#  $S$-arithmetic (co)homology and $p$-adic automorphic forms

Guillem Tarrach

###### Abstract.

We study the $S$-arithmetic (co)homology of reductive groups over number
fields with coefficients in (duals of) certain locally algebraic and locally
analytic representations for finite sets of primes $S$. We use our results to
construct eigenvarieties associated to parabolic subgroups at places in $S$
and certain classes of supercuspidal and algebraic representations of their
Levi factors. We show that these agree with eigenvarieties constructed using
overconvergent homology and that for definite unitary groups they are closely
related to the Bernstein eigenvarieties constructed by Breuil-Ding in [BD21].

###### Contents

  1. 1 Introduction
  2. 2 The structure of $p$-adic reductive groups
  3. 3 A vanishing criterion for Tor
  4. 4 The locally analytic representations
  5. 5 $S$-arithmetic (co)homology
  6. 6 Eigenvarieties
  7. A Orlik-Strauch functors

##  1\. Introduction

Let $p$ be a prime number. One notable feature of the theory of $p$-adic
automorphic forms is that it has more than one incarnation, depending on what
space of $p$-adic automorphic forms one is working with. Two of the most
successful constructions of such spaces are completed and overconvergent
cohomology. As their name might suggest, these are defined in different ways
using cohomology, more specifically the cohomology spaces of arithmetic groups
such as $\text{GL}_{n}(\mathbb{Z})$. These contain important arithmetic
information and are deeply connected to automorphic forms and representations:
if $\mathbf{G}$ is a reductive group over a number field $F$, the cohomology
of an arithmetic subgroup of $\mathbf{G}(F)$ with coefficients in a finite-
dimensional representation of $\mathbf{G}(F\otimes_{\mathbb{Q}}\mathbb{C})$
can be computed in terms of automorphic representations of
$\mathbf{G}(\mathbb{A}_{F})$.

In this article, we study a different construction involving instead the
cohomology of $S$-arithmetic groups (such as $\text{GL}_{n}(\mathbb{Z}[1/N])$
for a positive integer $N$) with coefficients in certain representations of
the group at primes in a finite set $S$. It is more convenient to work with
cohomology from the adelic point of view (as opposed to working directly with
$S$-arithmetic groups), so let us fix a level $K^{S}$ away from $S$, i.e. an
open compact subgroup of $\mathbf{G}(\mathbb{A}_{F}^{S,\infty})$, and assume
it is neat. The $K^{S}$-(co)homology groups below will be isomorphic to direct
sums of (co)homology groups of $S$-arithmetic groups. It is well-known that
the cohomology of such groups with coefficients in finite-dimensional
representations of $\mathbf{G}(F\otimes_{\mathbb{Q}}\mathbb{C})$ is finite-
dimensional and can also be computed in terms of automorphic representations
(cf. [BW00, Chapter XIV]). The following result gives an extension of this to
an infinite-dimensional setting.

######  Theorem 1.1.

Let $\mathbf{G}$ be a reductive group over a number field $F$. Set $S$ be a
finite set of primes of $F$. For each $v\in S$ let $\mathbf{P}_{v}$ be a
parabolic subgroup of
$\mathbf{G}_{v}:=\operatorname{Spec}(F_{v})\times_{\operatorname{Spec}(F)}\mathbf{G}$
with Levi factor $\mathbf{M}_{v}$. For all $v\in S$, let $\omega_{v}$ be a
supercuspidal representation of $\mathbf{M}_{v}(F_{v})$ that is compactly
induced from a finite-dimensional representation of an open compact-mod-center
subgroup, and let $W$ be an irreducible finite-dimensional algebraic
representation of $\mathbf{G}(F\otimes_{\mathbb{Q}}\mathbb{C})$ over
$\mathbb{C}$. Consider the $S$-arithmetic cohomology

$H^{*}\left(K^{S},\left(W\boxtimes_{\mathbb{C}}\operatorname*{\scalerel*{\boxtimes}{\sum}}_{v\in
S}\operatorname{Ind}_{\mathbf{P}_{v}(F_{v})}^{\mathbf{G}(F_{v})}(\omega_{v})^{\text{sm}}\right)^{\prime}\right),$

where
$\operatorname{Ind}_{\mathbf{P}_{v}(F_{v})}^{\mathbf{G}(F_{v})}({-})^{\text{sm}}$
denotes smooth parabolic induction and $({-})^{\prime}$ denotes the abstract
dual. This cohomology space is finite-dimensional and admits a decomposition
into _cuspidal_ and _Eisenstein_ direct summands, the first of which can be
described in terms of cuspidal automorphic representations for
$\mathbf{G}(\mathbb{A}_{F})$ satisfying certain conditions with respect to the
data $(\mathbf{P}_{v},\mathbf{M}_{v},\omega_{v})_{v\in S}$ and $W$.

See Theorem 5.21 for a more precise statement. One example of the conditions
that the cuspidal automorphic representations appearing in the cohomology
groups must satisfy is that its $v$-component must have supercuspidal support
equal to $\omega_{v}$ for all $v\in S$. The condition on the supercuspidal
representations $\omega_{v}$ in Theorem 1.1 is expected to be satisfied for
all supercuspidal representations of reductive groups in this setting, and
this is known for example when $\mathbf{M}_{v}$ is a product of terms of the
following forms:

  * •

A general linear group (cf. [BK93]) or an inner form of it (cf. [SS08]),

  * •

A unitary, symplectic or special orthogonal group, if $v$ does not lie above 2
(cf. [Ste08]),

  * •

A tamely ramified group, if $v$ does not divide the order of its Weyl group
(cf. [Fin21]).

In fact, an analogue of Theorem 1.1 can be proven even if the $\omega_{v}$ are
not supercuspidal, but we will only be interested in this case. Thus, for the
rest of the introduction we will assume that the hypotheses of Theorem 1.1 are
satisfied. By using well-known results of Zelevinsky [Zel80], the following
theorem can also be deduced from Theorem 1.1.

######  Theorem 1.2.

If for all $v\in S$, $\mathbf{G}_{v}$ is isomorphic to $\text{GL}_{n/F_{v}}$,
then for all finite-length smooth representations $\pi$ of $\prod_{v\in
S}\mathbf{G}_{v}(F_{v})$, the cohomology
$H^{*}(K^{S},(W\otimes_{\mathbb{C}}\pi)^{\prime})$ is finite-dimensional.

In order to study $p$-adic automorphic forms, we are led to consider similar
cohomology groups as we vary some of the supercuspidal data $p$-adically. In
order to make sense of this, we have to make a few changes. Fixing an
algebraic closure $\overline{\mathbb{Q}}_{p}$ of $\mathbb{Q}_{p}$ and an
isomorphism
$\overline{\mathbb{Q}}_{p}\stackrel{{\scriptstyle\sim}}{{\longrightarrow}}\mathbb{C}$,
we may change the field of coefficients from $\mathbb{C}$ to
$\overline{\mathbb{Q}}_{p}$, and in fact to some finite Galois extension $E$
of $\mathbb{Q}_{p}$ in $\overline{\mathbb{Q}}_{p}$ which we may assume is
large enough to split $\mathbf{M}_{v}$ for all $v\mid p$. Let $\mathbf{N}_{v}$
be the unipotent radical of $\mathbf{P}_{v}$. If $\Sigma\subseteq S$ is a
subset consisting only of places dividing $p$, then we will replace the smooth
induction by locally ($\mathbb{Q}_{p}$-)analytic induction at the places in
$\Sigma$. We may also view $W$ as an algebraic representation of
$\mathbf{G}(F\otimes_{\mathbb{Q}}\mathbb{Q}_{p})\subseteq\mathbf{G}(F\otimes_{\mathbb{Q}}\overline{\mathbb{Q}}_{p})$.
The representation $W$ is then a tensor product of irreducible algebraic
representations $W_{v}$ for $v\mid p$, and we will write $U$ for the tensor
product of those corresponding to places outside $\Sigma$. If
$\delta_{v}\colon\mathbf{M}_{v}(F_{v})\longrightarrow E^{\times}$ is a
continuous character for all $v\in\Sigma$, then we may consider the
$K^{S}$-cohomology of the _continuous_ dual of

(1.3) $\displaystyle
U\boxtimes_{E}\left(\widehat{\operatorname*{\scalerel*{\boxtimes}{\sum}}}_{v\in\Sigma}\operatorname{Ind}_{\mathbf{P}_{v}(F_{v})}^{\mathbf{G}(F_{v})}(W_{v}^{\mathbf{N}_{v}}\otimes\omega_{v}\otimes\delta_{v})^{\text{la}}\right)\widehat{\boxtimes}_{E}\left(\operatorname*{\scalerel*{\boxtimes}{\sum}}_{v\in
S\smallsetminus\Sigma}\operatorname{Ind}_{\mathbf{P}_{v}(F_{v})}^{\mathbf{G}(F_{v})}(\omega_{v})^{\text{sm}}\right),$

where now
$\operatorname{Ind}_{\mathbf{P}_{v}(F_{v})}^{\mathbf{G}(F_{v})}({-})^{\text{la}}$
denotes locally ($\mathbb{Q}_{p})$-analytic induction. In order to shorten
notation, let us set
$V_{\delta}=(\operatorname*{\scalerel*{\boxtimes}{\sum}}_{v\in\Sigma}W_{v}^{\mathbf{N}_{v}}\otimes\omega_{v}\otimes\delta_{v})\boxtimes(\operatorname*{\scalerel*{\boxtimes}{\sum}}_{v\in
S\smallsetminus\Sigma}\omega_{v})$, and $\operatorname{\Sigma\hskip
1.13809pt\text{-Ind}}_{P}^{G}(U,V_{\delta})$ for 1.3. If $U$ is trivial, the
groups $\mathbf{G}(F_{v})$ act smoothly on this representation for $v\in
S\smallsetminus\Sigma$ and locally analytically when $v\in\Sigma$, and for
brevity we will refer to it as a locally $\Sigma$-analytic representation of
$G:=\prod_{v\in S}\mathbf{G}(F_{v})$. By duality, studying the cohomology of
the dual of $\operatorname{\Sigma\hskip
1.13809pt\text{-Ind}}_{P}^{G}(U,V_{\delta})$ is equivalent to studying the
homology of $\operatorname{\Sigma\hskip
1.13809pt\text{-Ind}}_{P}^{G}(U,V_{\delta})$. The same methods that prove the
finite dimensionality in Theorem 1.1 show that this is finite-dimensional, and
can be used to give a criterion for its vanishing. Again, this also holds for
general representations that are compactly induced from open compact-mod-
center subgroups, without the supercuspidality assumption. It is also
interesting to study the $S$-arithmetic (co)homology of more general locally
$\Sigma$-analytic representations of $G$. A good source of such
representations are the Orlik-Strauch functors $\mathcal{F}_{P}^{G}$. We show
that the $S$-arithmetic homology of many of the representations in the image
of these functors is also finite-dimensional. Under the assumptions of Theorem
1.2, we can also show finite-dimensionality for _all_ such representations
when they are of finite length, and we show that this condition is necessary.

Using this to construct $p$-adic families of ($p$-adic) automorphic forms
implies checking that two conditions are satisfied. First, we must be able to
relate (co)homology with coefficients in locally $\Sigma$-analytic inductions
with (co)homology with coefficients in locally algebraic induction (i.e. those
in Theorem 1.1). This is addressed by the following result.

######  Theorem 1.4.

Assume that for all $v\in\Sigma$, the action of the maximal split torus in the
center of $\mathbf{M}_{v}$ acts numerically non-critically on $U\otimes V_{1}$
in the sense of Definition 5.24. Then, the inclusion of locally algebraic
induction into locally $\Sigma$-analytic induction induces an isomorphism

$H_{*}(K^{S},W\otimes\operatorname*{\scalerel*{\boxtimes}{\sum}}_{v\in
S}\operatorname{Ind}_{\mathbf{P}_{v}(F_{v})}^{\mathbf{G}(F_{v})}(\omega_{v})^{\text{sm}})\stackrel{{\scriptstyle\sim}}{{\longrightarrow}}H_{*}(K^{S},\operatorname{\Sigma\hskip
1.13809pt\text{-Ind}}_{P}^{G}(U,V_{1}))$

(and similarly for the cohomology of their duals).

Secondly, we must show that the homology groups behave nicely as we vary the
characters $\delta$ in rigid-analytic families. We show that this is true and
use this to construct eigenvarieties, i.e. rigid-analytic varieties
interpolating the automorphic representations appearing in Theorem 1.1 in the
setting of Theorem 1.4. More precisely, we have the following result.

######  Theorem 1.5.

Fix $W$ and $\omega=(\omega_{v})_{v\in S}$ as above. There exists a rigid-
analytic space $\mathcal{W}_{\omega}$ parametrizing locally analytic twists of
$(\omega_{v})_{v\in\Sigma}$ and a rigid-analytic space
$\mathscr{Y}_{K^{s},W,\omega}^{\Sigma}$ over $\mathcal{W}_{\omega}$ whose
fiber along a twist $(\omega_{v}\otimes\delta_{v})_{v\in\Sigma}$ is
canonically in bijection with the set of systems of eigenvalues for the action
of a certain Hecke algebra on $H_{*}(K^{S},\operatorname{\Sigma\hskip
1.13809pt\text{-Ind}}_{P}^{G}(U,V_{\delta}))$.

