Proof. The inclusion $ {\mathbb {I}}_{B}^{\operatorname {an}}({E})\hookrightarrow {\mathbb {I}}_{B}^{\operatorname {an}}(\tau _\lambda )$ induces the map

\[ \operatorname{H}^{q+d}(G(F),{\mathcal{C}}^{\operatorname{ct}}_{{E}}(K^{{\mathfrak {p}}},{\mathbb{I}}_{B}^{\operatorname{an}}(\tau_\lambda);{E}(\epsilon)))\longrightarrow\operatorname{H}^{q+d}(G(F),{\mathcal{C}}^{\operatorname{ct}}_{E}(K^{{\mathfrak {p}}},{\mathbb{I}}_{B}^{\operatorname{an}}({E});{E}(\epsilon))) \]

in cohomology. Its image is the kernel of the cup product with the extension class $\widehat {\mathcal {E}}^{\operatorname {an}}(\lambda )$ defined in (1.2).

Thus, by Lemma 2.1 our assumption implies that the $\pi$-isotypic component $\operatorname {H}^{q+d}(G(F), {\mathcal {C}}^{\operatorname {ct}}_{E}(K^{{\mathfrak {p}}}, {\mathbb {I}}_{B}^{\operatorname {an}}({E});{E}(\epsilon )))[\pi ^{ {\mathfrak {p}}}]$ is contained in the kernel of the cup product with $\widehat {\mathcal {E}}^{\operatorname {an}}(\lambda )$. We have the following commutative diagram.

The claim now follows from the first part of the next lemma and a simple diagram chase.

Proof. The Jordan–Hölder decomposition of $i^{\infty }_{P_J}({E})$ consists of all generalized Steinberg representations $v^{\infty }_{P_I}({E})$ with $J\subseteq I$, each occurring with multiplicity one. Thus, the second claim follows from [Reference GehrmannGeh21, Proposition 3.9].

Via the well-known resolution (see, for example, [Reference OrlikOrl05, Proposition 11])

\[ 0 \to i^{\infty}_{P_\Delta}({E})\to \bigoplus_{\substack{I\subseteq J\subseteq \Delta\\ |\Delta\setminus I|=1}}i^{\infty}_{P_I}({E}) \to \cdots \to \bigoplus_{\substack{J\subseteq I\subseteq \Delta\\ |I\setminus J|=1}}i^{\infty}_{P_I}({E}) \to i^{\infty}_{P_J}({E}) \longrightarrow v^{\infty}_{P_J}({E}) \to 0 \]

one can reduce the first claim to the following statement: Let $\mathcal {E}_{J,I}$ be any smooth extension of $v^{\infty }_{P_J}({E})$ by $v^{\infty }_{P_I}({E})$, where $J\subseteq I \subseteq \Delta$ with $|I|=|J|+1$. Then the map

\[ \operatorname{H}^{d}(G(F),{\mathcal{C}}_{E}(K^{{\mathfrak {p}}},v^{\infty}_{P_J}({E});{E}(\epsilon)))[\pi^{{\mathfrak{p}}}]\longrightarrow\operatorname{H}^{d}(G(F),{\mathcal{C}}_{E}(K^{{\mathfrak {p}}},\mathcal{E}_{J,I};{E}(\epsilon)))[\pi^{{\mathfrak{p}}}] \]

is injective for all $d$. Equivalently, it is enough to show that the cup product

\[ \operatorname{H}^{d}(G(F),{\mathcal{C}}_{E}(K^{{\mathfrak {p}}},v^{\infty}_{P_I}({E});{E}(\epsilon)))[\pi^{{\mathfrak{p}}}]\xrightarrow{\cup \mathcal{E}_{J,I}}\operatorname{H}^{d+1}(G(F),{\mathcal{C}}_{E}(K^{{\mathfrak {p}}},v^{\infty}_{P_J}({E});{E}(\epsilon)))[\pi^{{\mathfrak{p}}}] \]

is the zero map for all $d$. Let $\mathcal {E}_{I,J}$ be the unique up to scalar non-split smooth extension of $v^{\infty }_{P_I}({E})$ by $v^{\infty }_{P_J}({E})$. By [Reference GehrmannGeh21, Corollary 3.8], the cup product

\[ \operatorname{H}^{d+1}(G(F),{\mathcal{C}}_{E}(K^{{\mathfrak {p}}},v^{\infty}_{P_J}({E});{E}(\epsilon)))[\pi^{{\mathfrak{p}}}]\xrightarrow{\cup \mathcal{E}_{I,J}}\operatorname{H}^{d+2}(G(F),{\mathcal{C}}_{E}(K^{{\mathfrak {p}}},v^{\infty}_{P_I}({E});{E}(\epsilon)))[\pi^{{\mathfrak{p}}}] \]

is an isomorphism. Therefore, it is enough to prove that taking the cup product with $\mathcal {E}_{J,I} \cup \mathcal {E}_{I,J}$ induces the zero map on $\operatorname {H}^{\bullet }(G(F), {\mathcal {C}}_{E}(K^{{\mathfrak {p}}},v^{\infty }_{P_I}({E});{E}(\epsilon )))[\pi ^{ {\mathfrak {p}}}]$. This is true because $\mathcal {E}_{J,I} \cup \mathcal {E}_{I,J}$ is a smooth $2$-extension of $v^{\infty }_{P_I}({E})$ by itself and the space of all such extensions is zero by [Reference OrlikOrl05, Theorem 1].

2.3 The automorphic Colmez–Greenberg–Stevens formula

Let $A$ be an ${E}$-affinoid algebra and $\chi \colon B\to A^{\ast }$ a locally analytic character. The parabolic induction $ {\mathbb {I}}_B^{\operatorname {an}}(\chi )$ is naturally an $A[G_ {\mathfrak {p}}]$-module. Given an ideal $ {\mathfrak {m}}\subseteq A$ we let $\chi _ {\mathfrak {m}}\colon B\to (A/ {\mathfrak {m}})^{\ast }$, $x\longmapsto \chi (x) \bmod {\mathfrak {m}}$ denote the reduction of $\chi$ modulo $ {\mathfrak {m}}$. Similar to Proposition 2.2.1 of [Reference HansenHan17], one can prove that

\[ \operatorname{Hom}_{A}^{\operatorname{ct}}({\mathbb{I}}_B^{\operatorname{an}}(\chi),A)\otimes_A A/{\mathfrak{m}} \cong \operatorname{Hom}_{A/{\mathfrak{m}}}^{\operatorname{ct}}({\mathbb{I}}_B^{\operatorname{an}}(\chi_{\mathfrak{m}}),A/{\mathfrak{m}}). \]

Let $ {\mathfrak {m}}\in \operatorname {Spm} A$ be an ${E}$-rational point such that $\chi _ {\mathfrak {m}}={{\mathbb 1}}_{G {\mathfrak {p}}}$. Thus, the reduction map induces the maps

\[ \operatorname{red}_\chi^{d,\epsilon}\colon \operatorname{H}^{d}(G(F),{\mathcal{C}}^{\operatorname{ct}}_{A}(K^{{\mathfrak {p}}},{\mathbb{I}}_{B}^{\operatorname{an}}(\chi);A(\epsilon)))\longrightarrow\operatorname{H}^{d}(G(F),{\mathcal{C}}^{\operatorname{ct}}_{E}(K^{{\mathfrak {p}}},{\mathbb{I}}_{B}^{\operatorname{an}}({E});{E}(\epsilon))) \]

in cohomology.

