# Continuous group cohomology with coefficients in locally analytic vectors of
admissible $\mathbb{Q}_{p}$-Banach space representations

Paulina Fust

###### Abstract

We show that the continuous cohomology groups of a $p$-adic reductive group
with coefficients in the locally analytic vectors of an admissible
$\mathbb{Q}_{p}$-Banach space representation are homeomorphic to those with
coefficients in the Banach space representation itself. Moreover, we deduce
that the canonical topologies on those continuous cohomology groups are
Hausdorff and are the uniquely determined finest locally convex topologies.

##  1 Introduction

Let $G$ be a $p$-adic reductive group and $L$ a finite field extension of
$\mathbb{Q}_{p}$. For an admissible unitary $L$-Banach space representation
$\Pi$ of $G$, we study the continuous group cohomology of $G$ with
coefficients in the subrepresentation $\Pi^{\operatorname{la}}$ of $\Pi$
consisting of locally analytic vectors. In the case, where $G$ is a compact
group, Rodrigues Jacinto and Rodríguez Camargo proved in [6, Corollary 1.6],
that these cohomology groups $H^{i}(G,\Pi^{\operatorname{la}})$ are isomorphic
to the cohomology groups with coefficients in the representation $\Pi$ itself.
We use this result as key ingredient for our proof of the following Theorem:

######  Theorem 1.1 (Proposition 3.3).

Let $\Pi\in\operatorname{Ban}^{\operatorname{adm}}_{G}(L)$ be an admissible
$L$-Banach space representation of $G$ and let $\Pi^{\operatorname{la}}$ be
the subspace of locally analytic vectors in $\Pi$, equipped with its natural
topology. The inclusion $\Pi^{\operatorname{la}}\hookrightarrow\Pi$ induces
natural isomorphisms between continuous group cohomology groups

$H^{i}(G,\Pi^{\operatorname{la}})\cong H^{i}(G,\Pi),\ \forall i\geq 0.$

Here, we consider the subrepresentation $\Pi^{\operatorname{la}}$ equipped
with the topology defined by Emerton in Section 3 of [4], which is finer than
the subspace topology coming from the inclusion
$\Pi^{\operatorname{la}}\subseteq\Pi$. To prove Theorem 1.1, we construct for
any continuous representation of $G$ on an $L$-vector space $V$, a spectral
sequence converging to the continuous cohomology of $G$ with coefficients in
$V$, whose first page terms are certain continuous cohomology groups of
stabilizers of facets in the Bruhat–Tits building of $G$. This allows us to
deduce Theorem 1.1 from the statement for compact groups. In Corollary 6.10 in
[5], we proved that the continuous cohomology groups $H^{i}(G,\Pi)$, for $\Pi$
an admissible $L$-Banach space representation, are finite dimensional. This
combined with Theorem 1.1 implies the following corollary.

######  Corollary 1.2 (Corollary 3.8).

For every $\Pi\in\operatorname{Ban}^{\operatorname{adm}}_{G}(L)$, the
continuous cohomology groups $H^{i}(G,\Pi^{\operatorname{la}})$ are finite
dimensional over $L$ for all $i\geq 0$.

For a topological $G$-module $V$, the spaces of continuous $i$-cochains
$C^{i}(G,V)$ are equipped with the compact-open topology. We thus get a
quotient topology on the continuous cohomology groups $H^{i}(G,V)$, called the
canonical topology. Moreover, if the $G$-module $V$ is a locally convex
$L$-vector space, then we show that the compact-open topology on $C^{i}(G,V)$,
as well as the canonical topology on $H^{i}(G,V)$ is locally convex. The
canonical topology is in general not Hausdorff. However, Theorems 1.2 and 1.1
allow us to apply a criterion for the cohomology groups being Hausdorff in [3]
to deduce the following.

######  Corollary 1.3 (Corollary 3.9).

Suppose that $G$ is a $p$-adic reductive group and $\Pi$ an admissible
$L$-Banach space representation of $G$. Then for all $i\geq 0$, the canonical
topologies on the continuous cohomology groups $H^{i}(G,\Pi)$ and
$H^{i}(G,\Pi^{\operatorname{la}})$ are Hausdorff.

We use Corollary 1.3 to deduce that the isomorphisms in Theorem 1.1 are in
fact homeomorphisms.

######  Corollary 1.4 (Corollary 3.13).

The canonical topology on the continuous cohomology groups
$H^{i}(G,\Pi^{\operatorname{la}})$ and $H^{i}(G,\Pi)$ is the finest locally
convex topology. In particular, for all $i\geq 0$, we have homeomorphisms

$H^{i}(G,\Pi^{\operatorname{la}})\cong H^{i}(G,\Pi).$

######  Remark 1.5.

The finest locally convex topology on the finite dimensional cohomology groups
$H^{i}(G,\Pi)$ and $H^{i}(G,\Pi^{\operatorname{la}})$ can be described more
explicitly: Let $V$ be a finite dimensional $L$-vector space. Then there is
only one way to equip $V$ with a topology making it a locally convex Hausdorff
$L$-vector space, i.e. the finest locally convex topology. Namely, it is the
topology defined by the norm $\|\sum_{i=1}^{n}\lambda_{i}e_{i}\|:=\max_{1\leq
i\leq n}|\lambda_{i}|$, where $e_{1},\dots,e_{n}$ is an $L$-basis of $V$
(Proposition 4.13 [7]).

Acknowledgements: This work was written as part of my PhD thesis under the
supervision of Vytautas Paškūnas. I want to thank him for his support and
Nicolas Dupré and Jonas Franzel for their helpful comments and suggestions.
This work was funded by the DFG Graduiertenkolleg 2553.

##  2 Continuous group cohomology

For coherence, we briefly recall the definition of continuous group cohomology
for a locally profinite group $G$ and collect some basic properties needed in
later sections.

Following [3], we define the continuous group cohomology as follows: Let $V$
be a topological $G$-module, i.e. a topological abelian group $V$ with a
$G$-action such that $G$ acts on $V$ via group automorphisms and the map
$G\times V\rightarrow V$ is continuous. Then we can define the cochain complex

$C^{n}(G,V):=C(G^{n+1},V):=\\{f:G^{n+1}\rightarrow V\text{ continuous}\\},$

with differentials $d^{n}:C^{n}(G,V)\rightarrow C^{n+1}(G,V)$, defined by

$d^{n}f(g_{0},\dots,g_{n+1})=\sum_{i=0}^{n+1}(-1)^{i}f(g_{0},\dots,\widehat{g_{i}},\dots,g_{n+1}).$

We endow the spaces $C^{n}(G,V)$ with the compact-open topology. More
explicitly, a basis of the compact-open topology is given by finite
intersections of sets of the form $\Omega(K,U)=\\{f\in C^{n}(G,V)|\
f(K)\subset U\\}$, for compact subsets $K\subset G^{n+1}$ and open subsets
$U\subset V$. By defining a $G$-action via
$(gf)(g_{0},\dots,g_{n})=gf(g^{-1}g_{0},\dots,g^{-1}g_{n})$, for
$g,g_{0},\dots,g_{n}\in G$ and $f\in C^{n}(G,V)$, these will define
topological $G$-modules. Furthermore, there is a continuous $G$-equivariant
injection $V\hookrightarrow C(G,V)$, defined by $v\mapsto[g\mapsto v]$.