We also compare these eigenvarieties to previously constructed ones. If $S$ is
the set of places lying above $p$, and if for all $v\in S$, $\mathbf{G}_{v}$
is split and $\mathbf{P}_{v}$ is a Borel subgroup, then our construction
agrees with the construction of Hansen [Han17], which is built using
overconvergent cohomology (or more precisely, it agrees with the analogous
construction for overconvergent homology), and in the case where
$\mathbf{P}_{v}$ is a bigger parabolic our construction also generalizes that
of Barrera Salazar-Williams [SW21] (although the corresponding $\omega_{v}$
will not be supercuspidal). In the setting of [BD21] we also compare our
eigenvarieties to the (non-patched) Bernstein eigenvarieties constructed by
Breuil-Ding in loc. cit., which are defined using completed cohomology, and
show that the underlying reduced varieties are essentially the same. The first
comparison follows naturally from our methods, but the second requires the use
of $p$-adic interpolation theorems and adjunction theorems for Emerton’s
locally analytic Jacquet functors.

###  1.1. Methods

Our strategy for proving Theorem 1.1 and its analogue for locally
$\Sigma$-analytic representations are based on the work of Kohlhaase-Schraen
[KS12], who proved finite-dimensionality and a vanishing criterion for locally
analytic induction in a special case, namely when $F=\mathbb{Q}$,
$\mathbf{G}(\mathbb{R})$ is compact, $S=\\{p\\}$, $\mathbf{G}_{p}$ is split,
semisimple and adjoint and $\mathbf{P}_{p}$ is a Borel subgroup. Let us give
an outline of this strategy. By our compact induction assumption on
$\omega_{v}$, $W_{v}^{\mathbf{N}_{v}}\otimes\omega_{v}\otimes\delta_{v}$ is
isomorphic to a compact induction
$\operatorname{c-Ind}_{\mathcal{J}_{v,M}}^{\mathbf{M}_{v}(F_{v})}(\sigma_{v})$
for each $v\in S$, where $\mathcal{J}_{v,M}$ is a (not too small) compact-mod-
center subgroup of $\mathbf{M}_{v}(F_{v})$ and $\sigma_{v}$ is finite-
dimensional. The main idea is that one can relate the $S$-arithmetic homology
of locally $\Sigma$-analytic induction to (a twisted version of)
overconvergent homology $H_{*}(K^{S}J,\mathcal{A}^{\Sigma}_{\sigma,U})$. Here,
$J$ is a certain compact open subgroup of $G$ and
$\mathcal{A}^{\Sigma}_{\sigma,U}$ is a locally $\Sigma$-analytic
representation of $J$ (and in fact, of a larger monoid). Shapiro’s lemma shows
that there is an isomorphism between
$H_{*}(K^{S}J,\mathcal{A}^{\Sigma}_{\sigma,U})$ and the $S$-arithmetic
homology
$H_{*}(K^{S},\operatorname{c-Ind}_{J}^{G}(\mathcal{A}^{\Sigma}_{\sigma,U}))$.
The representation
$\operatorname{c-Ind}_{J}^{G}(\mathcal{A}^{\Sigma}_{\sigma,U})$ of $G$ is also
naturally a module for a Hecke algebra $\mathcal{H}_{S,E}^{-}$, which can also
be made to act trivially (in some sense) on $\operatorname{\Sigma\hskip
1.13809pt\text{-Ind}}_{P}^{G}(U^{\Sigma},V_{\delta})$ and on the coefficient
field $E$. We write $E(1)$ for the field $E$ equipped with this trivial action
of $\mathcal{H}_{S,E}^{-}$. The key technical result is the following.

######  Theorem 1.6.

There is a $G$-equivariant isomorphism in the derived category of
$\mathcal{H}_{S,E}^{-}$-modules

$E(1)\otimes^{\mathbb{L}}_{\mathcal{H}_{S,E}^{-}}\operatorname{c-Ind}_{J}^{G}(\mathcal{A}^{\Sigma}_{\sigma,U})\stackrel{{\scriptstyle\sim}}{{\longrightarrow}}\operatorname{\Sigma\hskip
1.13809pt\text{-Ind}}_{P}^{G}(U,V_{\delta}).$

One can formally deduce from this that a similar statement holds after
replacing these representations of $G$ by their $S$-arithmetic homology. A
rough way of stating this is to say that the $S$-arithmetic homology of
$\operatorname{\Sigma\hskip 1.13809pt\text{-Ind}}_{P}^{G}(U,V_{\delta})$ can
be identified with (derived) eigenspaces of 1 for the action of a Hecke
algebra at $S$ on this twisted version of overconvergent homology. In some
cases it is possible to “untwist” this action, and the identification is with
systems of eigenvalues in the usual overconvergent homology spaces that can be
read off from the action of the compact-mod-center groups $\mathcal{J}_{v,M}$
from which we are compactly inducing the $\sigma_{v}$. For example, if
$\mathbf{G}_{v}$ splits over $F_{v}$ and $\mathbf{P}_{v}$ is a Borel subgroup
for all $v$, then $\mathcal{H}_{S,E}^{-}$ has a basis consisting of Hecke
operators associated to antidominant cocharacters of some fixed maximal split
tori of the $\mathbf{G}(F_{v})$; in this case one can take $\mathcal{J}_{v,M}$
to be the group of $F_{v}$-points of these tori, and if $\sigma_{v}$ is
irreducible (i.e. a character), a choice of uniformizer $\varpi_{v}$ of
$F_{v}$ gives one such untwisting, and the eigenvalue of the Hecke operator
corresponding to a cocharacter $\beta$ is $\sigma_{v}(\beta(\varpi_{v}))$.

We use this identification to prove Theorem 1.1, as well as finite-
dimensionality in the locally $\Sigma$-analytic case, the vanishing criterion
we alluded to above, and the properties for representations in the image of
Orlik-Strauch functors. Theorem 1.4 can be deduced without much difficulty
from these properties and the properties of the BGG category $\mathcal{O}$.
This is an example of one of the advantages that the $S$-arithmetic approach
has over overconvergent (co)homology. In the latter setting, the analogue of
Theorem 1.4 is proven in [Urb11] and in [SW21] by constructing an ad hoc
version for locally analytic representations of $J$ of BGG resolutions in the
category $\mathcal{O}$. In the $S$-arithmetic setting, however, where we can
work with representations of the whole group $G$, the existence of such a
resolution of locally analytic representations can be deduced directly from
the BGG resolution in $\mathcal{O}$ by applying the Orlik-Strauch functors to
it.

Regarding $S$-arithmetic homology in families, the obvious reformulation of
Theorem 1.6 when the character $\delta$ takes values in affinoid algebras
remains true (and is also true for more general families of compactly induced
representations than the $\sigma_{v}\otimes\delta_{v}$). This again allows us
to relate our $S$-arithmetic homology to overconvergent homology, but it is
not as straightforward to deduce finiteness theorems from this relation, at
least without assuming extra hypotheses (such as the existence of slope
decompositions). However, we can use a spectral theorem for compact operators
on orthonormalizable Banach algebras to show these statements, and also to
show that it is possible to glue these $S$-arithmetic homology groups for
different affinoid algebras. This allows us to construct eigenvarieties
directly over the character space of the group
$\prod_{v\in\Sigma}\mathbf{M}_{v}(F_{v})$ and not via Fredholm hypersurfaces
over weight space, in a similar way to the constructions that use completed
cohomology and Emerton’s locally analytic Jacquet functor (cf. [Eme06a],
[Eme06b]). In fact, the spectral theorem we use is a variant of the spectral
theorem underlying the construction of locally analytic Jacquet modules. It is
often the case when studying eigenvarieties that one needs to rely on Fredholm
hypersurfaces in order to prove meaningful properties, and this situation is
no different. To do this, we also generalize the usual construction of
eigenvarieties using Fredholm hypersurfaces and overconvergent homology to our
more general setting and relate it to our other construction. It is important
to note that in order to make this work we have to work with the untwisted
(i.e. the usual) version of overconvergent homology mentioned above.

###  1.2. Organization of the article

In the first section, Section 2, we will collect most of the results on the
structure of $p$-adic reductive groups that we will need in the rest of the
article (especially in Section 4) putting a slight emphasis on some rigid-
analytic aspects. Next, in Section 3 we will write down some of the properties
of the Tor groups that will become ubiquitous in the rest of the article.
Especially, we will prove a vanishing criterion that underlies part of the
proof of Theorem 1.6. Section 4 contains the rest of the local part of the
argument: we will define the relevant locally ($\Sigma$-)analytically induced
representations and prove Theorem 1.6. In Section 5 we will move on to study
$S$-arithmetic homology and its relation to arithmetic homology. In it, we
prove Theorem 1.1, Theorem 1.2 and Theorem 1.4 among other results on these
$S$-arithmetic (co)homology groups. Finally, in Section 6 we construct
eigenvarieties, study some of their properties (in particular, proving Theorem
1.5), and compare them to previous constructions.

###  1.3. Acknowledgements

I am especially grateful to Jack Thorne for suggesting these topics and for
his continuous and invaluable help during the development of this article. I
am also thankful to David Schwein for helpful comments on an earlier version
of the manuscript. This research received funding from the European Research
Council (ERC) under the European Union’s Horizon 2020 research and innovation
programme (grant agreement No 714405).

###  1.4. Notation

If $C_{\bullet}$ is a chain complex, we write
$H_{*}(C_{\bullet})=\bigoplus_{i\in\mathbb{Z}}H_{i}(C_{\bullet})$ and
similarly for cohomology. If $V$ is a locally convex vector space over a
finite extension of $\mathbb{Q}_{p}$, we will write $V^{\prime}$ for its
continuous dual, which we always equip with its strong topology. All
topological tensor products are projective tensor products. By locally
analytic, we always mean locally $\mathbb{Q}_{p}$-analytic. We will usually
use the same symbols to denote representations of a group and the underlying
vector spaces or modules.

All rigid-analytic spaces appearing in this article are Tate rigid-analytic
spaces. If $A$ is an affinoid algebra over a finite extension $E$ of
$\mathbb{Q}_{p}$, we will write $\text{Sp}(A)$ for the corresponding affinoid
rigid $E$-analytic space. If $X$ is a rigid $E$-analytic space, we will write
$\mathcal{O}(X)$ to denote the algebra of global $E$-analytic functions on
$X$.

If $\mathbf{H}$ is an algebraic group over $E$, we will write
$H:=\mathbf{H}(E)$ for its groups of $E$-points, and similarly for all other
letters. We will write $X^{*}(\mathbf{H})$ (resp. $X_{*}(\mathbf{H})$) for the
group of $E$-rational characters (resp. cocharacters) of $\mathbf{H}$, or
$X^{*}_{E}(\mathbf{H})$ (resp. $X_{*}^{E}(\mathbf{H})$) if we want to
emphasize the field $E$. If $F$ is a field extension of $E$, we will write
$\mathbf{H}_{F}$ or $F\times_{E}\mathbf{H}$ for the base change of
$\mathbf{H}$ from $E$ to $F$. Similarly, if $F$ is a subfield of $E$ we will
write $\operatorname{Res}_{E/F}\mathbf{H}$ for the restriction of scalars.

##  2\. The structure of $p$-adic reductive groups

In this section we will recall some facts about reductive groups over $p$-adic
local fields. Throughout this section, except in Section 2.9, we let $L$ be a
finite extension of $\mathbb{Q}_{p}$ with normalized valuation $v\colon
L^{\times}\twoheadrightarrow\mathbb{Z}$. We normalize the absolute value
$|{-}|$ of $L$ so that it sends a uniformizer to $q^{-1}$, where $q$ is the
cardinality of the residue field of $L$.

###  2.1. Basic notation

Let $\mathbf{G}$ be a connected reductive group over $L$, $\mathbf{S}$ a
maximal $L$-split torus of $\mathbf{G}$, $\bm{\mathfrak{Z}}$ (resp.
$\bm{\mathfrak{N}}$) its centralizer (resp. normalizer) in $\mathbf{G}$, and
$\mathfrak{Z}=\bm{\mathfrak{Z}}(L)$ (resp.
$\mathfrak{N}=\bm{\mathfrak{N}}(L)$). Let $\Phi=\Phi(\mathbf{G},\mathbf{S})$
be the relative root system of $\mathbf{G}$ with respect to $\mathbf{S}$ and
$W$ be its Weyl group, which is naturally isomorphic to
$\mathfrak{N}/\mathfrak{Z}$. If $\alpha\in\Phi$ is a root, we will write
$s_{\alpha}\in W$ for the corresponding simple reflection and
$\mathbf{U}_{\alpha}$ for the corresponding root subgroup of $\mathbf{G}$. We
fix an ordering of the roots of $\Phi$ and let $\Phi^{+}$ (resp. $\Phi^{-}$)
be the subset of positive (resp. negative) roots and $\Delta\subseteq\Phi^{+}$
the corresponding root basis. Write $\Phi^{\text{ndiv}}$ for the set of non-
divisible roots in $\Phi$, i.e. the set of $\alpha\in\Phi$ such that
$\frac{1}{2}\alpha\not\in\Phi$, and
$\Phi^{\text{ndiv},+}=\Phi^{\text{ndiv}}\cap\Phi^{+}$ (resp.
$\Phi^{\text{ndiv},-}=\Phi^{\text{ndiv}}\cap\Phi^{-}$).

Fix a subset $\Delta_{M}\subseteq\Delta$. The subgroup $\mathbf{P}$ of
$\mathbf{G}$ generated by $\bm{\mathfrak{Z}}$ and the root subgroups
$\mathbf{U}_{\alpha}$, where $\alpha\in\Phi$ runs through the positive roots
and linear combinations of elements of $\Delta_{M}$, is a parabolic subgroup
containing $\mathbf{S}$, and all parabolic subgroups of $\mathbf{G}$ arise
this way (for possibly different choices of $\mathbf{S}$ and orderings of the
root system). Let $\mathbf{A}$ be the identity component of
$\bigcap_{\alpha\in\Delta_{M}}\ker(\alpha)\subseteq\mathbf{S}$ and
$\mathbf{M}$ its centralizer in $\mathbf{G}$. Then, $\mathbf{A}$ is the
maximal split torus in the center of $\mathbf{M}$, which is a Levi subgroup of
$\mathbf{P}$ containing $\mathbf{S}$. If $\mathbf{N}$ denotes the unipotent
radical of $\mathbf{P}$, then $\mathbf{P}$ admits a Levi decomposition
$\mathbf{P}=\mathbf{M}\ltimes\mathbf{N}$. We will write
$\overline{\mathbf{P}}$ for the opposite parabolic subgroup to $\mathbf{P}$
and $\overline{\mathbf{N}}$ for its unipotent radical.