Let $i^{\vee }$ be the coroot associated with $i$. We put $\chi _i=\chi \circ i^{\vee } \in \operatorname {Hom}(F_{ {\mathfrak {p}}}^{\ast },A)$. Suppose $v\colon \operatorname {Spec} {E}[\varepsilon ]\to \operatorname {Spm} A$ is an element of the tangent space of $\operatorname {Spm} A$ at $ {\mathfrak {m}}$. The pullback $\chi _{i,v}$ of $\chi _i$ along $v$ is of the form $\chi _{i,v}=1+ ({\partial }/{\partial v})\chi _{i}\cdot \varepsilon$ for a unique homomorphism $ ({\partial }/{\partial v})\chi _{i}\colon F_ {\mathfrak {p}}^{\ast }\to {E}.$

Lemma 2.2 immediately implies the following.

3.1 Koszul complexes

In [Reference Kohlhaase and SchraenKS12], Kohlhaase and Schraen construct a resolution of locally analytic principal series representations via a Koszul complex. We recall their construction in a slightly more general setup: instead of restricting to $p$-adic fields as coefficient rings we allow affinoid algebras. The proofs of [Reference Kohlhaase and SchraenKS12] carry over verbatim to this more general framework.

Let us fix some notation: we denote the Borel opposite of $B\subseteq {G_{F_ {\mathfrak {p}}}}$ by $\bar {B}$. Let $\bar {N}\subseteq \bar {B}$ be its unipotent radical. The chosen torus $T\subseteq B\subseteq G_{F_ {\mathfrak {p}}}$ determines an apartment in the Bruhat–Tits building of $G_ {\mathfrak {p}}$. We chose a chamber $C$ of that apartment and a special vertex $v$ of $C$ as in § 3.5. of [Reference CartierCar79]. The stabilizer $G_{ {\mathfrak {p}},0}\subseteq G_ {\mathfrak {p}}$ of $v$ is a maximal compact subgroup of $G_ {\mathfrak {p}}$ and the stabilizer $I_ {\mathfrak {p}}\subseteq G_{ {\mathfrak {p}},0}$ of $C$ is an Iwahori subgroup. Let $\mathfrak {G}_0$ be the Bruhat–Tits group scheme over $\mathcal {O}_ {\mathfrak {p}}$ associated with $G_{ {\mathfrak {p}},0}$. We define

\[ I_{\mathfrak{p}}^{n}=\operatorname{ker}(I_{\mathfrak{p}}\longrightarrow \mathfrak{G}_0(\mathcal{O}_{\mathfrak{p}}/{\mathfrak{p}}^{n})). \]

The open normal subgroups $I_ {\mathfrak {p}}^{n}\subseteq I_ {\mathfrak {p}}$ form a system of neighbourhoods of the identity in $I_ {\mathfrak {p}}$. The subgroup $T_0=T\cap G_{ {\mathfrak {p}},0}$ is maximal compact subgroup of $T$.

Let $X^{\ast }(T)$ (respectively, $X_{\ast }(T)$) denote the group of $F_ {\mathfrak {p}}$-rational characters (respectively, cocharacters) of $T$. The natural pairing

\[ \langle \cdot,\cdot \rangle\colon X^{\ast}(T) \times X_\ast(T)\longrightarrow {\mathbb{Z}} \]

is a perfect pairing. There is a natural isomorphism $T/T_0\cong X_{\ast }(T)$ characterized by

\[ \langle \chi,t\rangle=\operatorname{ord}_{{\mathfrak{p}}}(\chi(t)). \]

We denote by $\Phi ^{+}$ the set of positive roots with respect to $B$ and put

\[ T^{-}=\{t\in T\mid \operatorname{ord}_{{\mathfrak{p}}}(\alpha(t))\leqslant 0\ \forall \alpha\in\Phi^{+}\}. \]

Let $A$ be a $ {\mathbb {Q}}_p$-affinoid algebra. Restricting to $T_0$ gives a bijection between locally analytic characters $\chi \colon B\cap I_ {\mathfrak {p}}\to A^{\ast }$ and locally analytic characters $\chi \colon T_0\to A^{\ast }$. Given such a character $\chi$ we write

\[ {\mathcal{A}}_{\chi}=\{f\colon I_{\mathfrak{p}}\longrightarrow A\ \mbox{locally analytic}\mid f(bk)=\chi(b)f(k)\ \forall b\in B\cap I_{\mathfrak{p}},\ k\in I_p\} \]

for the locally analytic induction of $\chi$ to $I_ {\mathfrak {p}}$. It is naturally an $A[I_ {\mathfrak {p}}]$-module. Restricting a function $f\in {\mathcal {A}}_{\chi }$ to the intersection $I_ {\mathfrak {p}}\cap \bar {N}$ induces an isomorphism of $ {\mathcal {A}}_\chi$ with the space of all locally analytic functions from $I_ {\mathfrak {p}}\cap \bar {N}$ to $A$. There exists a minimal integer $n_\chi \geqslant 1$ such that $\chi$ restricted to $B\cap I_ {\mathfrak {p}}^{n_\chi }$ is rigid analytic. For any $n\geqslant n_\chi$ we define the $A[I_ {\mathfrak {p}}]$-submodule

\[ {\mathcal{A}}_{\chi}^{n}=\{f\in {\mathcal{A}}_{\chi} \mid f \mbox{ is rigid analytic on any coset in } I_{\mathfrak{p}}/I_{\mathfrak{p}}^{n}\}. \]

For later purposes, we define the dual spaces

\[ {\mathcal{D}}^{n}_{\chi}=\operatorname{Hom}_A^{\operatorname{ct}}({\mathcal{A}}^{n}_{\chi}, A). \]

By Frobenius reciprocity we can identify $\operatorname {End}_{A[ G_ {\mathfrak {p}}]}(\operatorname {c-ind}_{I_ {\mathfrak {p}}}^{G_ {\mathfrak {p}}} ( {\mathcal {A}}^{n}_{\chi }))$ with the space of all functions $\Psi \colon G_ {\mathfrak {p}}\to \operatorname {End}_{A}( {\mathcal {A}}_{\chi }^{n})$ such that:

1. (i) $\Psi$ is $I_ {\mathfrak {p}}$-biequivariant, i.e. $\Psi (k_1 g k_2)=k_1 \Psi (g)k_2$ for all $k_1 ,k_2\in I_ {\mathfrak {p}}$, $g\in G_ {\mathfrak {p}}$; and

2. (ii) for every $f\in {\mathcal {A}}_{\chi }^{n}$ the function $G_ {\mathfrak {p}}\to {\mathcal {A}}_{\chi }^{n}$, $g\mapsto \Psi (g)(f)$ is compactly supported.

Let $t$ be an element of $T^{-}$ and $f\in {\mathcal {A}}_{\chi }^{n}$. The function $I_ {\mathfrak {p}}\to A$, $u\mapsto f(tut^{-1})$ defines an element of $ {\mathcal {A}}_{\chi }^{n}$.