For a closed subgroup $H\leq G$ and a topological $G$-module $V$, define the
induced representation to be

$\operatorname{Ind}_{H}^{G}V:=\\{f\in C(G,V)\mid f(hg)=hf(g)\ \forall h\in H,\
\forall g\in G\\},$

with the subspace topology induced from $\operatorname{Ind}^{G}_{H}V\subseteq
C(G,V)$ and $G$-action $gf(g^{\prime}):=f(g^{\prime}g)$, for $g,g^{\prime}\in
G$. Similarly, the compact induction is defined as

$\operatorname{c-Ind}_{H}^{G}V:=\\{f\in\operatorname{Ind}_{H}^{G}V\mid\text{
the support of }f\text{ is compact modulo }H\\}$

with the same $G$-action.

######  Lemma 2.1.

For a topological $G$-module $V$, there are homeomorphisms of $G$-modules

$C^{n+1}(G,V)\cong C^{0}(G,C^{n}(G,V))\text{ and
}C^{0}(G,V)\cong\operatorname{Ind}^{G}_{1}V,$

for all $n\geq 0$.

###### Proof.

By [2, X.3.4 Corollaire 2], the map

$\displaystyle\phi:C^{n+1}(G,V)$ $\displaystyle\rightarrow
C^{0}(G,C^{n}(G,V))$ $\displaystyle f$ $\displaystyle\mapsto\phi(f),$

where $\phi(f)(g_{0})(g_{1},\dots,g_{n})=f(g_{0},\dots,g_{n})$, is a
homeomorphism. It is straightforward to check that this is $G$-equivariant,
making it an isomorphism of topological $G$-modules.

The assignment $f\mapsto[g\mapsto gf(g^{-1})]$ defines both maps
$C^{0}(G,V)\rightarrow\operatorname{Ind}^{G}_{1}V$ and its inverse. One can
check that both maps are continuous and $G$-equivariant, hence the modules are
homeomorphic. ∎

######  Lemma 2.2.

For two topological $G$-modules $V$ and $W$, one has the following
isomorphism:

$\operatorname{Hom}_{G}^{\operatorname{cts}}(V,\operatorname{Ind}^{G}_{1}W)\cong\operatorname{Hom}^{\operatorname{cts}}(V,W),$

where $\operatorname{Hom}^{\operatorname{cts}}(V,W)$ are continuous group
homomorphisms and on the left hand side, we take continuous $G$-equivariant
group homomorphisms.

###### Proof.

[3, Lemma 2]. ∎

######  Lemma 2.3.

The complex $0\rightarrow V\rightarrow C^{\bullet}(G,V)$ is an exact complex
of $G$-modules.

###### Proof.

[5, Lemma 2.3] ∎

######  Definition 2.4.

The $i^{\text{th}}$ continuous group cohomology group of $G$ with coefficients
in $V$ is defined to be $H^{i}(G,V):=H^{i}(C^{\bullet}(G,V)^{G})$.

A homomorphism of topological $G$-modules $\phi:V\rightarrow W$ is said to be
a _strong morphism_ , if the induced morphisms
$\operatorname{Ker}\phi\rightarrow V$ and $V/\operatorname{Ker}\phi\rightarrow
W$ each have a continuous left inverse in the category of topological abelian
groups. And if one has a short exact sequence of topological $G$-modules

$0\rightarrow V\rightarrow W\rightarrow U\rightarrow 0,$

in which all the morphisms are strong, then the induced sequence

$0\rightarrow C^{n}(G,V)\rightarrow C^{n}(G,W)\rightarrow
C^{n}(G,U)\rightarrow 0$

is again strong and then induces a long exact sequence in cohomology.

For example, any short exact sequence

$0\rightarrow V\rightarrow W\rightarrow U\rightarrow 0,$

of topological $G$-modules is automatically strong if $U$ is discrete.
Therefore it induces a long exact sequence in cohomology.

######  Definition 2.5.

We say that a topological $G$-module $V$ is acyclic (for the continuous group
cohomology), if

$H^{i}(G,V)=\begin{cases}V^{G},\text{ if }i=0\\\ 0,\text{
otherwise}.\end{cases}$

######  Remark 2.6.

  1. 1.

The $G$-modules $C^{n}(G,V)$ are acyclic for the continuous group cohomology.
Hence, the complex $0\rightarrow V\rightarrow C^{\bullet}(G,V)$ is an acyclic
resolution of $V$. (cf. [3, p. 201].)

  2. 2.

A topological $G$-module $V$ is said to be _continuously injective_ , if for
every strong $G$-injection $U\hookrightarrow W$ and for every morphism of
topological $G$-modules $\phi:U\rightarrow V$, the morphism $\phi$ extends to
a morphism from $W$ to $V$. An example of continuously injective $G$-modules
are the modules $C^{n}(G,V)$ (cf. [3, p. 201]).

######  Lemma 2.7.

For an open subgroup $K\leq G$ and a topological $G$-module $V$, the
$G$-modules $C^{n}(G,V)$ are acyclic for the continuous group cohomology
$H^{\bullet}(K,-)$.

Moreover, the cohomology of the complex $C^{\bullet}(G,V)^{K}$ is
$H^{\bullet}(K,V)$.

###### Proof.

A proof is given in [3, Proposition 4 (a)]. The idea is as follows: since the
quotient $G/K$ is discrete, one has a continuous section $s:G/K\rightarrow G$
of the natural projection $G\rightarrow G/K$ with which one can define the
$K$-invariant homeomorphism $K\times G/K\rightarrow G,\ (h,gK)\mapsto s(gK)h$.
Therefore, one gets an isomorphism of $K$-modules

$C(G,V)\cong C(K\times G/K,V)\cong C(K,C(G/K,V))$

showing that $C(G,V)$ is acyclic as a $K$-module.

To conclude that the complex $C^{\bullet}(G,V)^{K}$ computes the cohomology of
$K$, one also has to use that the resolution $V\hookrightarrow
C^{\bullet}(G,V)$ is a strong resolution of acyclic $K$-modules and hence can
be used to compute the cohomology of $K$ ([3, Proposition 1]). ∎

##  3 Locally analytic vectors of admissible Banach space representations

We fix the following notation: let $G$ be a $p$-adic reductive group and $L$
be a finite field extension of $\mathbb{Q}_{p}$. Since $G$ is a $p$-adic
reductive group, it is in particular a locally $\mathbb{Q}_{p}$-analytic
group. We want to study the continuous cohomology groups of $G$ with
coefficients in the representation of locally analytic vectors in an
admissible Banach space representation. For this, we use a spectral sequence,
which we will first construct in a more general setting.