The root subsystem $\Phi_{M}$ of $\Phi$ consisting of linear combinations of
elements of $\Delta_{M}$ is isomorphic to the root system of $\mathbf{M}$ with
respect to $\mathbf{S}$, and $\Delta_{M}$ forms a root basis for the induced
ordering of $\Phi_{M}$. Set $\Phi^{+}_{M}=\Phi^{+}\cap\Phi_{M}$, and define
$\Phi^{-}_{M},\Phi^{\text{ndiv},+}_{M}$ and $\Phi^{\text{ndiv},-}_{M}$
similarly. The Weyl group of $\Phi_{M}$ is isomorphic to the subgroup $W_{M}$
of $W$ generated by the simple reflections $s_{\alpha}$ with
$\alpha\in\Delta_{M}$.

The inclusion $\mathbf{S}\subseteq\bm{\mathfrak{Z}}$ induces an inclusion
$X^{*}(\bm{\mathfrak{Z}})\subseteq X^{*}(\mathbf{S})$ of finite index. There
is a homomorphism $\text{ord}\colon\mathfrak{Z}\longrightarrow
X_{*}(\mathbf{S})\otimes_{\mathbb{Z}}\mathbb{Q}$ sending $t$ to the
cocharacter $\text{ord}(t)$ defined by
$\langle\alpha,\text{ord}(t)\rangle=v(\alpha(t))$ for $\alpha\in
X^{*}(\bm{\mathfrak{Z}})$. The kernel $\mathfrak{Z}_{0}$ of this map is the
maximal compact open subgroup of $\mathfrak{Z}$, and
$S_{0}=S\cap\mathfrak{Z}_{0}$ and $A_{0}=A\cap\mathfrak{Z}_{0}$ are the
maximal compact subgroups of $S$ and $A$ respectively. The quotient $A/A_{0}$
is a free abelian group whose rank is the rank of $\mathbf{A}$. We fix once
and for all a section $A/A_{0}\longrightarrow A$ of the natural projection,
and call its image $\Lambda$.

Most of the results of this section are consequences of standard facts from
Bruhat-Tits theory [BT72], [BT84]. We refer the unfamiliar reader to [Tit79]
for a survey of the topic, and we also note that many of the properties we
will use are contained in the first two sections of [Mor99]. Let
$\mathscr{B}=\mathscr{B}(\mathbf{G},L)$ be the (enlarged) Bruhat-Tits building
of $\mathbf{G}$ over $L$. Recall that this is the product of the Bruhat-Tits
buildings of the adjoint group $\mathbf{G}^{\text{ad}}$ of $\mathbf{G}$ (that
is, the reduced building of $\mathbf{G}$) and of the cocenter
$\mathbf{G}/\mathbf{G}^{\text{der}}$ of $\mathbf{G}$, where
$\mathbf{G}^{\text{der}}$ is the derived subgroup of $\mathbf{G}$. Let
$\mathscr{A}=\mathscr{A}(\mathbf{G},\mathbf{S},L)$ be the apartment of
$\mathscr{B}$ corresponding to $\mathbf{S}$; this is an affine space under the
vector space $X_{*}(\mathbf{S})\otimes_{\mathbb{Z}}\mathbb{R}$ with an action
of $\mathfrak{N}/\mathfrak{Z}_{0}$. Let us also write $\mathscr{A}_{M}$ for
the apartment corresponding to $\mathbf{S}$ of the Bruhat-Tits building of
$\mathbf{M}$. There is a natural map
$\mathscr{A}\longrightarrow\mathscr{A}_{M}$ which is an isomorphism of affine
spaces (cf. [BT72, 7.6.4], [BT84, 4.2.18]). Let $\mathscr{F}_{M}$ be a facet
of $\mathscr{A}_{M}$ and $J_{M}$ be its pointwise stabilizer in $M$. The
preimage of $\mathscr{F}_{M}$ in $\mathscr{A}$ is a union of facets. Fix a
maximal facet $\mathscr{F}$ in this preimage, a chamber $\mathscr{C}$ whose
closure $\overline{\mathscr{C}}$ contains $\mathscr{F}$ and a special point
$x_{0}\in\overline{\mathscr{C}}$. Let $K$ be the stabilizer in $G$ of $x_{0}$,
and write $J$ for the pointwise stabilizer of $\mathscr{F}$. In particular,
$J_{M}=J\cap M$ (by [Mor99, Lemma 1.13]), and
$J\cap\mathfrak{Z}=K\cap\mathfrak{Z}=J_{M}\cap\mathfrak{Z}=\mathfrak{Z}_{0}$.
We also let $I$ be the pointwise stabilizer in $G$ of $\mathscr{C}$ and
$\widetilde{J}_{M}$ the normalizer of $J_{M}$ in $M$.

There exists a maximal torus $\mathbf{T}$ in $\bm{\mathfrak{Z}}$ defined over
$L$ containing $\mathbf{S}$ which splits over a finite Galois extension
$L^{\prime}/L$. For example, if $\mathbf{G}$ is quasi-split we may take
$\mathbf{T}=\bm{\mathfrak{Z}}$. We normalize the valuation and absolute value
of $L^{\prime}$ so that it extends those of $L$, and we also denote it by $v$
and $|{-}|$. Let $\Phi^{\prime}$ be the root system of the pair
$(\mathbf{G},\mathbf{T})$ and define $\Phi^{\prime}_{M}$ in the same way as
$\Phi_{M}$. We have restriction maps
$X^{*}_{L^{\prime}}(\mathbf{T})\longrightarrow X^{*}_{L}(\mathbf{S})$ and
$\Phi^{\prime}\longrightarrow\Phi\cup\\{0\\}$. Given $\beta\in\Phi^{\prime}$,
we will also write $\mathbf{U}_{\beta}$ for the corresponding root subgroup
defined over $L^{\prime}$. We may embed $\mathscr{B}$ into the points in the
Bruhat-Tits building of $\mathbf{G}$ over $L^{\prime}$ which are fixed under
the natural action of $\text{Gal}(L^{\prime}/L)$, and similarly for the
apartment $\mathscr{A}$ of the former and the apartment $\mathscr{A}^{\prime}$
of the latter corresponding to $\mathbf{T}$. These embeddings are bijections
if $L^{\prime}/L$ is tamely ramified (cf. [PY02]).

###  2.2. Root subgroups and Iwahori decompositions

Recall that we may associate to an affine function $\psi$ on $\mathscr{A}$
whose vector part is a root $\alpha\in\Phi$ a subgroup $U_{\psi}$ of
$U_{\alpha}$ as follows. Given $u\in U_{\alpha}$, then
$U_{-\alpha}uU_{-\alpha}\cap\mathfrak{N}$ consists of a single element $m(u)$,
whose action on $\mathscr{A}$ is a reflection around a hyperplane that is
perpendicular to $\alpha$. If $\Psi(u,\alpha)$ denotes the affine function
with vector part $\alpha$ that vanishes on this hyperplane, then $U_{\psi}$ is
the subgroup of $U_{\alpha}$ consisting of the identity and those $u\in
U_{\alpha}$ such that $\Psi(u,\alpha)\geq\psi$. Recall also that $\psi$ is
called an affine root if $U_{\psi}$ is not contained in
$U_{\psi+\varepsilon}\cdot U_{2\alpha}$ (where $U_{2\alpha}=1$ if
$2\alpha\not\in\Phi$) for any $\varepsilon>0$.

If $\alpha\in\Phi$ and $r\in\mathbb{R}$, we will write $\psi_{\alpha,r}$ for
the affine function on $\mathscr{A}$ with vector part $\alpha$ and
$\psi_{\alpha,r}(x_{0})=r$, and $U_{\alpha,r}=U_{\psi_{\alpha,r}}$. This is a
decreasing family of neighborhoods of the identity in $U_{\alpha}$. The group
$\mathfrak{N}$ acts on $\mathscr{A}$, and for each $n\in\mathfrak{N}$ and
$\psi$ as above we have $nU_{\psi}n^{-1}=U_{\psi\circ n^{-1}}$. An element
$t\in\mathfrak{Z}$ acts on the apartment
$\mathscr{A}(\mathbf{G}^{\text{ad}},\mathbf{S},L)$ by translation by
$-\langle\alpha,\text{ord}(t)\rangle$ in the direction of
$\alpha\in\Phi^{+,\text{ndiv}}$, so in particular we have
$tU_{\alpha,r}t^{-1}=U_{\alpha,r+\langle\alpha,\text{ord}(t)\rangle}$. If
$\langle\alpha,\text{ord}(t)\rangle>0$, then $tU_{\alpha,r}t^{-1}$ is strictly
contained in $U_{\alpha,r}$. More generally, we can write any element $n$ of
$\mathfrak{N}$ as a product $n=wt$, where $w\in\mathfrak{N}$ fixes the special
point $x_{0}$ and $t\in\mathfrak{Z}$ (see Lemma 2.25 below), and then
$nU_{\alpha,r}n^{-1}=U_{n\alpha,r+\langle\alpha,\text{ord}(t)\rangle}$.

Recall that we say that an open subgroup $H\subseteq G$ admits an Iwahori
decomposition with respect to $\mathbf{P}$ if the multiplication map

(2.1) $\displaystyle(H\cap\overline{N})\times(H\cap M)\times(H\cap
N)\longrightarrow H$

is an isomorphism of $p$-adic locally analytic manifolds (note that $H$ is
automatically a $p$-adic locally analytic manifold). As the multiplication map
$\overline{\mathbf{N}}\times\mathbf{M}\times\mathbf{N}\longrightarrow\mathbf{G}$
is an isomorphism of algebraic varieties onto an open subset of $\mathbf{G}$,
the map 2.1 is an isomorphism of $p$-adic locally analytic manifolds if and
only if it is surjective. The subgroups that we will encounter admitting
Iwahori decompositions will admit Iwahori decompositions of a special kind
(which is rather common when the subgroups arise from Bruhat-Tits theory),
which to avoid repetition we will call _rooted_. Recall from [BT72, Definition
6.4.3] that a function $f\colon\Phi\longrightarrow\mathbb{R}$ is called convex
if for any finite family $(\alpha_{i})_{i}$ of elements of $\Phi$ such that
$\sum_{i}\alpha_{i}\in\Phi$ we have
$f(\sum_{i}\alpha_{i})\leq\sum_{i}f(\alpha_{i})$. For example, if $\Omega$ is
a bounded subset of $\mathscr{A}$, then the function defined by
$f(\alpha)=-\inf\psi_{\alpha,0}(\Omega)$ is convex. We say that an open
subgroup $H$ admits a _rooted Iwahori decomposition_ if it admits an Iwahori
decomposition and, moreover, there exists a convex function
$f\colon\Phi\longrightarrow\mathbb{R}$ such that

(2.2)
$\displaystyle\prod_{\alpha\in\Phi^{\text{ndiv},+}\smallsetminus\Phi_{M}}U_{f,\alpha}$
$\displaystyle\longrightarrow H\cap N,$
$\displaystyle\prod_{\alpha\in\Phi^{\text{ndiv},-}\smallsetminus\Phi_{M}}U_{f,\alpha}$
$\displaystyle\longrightarrow H\cap\overline{N}$

are isomorphisms of $p$-adic locally analytic manifolds for some, or
equivalently any (by [BT72, Proposition 6.4.9 (ii)]) ordering of the terms in
the products, where $U_{f,\alpha}:=U_{\alpha,f(\alpha)}\cdot
U_{2\alpha,f(2\alpha)}$ is a subgroup of $U_{\alpha}$ for all
$\alpha\in\Phi^{\text{ndiv}}\smallsetminus\Phi_{M}$ (where
$U_{2\alpha,f(2\alpha)}=\\{1\\}$ if $2\alpha\not\in\Phi$). As above, the
condition that the maps in 2.2 be isomorphisms of $p$-adic locally analytic
manifolds is equivalent to them being surjective.

######  Lemma 2.3.

Assume that $H$ admits a rooted Iwahori decomposition with respect to a
standard parabolic $\mathbf{P}^{\prime}\subseteq\mathbf{P}$. Then, $H$ admits
a rooted Iwahori decomposition with respect to $\mathbf{P}$.

Proof. Let $\mathbf{M}^{\prime}$ (resp. $\mathbf{N}^{\prime}$, resp.
$\overline{\mathbf{N}}^{\prime}$) be the standard Levi subgroup of
$\mathbf{P}^{\prime}$ (resp. the unipotent radical of $\mathbf{P}^{\prime}$,
resp. the unipotent radical of the opposite parabolic
$\overline{\mathbf{P}}^{\prime}$). Let $\Phi_{M^{\prime}}\subseteq\Phi_{M}$ be
the root system of $\mathbf{M}^{\prime}$, and choose a convex function $f$
realizing the rooted Iwahori decomposition. We first show that the
multiplication map

$\prod_{\alpha\in\Phi^{\text{ndiv},+}\smallsetminus\Phi_{M}}U_{f,\alpha}\longrightarrow
H\cap N$

is surjective, and hence an isomorphism of $p$-adic locally analytic
manifolds, for any ordering of the factors in the product such where the first
roots are $\Phi^{\text{ndiv},+}\smallsetminus\Phi_{M}$. Any element $x$ of
$H\cap N\subseteq H\cap N^{\prime}$ can be written as a product
$\prod_{\alpha\in\Phi^{\text{ndiv},+}\smallsetminus\Phi_{M^{\prime}}}x_{\alpha}=(\prod_{\alpha\in\Phi^{\text{ndiv},+}\smallsetminus\Phi_{M}}x_{\alpha})\cdot(\prod_{\alpha\in\Phi_{M}^{\text{ndiv},+}\smallsetminus\Phi_{M^{\prime}}}x_{\alpha})$
with $x_{\alpha}\in U_{f,\alpha}$ for all $\alpha$. Both $x$ and the first
term in the last expression lie in $N$, while the second term lies in $M$. As
$M$ and $N$ have trivial intersection, this means that the second term is 1,
so $x$ must lie in
$\prod_{\alpha\in\Phi^{\text{ndiv},+}\smallsetminus\Phi_{M}}x_{\alpha}$ as
claimed. The same argument shows the corresponding statement for
$H\cap\overline{N}$.