For $t\in T^{-}$ we denote by $U_t$ the endomorphism of $\operatorname {c-ind}_{I_ {\mathfrak {p}}}^{G_ {\mathfrak {p}}} ( {\mathcal {A}}^{n}_{\chi })$ corresponding to $\Psi _t$. The following is a straightforward generalization of [Reference Kohlhaase and SchraenKS12, Lemma 2.3].

Now let us fix a character $\chi \colon T\to A^{\ast }$ and let $\chi _0$ be its restriction to $T_0$. Given an open subset $C\subseteq G_ {\mathfrak {p}}$, which is stable under multiplication with $B$ from the left, we denote by

\[ {\mathbb{I}}^{\operatorname{an}}_B(\chi) (C)\subseteq {\mathbb{I}}^{\operatorname{an}}_B(\chi) \]

the subset of all functions with support in $C$. Restricting functions to $I_ {\mathfrak {p}}$ gives an $I_ {\mathfrak {p}}$-equivariant $A$-linear isomorphism

\[ {\mathbb{I}}^{\operatorname{an}}_B(\chi)(BI_{\mathfrak{p}})\xrightarrow{\cong} {\mathcal{A}}_{\chi_0}. \]

Thus, by Frobenius reciprocity its inverse induces a $G_ {\mathfrak {p}}$-equivariant $A$-linear map

(3.1)\begin{equation} \operatorname{aug}_\chi\colon \operatorname{c-ind}_{I_{\mathfrak{p}}}^{G_{\mathfrak{p}}}({\mathcal{A}}^{n}_{\chi_0})\longrightarrow\operatorname{c-ind}_{I_{\mathfrak{p}}}^{G_{\mathfrak{p}}}({\mathcal{A}}_{\chi_0})\longrightarrow {\mathbb{I}}^{\operatorname{an}}_B(\chi) \end{equation}

for any integer $n\geqslant n_{\chi _0}$.

As the group $G_{F_ {\mathfrak {p}}}$ is adjoint, there exist elements $t_i\in T^{-}$, $i\in \Delta$, such that

\[ t_i\in \bigcap_{j\in\Delta\setminus\{i\}}\operatorname{ker}(j) \]

and

\[ \operatorname{ord}_{{\mathfrak{p}}}(i(t_i))=-1. \]

The element $t_i$ is uniquely determined by the value $i(t_i)^{-1}\in F_ {\mathfrak {p}}^{\ast }$, which is a uniformizer. Every element $t\in T$ can uniquely be written as $t=t_0\prod _{i\in \Delta }t_{i}^{n_i}$ with $t\in T_0$ and integers $n_{i}\in {\mathbb {Z}}$. Let us fix a choice of $t_i$, $i\in \Delta$, and put

\begin{align*} y_i=U_{t_i}-\chi(t_i). \end{align*}

By Lemma 3.1 the $G_ {\mathfrak {p}}$-representation $\operatorname {c-ind}_{I_ {\mathfrak {p}}}^{G_ {\mathfrak {p}}}( {\mathcal {A}}^{n}_{\chi _0})$ is a module over the polynomial algebra $A[X_i\mid i\in \Delta ]$, where $X_i$ acts through the operator $T_i$. The Koszul complex of $\operatorname {c-ind}_{I_ {\mathfrak {p}}}^{G_ {\mathfrak {p}}}( {\mathcal {A}}^{n}_{\chi _0})$ with respect to the endomorphisms $(y_i)_{i\in \Delta }$ is the complex $\Lambda ^{\bullet }_A(A^{\Delta }) \otimes \operatorname {c-ind}_{I_ {\mathfrak {p}}}^{G_ {\mathfrak {p}}}( {\mathcal {A}}_{\chi _0})$ with boundary maps

(3.2)\begin{equation} d_k(e_{i_1}\wedge \cdots \wedge e_{i_k}\otimes f)=\sum_{l=1}^{k}(-1)^{l+1} e_{i_1}\wedge \cdots \wedge \widehat{e_{i_l}}\wedge \cdots \wedge e_{i_k}\otimes y_{i_l}(f). \end{equation}

The following is the main technical theorem of Kohlhaase–Schraen (cf. [Reference Kohlhaase and SchraenKS12, Theorem 2.5]) generalized to affinoid coefficient rings.

Theorem 3.4 (Kohlhaase–Schraen)

For any $n\geqslant n_{\chi _0}$ the augmented Koszul complex

\[ \Lambda^{\bullet}_A(A^{\Delta}) \otimes_A \operatorname{c-ind}_{I_{\mathfrak{p}}}^{G_{\mathfrak{p}}}({\mathcal{A}}^{n}_{\chi_0})\longrightarrow {\mathbb{I}}^{\operatorname{an}}_B(\chi)\longrightarrow 0 \]

with boundary maps (3.2) and augmentation map (3.1) is exact.

Example 3.6 Suppose $G_ {\mathfrak {p}}=\operatorname {PGL}_n(F_ {\mathfrak {p}})$, $T$ is the torus of diagonal matrices and $B$ is the Borel subgroup of upper triangular matrices. The simple roots of $T$ with respect to $B$ are given by

\[ i(\operatorname{diag}(x_1,\ldots,x_n))=x_i x_{i+1}^{-1} \]

for $1\leqslant i \leqslant d-1$. For each simple root $1\leqslant i \leqslant d-1$, we might choose for $t_i$ the image of the diagonal matrix

\[ \operatorname{diag}(1,\ldots,1,\pi,\ldots,\pi), \]

where $\pi$ is a uniformizer and exactly the first $i$ entries are equal to one. For the Iwahori subgroup $I_ {\mathfrak {p}}$, we may choose the image in $G_ {\mathfrak {p}}$ of all matrices in $\operatorname {GL}_n(\mathcal {O}_ {\mathfrak {p}})$, which are upper triangular modulo $ {\mathfrak {p}}$. Then $I_ {\mathfrak {p}}^{n}$ consists of all matrices in $I_ {\mathfrak {p}}$ which are congruent to the identity modulo $ {\mathfrak {p}}^{n}$.

There is also a smooth variant of the above result, which is probably well-known. As we could not find a reference in the literature, we give a proof of said variant in the following.

Let $\Omega$ be field of characteristic zero. With the same formula as before, we can define commuting Hecke operators $U_t\in \operatorname {End}_{G_ {\mathfrak {p}}}(\operatorname {c-ind}_{I_ {\mathfrak {p}}}^{G_ {\mathfrak {p}}}({E}))$ for $t\in T^{-}$. Let ${{\mathbb 1}}\colon T_0\to \Omega ^{\times }$ denote the constant character on $T_0$. In case $\Omega ={E}$ we can identify ${E}$ with the subspace of constant functions in $ {\mathcal {A}}^{n}_{{{\mathbb 1}}}$ and the induced embedding

(3.3)\begin{equation} \operatorname{c-ind}_{I_{\mathfrak{p}}}^{G_{\mathfrak{p}}}({E})\hookrightarrow \operatorname{c-ind}_{I_{\mathfrak{p}}}^{G_{\mathfrak{p}}}({\mathcal{A}}^{n}_{{{\mathbb 1}}}) \end{equation}

is equivariant with respect to the operators $U_t$, $t\in T^{-}$.