###  3.1 Construction of a spectral sequence

In order to construct the spectral sequence, we use a resolution of the
trivial representation in terms of certain compactly induced representations
from stabilizer subgroups of vertices in the Bruhat–Tits building. Following
the notations of [8], we denote by $X$ the reduced Bruhat–Tits building
associated to $G$ and for any $q\geq 0$ we denote by $X_{q}$ the set of all
$q$-dimensional facets in $X$. Moreover, for a $q$-facet $F\in X_{q}$, we
define $P_{F}^{\dagger}$ to be the $G$-stabilizer of $F$. In [8], the authors
use the fact that the Bruhat–Tits tree is a contractible space to construct a
$G$-equivariant resolution of $\mathbb{Z}$. We use a reformulation of this
resolution as described in Section 6.4 of [5]:

$\dots\rightarrow\bigoplus_{F\in
R_{1}}\operatorname{c-Ind}^{G}_{P_{F}^{\dagger}}\delta_{F}\rightarrow\bigoplus_{F\in
R_{0}}\operatorname{c-Ind}^{G}_{P_{F}^{\dagger}}\delta_{F}\rightarrow\mathbb{Z}\rightarrow
0.$ (1)

In this section, we use the resolution (1) to construct a spectral sequence of
the form

$E_{1}^{i,j}=\bigoplus_{F\in
R_{i}}H^{j}(P_{F}^{\dagger},V\otimes_{L}\delta_{F}^{-1})\Rightarrow
H^{i+j}(G,V),$ (2)

where $\delta_{F}$ are characters on $P_{F}$, taking values in $\\{\pm 1\\}$.

A priori, the complex (1) is a resolution by smooth $G$-representations on
free $\mathbb{Z}$-modules, but by tensoring with $L$, we obtain a resolution
of the trivial representation by smooth $G$-representations on $L$-vector
spaces.

$\bigoplus_{F\in
R_{\bullet}}\operatorname{c-Ind}_{P_{F}^{\dagger}}^{G}\delta_{F}\rightarrow\mathds{1}\rightarrow
0.$

By abuse of notation, we now denote by $\delta_{F}$ the representation

$\delta_{F}:P_{F}^{\dagger}\rightarrow\\{\pm 1\\}\subset L^{\times}$

on the one-dimensional $L$-vector space.

Since the compactly induced representations in this resolution are smooth, we
can give them the structure of topological $G$-modules by equipping them with
the discrete topology.

Now consider the double complex

$C^{i,j}:=\operatorname{Hom}_{G}(\bigoplus_{F\in
R_{i}}\operatorname{c-Ind}_{P_{F}^{\dagger}}^{G}\delta_{F},C^{j}(G,V)),\
i,j\geq 0.$ (3)

Note that since we equipped the compactly induced representations with the
discrete topology, all the homomorphisms in $C^{i,j}$ are automatically
continuous and we can also write it as
$C^{i,j}=\operatorname{Hom}^{\operatorname{cts}}_{G}(\bigoplus_{F\in
R_{i}}\operatorname{c-Ind}_{P_{F}^{\dagger}}^{G}\delta_{F},C^{j}(G,V)).$

This induces two spectral sequences with the same limit term. One is coming
from a horizontal filtration and the other from the vertical one. To write
these down explicitly, we need to understand the cohomology of the rows and of
the columns of our double complex (3). We start by studying the rows:

Fixing an $i\geq 0$, we can write

$C^{i}(G,V)\cong
C^{0}(G,C^{i-1}(G,V))\cong\operatorname{Ind}_{1}^{G}C^{i-1}(G,V)$ (4)

(cf. Lemma 2.1), with the convention that $C^{-1}(G,V)=V$. In particular, we
have isomorphisms
$C^{i}(G,V)^{G}\cong(\operatorname{Ind}_{1}^{G}C^{i-1}(G,V))^{G}\cong
C^{i-1}(G,V)$. The isomorphism (4) allows us then to apply Frobenius
reciprocity for topological $G$-modules (cf. Lemma 2.2). We obtain

$\displaystyle C^{\bullet,i}$
$\displaystyle=\operatorname{Hom}^{\operatorname{cts}}_{G}(\bigoplus_{F\in
R_{\bullet}}\operatorname{c-Ind}_{P_{F}^{\dagger}}^{G}\delta_{F},C^{i}(G,V))$
$\displaystyle\cong\operatorname{Hom}^{\operatorname{cts}}_{L}(\bigoplus_{F\in
R_{\bullet}}\operatorname{c-Ind}_{P_{F}^{\dagger}}^{G}\delta_{F},C^{i-1}(G,V))$
$\displaystyle=\operatorname{Hom}_{L}(\bigoplus_{F\in
R_{\bullet}}\operatorname{c-Ind}_{P_{F}^{\dagger}}^{G}\delta_{F},C^{i-1}(G,V)).$

But now, the complex

$0\rightarrow\operatorname{Hom}_{L}(L,C^{i-1}(G,V))\rightarrow\operatorname{Hom}_{L}(\bigoplus_{F\in
R_{0}}\operatorname{c-Ind}_{P_{F}^{\dagger}}^{G}\delta_{F},C^{i-1}(G,V))\rightarrow\dots$

is exact, as we have taken $L$-linear homomorphisms of an exact sequence into
the $L$-vector space $C^{i-1}(G,V)$. Therefore, taking cohomology of the rows
$C^{\bullet,i}$ gives

$H^{j}(C^{\bullet,i})=\begin{cases}0,\text{ if }j\neq 0,\\\
\operatorname{Hom}_{L}(L,C^{i-1}(G,V))\cong C^{i-1}(G,V)\cong
C^{i}(G,V)^{G}\text{, if }j=0.\end{cases}$

The first page terms of the spectral sequence coming from the horizontal
filtration are precisely given by these cohomology groups:

$E_{h,1}^{i,j}=H^{j}(C^{\bullet,i}).$

Moreover, the first page differentials $d_{h,1}^{i,j}:E_{h,1}^{i,j}\rightarrow
E_{h,1}^{i+1,j}$ are zero whenever $j$ is non-zero and for $j=0$, they are
just the maps

$d_{h,1}^{i,0}:C^{i}(G,V)^{G}\rightarrow C^{i+1}(G,V)^{G},$

induced by the differential maps on the complex $C^{\bullet}(G,V)$. We can
easily compute the second page terms:

$E_{h,2}^{i,j}=H^{i,j}(E_{h,1}^{\bullet,\bullet})=\begin{cases}0\text{, if
}j\neq 0\\\ H^{i}(C^{\bullet}(G,V)^{G})=H^{i}(G,V)\text{, if }j=0.\end{cases}$

In conclusion, we know that the spectral sequence coming from the horizontal
filtration converges to the continuous cohomology groups of $G$ with
coefficients in $V$. The spectral sequence induced by the vertical filtration
also converges to $H^{\bullet}(G,V)$, since it has the same limit term.

On the other hand, we can fix an $i\geq 0$ and consider the columns of the
double complex

$C^{i,\bullet}=\operatorname{Hom}_{G}(\bigoplus_{F\in
R_{i}}\operatorname{c-Ind}_{P_{F}^{\dagger}}^{G}\delta_{F},C^{\bullet}(G,V)).$

By pulling out the direct sum and applying Frobenius reciprocity for the
compact induction, we obtain

$\displaystyle C^{i,\bullet}\cong$ $\displaystyle\bigoplus_{F\in
R_{i}}\operatorname{Hom}_{P_{F}^{\dagger}}(\delta_{F},C^{\bullet}(G,V))$
$\displaystyle\cong$ $\displaystyle\bigoplus_{F\in
R_{i}}\operatorname{Hom}_{P_{F}^{\dagger}}(\mathds{1},C^{\bullet}(G,V)\otimes_{L}\delta_{F}^{-1}).$
(5)

######  Lemma 3.1.

Let $V$ be a topological $G$-module on an $L$-vector space, let $H\leq G$ be
an open subgroup and $\delta:H\rightarrow L^{\times}$ a smooth character on
$H$. We have an $L$-linear isomorphism of topological $H$-modules

$C^{n}(G,V)\otimes_{L}\delta\overset{\cong}{\longrightarrow}C^{n}(G,V\otimes_{L}\delta),\
\forall n\geq 0.$

###### Proof.

As in Lemma 2.1, we have homeomorphisms $C^{n+1}(G,W)\rightarrow
C^{0}(G,C^{n}(G,W))$ for every topological space $W$. These homeomorphisms are
$H$-equivariant if $W$ is equipped with the structure of a topological
$H$-module. Hence, we may assume that $n=0$.