If $\Sigma$ is a subset of $\Phi^{\text{ndiv}}$, let us temporarily write
$U_{f,\Sigma}=\prod_{\alpha\in\Sigma}U_{f,\alpha}$ to simplify notation. To
show that $H$ admits an Iwahori decomposition with respect to $\mathbf{P}$,
it’s enough to show that the map

$(H\cap\overline{N})\times(H\cap M)\times(H\cap N)\longrightarrow H$

is surjective. This follows from the Iwahori decomposition of $H$ with respect
to $\mathbf{P}^{\prime}$ by writing

$\displaystyle(H\cap\overline{N}^{\prime})\times(H\cap M^{\prime})\times(H\cap
N^{\prime})$ $\displaystyle\simeq
U_{f,\Phi^{\text{ndiv},-}\smallsetminus\Phi_{M^{\prime}}}\times(H\cap
M^{\prime})\times U_{f,\Phi^{\text{ndiv},+}\smallsetminus\Phi_{M^{\prime}}}$
$\displaystyle\simeq U_{f,\Phi^{\text{ndiv},-}\smallsetminus\Phi_{M}}\times
U_{f,\Phi_{M}^{\text{ndiv},-}\smallsetminus\Phi_{M^{\prime}}}\times(H\cap
M^{\prime})\times
U_{f,\Phi_{M}^{\text{ndiv},+}\smallsetminus\Phi_{M^{\prime}}}\times
U_{f,\Phi^{\text{ndiv},+}\smallsetminus\Phi_{M}}$
$\displaystyle\simeq(H\cap\overline{N})\times\left(U_{f,\Phi_{M}^{\text{ndiv},-}\smallsetminus\Phi_{M^{\prime}}}\times(H\cap
M^{\prime})\times
U_{f,\Phi_{M}^{\text{ndiv},+}\smallsetminus\Phi_{M^{\prime}}}\right)\times(H\cap
N)$ $\displaystyle\subseteq(H\cap\overline{N})\times(H\cap M)\times(H\cap
N).\qed$

######  Lemma 2.4.

Assume that $H$ admits a rooted Iwahori decomposition and let
$t\in\mathfrak{Z}$. The following are equivalent:

  1. (i)

$\langle\alpha,\text{ord}(t)\rangle\leq 0\text{ for all
}\alpha\in\Phi^{+}\smallsetminus\Phi_{M}$,

  2. (ii)

$t(H\cap\overline{N})t^{-1}\subseteq H\cap\overline{N}$,

  3. (iii)

$t^{-1}(H\cap N)t\subseteq H\cap N$.

Similarly, the following are equivalent:

  1. (i)

$\langle\alpha,\text{ord}(t)\rangle<0\text{ for all
}\alpha\in\Phi^{+}\smallsetminus\Phi_{M}$,

  2. (ii)

$\bigcap_{n\geq 0}t^{n}(H\cap\overline{N})t^{-n}=\\{1\\}$,

  3. (iii)

$\bigcap_{n\geq 0}t^{-n}(H\cap N)t^{n}=\\{1\\}$.

Proof. If $t\in\mathfrak{Z}$ and $\alpha\in\Phi$, then
$tU_{\alpha,r}t^{-1}=U_{\alpha,r+\langle\alpha,\text{ord}(t)\rangle}$, and if
$\langle\alpha,\text{ord}(t)\rangle\neq 0$ then these are not equal to
$U_{\alpha,r}$. Thus, if $t\in\mathfrak{Z}$ and
$\alpha\in\Phi^{\text{ndiv},+}$, for any convex function $f$ one has
$tU_{f,-\alpha}t^{-1}\subseteq U_{f,-\alpha}$ if and only if
$\langle\alpha,\text{ord}(t)\rangle\leq 0$, if and only if
$t^{-1}U_{f,\alpha}t\subseteq U_{f,\alpha}$. Using that $H$ admits a rooted
Iwahori decomposition, we see that in the first case above, (2) and (3) are
equivalent to (1). The second case follows similarly. ∎

As explained in [RTW14, Remark 3.23], when $r\in\mathbb{Q}$ we may define
affinoid subspaces $\mathbf{U}_{\alpha,r}$ of (the rigid-analytification of)
$\mathbf{U}_{\alpha}$ such that each $U_{\alpha,r}$ is the group of $L$-points
of $\mathbf{U}_{\alpha,r}$. Let us recall this construction. Recall that we
have fixed maximal torus $\mathbf{T}$ defined over $L$ containing $\mathbf{S}$
which splits over a finite extension $L^{\prime}$ of $L$. There exists an
isomorphism of algebraic groups

(2.5) $\displaystyle
L^{\prime}\times_{L}\mathbf{U}_{\alpha}\simeq\prod_{\begin{subarray}{c}\beta\in\Phi^{\prime}\\\
\beta|_{\mathbf{S}}=\alpha\end{subarray}}\mathbf{U}_{\beta}\times\prod_{\begin{subarray}{c}\beta\in\Phi^{\prime}\\\
\beta|_{\mathbf{S}}=2\alpha\end{subarray}}\mathbf{U}_{\beta}$

such that $U_{\alpha,r}$ corresponds to the group of
$\text{Gal}(L^{\prime}/L)$-invariants of

(2.6) $\displaystyle\prod_{\begin{subarray}{c}\beta\in\Phi^{\prime}\\\
\beta|_{\mathbf{S}}=\alpha\end{subarray}}U_{\beta,r}\times\prod_{\begin{subarray}{c}\beta\in\Phi^{\prime}\\\
\beta|_{\mathbf{S}}=2\alpha\end{subarray}}U_{\beta,2r}.$

Here, the indexing on the groups $U_{\beta}$ is given by the image
$x^{\prime}_{0}$ of $x_{0}$ in $\mathscr{A}^{\prime}$ and the valuation $v$ of
$L^{\prime}$ extending that of $L$. The subgroups $U_{\beta,r}$ for
$\beta\in\Phi^{\prime}$ are naturally the set of $L^{\prime}$-points of rigid
$L^{\prime}$-analytic spaces $\mathbf{U}_{\beta,r}$. Namely, recall that a
choice of special vertex $y_{0}$ in $\mathscr{A}^{\prime}$ determines
isomorphisms from the additive group over $L^{\prime}$ to
$\mathbf{U}_{\beta}$, and we may define $\mathbf{U}_{\beta,r}$ as the preimage
under these isomorphisms of the closed ball of radius
$q^{-r-\beta(y_{0}-x^{\prime}_{0})}$ centered at the origin. This definition
is independent of the choice of $y_{0}$. The product

$\prod_{\begin{subarray}{c}\beta\in\Phi^{\prime}\\\
\beta|_{\mathbf{S}}=\alpha\end{subarray}}\mathbf{U}_{\beta,r}\times\prod_{\begin{subarray}{c}\beta\in\Phi^{\prime}\\\
\beta|_{\mathbf{S}}=2\alpha\end{subarray}}\mathbf{U}_{\beta,2r}.$

is stable under the action of $\text{Gal}(L^{\prime}/L)$ on
$L^{\prime}\times_{L}\mathbf{U}_{\alpha}$ (it’s enough to check this on points
over finite extensions of $L$ by the argument used in the proof of Lemma 2.24
below) and the affinoid subspace $\mathbf{U}_{\alpha,r}$ is defined as its
quotient by this action. Note that it is possible if $r\neq s$ that
$U_{\alpha,r}=U_{\alpha,s}$ but
$\mathbf{U}_{\alpha,r}\neq\mathbf{U}_{\alpha,s}$. More generally, if
$f\colon\Phi\longrightarrow\mathbb{R}$ is convex, we may define an affinoid
subspace $\mathbf{U}_{f,\alpha}$ of $\mathbf{U}_{\alpha}$ such that
$U_{f,\alpha}=\mathbf{U}_{f,\alpha}(L)$. Indeed, by our characterization of
$U_{\alpha,r}$ above and the definition of $U_{f,\alpha}$, we have

(2.7) $\displaystyle
U_{f,\alpha}=\left(\prod_{\begin{subarray}{c}\beta\in\Phi^{\prime}\\\
\beta|_{\mathbf{S}}=\alpha\end{subarray}}U_{\beta,f(\alpha)}\times\prod_{\begin{subarray}{c}\beta\in\Phi^{\prime}\\\
\beta|_{\mathbf{S}}=2\alpha\end{subarray}}U_{\beta,f(2\alpha)}\right)^{\text{Gal}(L^{\prime}/L)}.$

Note that the product inside the parentheses is a subgroup of
$\mathbf{U}_{\alpha}(L^{\prime})$ by [BT72, 6.1.6]. Thus, we may define
$\mathbf{U}_{f,\alpha}$ as the quotient by $\text{Gal}(L^{\prime}/L)$ of the
$\text{Gal}(L^{\prime}/L)$-stable affinoid subspace of 2.5

(2.8) $\displaystyle\prod_{\begin{subarray}{c}\beta\in\Phi^{\prime}\\\
\beta|_{\mathbf{S}}=\alpha\end{subarray}}\mathbf{U}_{\beta,f(\alpha)}\times\prod_{\begin{subarray}{c}\beta\in\Phi^{\prime}\\\
\beta|_{\mathbf{S}}=2\alpha\end{subarray}}\mathbf{U}_{\beta,f(2\alpha)}.$

######  Lemma 2.9.

Let $\alpha\in\Phi^{\text{ndiv}}$ and $f,g\colon\Phi\longrightarrow\mathbb{R}$
be convex functions with $f(\alpha)<g(\alpha)$ and $f(2\alpha)<g(2\alpha)$ if
$2\alpha\in\Phi$. Then, the map of $L$-affinoid algebras
$\mathcal{O}(\mathbf{U}_{f,\alpha})\longrightarrow\mathcal{O}(\mathbf{U}_{g,\alpha})$
induced by the inclusion
$\mathbf{U}_{g,\alpha}\xhookrightarrow{}\mathbf{U}_{f,\alpha}$ is compact as a
map of $L$-Banach spaces.

Proof. Let $R_{f}$ be the $L^{\prime}$-algebra of rigid $L^{\prime}$-analytic
functions on the affinoid space 2.8 for some fixed $\alpha$. The inclusion
$\mathbf{U}_{g,\alpha}\xhookrightarrow{}\mathbf{U}_{f,\alpha}$ then
corresponds to the bottom row of the diagram of $L$-algebras

${R_{f}}$${R_{g}}$${(R_{f})^{\text{Gal}(L^{\prime}/L)}}$${(R_{g})^{\text{Gal}(L^{\prime}/L)}.}$

The map in the top row is compact, since it corresponds to a product of
inclusions of closed balls with different radii. In this case, the rightmost
vertical arrow admits a section as a map of $L$-Banach spaces by [Sch02,
Proposition 10.5], so the bottom arrow factors through the top arrow, and
therefore will also be compact. ∎

###  2.3. The groups $K$, $J$ and $\widetilde{J}_{M}$

######  Lemma 2.10.

The subgroup $J$ admits a rooted Iwahori decomposition with respect to
$\mathbf{P}$.

Proof. Let $\mathbf{P}^{\prime}$ be the parabolic subgroup of $\mathbf{G}$
associated to $\mathscr{F}$ by [Mor99, Theorem 2.1], whose definition we now
recall. We can associate to the facet $\mathscr{F}$ the subroot system
$\Phi_{\mathscr{F}}$ of $\Phi$ consisting of the vector parts of affine roots
of $G$ that vanish at $\mathscr{F}$, and let $\mathfrak{M}$ be the reductive
subgroup of $\mathbf{G}$ containing $\mathbf{S}$ whose root system is the
closure ${}^{c}\Phi_{\mathscr{F}}$ of $\Phi_{\mathscr{F}}$ in $\Phi$. If
$\mathbf{A}^{\prime}$ is its connected center, then $\mathbf{P}^{\prime}$ is
the standard parabolic subgroup of $\mathbf{G}$ whose standard Levi subgroup
is the centralizer $\mathbf{M}^{\prime}$ of $\mathbf{A}^{\prime}$ in
$\mathbf{G}$. The condition that the facet $\mathscr{F}$ is maximal in the
preimage of $\mathscr{F}_{M}$ in $\mathscr{A}$ ensures that
$\Phi_{\mathscr{F}}$, and hence also ${}^{c}\Phi_{\mathscr{F}}$, is contained
in $\Phi_{M}$. In particular, $\mathbf{P}^{\prime}\subseteq\mathbf{P}$. By
[Mor99, Theorem 2.1] and its proof, $J$ admits a rooted Iwahori decomposition
with respect to $\mathbf{P}^{\prime}$, and hence also with respect to
$\mathbf{P}$ by Lemma 2.3. ∎

In fact, [Mor99, Theorem 2.1] shows that $J\cap\overline{N}$ and $J\cap N$ are
contained in the pro-$p$ radical of the parahoric subgroup of $J$
corresponding to $\mathscr{F}$ (see [HV15, 3.6 Definition and 3.7
Proposition]), which in turn is contained in (the Iwahori subgroup of) $I$.
The group $K$ will not in general admit an Iwahori decomposition, but the
following result still holds.