The operators $U_t\in \operatorname {End}_{G_ {\mathfrak {p}}}(\operatorname {c-ind}_{I_ {\mathfrak {p}}}^{G_ {\mathfrak {p}}}(\Omega ))$ are invertible. Moreover, by the Bernstein decomposition the map

\[ \Omega[X_i^{\pm 1} | i\in \Delta]\longrightarrow \operatorname{End}_{G_{\mathfrak{p}}}(\operatorname{c-ind}_{I_{\mathfrak{p}}}^{G_{\mathfrak{p}}}(\Omega)),\ X_i\longmapsto U_{t_i} \]

is injective and $\operatorname {End}_{G_ {\mathfrak {p}}}(\operatorname {c-ind}_{I_ {\mathfrak {p}}}^{G_ {\mathfrak {p}}}(\Omega ))$ is a free module of finite rank (and, therefore, flat) over $\Omega [X_i^{\pm 1} | i\in \Delta ]$ (see, for example, [Reference ReederRee92] for more details). As $\operatorname {c-ind}_{I_ {\mathfrak {p}}}^{G_ {\mathfrak {p}}}(\Omega )$ is a flat module over its endomorphism algebra by a theorem of Borel (see [Reference BorelBor76, Theorem 4.10]), it is thus also flat as a $\Omega [X_i^{\pm 1} | i\in \Delta ]$-module.

Therefore, for any choice of elements $a_i\in \Omega ^{\times }$ the Koszul complex

\[ \Lambda^{\bullet}_\Omega(\Omega^{\Delta}) \otimes_\Omega \operatorname{c-ind}_{I_{\mathfrak{p}}}^{G_{\mathfrak{p}}}(\Omega) \]

associated with the regular sequence $y_i=U_{t_i}-a_i$, $i\in \Delta$, is a resolution of the $G_ {\mathfrak {p}}$-representation

\[ M_{\underline{a}}=\operatorname{coker} \Bigl(\bigoplus_{i \in \Delta} \operatorname{c-ind}_{I_{\mathfrak{p}}}^{G_{\mathfrak{p}}}(\Omega) \xrightarrow{(y_{i})_{i\in \Delta}}\operatorname{c-ind}_{I_{\mathfrak{p}}}^{G_{\mathfrak{p}}}(\Omega)\Bigr). \]

Let $\chi _{\underline {a}}\colon T\to \Omega ^{\ast }$ be the unique smooth unramified character such that $\chi _{\underline {a}}(t_i)=a_i$. As before, we extend $\chi _{\underline {a}}$ to a character of $B$. Let $\phi \in i^{\infty }_B(\chi _{\underline {a}})$ be the unique element such that:

1. (i) $\phi$ is invariant under $I_ {\mathfrak {p}}$;

2. (ii) the support of $\phi$ is $BI_ {\mathfrak {p}}$; and

3. (iii) $\phi (1)=1$.

Note that in case $\chi _{\underline {a}}={{\mathbb 1}}$ is the trivial character the image of $\phi$ under the quotient map $i^{\infty }_B(\Omega )\twoheadrightarrow \operatorname {St}^{\infty }(\Omega )$ is non-zero and, thus, generates the space of Iwahori invariants of the Steinberg representation.

By Frobenius reciprocity $\phi$ induces a $G_ {\mathfrak {p}}$-equivariant homomorphism

(3.4)\begin{equation} \operatorname{c-ind}_{I_{\mathfrak{p}}}^{G_{\mathfrak{p}}}\Omega \longrightarrow i^{\infty}_B(\chi_{\underline{a}}). \end{equation}

One can argue as in the proof of [Reference OllivierOll14, Proposition 4.4] that the map (3.4) induces an isomorphism

\[ M_{\underline{a}}\longrightarrow i^{\infty}_B(\chi_{\underline{a}}). \]

In conclusion, we see that the augmented Koszul complex

(3.5)\begin{equation} \Lambda^{\bullet}_A(A^{\Delta}) \otimes_A \operatorname{c-ind}_{I_{\mathfrak{p}}}^{G_{\mathfrak{p}}}(\Omega)\longrightarrow i^{\infty}_B(\chi_{\underline{a}})\longrightarrow 0 \end{equation}

is exact.

3.2 Overconvergent cohomology

We show that the result from the previous section together with the theory of overconvergent cohomology allows us to construct cohomology classes with values in duals of locally analytic representations.

Before sticking to the case which is most relevant for our applications let us consider the general case: let $A$ be a $ {\mathbb {Q}}_p$-affinoid algebra and $\chi \colon T\to A$ a locally analytic character. Denote by $\chi _0$ its restriction to $T_0$ and fix elements $t_i\in T^{-}$, $i\in \Delta$, as before. For $n\geqslant n_{\chi _0}$ the continuous dual of the augmentation map (3.1) of the previous section yields the map

\[ \operatorname{aug}_\chi^{\ast}\colon\operatorname{H}^{d}(G(F),{\mathcal{C}}^{\operatorname{ct}}_{E}(K^{{\mathfrak {p}}},{\mathbb{I}}^{\operatorname{an}}_{B}(\chi);A(\epsilon)))\longrightarrow \operatorname{H}^{d}(X_{K^{{\mathfrak {p}}}\times I_{\mathfrak{p}}},{\mathcal{D}}^{n}_{\chi_0})^{\epsilon} \]

for every sign character $\epsilon$ and every integer $d\geqslant 0$. We denote the operator induced by $U_{t_i}$, $i\in \Delta$, on the right-hand side also by $U_{t_i}$ and similar for $U_{\tilde {t}}$. Then, Proposition 3.3 implies the following.

Corollary 3.7 We have

\[ \operatorname{im} (\operatorname{aug}_\chi^{\ast})\subseteq \bigcap_{i \in \Delta}\operatorname{ker} (U_{t_i}-\chi(t_i)). \]

For the remainder of the section we study the case of the trivial character ${{\mathbb 1}}={{\mathbb 1}}_{T}\colon T\to {E}^{\ast }$ of $T$ with values in the units of the $p$-adic field ${E}$. We denote its restriction to $T_0$ also by ${{\mathbb 1}}$. Let $n\geqslant 1$ be any integer. The map (3.3) induces a homomorphism

(3.6)\begin{equation} \operatorname{H}^{d}(X_{K^{{\mathfrak {p}}}\times I_{\mathfrak{p}}},{\mathcal{D}}^{n}_{{{\mathbb 1}}})^{\epsilon} \longrightarrow \operatorname{H}^{d}(X_{K^{{\mathfrak {p}}}\times I_{\mathfrak{p}}},{E})^{\epsilon} \end{equation}

in cohomology, that is, equivariant with respect to the commuting actions of the Hecke algebra $ {\mathbb {T}}^{ {\mathfrak {p}}}$ and the operators $U_t$, $t \in T^{-}$.

In the following, we study the finite slope parts of the above cohomology groups (see, for example, [Reference HansenHan17, § 2.3], for definitions and notation).