Let $\widetilde{\delta}:G\rightarrow L^{\times}$ be the map that sends an
element $g\in G$ to $\delta(g)$ if $g$ lies in $H$ and to $1$ otherwise. Since
$H$ is open in $G$, this map is locally constant (but not necessarily a
character). Therefore the map

$\alpha:\ C^{0}(G,V)\otimes_{L}\delta\rightarrow C^{0}(G,V\otimes_{L}\delta),$

where

$\alpha(f\otimes 1)(g):=f(g)\otimes\widetilde{\delta}(g)$

is a well-defined $L$-linear isomorphism. Moreover, it is $H$-equivariant,
because for any $h\in H$ and $g\in G$ we have

$\displaystyle\alpha(h(f\otimes 1))(g)$
$\displaystyle=\alpha((hf)\otimes\delta(h))(g)=f(gh)\otimes\widetilde{\delta}(g)\delta(h)$
$\displaystyle=f(gh)\otimes\widetilde{\delta}(gh)=\alpha(f\otimes
1)(gh)=h\alpha(f\otimes 1)(g).$

Continuity can be checked on a base of the compact-open topology on the right
hand side. Such a base is given by the sets $\Omega(K,U)=\\{f\in
C^{0}(G,V\otimes_{L}\delta)|\ f(K)\subset U\\}$ for compact subsets $K\subset
G$ and open subsets $U\subset V\otimes_{L}\delta$. Fixing such a $K$ and $U$,
we see that $\alpha^{-1}(\Omega(K,U))$ consists of functions $f$ such that for
any $k\in K$, $\widetilde{\delta}(k)f(k)$ lies in the open subset $U$. But
since $\widetilde{\delta}$ is locally constant and $K$ compact, we can cover
$K$ by finitely many open sets of the form
$\widetilde{\delta}^{-1}(\lambda_{i})$ for some $\lambda_{i}\in L$,
$i=1,\dots,s$. Since the intersections
$K\cap\widetilde{\delta}^{-1}(\lambda_{i})$ are again compact, we obtain that
$\alpha^{-1}(\Omega(K,U))$ is the finite intersection

$\alpha^{-1}(\Omega(K,U))=\bigcap_{i=1}^{s}\Omega(K_{i},\lambda_{i}^{-1}U)$

of open sets and therefore is open itself. Thus, $\alpha$ is continuous. ∎

We apply Lemma 3.1 to the character $\delta_{F}^{-1}$ in (5), and obtain

$\displaystyle C^{i,\bullet}\cong$ $\displaystyle\bigoplus_{F\in
R_{i}}\operatorname{Hom}_{P_{F}^{\dagger}}(\mathds{1},C^{\bullet}(G,V\otimes_{L}\delta_{F}^{-1}))$
$\displaystyle\cong$ $\displaystyle\bigoplus_{F\in
R_{i}}C^{\bullet}(G,V\otimes_{L}\delta_{F}^{-1})^{P_{F}^{\dagger}}.$

As we have seen in Lemma 2.7, the complex $C^{\bullet}(G,V)$ is acyclic for
the continuous cohomology of $H$ for any open subgroup $H\leq G$ and hence, it
can be used to compute the cohomology groups of $H$. Therefore, taking the
cohomology of this complex gives us the following:

$\displaystyle H^{j}(C^{i,\bullet})$ $\displaystyle=\bigoplus_{F\in
R_{i}}H^{j}(C^{\bullet}(G,V\otimes_{L}\delta_{F}^{-1})^{P_{F}^{\dagger}})$
$\displaystyle=\bigoplus_{F\in
R_{i}}H^{j}(P_{F}^{\dagger},V\otimes_{L}\delta_{F}^{-1}).$

We thus get a spectral sequence, converging to the continuous cohomology of
$G$ with coefficients in $V$, where the first page terms are given by the
cohomology of the columns $C^{i,\bullet}$ of our double complex. More
precisely, we obtain the spectral sequence (2).

###  3.2 Locally analytic vectors of admissible Banach space representations

The spectral sequence (2) can be very useful to generalize results that are
known for compact groups to arbitrary $p$-adic reductive groups. In this
section, we will show an application of this kind, by comparing the continuous
cohomology with coefficients in an admissible Banach space representation to
the continuous cohomology with coefficients in its subrepresentation given by
locally analytic vectors. We start with a short reminder on the definition of
locally analytic representations. For more details on this matter, see for
example Section 3 in [4].

Let $V$ be a barrelled locally convex Hausdorff $L$-vector space, meaning that
it is a Hausdorff topological $L$-vector space whose topology is defined by a
family of seminorms $q_{i}:V\rightarrow\mathbb{R}_{\geq 0}$, $i\in I$, in the
sense that the topology on $V$ is the coarsest topology, making all seminorms
continuous maps, and in which every closed lattice is open (for more details,
see Chapter 1, Section 6 in [7]). For such a space, we denote by
$C^{\operatorname{la}}(G,V)$ the set of locally analytic $V$-valued functions
on $G$.

######  Definition 3.2.

A locally analytic representation of $G$ over $L$ is a representation of $G$
on a barrelled locally convex Hausdorff $L$-vector space, such that $G$ is
acting on $V$ via continuous endomorphisms and for each $v\in V$, the orbit
map $\rho_{v}:G\rightarrow V$, $g\mapsto gv$, is a locally analytic function
$\rho_{v}\in C^{\operatorname{la}}(G,V)$.

Let $\Pi$ be an admissible $L$-Banach space representation of $G$. By
definition, $\Pi$ is a normed $L$-vector space, making it in particular a
locally convex $L$-vector space, so that it makes sense to talk about the
locally analytic vectors in $\Pi$, meaning the vectors $v\in\Pi$, whose orbit
maps $\rho_{v}$ are locally analytic functions. The locally analytic vectors
form a locally analytic subrepresentation of $\Pi$, denoted by
$\Pi^{\operatorname{la}}$. The subrepresentation $\Pi^{\operatorname{la}}$ is
equipped with a topology as described in [4], making it a topological
$G$-module. The topology on $\Pi^{\operatorname{la}}$ is finer than the
subspace topology coming from $\Pi^{\operatorname{la}}\subset\Pi$, so that the
injection $\Pi^{\operatorname{la}}\hookrightarrow\Pi$ is a continuous map. An
explanation for this is given for example in the beginning of Section 3.5 in
[4].

By Corollary 1.6 in [6], for a compact $p$-adic Lie group $K$, we have
isomorphisms

$H^{i}(K,\Pi^{\operatorname{la}})\cong H^{i}(K,\Pi).$

Note that in [6], the condition for the existence of such isomorphism is that
the representation $\Pi$ has no higher locally analytic vectors, which is true
for admissible Banach space representations (cf. Theorem 7.1 [9]).