######  Lemma 2.11.

The multiplication map $(K\cap M)\times(K\cap N)\longrightarrow K\cap P$ is an
isomorphism of $p$-adic locally analytic manifolds.

Proof. This is proven in [HV15, Theorem 6.5]. ∎

######  Remark 2.12.

The group $J_{M}$ is the intersection of its normalizer $\widetilde{J}_{M}$
and the subgroup ${}^{0}M$ of $M$ consisting of elements $m$ such that
$v(\chi(m))=0$ for all $\chi\in X^{*}(\mathbf{M})$, since these are the groups
fixing pointwise the image of $\mathscr{F}_{M}$ in the buildings of
$\mathbf{M}^{\text{ad}}$ and $\mathbf{M}/\mathbf{M}^{\text{der}}$
respectively. As $X^{*}(\mathbf{M})$ can be naturally identified with a finite
index subgroup of $X^{*}(\mathbf{A})$, we deduce from the commutative diagram

${A/A_{0}}$${\widetilde{J}_{M}/J_{M}}$${M/{}^{\circ}M}$${X_{*}(\mathbf{A})}$${\text{Hom}(X^{*}(\mathbf{A}),\mathbb{Z})}$${\text{Hom}(X^{*}(\mathbf{M}),\mathbb{Z})}$$\scriptstyle{\sim}$$\scriptstyle{\sim}$

that $\widetilde{J}_{M}/J_{M}$ is a free abelian group of rank equal to the
rank of the torus $\mathbf{A}$ and that $AJ_{M}$ is a subgroup of
$\widetilde{J}_{M}$ of finite index. Note that this inclusion may be strict,
for example this is the case when $\mathbf{M}$ is $\text{GSp}_{4}$ and
$\mathscr{F}$ is a non-special minimal facet. Inclusion
$\mathfrak{N}\cap\widetilde{J}_{M}\subseteq\widetilde{J}_{M}$ induces an
isomorphism $(\mathfrak{N}\cap\widetilde{J}_{M})/(\mathfrak{N}\cap
J_{M})\stackrel{{\scriptstyle\sim}}{{\longrightarrow}}\widetilde{J}_{M}/J_{M}$
(this follows, for example, from [BT72, 7.4.15]).

Consider the submonoids of $M$ defined by

$\displaystyle M^{-}$ $\displaystyle:=\\{m\in
M:m(J\cap\overline{N})m^{-1}\subseteq J\cap\overline{N}\text{ and
}m^{-1}(J\cap N)m\subseteq J\cap N\\}$

and $M^{+}=(M^{-})^{-1}$. For any subgroup $H$ of $M$, we will also write
$H^{-}$ for $H\cap M^{-}$ and $H^{+}=H\cap M^{+}$. Lemma 2.4 and Lemma 2.10
implies that

(2.13) $\displaystyle\mathfrak{Z}^{-}$
$\displaystyle=\\{t\in\mathfrak{Z}:\langle\alpha,\text{ord}(t)\rangle\leq
0\text{ for all }\alpha\in\Phi^{+}\smallsetminus\Phi_{M}\\}.$

Similarly,

(2.14) $\displaystyle A^{-}$ $\displaystyle=\\{a\in A:v(\alpha(a))\leq 0\text{
for all }\alpha\in\Delta\smallsetminus\Delta_{M}\\}$

since $v(\alpha(a))=\langle\alpha,\text{ord}(a)\rangle$ for all $\alpha$ and
$\alpha(z)=1$ for all $\alpha\in\Phi_{M}$. Define also

$\displaystyle A^{--}$ $\displaystyle=\\{a\in A:v(\alpha(a))<0\text{ for all
}\alpha\in\Delta\smallsetminus\Delta_{M}\\}.$

This subsemigroup of $A$ will usually not contain the identity element, but it
is always non-empty. Note that if $\mu\in\widetilde{J}_{M}^{-}$, then

$\mu^{-1}(J\cap P)\mu=(J\cap M)\mu^{-1}(J\cap N)\mu\subseteq(J\cap M)(J\cap
N)=J\cap P.$

The following two lemmas are straight-forward generalizations of results of
Schneider-Stuhler [SS91, Lemma 4.10 and Proposition 4.7].

######  Lemma 2.15.

  1. (i)If $m\in M^{-}$ and $\mu\in\widetilde{J}_{M}^{-}$, then $JmJ\mu J=Jm\mu J$.

  2. (ii)

Let $\mu,\widetilde{\mu}\in\widetilde{J}_{M}^{-}$. If $x,\widetilde{x}$ run
through coset representatives of
$(J\cap\overline{N})/\mu(J\cap\overline{N})\mu^{-1}$ and
$(J\cap\overline{N})/\widetilde{\mu}(J\cap\overline{N})\widetilde{\mu}^{-1}$
respectively, then
$J\mu\widetilde{\mu}J=\coprod_{x,\widetilde{x}}x\mu\widetilde{x}\widetilde{\mu}J$.

Proof. Statement (1) follows from the fact that

$\displaystyle mJ\mu J$
$\displaystyle=m(J\cap\overline{N})\mu\cdot\mu^{-1}(J\cap P)\mu\cdot J$
$\displaystyle=m(J\cap\overline{N})m^{-1}\cdot m\mu J\subseteq Jm\mu J.$

For (2), note that $J\mu J=(J\cap\overline{N})\mu J=\coprod_{x}x\mu J$ and
similarly for $J\widetilde{\mu}J$, so by part (1) it is clear that
$J\mu\widetilde{\mu}J=\bigcup_{x,\widetilde{x}}x\mu\widetilde{x}\widetilde{\mu}J$,
so it’s enough to check that the cosets in the right-hand side are pairwise
disjoint. For a fixed $x$, it is clear that the cosets
$x\mu\widetilde{x}\widetilde{\mu}J$ are pairwise disjoint for the various
$\widetilde{x}$. Thus, it’s enough to show that if $x\mu
J\widetilde{\mu}J\cap\mu J\widetilde{\mu}J\neq\emptyset$ then
$x\in\mu(J\cap\overline{N})\mu^{-1}$. As $\mu
J\widetilde{\mu}J=\mu(J\cap\overline{N})\widetilde{\mu}J$, there exist
$\overline{n}_{0},\overline{n}_{1}\in\mu(J\cap\overline{N})\mu^{-1}$ such that
$x\overline{n}_{0}\mu\widetilde{\mu}J=\overline{n}_{1}\mu\widetilde{\mu}J$.
Since $x\overline{n}_{0}$ and $\overline{n}_{1}$ are in
$\mu(J\cap\overline{N})\mu^{-1}$, we deduce from the Iwahori decomposition
that
$x\overline{n}_{0}\cdot\mu\widetilde{\mu}(J\cap\overline{N})\widetilde{\mu}^{-1}\mu^{-1}=\overline{n}_{1}\cdot\mu\widetilde{\mu}(J\cap\overline{N})\widetilde{\mu}^{-1}\mu^{-1}$.
In particular, $x\in\mu(J\cap\overline{N})\mu^{-1}$ as required. ∎

In particular, the subsets $J\widetilde{J}_{M}^{-}J$ and $JA^{-}J$ of $G$ (or
more generally, $J\mathcal{J}_{M}^{-}J$ for any finite index subgroup
$\mathcal{J}_{M}$ of $\widetilde{J}_{M}$) are submonoids of $G$.

######  Lemma 2.16.

Let $g,\widetilde{g}\in J$, $\mu\in\widetilde{J}_{M}^{-}$ such that $g\mu
JP\cap\widetilde{g}\mu JP\neq\emptyset$. Then, $g\mu J=\widetilde{g}\mu J$.

Proof. We may assume that $\widetilde{g}=1$. Write $g=\overline{n}p$ with
$\overline{n}\in J\cap\overline{N}$ and $p\in J\cap P$. Since $\mu^{-1}(J\cap
P)\mu\subseteq J\cap P$, we have

$g\mu J=\overline{n}p\mu J=\overline{n}\mu(\mu^{-1}p\mu)J=\overline{n}\mu J.$

Hence,
$\overline{n}\mu(J\cap\overline{N})\mu^{-1}P\cap\mu(J\cap\overline{N})\mu^{-1}P\neq\emptyset$.
As $\mu(J\cap\overline{N})\mu^{-1}\subseteq J\cap\overline{N}$, it follows
from the injectivity of multiplication $\overline{N}\times P\longrightarrow G$
that
$\overline{n}\mu(J\cap\overline{N})\mu^{-1}\cap\mu(J\cap\overline{N})\mu^{-1}\neq\emptyset$.
In other words, $\overline{n}\in\mu(J\cap\overline{N})\mu^{-1}$, so $g\mu
J=\overline{n}\mu J=\mu J$. ∎

######  Lemma 2.17.

The group $\widetilde{J}_{M}$ is generated by its submonoid
$\widetilde{J}_{M}^{-}$. In fact, $\widetilde{J}_{M}$ is generated as a monoid
by $\widetilde{J}_{M}^{-}$ and $A^{+}$.

Proof. It suffices to prove the second statement. Let $\mu$ be an element of
$\widetilde{J}_{M}$. As $\mu\in M$, conjugation by $\mu$ preserves both $N$
and $\overline{N}$. The group $\mu(J\cap\overline{N})\mu^{-1}$ is contained in
the subgroup generated by $U_{\alpha,r_{\alpha}}$ with
$\alpha\in\Phi^{-}\smallsetminus\Phi_{M}$ for some choices of $r_{\alpha}$,
and similarly for $\mu^{-1}(J\cap N)\mu$. We may choose $s$ sufficiently large
so that for all $\alpha$, $\mathbf{U}_{\alpha,r_{\alpha}+s}$ is contained in
$\mathbf{U}_{\alpha,f(\alpha)}$, where $f$ is a convex function appearing in
the rooted Iwahori decomposition of $J$. Choose $a\in A^{-}$ satisfying
$v(\alpha(a))\leq-s$ for all $\alpha\in\Phi^{+}\smallsetminus\Phi_{M}$. Then,
we have $a\mu(J\cap\overline{N})\mu^{-1}a^{-1}\subseteq J\cap\overline{N}$ and
$\mu^{-1}a^{-1}(J\cap N)a\mu\subseteq J\cap N$. In other words,
$a\mu\in\widetilde{J}_{M}^{-}$. ∎

###  2.4. Congruence subgroups of $J$

Next, we will introduce certain families of subgroups of $J$. Let $s\geq 0$.
Consider the subset $T^{\prime}_{s}$ of $\mathbf{T}(L^{\prime})$ consisting of
elements $t$ such that $v(\chi(t)-1)\geq s$ for all $\chi\in
X^{*}_{L^{\prime}}(\mathbf{T})$. Choose an ordering of $\Phi^{\prime}$
compatible with that of $\Phi$ (so that the positive roots of $\Phi^{\prime}$
restrict to elements of $\Phi^{+}\cup\\{0\\}$) and let $\mathbf{B}^{\prime}$
be the associated Borel subgroup of $\mathbf{G}_{L^{\prime}}$ containing
$\mathbf{T}_{L^{\prime}}$ over $L^{\prime}$, $\mathbf{U}^{\prime}$ its
unipotent radical and $\overline{\mathbf{U}}^{\prime}$ the unipotent radical
of the opposite Borel. Let $N^{\prime}_{s}$ (resp.
$\overline{N}^{\prime}_{s}$, resp. $U^{\prime}_{s}$, resp
$\overline{U}^{\prime}_{s}$) be the subgroup generated by $\mathbf{U}_{\psi}$
where $\psi$ runs over all affine functions of $\mathscr{A}^{\prime}$ such
that $\inf\psi|_{\mathscr{A}}(\mathscr{F})\geq s$ and the image in
$X^{*}(\mathbf{S})\otimes_{\mathbb{Z}}\mathbb{R}$ of the vector part of $\psi$
(resp. the image in $X^{*}(\mathbf{S})\otimes_{\mathbb{Z}}\mathbb{R}$ of the
vector part of $\psi$, resp. the vector part of $\psi$, resp. the vector part
of $\psi$) lies in $\Phi^{+}\smallsetminus\Phi_{M}$ (resp.
$\Phi^{-}\smallsetminus\Phi_{M}$, resp. $(\Phi^{\prime})^{+}$, resp.
$(\Phi^{\prime})^{-}$). Write also
$U^{\prime}_{M,s}=U^{\prime}_{s}\cap\mathbf{M}(L^{\prime})$ and
$\overline{U}^{\prime}_{M,s}=\overline{U}^{\prime}_{s}\cap\mathbf{M}(L^{\prime})$.
Let $J^{\prime}_{s}$ (resp. $J^{\prime}_{M,s}$) be the subgroup of
$\mathbf{G}(L^{\prime})$ generated by $T^{\prime}_{s}$, $U^{\prime}_{s}$ and
$\overline{U}^{\prime}_{s}$ (resp. by $T^{\prime}_{s}$, $U^{\prime}_{M,s}$ and
$\overline{U}^{\prime}_{M,s}$). The groups
$J^{\prime}_{s},J^{\prime}_{M,s},N^{\prime}_{s}$ and
$\overline{N}^{\prime}_{s}$ are all stable under the action of
$\text{Gal}(L^{\prime}/L)$. Define $J_{s},J_{M,s},N_{s}$ and
$\overline{N}_{s}$ as the subgroups of $\text{Gal}(L^{\prime}/L)$-fixed points
of the respective subgroups of $\mathbf{G}(L^{\prime})$. Finally, let
$J_{0,s}$ be the subgroup of $J$ generated by $N_{s}$, $J_{M}$ and
$J\cap\overline{N}$ and let $J_{1,s}$ be the subgroup of $J$ generated by
$N_{s}$, $J_{M,s}$ and $J\cap\overline{N}$.