Composing the dual of the augmentation map

(3.7)\begin{equation} \operatorname{aug}_{{\mathbb 1}}^{\ast}\colon \operatorname{H}^{d}(G(F),{\mathcal{C}}^{\operatorname{ct}}_{E}(K^{{\mathfrak {p}}},{\mathbb{I}}^{\operatorname{an}}_{B}({E});{E}(\epsilon)))\longrightarrow \operatorname{H}^{d}(X_{K^{{\mathfrak {p}}}\times I_{\mathfrak{p}}},{\mathcal{D}}^{n}_{{{\mathbb 1}}})^{\epsilon} \end{equation}

with the map (3.6) yields the map

(3.8)\begin{equation} \operatorname{H}^{d}(G(F),{\mathcal{C}}^{\operatorname{ct}}_{E}(K^{{\mathfrak {p}}},{\mathbb{I}}^{\operatorname{an}}_{B}({E});{E}(\epsilon)))\longrightarrow \operatorname{H}^{d}(X_{K^{{\mathfrak {p}}}\times I_{\mathfrak{p}}},{E})^{\epsilon}. \end{equation}

As the diagram

is commutative up to multiplication with a non-zero constant, which comes from the choice of a Iwahori-fixed vector in (1.7), so is the induced diagram in cohomology as follows.

Proposition 3.9 Let $a_i\in E^{\times }$, $i\in \Delta$, be elements with $p$-adic valuation equal to $0$ and $\chi _{\underline {a}}\colon T\to E^{\times }$ the unique smooth unramified character such that $\chi _{\underline {a}}(t_i)=a_i$. The natural inclusion $i^{\infty }_{B}(\chi _{\underline {a}})\hookrightarrow {\mathbb {I}}^{\operatorname {an}}_{B}(\chi _{\underline {a}})$ induces an isomorphism

\[ \operatorname{H}^{d}(G(F),{\mathcal{C}}^{\operatorname{ct}}_{E}(K^{{\mathfrak {p}}},{\mathbb{I}}^{\operatorname{an}}_{B}(\chi_{\underline{a}});{E}(\epsilon)))\xrightarrow{\cong} \operatorname{H}^{d}(G(F),{\mathcal{C}}_{E}(K^{{\mathfrak {p}}},i^{\infty}_{B}(\chi_{\underline{a}});{E}(\epsilon))) \]

for all $d\geqslant 0$. In particular, the map (3.8) induces an isomorphism

\[ \operatorname{H}^{q}(G(F),{\mathcal{C}}^{\operatorname{ct}}_{E}(K^{{\mathfrak {p}}},{\mathbb{I}}^{\operatorname{an}}_{B}({E});{E}(\epsilon)))[\pi^{{\mathfrak {p}}}]\xrightarrow{\cong} \operatorname{H}^{q}(X_{K^{{\mathfrak {p}}}\times I_{\mathfrak{p}}},{E})^{\epsilon}[\pi] \]

on $\pi$-isotypic components in degree $q$.

Proof. The second claim is a direct consequence of the first claim and Lemma 2.3.

Given an $I_ {\mathfrak {p}}$-representation $M$ we denote by $\operatorname {Coind}_{I_ {\mathfrak {p}}}^{G_ {\mathfrak {p}}}M$ the coinduction of $M$ to $G_ {\mathfrak {p}}$, i.e. the module of all functions $f\colon G_ {\mathfrak {p}}\to M$ such that $f(kg)=kf(g)$ for all $k\in I_ {\mathfrak {p}}$, $g\in G_ {\mathfrak {p}}$. By taking continuous duals the Koszul complex of Theorem 3.4 with $y_i=U_{t_i}-a_i$ yields a quasi-isomorphism

\[ \operatorname{Hom}_E^{\operatorname{ct}}({\mathbb{I}}^{\operatorname{an}}_{B}(\chi_{\underline{a}}),E) \xrightarrow{\cong} \Lambda^{\bullet}_{E}({E}^{\Delta}) \otimes_{E} \operatorname{Coind}_{I_{\mathfrak{p}}}^{G_{\mathfrak{p}}}({\mathcal{D}}^{n}_{{{\mathbb 1}}}) \]

of complexes. Note that taking continuous duals in this case is exact by the Hahn–Banach theorem. Similarly by the smooth Koszul resolution (3.5) we have a quasi-isomorphism

\[ \operatorname{Hom}_E(i^{\infty}_{B}(\chi_{\underline{a}}),E) \xrightarrow{\cong} \Lambda^{\bullet}_{E}({E}^{\Delta}) \otimes_{E} \operatorname{Coind}_{I_{\mathfrak{p}}}^{G_{\mathfrak{p}}}({E}) \]

and the canonical diagram

is commutative.

From the associated spectral sequences for double complex, we deduce that it is enough to prove that the map of complexes

(3.9)\begin{equation} \Lambda^{\bullet}_{E}({E}^{\Delta})\otimes_{E} \operatorname{H}^{d}(X_{K^{{\mathfrak {p}}}\times I_{\mathfrak{p}}},{\mathcal{D}}^{n}_{{{\mathbb 1}}}) \longrightarrow \Lambda^{\bullet}_{E}({E}^{\Delta})\otimes_{E} \operatorname{H}^{d}(X_{K^{{\mathfrak {p}}}\times I_{\mathfrak{p}}},{E}) \end{equation}

is a quasi-isomorphism for all $d\geqslant 0.$ The modules on the left-hand side admit a slope decomposition for $U_{\tilde {t}}$ by Theorem 3.8(i), whereas the modules on the right-hand side admit a slope decomposition because they are finite-dimensional ${E}$-vector spaces.

As the operators $y_i$ commute with $U_{\tilde {t}}$ they respect the slope decomposition on both sides (see [Reference UrbanUrb11, Lemma 2.3.2]). On the slope less than or equal to zero part the map (3.9) is even an isomorphism of complexes by Theorem 3.8(ii).

Thus, we are reduced to proving that

\[ \Lambda^{\bullet}_{E}({E}^{\Delta})\otimes_{E} \operatorname{H}^{d}(X_{K^{{\mathfrak {p}}}\times I_{\mathfrak{p}}},{\mathcal{D}}^{n}_{{{\mathbb 1}}})^{>h} \longrightarrow \Lambda^{\bullet}_{E}({E}^{\Delta})\otimes_{E} \operatorname{H}^{d}(X_{K^{{\mathfrak {p}}}\times I_{\mathfrak{p}}},{E})^{>h} \]

is a quasi-isomorphism for all $d\geqslant 0$.

In fact, both sides are acyclic: standard properties of Koszul complexes imply that the operators $y_i$ act as multiplication by zero on the cohomology of the complexes. By the definition of a slope decomposition the operator $U_{\tilde {t}}-\prod _{i\in \Delta } a_i$ acts via isomorphisms on both complexes and, thus, on their cohomology. However, because $U_{\tilde {t}}-\prod _{i\in \Delta } a_i$ lies in the ideal generated by the operators $y_i$, it also acts via multiplication by zero, which proves the claim.

3.3 Overconvergent families

Let $ {\mathcal {W}}$ be the weight space of $T$, i.e. it is the rigid space over ${E}$ such that for every ${E}$-affinoid algebra $A$ its $A$-points are given by

\[ {\mathcal{W}}(A)=\operatorname{Hom}^{\operatorname{ct}}(T,A^{\ast}). \]

It is smaller than the usual weight space as we only consider characters of the torus in $G_{F_ {\mathfrak {p}}}$. For any open affinoid $\mathcal {U} \subset {\mathcal {W}}$ and we let $\chi _{\mathcal {U}}$ be the corresponding universal weight. As in § 3.1 we define the space of $n$-analytic functions and distributions $ {\mathcal {A}}_{\chi _{\mathcal {U}}}^{n}$ and $ {\mathcal {D}}^{n}_{\chi _{\mathcal {U}}}$ for $n\gg 0$.