We want to use the spectral sequence (2) to generalize this to the following:

######  Proposition 3.3.

Let $\Pi$ be an admissible $L$-Banach space representation of $G$. Then for
every $i\geq 0$, the maps

$H^{i}(G,\Pi^{\operatorname{la}})\overset{\cong}{\longrightarrow}H^{i}(G,\Pi),$

induced by the inclusion $\Pi^{\operatorname{la}}\hookrightarrow\Pi$, are
isomorphisms.

Before we can apply the spectral sequence argument, recall that the stabilizer
groups $P_{F}^{\dagger}$ appearing in the sequence are not compact. We
therefore need the following lemmas first.

######  Lemma 3.4.

Let $\Gamma$ be a discrete group and let $0\rightarrow
M^{0}\overset{d^{0}}{\rightarrow}M^{1}\overset{d^{1}}{\rightarrow}M^{2}\rightarrow\dots$
be a complex of $\Gamma$-modules. Then given an exact functor
$F:\operatorname{Mod}_{\Gamma}\rightarrow\operatorname{Mod}_{\mathbb{Z}}$, we
have isomorphisms

$H^{i}(F(M^{\bullet}))\cong F(H^{i}(M^{\bullet}))$

for every $i\geq 0$.

###### Proof.

We fix the following notation: for any complex

$0\rightarrow
C^{0}\overset{\delta^{0}}{\rightarrow}C^{1}\overset{\delta^{1}}{\rightarrow}C^{2}\rightarrow\dots,$

let $Z^{i}(C^{\bullet}):=\ker(\delta^{i})$ be the $i$-cocycles and
$B^{i}(C^{\bullet}):=\operatorname{im}(\delta^{i-1})\cong
C^{i-1}/Z^{i-1}(C^{\bullet})$ the $i$-coboundaries.

Applying the exact functor $F$ to the exact sequence

$0\rightarrow Z^{i}(M^{\bullet})\rightarrow M^{i}\rightarrow M^{i+1}$

gives the isomorphism

$Z^{i}(F(M^{\bullet}))=F(Z^{i}(M^{\bullet})),\ \forall i\geq 0.$ (6)

Using this identification (6) and the exactness of $F$, we can also rewrite
the coboundaries as

$B^{i}(F(M^{\bullet}))\cong\frac{F(M^{i-1})}{Z^{i-1}(F(M^{\bullet}))}\cong\frac{F(M^{i-1})}{F(Z^{i-1}(M^{\bullet}))}\cong
F\left(\frac{M^{i-1}}{Z^{i-1}(M^{\bullet})}\right)\cong
F(B^{i}(M^{\bullet})).$ (7)

Combining these isomorphisms (6) and (7) gives us

$H^{i}(F(M^{\bullet}))\cong\frac{F(Z^{i}(M^{\bullet}))}{F(B^{i}(M^{\bullet}))}\cong
F\left(\frac{Z^{i}(M^{\bullet})}{B^{i}(M^{\bullet})}\right)\cong
F(H^{i}(M^{\bullet})),$

as claimed. ∎

######  Lemma 3.5.

Let $\Gamma$ be a discrete group, then for every $i\geq 0$, the functor
$C^{i}(\Gamma,-)^{\Gamma}$ is exact.

In particular, for any complex $0\rightarrow
M^{0}\overset{d^{0}}{\rightarrow}M^{1}\overset{d^{1}}{\rightarrow}M^{2}\rightarrow\dots$
of $\Gamma$-modules, we have

$H^{j}(C^{i}(\Gamma,M^{\bullet})^{\Gamma})\cong
C^{i}(\Gamma,H^{j}(M^{\bullet}))^{\Gamma},\ \forall i,j\geq 0.$

###### Proof.

To show that the functor $C^{i}(\Gamma,-)^{\Gamma}$ is exact, let

$0\rightarrow N\rightarrow M\rightarrow Q\rightarrow 0$

be a short exact sequence of $\Gamma$-modules. Note that, since $\Gamma$ is
discrete, any function on $\Gamma$ (or on a product $\Gamma^{i+1}$ of copies
of $\Gamma$) is automatically continuous and hence, the functor
$C^{i}(\Gamma,-)$ is exact and we obtain a short exact sequence

$0\rightarrow C^{i}(\Gamma,N)\rightarrow C^{i}(\Gamma,M)\rightarrow
C^{i}(\Gamma,Q)\rightarrow 0.$

Applying the functor $(-)^{\Gamma}$ to this, induces a long exact sequence in
cohomology

$0\rightarrow C^{i}(\Gamma,N)^{\Gamma}\rightarrow
C^{i}(\Gamma,M)^{\Gamma}\rightarrow C^{i}(\Gamma,Q)^{\Gamma}\rightarrow
H^{1}(\Gamma,C^{i}(\Gamma,N)),$

where the first cohomology group $H^{1}(\Gamma,C^{i}(\Gamma,N))$ vanishes,
because $C^{i}(\Gamma,N)$ is acyclic, since by Lemma 2.1 and Shapiro’s Lemma
[3, Proposition 3], we have

$H^{i}(\Gamma,C^{i}(\Gamma,N))\cong
H^{i}(\Gamma,\operatorname{Ind}_{\\{1\\}}^{\Gamma}C^{i-1}(\Gamma,N))\cong
H^{i}(\\{1\\},C^{i-1}(\Gamma,N))=0$

for $i>0$. This proves exactness of $C^{i}(\Gamma,-)^{\Gamma}$.

We can thus apply Lemma 3.4 to the functor $C^{i}(\Gamma,-)^{\Gamma}$ and the
complex $M^{\bullet}$, to obtain
$H^{j}(C^{i}(\Gamma,M^{\bullet})^{\Gamma})\cong
C^{i}(\Gamma,H^{j}(M^{\bullet}))^{\Gamma}$. ∎

######  Lemma 3.6.

Let $V$ be a topological $G$-module and let $H\leq G$ be an open normal
subgroup of $G$. Then there exists a Hochschild–Serre spectral sequence:

$E_{2}^{i,j}\cong H^{i}(G/H,H^{j}(H,V))\Rightarrow H^{i+j}(G,V).$

###### Proof.

We mimic the proof of Proposition 5 in [3], where the authors prove the
existence of a Hochschild–Serre spectral sequence for a closed normal subgroup
$H$, assuming some additional conditions that are not needed for the case of
an open normal subgroup. We consider the double complex

$C^{i,j}=C^{i}(G/H,C^{j}(G,V)^{H})^{G/H},\ i,j\geq 0.$

This double complex induces two spectral sequences with the same limit term.
The first spectral sequence comes from the horizontal filtration of the double
complex and its $E_{1}$-terms are given by:

$\displaystyle
E_{1,h}^{i,j}=H^{j}(C^{\bullet,i})=H^{j}(C^{\bullet}(G/H,C^{i}(G,V)^{H})^{G/H})=H^{j}(G/H,C^{i}(G,V)^{H})$

Note that the groups $C^{i}(G,V)^{H}$ are acyclic for the cohomology of $G/H$.
Indeed, since $G/H$ is discrete, the continuous cohomology of $G/H$ is just
group cohomology. Now for any $G/H$-module $W$, we can give it a topological
$G/H$-module structure by equipping it with the discrete topology and obtain

$\displaystyle\operatorname{Hom}_{G/H}(W,C^{i}(G,V)^{H})$
$\displaystyle=\operatorname{Hom}^{\operatorname{cts}}_{G/H}(W,C^{i}(G,V)^{H})$
$\displaystyle\cong\operatorname{Hom}^{\operatorname{cts}}_{G}(W,C^{i}(G,V))$
$\displaystyle\cong\operatorname{Hom}^{\operatorname{cts}}(W,C^{i-1}(G,V))$
$\displaystyle=\operatorname{Hom}(W,C^{i-1}(G,V)),$

using Lemma 2.2. Since the functor $\operatorname{Hom}(-,C^{i-1}(G,V))$ is
exact, so is the functor $\operatorname{Hom}_{G/H}(-,C^{i}(G,V)^{H})$, showing
that $C^{i}(G,V)^{H}$ is acyclic for the cohomology of $G/H$.