######  Lemma 2.18.

Let $f\colon\Phi\longrightarrow\mathbb{R}$ be the convex function defined by
$f(\alpha)=-\inf\psi_{\alpha,0}(\mathscr{F})$. Then, the multiplication maps

$\displaystyle\prod_{\alpha\in\Phi^{\text{ndiv},+}\smallsetminus\Phi_{M}}U_{f+s,\alpha}$
$\displaystyle\longrightarrow N_{s},$
$\displaystyle\prod_{\alpha\in\Phi^{\text{ndiv},-}\smallsetminus\Phi_{M}}U_{f+s,\alpha}$
$\displaystyle\longrightarrow\overline{N}_{s}$

are isomorphisms of locally $L$-analytic manifolds for any ordering of the
terms in the products.

Proof. We will only prove the statement for $N_{s}$ since the other case is
analogous. As usual, it is enough to show that the map above is surjective.
Let $f^{\prime}\colon\Phi^{\prime}\longrightarrow\mathbb{R}$ be the function
defined by $f^{\prime}(\beta)=f(\beta|_{\mathbf{S}})$ is
$\beta|_{\mathbf{S}}\neq 0$ and $f^{\prime}(\beta)=0$ otherwise. It is a
convex function of $\Phi^{\prime}$. Thus, for an affine root $\psi$ on
$\mathscr{A}^{\prime}$ with vector part $\beta$ we have $U_{\psi}\subseteq
U_{\beta,f^{\prime}(\beta)+s}$ if and only if
$\inf\psi|_{\mathscr{A}}(\mathscr{F})\geq s$, so $N^{\prime}_{s}$ is generated
by $U_{\beta,f^{\prime}(\beta)+s}$ as $\beta$ runs through
$\Phi^{\prime+}\smallsetminus\Phi^{\prime}_{M}$. Hence, by [BT72, 6.1.6]
multiplication
$\prod_{\beta\in\Phi^{\prime+}\smallsetminus\Phi^{\prime}_{M}}U_{f^{\prime}+s,\beta}\longrightarrow
N^{\prime}_{s}$ is a bijection. Taking $\text{Gal}(L^{\prime}/L)$-invariants,
the result follows from 2.7. ∎

In particular, $N_{0}=J\cap N$ and $\overline{N}_{0}=J\cap\overline{N}$, and
similar arguments show that $J_{M,0}=J_{M}$ and $J_{0}=J_{0,0}=J_{1,0}=J$.

######  Lemma 2.19.

Let $s\geq 0$.

  1. (i)

$J_{M,s}$ is normal in $\widetilde{J}_{M}$.

  2. (ii)

$J_{s}$ is normal in $J$.

  3. (iii)

$J_{1,s}$ is normal in $J_{0,s}$.

  4. (iv)

If $\mu\in\widetilde{J}_{M}^{-}$, then
$\mu\overline{N}_{s}\mu^{-1}\subseteq\overline{N}_{s}$ and
$\mu^{-1}N_{s}\mu\subseteq N_{s}$.

  5. (v)

$(J_{M,s})_{s}$ is a basis of neighborhoods of the identity in $J_{M}$ by open
subgroups.

Proof. Let us start by proving (2) and (3). First, we make the following
observations for $s,\widetilde{s}\geq 0$,
$\psi|_{\mathscr{A}}(\mathscr{F})\geq s$,
$\widetilde{\psi}|_{\mathscr{A}}(\mathscr{F})\geq\widetilde{s}$, $u\in
U_{\psi},\widetilde{u}\in U_{\widetilde{\psi}}$ and $t\in
T^{\prime}_{\widetilde{s}}$:

  1. (a)

$u\widetilde{u}u^{-1}$ lies in the subgroup generated by
$U_{\widetilde{\psi}}$ and $J^{\prime}_{s+\widetilde{s}}$,

  2. (b)

if the vector parts of $\psi$ and $\widetilde{\psi}$ are negative, then
$u\widetilde{u}u^{-1}\in\overline{U}^{\prime}_{\widetilde{s}}$,

  3. (c)

if the vector part of $\psi$ and $\widetilde{\psi}$ have images in
$\Phi_{M}\cup\Phi^{-}\cup\\{0\\}$ and $\Phi^{-}\smallsetminus\Phi_{M}$
respectively, then $u\widetilde{u}u^{-1}\in\overline{N}^{\prime}_{0}$,

  4. (d)

$utu^{-1}$ is contained in the subgroup generated by $U_{\psi+\widetilde{s}}$
and $T^{\prime}_{\widetilde{s}}$,

  5. (e)

$tut^{-1}\in U_{\psi}$.

Indeed, (a) and (b) follow from [Tit79, 1.4.2], (c) follows from the same
reference and the fact that $\overline{\mathbf{N}}$ is normalized by
$\overline{\mathbf{P}}$, and (d) and (e) follow from [BT72, Example 6.4.16
(b)]. Combining these facts and taking $\text{Gal}(L^{\prime}/L)$-invariants,
(2) and (3) follow easily.

Next, let us prove (1) and (4). Let $\mu\in\widetilde{J}_{M}^{-}$. If $\mu\in
J_{M}$, then $\mu$ is also contained in $J^{\prime}_{M,0}$, which the argument
above shows normalizes $\overline{N}^{\prime}_{s}$, $N^{\prime}_{s}$ and
$J^{\prime}_{M,s}$. Thus, by Remark 2.12 we may assume that
$\mu\in\mathfrak{N}\cap\widetilde{J}_{M}^{-}$. It is clear that $\mu$
normalizes $T^{\prime}_{s}$. Since $\mu
U_{\psi}\mu^{-1}=U_{\psi\circ\mu^{-1}}$ (resp.
$\mu^{-1}U_{\psi}\mu=U_{\psi\circ\mu}$), it’s enough to show that if $\psi$ is
an affine function on $\mathscr{A}^{\prime}$ whose vector part restricts to an
element of $\Phi^{-}\cup\\{0\\}$ (resp. $\Phi^{+}\cup\\{0\\}$) and
$\inf\psi(\mathscr{F})\geq s$, then $\psi\circ\mu^{-1}$ (resp. $\psi\circ\mu$)
satisfies the same condition. By changing $\psi$, it’s enough to show this for
$s=0$, which follows from the fact that
$\mu\overline{N}_{0}\mu^{-1}\subseteq\overline{N}_{0}$ (resp.
$\mu^{-1}N_{0}\mu\subseteq N_{0}$) and $\mu$ normalizes $J_{M}$. Finally,
statement (5) follows from the corresponding fact for $J^{\prime}_{M,s}$ and
$J^{\prime}_{M}$. ∎

######  Proposition 2.20.

Let $s\geq 0$. The groups $J_{s},J_{1,s}$ and $J_{0,s}$ admit rooted Iwahori
decompositions

$\displaystyle\overline{N}_{s}\times J_{M,s}\times N_{s}$
$\displaystyle\stackrel{{\scriptstyle\sim}}{{\longrightarrow}}J_{s},$
$\displaystyle(J\cap\overline{N})\times J_{M,s}\times N_{s}$
$\displaystyle\stackrel{{\scriptstyle\sim}}{{\longrightarrow}}J_{1,s},$
$\displaystyle(J\cap\overline{N})\times J_{M}\times N_{s}$
$\displaystyle\stackrel{{\scriptstyle\sim}}{{\longrightarrow}}J_{0,s}.$

Proof. The case $s=0$ was proven in Lemma 2.10, so assume $s>0$. To obtain the
rooted Iwahori decomposition for $J_{1,s}$ (resp. $J_{s}$) we can apply [BT72,
6.4.9 and 6.4.48] to obtain a similar decomposition for a minimal parabolic
for $J_{1,s}$ (resp. for $J^{\prime}_{s}$, and then take
$\text{Gal}(L^{\prime}/L)$-invariants to obtain one for $J_{s}$), and apply
Lemma 2.3 and Lemma 2.18. The decomposition for $J_{0,s}$ is proved in the
same way as [Mor99, Theorem 2.1 (ii)], as we now show. As usual, it’s enough
to show that multiplication $(J\cap\overline{N})\times J_{M}\times
N_{s}\longrightarrow J_{0,s}$ is surjective. Let $U$ be the pro-$p$ radical of
the parahoric subgroup of $J$ corresponding to $\mathscr{F}$. Recall from the
comment below Lemma 2.10 that $J\cap\overline{N}=U\cap\overline{N}$ and $J\cap
N=U\cap N$. For each $x\in J_{0,s}$, there exists $m_{0}\in J_{M}$ such that
$y:=m_{0}^{-1}x\in U$. If we show that $y=\overline{n}mn$ with
$\overline{n}\in U\cap\overline{N},m\in J_{M}$ and $n\in N_{s}$, then
$x=(m_{0}\overline{n}m_{0}^{-1})(m_{0}m)n$ will be in the image of the
multiplication map above. In other words, it’s enough to show that $U\cap
J_{0,s}$ admits an Iwahori decomposition. Let
$f\colon\Phi\longrightarrow\mathbb{R}$ be the function defined by

$f(\alpha)=\begin{cases}r_{\alpha}&\text{ if
}\alpha\in\Phi^{-}\cup\Phi_{M},\\\ r_{\alpha}+s&\text{ if
}\alpha\in\Phi^{+}\smallsetminus\Phi_{M},\end{cases}$

where $r_{\alpha}=-\inf\psi_{\alpha,0}(\mathscr{F})$, and define $f^{*}$ as in
[BT72, 6.4.23]. Then, the Iwahori decomposition for $U\cap J_{0,s}$ with
respect to a minimal parabolic follows from Lemma 2.3 and [BT72, 6.4.9 and
6.4.48] applied to $f^{*}$, and the decomposition with respect to $P$ follows
by Lemma 2.3 again. ∎

From this, it is easy to check that there is the inclusion
$J_{M}\longrightarrow J_{0,s}$ induces an isomorphism
$J_{M}/J_{M,s}\stackrel{{\scriptstyle\sim}}{{\longrightarrow}}J_{0,s}/J_{1,s}$.
We will need the following refined version of this fact.

######  Corollary 2.21.

The inclusion $\widetilde{J}_{M}^{-}\longrightarrow J_{0,s}$ induces a monoid
isomorphism

$\widetilde{J}_{M}^{-}/J_{M,s}\stackrel{{\scriptstyle\sim}}{{\longrightarrow}}J_{0,s}\widetilde{J}_{M}^{-}J_{0,s}/J_{1,s}.$

Proof. It follows from Lemma 2.19 (4) and Proposition 2.20 that the
multiplication map $\overline{N}_{0}\times\widetilde{J}_{M}^{-}\times
N_{s}\longrightarrow J_{0,s}\widetilde{J}_{M}^{-}J_{0,s}$ is bijective. In
order to prove the theorem, we must show that if
$\overline{n},\overline{n}^{\prime},\overline{n}^{\prime\prime}\in\overline{N}_{0},\mu,\mu^{\prime},\mu^{\prime\prime}\in\widetilde{J}_{M}^{-}$
and $n,n^{\prime},n^{\prime\prime}\in N_{s}$ satisfy $\overline{n}\mu
n\overline{n}^{\prime}\mu^{\prime}n^{\prime}=\overline{n}^{\prime\prime}\mu^{\prime\prime}n^{\prime\prime}$,
then $\mu^{\prime\prime}J_{M,s}=\mu\mu^{\prime}J_{M,s}$. This can be seen from
Lemma 2.19 (1) and (4) and the fact that $n\overline{n}^{\prime}\in J_{1,s}$.
∎

We also have the following version of Lemma 2.15 for $J_{0,s}$.

######  Lemma 2.22.

Let $\mu,\widetilde{\mu}\in\widetilde{J}_{M}^{-}$. Then
$J_{0,s}\mu\widetilde{\mu}J_{0,s}=\coprod_{x,\widetilde{x}}x\mu\widetilde{x}\widetilde{\mu}J_{0,s}$,
where $x,\widetilde{x}$ run through coset representatives of
$(J\cap\overline{N})/\mu(J\cap\overline{N})\mu^{-1}$ and
$(J\cap\overline{N})/\widetilde{\mu}(J\cap\overline{N})\widetilde{\mu}^{-1}$
respectively.

Proof. As in the proof of Lemma 2.15 we have
$J_{0,s}\mu\widetilde{\mu}J_{0,s}=\cup_{x,\widetilde{x}}x\mu\widetilde{x}\widetilde{\mu}J_{0,s}$,
and the cosets in the union are disjoint because the (bigger) corresponding
$J$-cosets are. ∎

Using Lemma 2.18 and the paragraph above Lemma 2.9, we may view
$\overline{N}_{s}$ as the set of $L$-points of the rigid $L$-analytic space

$\overline{\mathbf{N}}_{s}=\prod_{\alpha\in\Phi^{\text{ndiv},-}\smallsetminus\Phi_{M}}\mathbf{U}_{f+s,\alpha},$

and similarly for $N_{s}$. In fact, the Iwahori decompositions from
Proposition 2.20 allow us to canonically define rigid $L$-analytic spaces
whose sets of $L$-points are $J_{s}$ and $J_{M,s}$ when $s>0$. In order to do
this, define
$\overline{\mathbf{U}}^{\prime}_{s},\mathbf{U}^{\prime}_{s},\overline{\mathbf{N}}^{\prime}_{s}$
and $\mathbf{N}^{\prime}_{s}$ in the same way as $\overline{\mathbf{N}}_{s}$
and $\mathbf{N}_{s}$, and let $\mathbf{T}^{\prime}_{s}$ be the open affinoid
subspace of (the rigid $L^{\prime}$-analytification of) $\mathbf{T}^{\prime}$
defined by the same inequalities as $T^{\prime}_{s}$. Define
$\mathbf{J}^{\prime}_{s}=\overline{\mathbf{U}}^{\prime}_{s}\times\mathbf{T}^{\prime}_{s}\times\mathbf{U}^{\prime}_{s}$,
so that we have a natural inclusion
$\mathbf{J}^{\prime}_{s}\xhookrightarrow{}\mathbf{G}_{L^{\prime}}$ that is an
isomorphism onto its image, and
$\mathbf{J}^{\prime}_{s}(L^{\prime})=J^{\prime}_{s}$ by [BT72, 6.4.9 and
6.4.48]. Then $\mathbf{J}^{\prime}_{s}$ is a
$\text{Gal}(L^{\prime}/L)$-invariant subspace of $\mathbf{G}_{L^{\prime}}$, so
we may define $\mathbf{J}_{s}$ as its quotient by this action, and define
$\mathbf{J}_{M,s}$ similarly.