We suppose now that the group $G_\infty$ has discrete series or, equivalently, that $\delta =0$ (see [Reference KnappKna01, Theorem 12.20] for the equivalence of these two properties). Thus, by Proposition 1.6, the representation $\pi ^{\infty }$ appears only in the middle degree cohomology $\operatorname {H}^{q}(X_{K^{ {\mathfrak {p}}}\times I_{ {\mathfrak {p}}}}, {E})$. Hence, we put $\mathcal {L}_{i}(\pi, {\mathfrak {p}})^{\epsilon }=\mathcal {L}_{i}^{(q)}(\pi, {\mathfrak {p}})^{\epsilon }$.

Let $ {\mathbb {T}}_{\operatorname {sph}}^{ {\mathfrak {p}}}\subseteq {\mathbb {T}}^{ {\mathfrak {p}}}$ be the spherical Hecke algebra of level $K^{ {\mathfrak {p}}}\times I_{ {\mathfrak {p}}}$ over ${E}$, i.e. the commutative Hecke algebra generated by all Hecke operators at finite places $v$ such that $K_v$ is hyperspecial. We define $ {\mathbb {T}}_{\operatorname {sph}}$ to be the commutative algebra generated by $ {\mathbb {T}}_{\operatorname {sph}}$ and all $U_t$-operators for $t\in T^{-}$. Let $ {\mathfrak {m}}_\pi \subseteq {\mathbb {T}}_{\operatorname {sph}}$ (respectively, $ {\mathfrak {m}}_\pi ^{ {\mathfrak {p}}}\subseteq {\mathbb {T}}_{\operatorname {sph}}^{ {\mathfrak {p}}}$) the maximal ideal associated with $\pi$. We assume the following weak non-Eisenstein assumption on the maximal ideal $ {\mathfrak {m}}_\pi$ throughout this section.

Hypothesis (NE)

We have

\begin{align*} \operatorname{H}^{d}(X_{K^{{\mathfrak{p}}}\times I_{{\mathfrak{p}}}}, {E})_{{\mathfrak{m}}_\pi}=0 \end{align*}

unless $d=q$.

Let $\mathcal {U}$ be an open affinoid of $ {\mathcal {W}}$ and define $\mathcal {O}_{\mathcal {U},{{\mathbb 1}}}$ to be the rigid localization of $\mathcal {O}_{ {\mathcal {W}}}(\mathcal {U})$ at the weight ${{\mathbb 1}}$ (which is the cohomological weight of $\pi$), i.e. 

\[ \mathcal{O}_{\mathcal{U},{{\mathbb 1}}}= \varinjlim_{{{\mathbb 1}} \in \mathcal{U}' \subset \mathcal{U}} \mathcal{O}_{{\mathcal{W}}}(\mathcal{U}'). \]

The following theorem is instrumental in calculating $\mathcal {L}$-invariants.

Theorem 3.12 After localization at the ideal $ {\mathfrak {m}}_{\pi }$ of $ {\mathbb {T}}_{\operatorname {sph}}$ and restricting to a small enough open affinoid $\mathcal {U}$ containing ${{\mathbb 1}}$, the canonical reduction map

\[ \operatorname{H}^{q} (X_{K^{{\mathfrak{p}}}\times I_{{\mathfrak{p}}}}, {\mathcal{D}}^{n}_{\chi_{\mathcal{U}}} )^{\epsilon}_{{\mathfrak{m}}_{\pi}}\rightarrow \operatorname{H}^{q}(X_{K^{{\mathfrak{p}}}\times I_{{\mathfrak{p}}}}, {E})^{\epsilon}_{{\mathfrak{m}}_{\pi}} \]

is surjective. Moreover, $\operatorname {H}^{q} (X_{K^{ {\mathfrak {p}}}\times I_{ {\mathfrak {p}}}}, {\mathcal {D}}^{n}_{\chi _{\mathcal {U}}} )^{\epsilon }_{ {\mathfrak {m}}_\pi }$ is a free $\mathcal {O}_{\mathcal {U},{{\mathbb 1}}}$-module of rank equal to the dimension of $\operatorname {H}^{q}(X_{K^{ {\mathfrak {p}}}\times I_{ {\mathfrak {p}}}}, {E})^{\epsilon }_{ {\mathfrak {m}}_{\pi }}$.

In addition, we have $\operatorname {H}^{d} (X_{K^{ {\mathfrak {p}}}\times I_{ {\mathfrak {p}}}}, {\mathcal {D}}^{n}_{\chi _{\mathcal {U}}} )^{\epsilon }_{ {\mathfrak {m}}_{\pi }}=0$ for all $d \neq q$.

Theorem 3.12 immediately implies the following.

Let $t_i\in T^{-}$, $i\in \Delta$, be a choice of elements as in § 3.1. Suppose $ {\mathfrak {m}}_\pi$ is $ {\mathfrak {p}}$-étale (with respect to $\epsilon$) with associated eigenvalues $\alpha _{U_{t_i}}\in \mathcal {O}_{\mathcal {U},{{\mathbb 1}}}^{\ast }$, $i \in \Delta$. By possibly shrinking $\mathcal {U}$, we may assume that $\alpha _{U_{t_i}}\in \mathcal {O}_{\mathcal {U}}^{\ast }$. Every element $t\in T$ can be written uniquely as a product $t=t_0 \prod _{i\in \Delta } t_i^{n_i}$ with $t_0 \in T_0$, $n_i\in {\mathbb {Z}}$. Hence, there exists a unique character $\chi _\alpha \colon T\to \mathcal {O}_{\mathcal {U}}^{\ast }$ such that

\[ \left.\chi_{\alpha}\right|_{T_0}=\chi_{U} \]

and

\begin{align*} \chi_\alpha(t_i)=\alpha_{U_{t_i}}. \end{align*}

Theorem 3.16 Suppose that $ {\mathfrak {m}}_\pi$ is $ {\mathfrak {p}}$-étale (with respect to $\epsilon$) with associated character $\chi _\alpha \colon T \to \mathcal {O}_{\mathcal {U}}^{\ast }$. Then for every tangent vector $v$ of $\mathcal {U}$ at ${{\mathbb 1}}$ we have:

\[ \frac{\partial}{\partial v}\chi_{\alpha,i}\in \mathcal{L}_{i}(\pi,{\mathfrak{p}})^{\epsilon} \]

where $\chi _{\alpha,i}$ denotes the composition $\chi _{\alpha }\circ i^{\vee }.$ Moreover, the subspace $\mathcal {L}_{i}(\pi, {\mathfrak {p}})^{\epsilon }\subseteq \operatorname {Hom}^{\operatorname {ct}}(F_ {\mathfrak {p}}^{\ast },{E})$ has codimension one.