Hence, we obtain

$E_{1,h}^{i,j}=H^{j}(G/H,C^{i}(G,V)^{H})=\begin{cases}0\text{, if }j\neq 0\\\
(C^{i}(G,V)^{H})^{G/H}\cong C^{i}(G,V)^{G}\text{, if }j=0.\end{cases}$

And the $E_{2}$-terms are given by

$E_{2,h}^{i,j}=\begin{cases}0\text{, if }j\neq 0\\\
H^{i}(C^{\bullet}(G,V)^{G})=H^{i}(G,V)\text{, if }j=0.\end{cases}$

Therefore, the two spectral sequences of the double complex $C^{i,j}$ converge
to the continuous cohomology of $G$ with coefficients in $V$.

The $E_{1}$-terms of the spectral sequence coming from the vertical filtration
are given by

$\displaystyle
E_{1,v}^{i,j}=H^{j}(C^{i,\bullet})=H^{j}(C^{i}(G/H,C^{\bullet}(G,V)^{H})^{G/H}).$

We can apply Lemma 3.5 to the discrete group $\Gamma=G/H$ and the complex
$M^{\bullet}=C^{\bullet}(G,V)^{H}$ and get

$H^{j}(C^{i}(G/H,C^{\bullet}(G,V)^{H})^{G/H})\cong
C^{i}(G/H,H^{j}(C^{\bullet}(G,V)^{H}))^{G/H}.$

By Lemma 2.7, the resolution $C^{\bullet}(G,V)$ can be used to compute the
continuous cohomology of $H$, so that

$E_{1,v}^{i,j}=C^{i}(G/H,H^{j}(C^{\bullet}(G,V)^{H}))^{G/H}=C^{i}(G/H,H^{j}(H,V))^{G/H}.$

Then the $E_{2}$-terms are given by

$E_{2,v}^{i,j}=H^{i,j}(E_{1,v}^{\bullet,\bullet})=H^{i}(C^{\bullet}(G/H,H^{j}(H,V))^{G/H})=H^{i}(G/H,H^{j}(H,V)).$

Since we know that this spectral sequence has the same limit term as the one
from the horizontal filtration, we obtain the wanted spectral sequence. ∎

######  Lemma 3.7.

Let $F\in X_{q}$ be a $q$-dimensional facet of the Bruhat–Tits building of
$G$. Then for every $i\geq 0$, we have isomorphisms

$H^{i}(P_{F}^{\dagger},\Pi^{\operatorname{la}})\cong
H^{i}(P_{F}^{\dagger},\Pi).$

###### Proof.

Let us fix a facet $F\in X_{q}$. Recall that there exists a compact open
subgroup $R_{F}\leq G$, which is open and normal in $P_{F}^{\dagger}$ (for a
construction, see Section 1.2 in [8]).

Since the subgroup $R_{F}$ is compact open in $P_{F}^{\dagger}$, we can apply
Lemma 3.6, which proves the existence of a Hochschild–Serre spectral sequence

$E_{2}^{i,j}=H^{i}(P_{F}^{\dagger}/R_{F},H^{j}(R_{F},\Pi))\Rightarrow
H^{i+j}(P_{F}^{\dagger},\Pi).$

By the same argument, we also obtain a spectral sequence for the locally
analytic subrepresentation

$\widetilde{E}_{2}^{i,j}=H^{i}(P_{F}^{\dagger}/R_{F},H^{j}(R_{F},\Pi^{\operatorname{la}}))\Rightarrow
H^{i+j}(P_{F}^{\dagger},\Pi^{\operatorname{la}}).$

On the other hand, the inclusion $\Pi^{\operatorname{la}}\hookrightarrow\Pi$
induces maps between the limit terms
$H^{i}(P_{F}^{\dagger},\Pi^{\operatorname{la}})\rightarrow
H^{i}(P_{F}^{\dagger},\Pi)$, as well as a map between all second page terms of
the spectral sequence

$\widetilde{E}_{2}^{i,j}=H^{i}(P_{F}^{\dagger}/R_{F},H^{j}(R_{F},\Pi^{\operatorname{la}}))\rightarrow
H^{i}(P_{F}^{\dagger}/R_{F},H^{j}(R_{F},\Pi))=E_{2}^{i,j}.$

We thus obtain a map of spectral sequences $\widetilde{E}_{r}^{i,j}\rightarrow
E_{r}^{i,j}.$

But since $R_{F}$ is compact, Corollary 1.6 in [6] tells us that these are
isomorphisms for $r=2$ and for all $i,j\geq 0$. By the Comparison Theorem
5.2.12 in [10], this implies that also the map of the limit terms

$H^{i}(P_{F}^{\dagger},\Pi^{\operatorname{la}})\rightarrow
H^{i}(P_{F}^{\dagger},\Pi)$

is an isomorphism. ∎

We can now prove Proposition 3.3.

###### Proof of Proposition 3.3.

The argumentation is similar to the one in the proof of Lemma 3.7. By the
discussion in Section 3.1, we obtain spectral sequences like in (2) for both,
$\Pi$ and $\Pi^{\operatorname{la}}$:

$\displaystyle\widetilde{E}_{1}^{i,j}=$ $\displaystyle\bigoplus_{F\in
R_{i}}H^{j}(P_{F}^{\dagger},\Pi^{\operatorname{la}}\otimes_{L}\delta_{F}^{-1})\Rightarrow
H^{i+j}(G,\Pi^{\operatorname{la}}),$ $\displaystyle E_{1}^{i,j}=$
$\displaystyle\bigoplus_{F\in
R_{i}}H^{j}(P_{F}^{\dagger},\Pi\otimes_{L}\delta_{F}^{-1})\Rightarrow
H^{i+j}(G,\Pi).$

But again, the inclusion $\Pi^{\operatorname{la}}\hookrightarrow\Pi$ induces
maps between all the terms. And by Lemma 3.7, those maps between the $E_{1}$
terms are isomorphisms. Theorem 5.2.12 in [10] implies that the maps

$H^{i}(G,\Pi^{\operatorname{la}})\rightarrow H^{i}(G,\Pi)$

are isomorphisms. ∎

######  Corollary 3.8.

For any admissible $L$-Banach space representation $\Pi$ of $G$, the
cohomology groups $H^{i}(G,\Pi^{\operatorname{la}})$ with coefficients in the
locally analytic vectors of $\Pi$ are finite dimensional over $L$.