######  Lemma 2.23.

Let $\mu\in\widetilde{J}_{M}^{-}$. Then, for all finite extensions
$L^{\prime\prime}$ of $L$,
$\mu\overline{\mathbf{N}}_{s}(L^{\prime\prime})\mu^{-1}\subseteq\overline{\mathbf{N}}_{s}(L^{\prime\prime})$.

Proof. By enlarging $L^{\prime\prime}$ and $L^{\prime}$ if necessary, we may
assume that $L^{\prime\prime}=L^{\prime}$. We have already proved this case in
Lemma 2.19. ∎

######  Lemma 2.24.

  1. (i)Conjugation by an element of $J$ induces a rigid $L$-analytic endomorphism of $\mathbf{J}_{s}$.

  2. (ii)

There is a rigid $L$-analytic isomorphism

$\overline{\mathbf{N}}_{s}\times\mathbf{J}_{M,s}\times\mathbf{N}_{s}\stackrel{{\scriptstyle\sim}}{{\longrightarrow}}\mathbf{J}_{s}.$

  3. (iii)

Let $\mu\in\widetilde{J}_{M}^{-}$. Then, conjugation by $\mu$ induces a
morphism of rigid $L$-analytic spaces
$\overline{\mathbf{N}}_{s}\longrightarrow\overline{\mathbf{N}}_{s}$.

Proof. It’s enough to prove the corresponding statements for the rigid
analytic spaces over $L^{\prime}$. Statement (2) follows from the definitions
and the argument in Lemma 2.3. For (3), given $\mu\in\widetilde{J}_{M}^{-}$ we
have an inclusion
$\mu\overline{\mathbf{N}}^{\prime}_{s}\mu^{-1}\subseteq\overline{\mathbf{N}}^{\prime}_{s}$
_on rigid-analytic points_ by Lemma 2.23. There exists some affinoid open
$\mathcal{U}$ of $\overline{\mathbf{N}}^{\prime}$ through which the inclusion
and conjugation by $\mu$ maps
$\overline{\mathbf{N}}^{\prime}_{s}\longrightarrow\overline{\mathbf{N}}^{\prime}$
factor (since $\overline{\mathbf{N}}^{\prime}$ is isomorphic to affine space),
and the previous sentence then implies that conjugation by $\mu$ on
$\overline{\mathbf{N}}^{\prime}_{s}$ factors through
$\overline{\mathbf{N}}^{\prime}_{s}$, as $\overline{\mathbf{N}}^{\prime}_{s}$
is an affinoid subspace of $\mathcal{U}$ by [BGR84, Corollary 8.1.2/4]. The
same argument shows (1) by taking into account that the inclusion and
conjugation by elements of $J$, $\mathbf{J}_{s}\xhookrightarrow{}\mathbf{G}$,
also factor through some affinoid open (because we can embed $\mathbf{G}$ into
affine space). ∎

###  2.5. Bruhat-Tits decomposition

Let $\widetilde{W}=\mathfrak{N}/\mathfrak{Z}_{0}$. If $H$ is a subgroup of $G$
or of $M$ containing $\mathfrak{Z}_{0}$, we will write $\widetilde{W}^{H}$ for
the group $(\mathfrak{N}\cap H)/\mathfrak{Z}_{0}$.

######  Lemma 2.25.

  1. (i)The natural homomorphism $\widetilde{W}^{K}\longrightarrow W$ is an isomorphism.

  2. (ii)

The group $\widetilde{W}$ is isomorphic to the semidirect product
$\widetilde{W}^{K}\ltimes(\mathfrak{Z}/\mathfrak{Z}_{0})$.

  3. (iii)

We have $\mathfrak{N}\cap J=\mathfrak{N}\cap J_{M}$. In particular, the
inclusion $J_{M}\subseteq J$ induces an isomorphism
$\widetilde{W}^{J_{M}}\stackrel{{\scriptstyle\sim}}{{\longrightarrow}}\widetilde{W}^{J}$.

Proof. The map in (1) is injective, and the fact that $x_{0}$ is special
implies that it is also surjective. Part (2) follows from part (1) together
with the exact sequence

$1\longrightarrow\mathfrak{Z}/\mathfrak{Z}_{0}\longrightarrow\widetilde{W}\longrightarrow
W\longrightarrow 1$

For (3), note that the groups $\mathfrak{N}\cap J$ and $\mathfrak{N}\cap
J_{M}$ are the subgroups of $\mathfrak{N}$ fixing the points of $\mathscr{F}$
and $\mathscr{F}_{M}$ respectively. As $\mathfrak{N}$ acts by affine
transformations and both facets generate the same affine subspace of
$\mathscr{A}$, both subgroups are the same. ∎

Consider the following right action of $\widetilde{W}$ on
$\mathfrak{Z}_{0}\backslash\mathfrak{Z}$. If $\mu\in\mathfrak{N}$ and
$t\in\mathfrak{Z}$, write $\mu=n\widetilde{t}$ with
$\widetilde{t}\in\mathfrak{Z}$ and $n\in\mathfrak{N}\cap K$ according to the
decomposition in Lemma 2.25 (2), and define
$(\mathfrak{Z}_{0}t)\cdot\mu:=\mathfrak{Z}_{0}\widetilde{t}\mu^{-1}t\mu=\mathfrak{Z}_{0}n^{-1}t\mu$.
The following proposition generalizes [Her11, Lemma 2.20].

######  Proposition 2.26.

  1. (i)Inclusion $\mathfrak{Z}\xhookrightarrow{}G$ induces a bijection

$\mathfrak{Z}_{0}\backslash\mathfrak{Z}/\widetilde{W}^{J_{M}}\stackrel{{\scriptstyle\sim}}{{\longrightarrow}}K\backslash
G/J,$

where the action of $\widetilde{W}^{J_{M}}$ on
$\mathfrak{Z}_{0}\backslash\mathfrak{Z}$ is induced by that of
$\widetilde{W}$. In particular, $G=K\mathfrak{Z}J$.

  2. (ii)

For all $t\in\mathfrak{Z}$, there exist $t_{1},...,t_{r}\in\mathfrak{Z}$ such
that $tJ\mu\subseteq\bigcup_{i}Kt_{i}\mu J$ for all
$\mu\in\widetilde{J}_{M}^{-}$.

  3. (iii)

Let $\mathfrak{Z}^{\sim}$ be a subsemigroup of $\mathfrak{Z}$ defined by
$\langle\alpha,\text{ord}(t)\rangle\leq c_{\alpha}$ for
$\alpha\in\Phi^{+}\smallsetminus\Phi_{M}$ and some $c_{\alpha}\in\mathbb{R}$.
Then, for all $g\in G$, there exists $a\in A^{-}$ such that $gJaJ\subseteq
K\mathfrak{Z}^{\sim}J$.

Proof. The Bruhat-Tits decomposition [BT72, 7.4.15] for $\mathscr{F}$ and
$x_{0}$ implies that inclusion $\mathfrak{N}\xhookrightarrow{}G$ induces a
bijection $\widetilde{W}^{K}\backslash\widetilde{W}/\widetilde{W}^{J}\simeq
K\backslash G/J$. Now,
$\widetilde{W}^{K}\backslash\widetilde{W}\simeq\mathfrak{Z}_{0}\backslash\mathfrak{Z}$
by Lemma 2.25 (2) and it is straightforward to check that the right action of
$\widetilde{W}$ by multiplication corresponds to the action on
$\mathfrak{Z}_{0}\backslash\mathfrak{Z}$ defined above. Thus, (1) follows from
Lemma 2.25 (3).

Recall that $I$ is the pointwise stabilizer of $\mathscr{C}$ in $G$. As
explained in [Her11, Proof of Lemma 2.20], one can deduce from [BT72, Théorème
6.5] that $(G,I,N)$ is a generalized $(B,N)$-pair, and it follows easily from
the axioms of generalised $(B,N)$-pairs that if
$\widetilde{n}\in\mathfrak{N}$, there exist $n_{1},...,n_{r}\in\mathfrak{N}$
such that for all $n\in\mathfrak{N}$,
$nI\widetilde{n}\subseteq\bigcup_{i=1}^{r}Inn_{i}I$. Applying this to
$\widetilde{n}=t^{-1}$ shows that there exist $n_{1},...,n_{r}\in\mathfrak{N}$
such that

$tJ\mu=t(J\cap\overline{N})\mu\mu^{-1}(J\cap P)\mu\subseteq tI\mu
J\subseteq\bigcup_{j}In_{j}\mu J\subseteq\bigcup_{j}Kn_{j}\mu J.$

By Lemma 2.25 (1) and (2), there exist $t_{1},...,t_{r}\in\mathfrak{Z}$ such
that $Kn_{j}=Kt_{j}$ for all $j$, so (2) follows.

Finally, in (3) the statement for $g$ depends only on the double coset $KgJ$,
so by (1) we may assume without loss of generality that $g=t\in\mathfrak{Z}$.
Let $t_{1},...,t_{r}$ be as in (2). If $v_{p}(\alpha(a))$ is small enough for
$\alpha\in\Phi^{+}\smallsetminus\Phi_{M}$, then
$\langle\alpha,\text{ord}(t_{j}a)\rangle\leq c_{\alpha}$ for all
$\alpha\in\Phi^{+}\smallsetminus\Phi_{M}$, so $t_{j}a\in\mathfrak{Z}^{\sim}$
and $tJa\subseteq\bigcup_{j}Kt_{j}aJ$. ∎

######  Remark 2.27.

When applying Proposition 2.26 (3), we will often choose the $c_{\alpha}$
small enough so that any $t\in\mathfrak{Z}^{\sim}$ satisfies
$t(J\cap\overline{N})t^{-1}\subseteq K\cap\overline{N}$. Such $c_{\alpha}$
always exist by the rooted Iwahori decomposition of $J$ and the fact that
$tU_{\alpha,r}t^{-1}=U_{\alpha,r+\langle\alpha,\text{ord}(t)\rangle}$. In this
case, for all $\mu\in\widetilde{J}_{M}^{-}$, we have

$KtJ\cdot J\mu J\subseteq K\cdot t(J\cap\overline{N})t^{-1}\cdot
t\mu\cdot\mu^{-1}(J\cap P)\mu\cdot J\subseteq Kt\mu J.$

######  Corollary 2.28.

If $\mu,\widetilde{\mu}\in\widetilde{J}_{M}$ and $t\in\mathfrak{Z}$ satisfy
$Kt\mu J=Kt\widetilde{\mu}J$, then $\mu J_{M}=\widetilde{\mu}J_{M}$.

Proof. We may assume without loss of generality that
$\mu,\widetilde{\mu}\in\mathfrak{N}\cap\widetilde{J}_{M}$. By the proof of
Proposition 2.26 (1) and Lemma 2.25 (3), we have $(\mathfrak{N}\cap
K)t\mu(\mathfrak{N}\cap J_{M})=(\mathfrak{N}\cap
K)t\widetilde{\mu}(\mathfrak{N}\cap J_{M})$. If $t\mu=kt\widetilde{\mu}m$ with
$k\in\mathfrak{N}\cap K$ and $m\in\mathfrak{N}\cap J_{M}$, then $k=t\mu
m^{-1}\widetilde{\mu}^{-1}t^{-1}\in K\cap M$ and
$\mu\widetilde{\mu}^{-1}=t^{-1}kt\widetilde{\mu}m\widetilde{\mu}^{-1}\in\widetilde{J}_{M}\cap(t^{-1}(K\cap
M)t\cdot\widetilde{\mu}J_{M}\widetilde{\mu}^{-1})$. Thus, it suffices to show
that the set in the right-hand side is contained in $J_{M}$. The subgroup
${}^{0}M$ of $M$ defined in Remark 2.12 contains all the compact subgroups of
$M$, and hence also $t^{-1}(K\cap
M)t\widetilde{\mu}J_{M}\widetilde{\mu}^{-1}$. Therefore, the result follows
from the fact that ${}^{0}M\cap\widetilde{J}_{M}=J_{M}$. ∎

###  2.6. Hecke algebras

Given a locally profinite group $H$, a compact open subgroup $H_{0}$ and a
commutative ring $R$, we will write $\mathcal{H}(H,H_{0})_{R}$ for the
convolution $R$-algebra of $H_{0}$-biinvariant compactly supported functions
$f\colon H\longrightarrow R$. Explicitly, multiplication is given by

$(f*\widetilde{f})(h)=\sum_{x\in H/H_{0}}f(x)\widetilde{f}(x^{-1}h).$

Given $h\in H$, write $[H_{0}hH_{0}]$ for the characteristic function on the
double coset $H_{0}hH_{0}$. These form an $R$-basis of
$\mathcal{H}(H,H_{0})_{R}$. If $H^{-}$ is a submonoid of $H$ containing
$H_{0}$, we will write $\mathcal{H}(H^{-},H_{0})_{R}$ for the subalgebra of
functions whose support is contained in $H^{-}$. All the modules for
$\mathcal{H}(H^{-},H_{0})_{R}$ that we will be interested in arise in the
following way.