Proof. Let $ {\mathfrak {m}}_{{\mathbb 1}}\subseteq \mathcal {O}_{\mathcal {U}}$ be the maximal ideal corresponding to the trivial character. We put $A=\mathcal {O}_{\mathcal {U}}/ {\mathfrak {m}}_{{{\mathbb 1}}}^{2}$ and $\chi _0 = \chi _{\mathcal {U}} \bmod {\mathfrak {m}}_{{{\mathbb 1}}}^{2}$. Let us write $\widetilde {\chi }_\alpha = \chi _\alpha \bmod {\mathfrak {m}}_{{{\mathbb 1}}}^{2}\colon T \to A$. Using Theorem 3.12 and arguments with Koszul complexes as in the proof of Proposition 3.9 one shows that the image of the map

\[ \operatorname{H}^{q}(G(F),{\mathcal{C}}^{\operatorname{ct}}_{A}(K^{{\mathfrak {p}}},{\mathbb{I}}_{B}^{\operatorname{an}}(\widetilde{\chi}_\alpha);A(\epsilon)))_{{\mathfrak{m}}_{\pi}^{{\mathfrak{p}}}}\longrightarrow \operatorname{H}^{q} (X_{K^{{\mathfrak{p}}}\times I_{{\mathfrak{p}}}}, E )^{\epsilon}_{{\mathfrak{m}}_{\pi}^{{\mathfrak{p}}}} \]

is the intersection of the kernels of the homomorphisms $U_{t_i}-1$. Therefore, the first claim follows from Proposition 2.4.

We may assume that ${E}$ is large enough. We explain how to choose appropriate tangent vectors so that $\mathcal {L}_{i}(\pi, {\mathfrak {p}})^{\epsilon }$ contains elements of the form $\log _{p,\sigma }-\mathcal {L}^{\sigma }\operatorname {ord}_ {\mathfrak {p}}$ for every embedding $\sigma \colon F_ {\mathfrak {p}} \to {E}$. This will show that the $\mathcal {L}$-invariant has codimension at most one.

Let ${E} \langle k_\sigma \rangle$ be the ring of power series in the $k_\sigma$ and let $\mathcal {O}_{ {\mathfrak {p}},1}^{\ast }$ denote the free part of $\mathcal {O}_{ {\mathfrak {p}}}^{\ast }$. By smoothness of the weight space, we can embed $\mathcal {O}_{\mathcal {U}}^{\ast }$ in ${E} \langle k_\sigma \rangle$ and the structural morphism of ${E} [\kern-1pt[ \mathcal {O}_{ {\mathfrak {p}},1}^{\ast } ]\kern-1pt] $ into $\mathcal {O}_{\mathcal {U}}^{\ast }$ can be described via the map

\[ \mathcal{O}_{\mathfrak{p}}^{\ast} \ni \langle u \rangle\mapsto \prod_{\sigma } \sigma(u)^{k_{\sigma}}. \]

We can, hence, identify $\chi _{\alpha,i}$ with a map from $F_{ {\mathfrak {p}}}^{\ast }$ to $\mathcal {O}_{\mathcal {U}}^{\ast }$. For all $n \in \mathbb {Z}$ and $u \in \mathcal {O}_{ {\mathfrak {p}},1}^{\ast }$ we have

\[ \chi_{\alpha,i}(\pi^{n} u)=\alpha_{U_{t_i}}^{n} \prod_{\sigma } \sigma(u)^{k_{\sigma}}. \]

Note that

\[ \frac{{d}}{{d}k_{\sigma'}} (\sigma(u)^{k_{\sigma}} )=\delta_{\sigma,\sigma'} \log_p(\sigma(u))\sigma(u)^{k_{\sigma}}, \]

where

\[ \delta_{\sigma,\sigma'} = \begin{cases} 1 & \mbox{if}\ \sigma=\sigma' \\ 0 & \mbox{otherwise.}\end{cases} \]

Take for $v$ the direction where only $k_{\sigma }$ varies, derive $\chi _{\alpha,i}$ along $v$, and evaluate at $k_{\sigma '}=0$ for all $\sigma '$ (corresponding to the weight ${{\mathbb 1}}$) to obtain

\[ \frac{\partial}{\partial v}\chi_{\alpha,i}(\pi^{n} u)= \frac{{d}}{ {d}k_{\sigma}}\alpha_{U_{t_i}}({{\mathbb 1}})\operatorname{ord}_{\mathfrak{p}}(\pi^{n})+\alpha_{U_{t_i}}({{\mathbb 1}})\log_p(\sigma(u)). \]

By the Steinberg hypothesis, $\alpha _{U_{t_i}}({{\mathbb 1}})$ is not vanishing and we are done.

By Proposition 1.8, the $\mathcal {L}$-invariant has codimension at least one and, hence, the second claim follows.

4.1 Hilbert modular forms

We want to study the case of inner forms of $\operatorname {PGL}_2$, which are split at $ {\mathfrak {p}}$, over a totally real number field $F$ in detail. As there is only one simple root we drop it from the notation. If $G$ is equal to $\operatorname {PGL}_2$, one can attach $2^{[F: {\mathbb {Q}}]}$ a priori different $\mathcal {L}$-invariants $\mathcal {L}(\pi,p)^{\epsilon }$ to $\pi$; in this case, a conjecture of Spieß (cf. [Reference SpießSpi14, Conjecture 6.4]) states that the definition does not depend on $\epsilon$, i.e.

\[ \mathcal{L}(\pi,{\mathfrak{p}})^{\epsilon}=\mathcal{L}(\pi,{\mathfrak{p}})^{\epsilon'} \]

for all choices of sign characters $\epsilon$ and $\epsilon '$. In the same paper, Remark 6.6(b), Spieß also states that ‘an interesting and difficult problem’ is to show that the $\mathcal {L}$-invariant is invariant under Jacquet–Langlands transfers. In this section, thanks to Theorem 3.16, we settle both Spieß’ conjecture and his question.

Moreover, let $\rho _{\pi }\colon \operatorname {Gal}(\overline { {\mathbb {Q}}}/F)\to \operatorname {GL}_2({E})$ be the Galois representation associated with $\pi$ (or rather associated with the Jacquet–Langlands transfer $\operatorname {JL}(\pi )$ of $\pi$ to $\operatorname {PGL}_2$), which exists by work of Taylor (see [Reference TaylorTay89]). As $\pi$ is $ {\mathfrak {p}}$-ordinary, by Theorem 2 of [Reference WilesWil88] we know that the restriction $\rho _{\pi, {\mathfrak {p}}}\colon \operatorname {Gal}(\overline { {\mathbb {Q}}_p}/F_ {\mathfrak {p}})$ of $\rho _{\pi }$ to a decomposition group at $ {\mathfrak {p}}$ is ordinary. Moreover, because $\pi _ {\mathfrak {p}}$ is Steinberg, a result of Saito (see [Reference SaitoSai09]) implies that this restriction is a non-split extension of the trivial character by the cyclotomic character. Therefore, it gives a class $\langle \rho _{\pi, {\mathfrak {p}}}\rangle \in \operatorname {H}^{1}(\operatorname {Gal}(\overline { {\mathbb {Q}}_p}/F_ {\mathfrak {p}}),{E}(1))$, where ${E}(1)$ denotes the Tate twist of ${E}$. The Fontaine–Mazur $\mathcal {L}$-invariant $\mathcal {L}^{\operatorname {FM}}(\rho _{\pi, {\mathfrak {p}}})$ of $\rho _{\pi, {\mathfrak {p}}}$ is the orthogonal complement of $\langle \rho _{\pi, {\mathfrak {p}}}\rangle$ with respect to the local Tate pairing

\[ \operatorname{H}^{1}(\operatorname{Gal}(\overline{{\mathbb{Q}}_p}/F_{\mathfrak{p}}),{E}(1)) \times \operatorname{H}^{1}(\operatorname{Gal}(\overline{{\mathbb{Q}}_p}/F_{\mathfrak{p}}),{E}) \longrightarrow {E}. \]

By local class field theory we have a canonical isomorphism

\[ \operatorname{H}^{1}(\operatorname{Gal}(\overline{{\mathbb{Q}}_p}/F_{\mathfrak{p}}),{E})\cong \operatorname{Hom}^{\operatorname{ct}}(F_{\mathfrak{p}}^{\ast},{E}) \]

and, thus, we consider $\mathcal {L}^{\operatorname {FM}}(\rho _{\pi, {\mathfrak {p}}})$ as a codimension-one subspace of $\operatorname {Hom}^{\operatorname {ct}}(F_ {\mathfrak {p}}^{\ast },{E})$. We show that automorphic $\mathcal {L}$-invariants equal the Fontaine–Mazur $\mathcal {L}$-invariant of the associated Galois representation.