###### Proof.

This follows immediately from Proposition 3.3 combined with the fact that the
continuous cohomology groups $H^{i}(G,\Pi)$ for an admissible $L$-Banach space
representation $\Pi$ is finite dimensional (cf. Corollary 6.10 in [5]). ∎

###  3.3 Topology on the cohomology groups

Let $V$ be a Hausdorff topological $G$-module. Note that the continuous
cohomology groups $H^{i}(G,V)$ inherit a quotient topology from the compact-
open topology on the cochains. We call this the _canonical topology_. The
compact-open topology on the cochains $C^{i}(G,V)$ is Hausdorff, since for any
pair of continuous maps $f,f^{\prime}:G^{i+1}\rightarrow V$ in $C^{i}(G,V)$
with $f\neq f^{\prime}$, we can find disjoint open neighborhoods of $f$,
$f^{\prime}$, respectively, as follows: There is an element $g\in G^{i+1}$
such that $f(g)\neq f^{\prime}(g)$ and since $V$ is Hausdorff by assumption,
we can find open disjoint neighborhoods $U\ni f(g)$, $U^{\prime}\ni
f^{\prime}(g)$, $U\cap U^{\prime}=\emptyset$. But then we have open disjoint
neighborhoods $\Omega(\\{g\\},U)\ni f$ and $\Omega(\\{g\\},U^{\prime})\ni
f^{\prime}$.

Since $C^{i}(G,V)$ is Hausdorff, so is $C^{i}(G,V)^{G}$ as a subspace of a
Hausdorff space. And the space of $i$-cocycles
$Z^{i}(G,V)=\ker(d^{i}:C^{i}(G,V)^{G}\rightarrow C^{i+1}(G,V)^{G})$ is also
Hausdorff and moreover, it is a closed subspace of $C^{i}(G,V)^{G}$ as it is
the kernel of a continuous map between Hausdorff spaces. This also implies
that the quotient topology on the $i$-coboundaries given by $B^{i}(G,V)\cong
C^{i-1}(G,V)^{G}/Z^{i-1}(G,V)$ is Hausdorff.

The quotient topology on the cohomology groups is in general not Hausdorff.
However, Proposition 6 in [3] gives a nice criterion for the cohomology groups
being Hausdorff, which almost fits our setting. We need to assume though that
the group $G$ is $\sigma$-compact, which means that it can be written as a
countable union of compact subsets. If $F$ is a finite extension of
$\mathbb{Q}_{p}$, then the group $\operatorname{GL}_{n}(F)$ is
$\sigma$-compact. This can be shown by using the Cartan decomposition (cf.
Section 3.2 in [1]).

As a consequence, we know that any $p$-adic reductive group $G=\mathbb{G}(F)$,
where $\mathbb{G}$ is a linear algebraic group, is also $\sigma$-compact,
since it embeds into $\operatorname{GL}_{n}(F)$ as a closed subgroup for some
$n$. We then obtain the following:

######  Corollary 3.9.

Assume that $G$ is a $p$-adic reductive group and $\Pi$ is an admissible
$L$-Banach space representation of $G$. Then the canonical topologies on the
cohomology groups $H^{i}(G,\Pi)$ and $H^{i}(G,\Pi^{\operatorname{la}})$ are
Hausdorff for all $i\geq 0$.

###### Proof.

Corollary 6.10 in [5] implies that the cohomology groups are finite
dimensional. Moreover, since $\Pi$ is assumed to be an $L$-Banach space
representation, it is a complete metrizable $L$-vector space. We can hence
apply Proposition 6 in [3], which states that the cohomology groups are
strongly Hausdorff, i.e. the inclusion of $i$-coboundaries $B^{i}(G,\Pi)$ into
$i$-cocycles $Z^{i}(G,\Pi)$ has a continuous section, say
$s:Z^{i}(G,\Pi)\rightarrow B^{i}(G,\Pi)$. We claim that the continuous
$L$-linear bijection

$\displaystyle\ker(s)\hookrightarrow
Z^{i}(G,\Pi)\overset{\pi}{\rightarrow}H^{i}(G,\Pi)$

is a homeomorphism. It suffices to show that the map is open. For this, let
$U\subset\ker(s)$ be an open subset. This means that there is an open subset
$V\subset Z^{i}(G,\Pi)$ such that $U=V\cap\ker(s)$. Since the map
$\operatorname{id}-s:Z^{i}(G,\Pi)\rightarrow Z^{i}(G,\Pi)$ is continuous, the
set $V^{\prime}:=(\operatorname{id}-s)^{-1}(V)=\\{z\in Z^{i}(G,\Pi)|z-s(z)\in
V\\}$ is open in $Z^{i}(G,\Pi)$. Moreover, the intersection

$\displaystyle V^{\prime}\cap\ker(s)$ $\displaystyle=\\{z\in
Z^{i}(G,\Pi)|z-s(z)\in V\text{ and }s(z)=0\\}$ $\displaystyle=\\{z\in
Z^{i}(G,\Pi)|s(z)=0\text{ and }z\in V\\}$ $\displaystyle=V\cap\ker(s)$

coincides with the intersection $V\cap\ker(s)=U$. But then we have

$\pi(U)=\pi(V\cap\ker(s))=\pi(V^{\prime}\cap\ker(s))=\pi(V^{\prime})\text{
open,}$

since for any $z\in V^{\prime}$, $\pi(z)=\pi(z)-\pi(s(z))=\pi(z-s(z))$ with
$z-s(z)\in V\cap\ker(s)$.

As the kernel of a continuous map between Hausdorff spaces, $\ker(s)$ is
Hausdorff and thus so is $H^{i}(G,\Pi)$.

By Proposition 3.3, the continuous $L$-linear map
$H^{i}(G,\Pi^{\operatorname{la}})\rightarrow H^{i}(G,\Pi)$ is a bijection.
This implies that the topology on $H^{i}(G,\Pi^{\operatorname{la}})$ is finer
than (or equal to) the topology on $H^{i}(G,\Pi)$ which is Hausdorff, so that
the groups $H^{i}(G,\Pi^{\operatorname{la}})$ are also Hausdorff. ∎

######  Lemma 3.10.

Let $V$ and $W$ be topological $L$-vector spaces and
$\phi:V\overset{\cong}{\rightarrow}W$ be an $L$-linear isomorphism. Assume
that $\mathcal{B}$ is a subbasis of open neighborhoods of $0\in V$, such that
the set $\phi(\mathcal{B})=\\{\phi(U)|\ U\in\mathcal{B}\\}$ is a subbasis of
open neighborhoods of $0\in W$. Then the isomorphism $\phi$ is a
homeomorphism.

###### Proof.

By symmetry, it is enough to show that the map $\phi$ is continuous. For this,
let $U\subset W$ be an open set. We show that $\phi^{-1}(U)$ is open in $V$,
by showing that each element $v\in\phi^{-1}(U)$ has an open neighborhood that
is contained in $\phi^{-1}(U)$.

Since $W$ is a topological vector space, the translated set $-\phi(v)+U$ is
again open in $W$ and contains the zero element. By assumption, we can find
finitely many open neighborhoods of $0$ in $V$,
$V_{1},\dots,V_{n}\in\mathcal{B}$, such that the intersection
$\bigcap_{i=1}^{n}\phi(V_{i})$ is contained in $-\phi(v)+U$. This implies that
the intersection of open sets $\bigcap_{i=1}^{n}(v+V_{i})$ is contained in the
inverse image $\phi^{-1}(U)$ and contains $v$. Therefore, the map is indeed
continuous. ∎

######  Proposition 3.11.

Let $V$ be a topological $G$-module on a locally convex $L$-vector space. Then
the compact-open topology on the space of cochains $C^{i}(G,V)$ is locally
convex, for every $i\geq 0$.