4.2 Unitary groups

Let $\tilde {F}$ be a CM field with totally real subfield $F$. We assume that the prime $ {\mathfrak {p}}$ of $F$ is split in $\tilde {F}$. Let $U$ be the unitary group attached to a positive definite hermitian space over $\tilde {F}$ and $G$ the associated adjoint group. By construction, $G_ {\mathfrak {p}}$ is isomorphic to $\operatorname {PGL}_n(F_ {\mathfrak {p}})$. We can identify the simple roots with respect to the upper triangular Borel with the set $\{1,\ldots, n-1\}$. Let $\pi$ be an automorphic representation of $G$ such that $\pi _\infty = {\mathbb {C}}$ and $\pi _ {\mathfrak {p}}$ is the Steinberg representation of $\operatorname {PGL}_n(F_ {\mathfrak {p}})$. In this case the only sign character is the trivial character and, therefore, we drop it from the notation.

By Shin's appendix to [Reference GoldringGol14] the base change $\operatorname {BC}(\pi )$ of $\pi$ to $\operatorname {PGL}_n$ over $\tilde {F}$ exists and it is Steinberg at both primes of $\tilde {F}$ lying above $ {\mathfrak {p}}$. By the work of many people (see Theorem 2.1.1 of [Reference Barnet-Lamb, Gee, Geraghty and TaylorBGGT14] for a detailed discussion) we can attach a $p$-adic Galois representation

\[ \rho_{\pi}=\rho_{\operatorname{BC}(\pi)}\colon \operatorname{Gal}(\overline{{\mathbb{Q}}}/\tilde{F})\longrightarrow \operatorname{GL}_n({E}) \]

to $\operatorname {BC}(\pi )$. As we consider the trivial coefficient system the Steinberg representation is ordinary (cf. [Reference GeraghtyGer19, Lemma 5.6]). Therefore, as shown in [Reference ThorneTho15, Theorem 2.4], one deduces from the local–global compatibility theorem of Caraiani (see [Reference CaraianiCar14]) that the restriction $\rho _{\pi, {\mathfrak {p}}}$ of $\rho _{\pi }$ to the decomposition group of a prime above $ {\mathfrak {p}}$ can be brought in the following form: it is upper triangular and the $i$th diagonal entry is the $(1-i)$-power of the cyclotomic character.

Therefore, we have $n-1$ canonical two-dimensional subquotients $\rho _{\pi, {\mathfrak {p}},i}$ which are extensions of ${E}(-n+i)$ by ${E}(-n+i+1)$. We can consider the associated Fontaine–Mazur $\mathcal {L}$-invariant $\mathcal {L}^{\operatorname {FM}}_{i}(\rho _{\pi, {\mathfrak {p}}})=\mathcal {L}^{\operatorname {FM}}(\rho _{\pi,\pi,i}(n-i))$. Recall that the Weil–Deligne representation associated with the Steinberg has always maximal monodromy so the corresponding $(\varphi,\Gamma )$-module is non-critical split à la Ding, see § 3.1 of [Reference DingDin19]. Replacing the local–global compatibility results of Saito by those of Caraiani (see [Reference CaraianiCar14]), we can apply Theorem 3.4 of [Reference DingDin19] to the $(\varphi,\Gamma )$-module associated with the Galois representation $\rho _{\pi }$ and then the same proof as that of Theorem 4.1 yields the following result.

Theorem 4.3 Let $F$ be a totally real number field and $G$ the adjoint of a unitary group over $F$ compact at infinity and split at $ {\mathfrak {p}}$. Let $\pi$ be an automorphic representation of $G$ such that $\pi _\infty = {\mathbb {C}}$ and $\pi _{ {\mathfrak {p}}}$ is Steinberg. Suppose $\pi$ satisfies (SMO), that we can choose the tame level $K^{ {\mathfrak {p}}}$ such that $ {\mathfrak {m}}_{\pi }$ is $ {\mathfrak {p}}$-étale and that the Galois representation $\rho _{\pi }$ attached to $\pi$ is irreducible. Then we have

\[ \mathcal{L}^{\operatorname{FM}}_{i}(\rho_{\pi}) = \mathcal{L}_{i}(\pi,{\mathfrak{p}}) \]

for every $i=1,\ldots,n-1$.

Remark 4.4 Something can be said also when $F =\mathbb {Q}$ and $G=\operatorname {Sp}_{2g}$. In this case the Galois representations have been constructed by Scholze [Reference ScholzeSch15] but local–global compatibility is not known. Still, if one supposes that the $2g+1$-dimensional Galois representation $\operatorname {Std}(\rho _{\pi })$ associated with $\pi$ is semistable with maximal monodromy, then the Greenberg–Benois $\mathcal {L}(\operatorname {Std}(\rho _{\pi }))$-invariant has been calculated in [Reference RossoRos15, Theorem 1.3].

In this case the root system of $\operatorname {Sp}_{2g}$ can be identified with the set $\{1,\ldots,g\}$ via the identification with the root system of $\operatorname {GL}_g$, which is embedded in $\operatorname {Sp}_{2g}$ by

\[ \operatorname{GL}_g\longrightarrow \operatorname{Sp}_{2g},\quad A\longmapsto \biggl(\!\!\!\!\begin{array}{cc} \phantom{e}{}^{\rm t} A^{-1} & 0\\ 0 & A \end{array}\!\!\biggr), \]

and we see that the automorphic $\mathcal {L}$-invariant $\mathcal {L}_{1}(\pi,p)$ coincides with that of [Reference RossoRos15].

The same calculation in § 4.2 of [Reference RossoRos15] gives us that $\mathcal {L}_{i}(\pi,p)$ is the Greenberg–Benois $\mathcal {L}$-invariant for $\operatorname {Std}(\rho _{\pi })(i-1)$. (This case has not been treated in [Reference RossoRos15] as the $L$-values are not Deligne-critical, but Benois’ definition applies also in this case, see formula (96) in [Reference BenoisBen21].)

If $F_ {\mathfrak {p}} \neq {\mathbb {Q}}_p$, then the comparison is more subtle, as there is only just one Greenberg–Benois $\mathcal {L}$-invariant per $p$-adic place, and from the Galois side one needs to consider Galois invariant characters of $F^{*}_ {\mathfrak {p}}$.