###### Proof.

By the heomeomorphism $C^{i}(G,V)\cong C^{0}(G,C^{i-1}(G,V))$ (cf. Lemma 2.1),
it is enough to show that $C(G,V)=C^{0}(G,V)$ is locally convex.

Say the locally convex topology on $V$ is defined by the family of seminorms
$\\{q_{i}\\}_{i\in I}$. For every compact subset $K\subset G$, and $i\in I$,
we define a seminorm on $C(G,V)$ by

$\displaystyle q_{K,i}:C(G,V)$ $\displaystyle\rightarrow\mathbb{R},$
$\displaystyle f$ $\displaystyle\mapsto\sup_{k\in K}q_{i}(f(k))$

Denote by $C(G,V)_{\operatorname{lc}}$ the space $C(G,V)$ equipped with the
locally convex topology given by the family $\\{q_{K,i}\\}_{K,i}$ of seminorms
and by $C(G,V)_{\operatorname{co}}$ the space $C(G,V)$ equipped with the
compact-open topology. We claim that the identity map
$C(G,V)_{\operatorname{co}}\rightarrow C(G,V)_{\operatorname{lc}}$ is a
homeomorphism.

By Lemma 3.10, it is enough to find a set of neighborhoods of $0\in C(G,V)$
that forms a subbasis of open neighborhoods in both topologies. By definition,
a subbasis of open neighborhoods of $0\in C(G,V)_{\operatorname{lc}}$ is given
by sets of the form $q_{K,i}^{-1}(B_{\epsilon})$, for $K\subset G$ compact,
$i\in I$, $\epsilon>0$. Here, $B_{\epsilon}$ denotes the open ball of radius
$\epsilon$ around $0\in\mathbb{R}$. We can rewrite these sets as follows:

$\displaystyle q_{K,i}^{-1}(B_{\epsilon})$ $\displaystyle=\\{f\in C(G,V)|\
q_{K,i}(f)<\epsilon\\}$ $\displaystyle=\\{f\in C(G,V)|\ \sup_{k\in
K}q_{i}(f(k))<\epsilon\\}$ $\displaystyle=\\{f\in C(G,V)|\
q_{i}(f(k))<\epsilon,\ \forall k\in K\\}$ $\displaystyle=\\{f\in C(G,V)|\
f(k)\in q_{i}^{-1}(B_{\epsilon})\ \forall k\in K\\}$
$\displaystyle=\Omega(K,q_{i}^{-1}(B_{\epsilon})).$

These sets are open neighborhoods of $0$ in $C(G,V)_{\operatorname{co}}$. We
want to show that they form a subbasis of open neighborhoods of $0$. Indeed,
by definition of the compact-open topology, a subbasis of open neighborhoods
of $0$ is given by sets of the form $\Omega(K,U)$ for $K\subset G$ compact and
$U\subset V$ open. But since $V$ is locally convex, we can find finitely many
open sets of the form $q_{i_{j}}^{-1}(B_{\epsilon_{j}})$, for $j=1,\dots,m$,
with the property that

$\bigcap_{j=1}^{m}q_{i_{j}}^{-1}(B_{\epsilon_{j}})\subset U.$

We thus obtain

$\bigcap_{j=1}^{m}\Omega(K,q_{i_{j}}^{-1}(B_{\epsilon_{j}}))\subset\Omega(K,U),$

and we see that the sets $\Omega(K,q_{i}^{-1}(B_{\epsilon}))$ form a subbasis
of open neighborhoods of $0$ in $C(G,V)_{\operatorname{co}}$ as stated. ∎

Since subspaces and quotients of locally convex vector spaces are again
locally convex (see Section 5 in [7]), Proposition 3.11 implies the following
corollary.

######  Corollary 3.12.

For every topological $G$-module on a locally convex $L$-vector space, the
canonical topology on the continuous cohomology groups $H^{i}(G,V)$ is locally
convex.

######  Corollary 3.13.

For every admissible $L$-Banach space representation $\Pi$ of $G$, the
continuous cohomology groups $H^{i}(G,\Pi)$ and
$H^{i}(G,\Pi^{\operatorname{la}})$, equipped with their canonical topologies,
are finite dimensional locally convex Hausdorff $L$-vector spaces. In
particular, their topology is uniquely determined, namely it is the finest
locally convex topology.

In particular, the isomorphisms $H^{i}(G,\Pi^{\operatorname{la}})\cong
H^{i}(G,\Pi)$ are homeomorphisms.

###### Proof.

By Corollary 6.10 in [5] and Corollaries 3.9 and 3.12, we know that the
cohomology groups are finite dimensional, Hausdorff and locally convex.
Therefore, we can apply Proposition 4.13 in [7], which proves that there is
only one choice of locally convex Hausdorff topology on a finite dimensional
$L$-vector spaces. The author gives an explicit description of this topology
as the one defined by the norm
$\|\sum_{i=1}^{n}\lambda_{i}e_{i}\|:=\max_{1\leq i\leq n}|\lambda_{i}|$, for
any choice of $L$-basis $e_{1},\dots,e_{n}$ of the vector space.

Since the canonical topologies on $H^{i}(G,\Pi^{\operatorname{la}})$ and
$H^{i}(G,\Pi)$ are uniquely determined and the groups are isomorphic as
$L$-vector spaces (Proposition 3.3), the topologies must agree and the
isomorphism is a homeomorphism of topological $L$-vector spaces. ∎

## References

  * [1] I. N. Bernšteĭn and A. V. Zelevinskiĭ, Representations of the group $GL(n,F),$ where $F$ is a local non-Archimedean field, Uspekhi Mat. Nauk, 31 (1976), pp. 5–70.
  * [2] N. Bourbaki, Éléments de mathématique. X. Première partie: Les structures fondamentales de l’analyse. Livre III: Topologie générale. Chapitre X: Espaces fonctionnels; dictionnaire, Actualités Scientifiques et Industrielles [Current Scientific and Industrial Topics], No. 1084, Hermann et Cie., Paris, 1949.
  * [3] W. Casselman and D. Wigner, Continuous cohomology and a conjecture of Serre’s, Invent. Math., 25 (1974), pp. 199–211.
  * [4] M. Emerton, Locally analytic vectors in representations of locally $p$-adic analytic groups, vol. 248, American Mathematical Society, 2017.
  * [5] P. Fust, Continuous group cohomology and Ext-groups, Münster Journal of Mathematics, 15 (2022), pp. 279–304.
  * [6] J. Rodrigues Jacinto and J. Rodríguez Camargo, Solid locally analytic representations of $p$-adic Lie groups, Representation Theory of the American Mathematical Society, 26 (2022), pp. 962–1024.
  * [7] P. Schneider, Nonarchimedean functional analysis, Springer Science & Business Media, 2001.
  * [8] P. Schneider and U. Stuhler, Representation theory and sheaves on the Bruhat-Tits building, Inst. Hautes Études Sci. Publ. Math., (1997), pp. 97–191.
  * [9] P. Schneider and J. Teitelbaum, Algebras of p-adic distributions and admissible representations, Inventiones mathematicae, 153 (2002), pp. 145–196.
  * [10] C. A. Weibel, An introduction to homological algebra, vol. 38 of Cambridge Studies in Advanced Mathematics, Cambridge University Press, Cambridge, 1994.