######  Lemma 2.29.

  1. (i)If $\mathcal{M}$ is a right $R[H^{-}]$-module, then the $H_{0}$-coinvariants $\mathcal{M}_{H_{0}}$ have a natural right action of $\mathcal{H}(H^{-},H_{0})_{R}$ via

$[m][H_{0}\mu H_{0}]=\left[\sum_{x\mu H_{0}\in H_{0}\mu
H_{0}/H_{0}}mx\mu\right],$

where $[{-}]$ denotes the canonical map
$\mathcal{M}\longrightarrow\mathcal{M}_{H_{0}}$.

  2. (ii)

If $\mathcal{M}$ is a left $R[H^{-}]$-module, then the $H_{0}$-invariants
$\mathcal{M}^{H_{0}}$ have a natural left action of
$\mathcal{H}(H^{-},H_{0})_{R}$ via

$[H_{0}\mu H_{0}]m=\sum_{x\mu H_{0}\in H_{0}\mu H_{0}/H_{0}}x\mu m.$

Proof. This is well-known when $H^{-}$ is a group, and the same arguments show
that this holds for submonoids. ∎

######  Lemma 2.30.

Let $\mu,\widetilde{\mu}\in\widetilde{J}_{M}^{-}$ and $s\geq 0$. Then, in
$\mathcal{H}(G,J_{0,s})_{R}$ one has

$[J_{0,s}\mu
J_{0,s}]*[J_{0,s}\widetilde{\mu}J_{0,s}]=[J_{0,s}\mu\widetilde{\mu}J_{0,s}].$

Proof. For $g\in G$, one computes

$\displaystyle([J_{0,s}\mu J_{0,s}]*[J_{0,s}\widetilde{\mu}J_{0,s}])(g)$
$\displaystyle=\sum_{x\in J_{0,s}\mu
J_{0,s}/J_{0,s}}[J_{0,s}\widetilde{\mu}J_{0,s}](x^{-1}g)$
$\displaystyle=\sum_{x^{\prime}\in
J\cap\overline{N}/\mu(J\cap\overline{N})\mu^{-1}}[J_{0,s}\widetilde{\mu}J_{0,s}]((x^{\prime}\mu)^{-1}g),$

A necessary condition for this sum to be non-zero is that $g\in J_{0,s}\mu
J_{0,s}(x^{\prime}\mu)^{-1}$ for some $x^{\prime}$, and this is contained in
$J_{0,s}\mu J_{0,s}\widetilde{\mu}J_{0,s}=J_{0,s}\mu\widetilde{\mu}J_{0,s}$.
Conversely, if $g\in J_{0,s}\mu\widetilde{\mu}J_{0,s}$, then by Lemma 2.22 we
may write $g=x\mu\widetilde{x}\widetilde{\mu}y$, where $x,\widetilde{x}\in
J\cap\overline{N}$ and $y\in J_{0,s}$. If $x^{\prime}\in J\cap\overline{N}$,
then the summand corresponding to $x^{\prime}$ is non-zero if and only if
$\mu^{-1}(x^{\prime})^{-1}x\mu\widetilde{x}\widetilde{\mu}y\in
J_{0,s}\widetilde{\mu}J_{0,s}$, if and only if $\mu^{-1}(x^{\prime})^{-1}x\mu$
lies in $J\cap\overline{N}$, if and only if $x$ and $x^{\prime}$ belong to the
same coset of $\mu(J\cap\overline{N})\mu^{-1}$ in $J\cap\overline{N}$. Thus,
all the terms in the sum are zero except one, which is 1. ∎

######  Proposition 2.31.

Restriction to $M$ induces an isomorphism

$\mathcal{H}(J_{0,s}\widetilde{J}_{M}^{-}J_{0,s},J_{0,s})_{R}\stackrel{{\scriptstyle\sim}}{{\longrightarrow}}\mathcal{H}(\widetilde{J}_{M}^{-},J_{M})_{R}.$

Proof. Let $\mu\in\widetilde{J}_{M}^{-}$. The Iwahori decomposition of
$J_{0,s}$ implies $J_{0,s}\mu J_{0,s}=(J\cap\overline{N})\mu J_{M}N_{s}$, so
$J_{0,s}\mu J_{0,s}\cap M=\mu J_{M}$. Thus, there is a bijection
$\widetilde{J}_{M}^{-}/J_{M}\longrightarrow J_{0,s}\backslash
J_{0,s}\widetilde{J}_{M}^{-}J_{0,s}/J_{0,s}$ and $[J_{0,s}\mu
J_{0,s}]|_{M}=[J_{M}\mu J_{M}]$. This implies that restriction to $M$ gives an
isomorphism of $R$-modules. By Lemma 2.30, restriction is compatible with
convolutions, and hence it is an isomorphism of $R$-algebras. ∎

######  Corollary 2.32.

Restriction induces an $R$-algebra isomorphism

$\mathcal{H}(J\widetilde{J}_{M}^{-}J,J)_{R}\longrightarrow\mathcal{H}(J_{0,s}\widetilde{J}_{M}^{-}J_{0,s},J_{0,s})_{R}.$

Proof. The composition
$\mathcal{H}(J\widetilde{J}_{M}^{-}J,J)_{R}\longrightarrow\mathcal{H}(J_{0,s}\widetilde{J}_{M}^{-}J_{0,s},J_{0,s})_{R}\longrightarrow\mathcal{H}(\widetilde{J}_{M}^{-},J_{0,s})_{R}$
is an isomorphism (by Proposition 2.31 applied to $s=0$), and so is the second
map. ∎

######  Corollary 2.33.

The algebra $\mathcal{H}(J\widetilde{J}_{M}^{-}J,J)_{R}$ is isomorphic to the
monoid ring $R[\widetilde{J}_{M}^{-}/J_{M}]$, in particular it is commutative.
In fact, it is isomorphic to a subalgebra of $R[X_{1}^{\pm
1},...,X_{\operatorname{rank}(\mathbf{A})}^{\pm 1}]$.

Proof. It’s enough to show this for
$\mathcal{H}(\widetilde{J}_{M}^{-},J_{M})_{R}$, which follows from Remark
2.12. ∎

######  Remark 2.34.

All of these results remain true if one replaces $\widetilde{J}_{M}$ with a
subgroup containing $AJ_{M}$.

###  2.7. A lemma on abelianizations

In this section we will prove a lemma on the topological abelianizations of
$M$ and some of its compact-mod-center open subgroups. The results on this
section do not depend on the larger group $\mathbf{G}$, so we may regard
$\mathbf{M}$ as a general reductive group over $L$ and $\mathbf{A}$ as the
maximal $L$-split torus in its center. If $H$ is a closed subgroup of $M$, let
us write $\overline{[H,H]}$ for the closure its commutator subgroup.

######  Lemma 2.35.

Let $H$ be an open subgroup of $M$ containing $AH_{0}$ as a subgroup of finite
index for some compact open subgroup $H_{0}$ of $M$. Then, the map
$H/\overline{[H,H]}\longrightarrow M/\overline{[M,M]}$ has finite kernel and
cokernel.

Proof. Let $\mathbf{M}^{\text{der}}$ be the derived subgroup of $\mathbf{M}$
and
$\varphi\colon\mathbf{M}^{\text{sc}}\longrightarrow\mathbf{M}^{\text{der}}$ be
the simply connected cover of $\mathbf{M}^{\text{der}}$. Then, the commutator
map $\mathbf{M}\times\mathbf{M}\longrightarrow\mathbf{M}$ factors through
$\varphi$. The simply connected group $\mathbf{M}^{\text{sc}}$ is isomorphic
to the product of its simple factors $\mathbf{M}_{i}$. We claim that
$\overline{[\mathbf{M}_{i}(L),\mathbf{M}_{i}(L)]}$ has finite index in
$\mathbf{M}_{i}(L)$. If $\mathbf{M}_{i}$ is $L$-isotropic, then it follows
from the Kneser-Tits conjecture [Pla69, Theorem B] that each group
$\mathbf{M}_{i}(L)$ is perfect, and the index is 1. If $\mathbf{M}_{i}$ is
$L$-anisotropic, then $\mathbf{M}_{i}(L)$ is compact, so
$\mathbf{M}_{i}(L)/\overline{[\mathbf{M}_{i}(L),\mathbf{M}_{i}(L)]}$ is an
abelian, compact $p$-adic locally analytic group. Its Lie algebra is an
abelian quotient of the Lie algebra of $\mathbf{M}_{i}(L)$, which is
semisimple, so it must be trivial. Thus, the quotient
$\mathbf{M}_{i}(L)/\overline{[\mathbf{M}_{i}(L),\mathbf{M}_{i}(L)]}$ is a
finite union of points, as required. This implies that the inclusion
$\overline{[\mathbf{M}^{\text{sc}}(L),\mathbf{M}^{\text{sc}}(L)]}\subseteq\mathbf{M}^{\text{sc}}(L)$
has finite index, and since

$\varphi(\overline{[\mathbf{M}^{\text{sc}}(L),\mathbf{M}^{\text{sc}}(L)]})\subseteq\overline{[M,M]}\subseteq\varphi(\mathbf{M}^{\text{sc}}(L))$

the second inclusion has finite index as well. As the kernel of $\varphi$ is
finite, its Galois cohomology in degree 1 is finite as well, so we conclude
that $\overline{[M,M]}$ has finite index in $\mathbf{M}^{\text{der}}(L)$.
Hence, the map $M/\overline{[M,M]}\longrightarrow
M/\mathbf{M}^{\text{der}}(L)$ has finite kernel and cokernel, and it suffices
to prove the statement of the lemma with the source replaced by the target.

Next, note that we may replace $H$ by $AH_{0}$. Indeed, if the composition

$AH_{0}/\overline{[H_{0},H_{0}]}\longrightarrow
H/\overline{[H,H]}\longrightarrow M/\mathbf{M}^{\text{der}}(L)$

has finite kernel and cokernel, then the first map has finite kernel and the
second has finite cokernel. But $AH_{0}$ has finite index in $H$, so the first
map also has finite cokernel, and hence the second has finite kernel. In fact,
the same argument shows that we may replace $H_{0}$ with any open compact
subgroup of it. In particular, we are free to replace $H_{0}$ with a subgroup
of the form $H_{1}H_{2}$ where $H_{1}$ is contained in the center of $M$,
$H_{2}$ is contained in $\mathbf{M}^{\text{der}}(L)$, and $H_{1}$ and $H_{2}$
have trivial intersection.

Let us first show that the cokernel is finite. It’s enough to check that the
subgroup of $M$ generated by $\mathbf{M}^{\text{der}}(L)$, $A$ and $H_{1}$ has
finite index. The map
$M/A\mathbf{M}^{\text{der}}(L)\longrightarrow(\mathbf{M}/\mathbf{A}\mathbf{M}^{\text{der}})(L)$
has finite kernel and cokernel (since the intersection of
$\mathbf{M}^{\text{der}}$ and the center of $\mathbf{M}$ is finite), so it’s
enough to check that the image of $H_{1}$ in the target has finite index. But
the target is compact and the image of $H_{1}$ in it is an open subgroup, so
the result follows. Let us now show that the kernel
$(AH_{1}H_{2}\cap\mathbf{M}^{\text{der}}(L))/\overline{[H_{2},H_{2}]}$ is
finite. Since

$AH_{1}H_{2}\cap\mathbf{M}^{\text{der}}(L)=(AH_{1}\cap\mathbf{M}^{\text{der}}(L))H_{2}$

and $AH_{1}\cap\mathbf{M}^{\text{der}}(L)$ is finite, we are reduced to
showing that $\overline{[H_{2},H_{2}]}$ has finite index in $H_{2}$. This
follows from the same Lie algebra argument as in the first paragraph. ∎

###  2.8. A lemma on roots

Write ${}^{M}W$ for the set of minimal length coset representatives of cosets
in $W_{M}\backslash W$. Alternatively, an element $w$ of $W$ belongs to
${}^{M}W$ if and only if $w^{-1}\Phi^{+}_{M}\subseteq\Phi^{+}$. Write $\ell$
for the length function on $W$.

######  Lemma 2.36.

Let $s\in S^{-}$ and $\lambda\in X^{*}(\mathbf{S})$ be a dominant weight. The
minimum of the quantities $v_{L}((w\lambda)(s))$ for $w\in{}^{M}W$ is achieved
at $w=s_{\alpha}$ for some $\alpha\in\Delta\smallsetminus\Delta_{M}$.

Proof. Let $w\in{}^{M}W$ be any non-trivial element. We claim that there
exists $\alpha\in\Delta\smallsetminus\Delta_{M}$ such that
$\ell(s_{\alpha}w)<\ell(w)$. Indeed, this is equivalent to $w^{-1}\alpha<0$,
so if this is false we would have
$w^{-1}(\Delta\smallsetminus\Delta_{M})\subseteq\Phi^{+}$, and hence
$w^{-1}\Phi^{+}\subseteq\Phi^{+}$, so
$\ell(w^{-1})=w^{-1}\Phi^{+}\cap\Phi^{-}=0$, which contradicts our assumption
that $w$ is non-trivial. So let $\alpha\in\Delta\smallsetminus\Delta_{M}$ be
such a root. Clearly, $s_{\alpha}\in{}^{M}W$, so it’s enough to show that
$v_{L}((s_{\alpha}\lambda)(s))\leq v_{L}((w\lambda)(s))$. As $s\in S^{-}$, by
2.13 it’s enough to check that $s_{\alpha}\lambda\geq w\lambda$, which follows
from the fact that $s_{\alpha}\leq w$ in the Bruhat order. ∎

###  2.9. Direct products of groups

When we apply the results we have proven so far in Section 4, we will do so
not for a reductive group over a finite extension of $\mathbb{Q}_{p}$, but
rather a product of reductive groups over finite extensions of
$\mathbb{Q}_{\ell}$ for different primes $\ell$. We will now introduce this
setting.

Let $\ell_{1},...,\ell_{n}$ be prime numbers (each of which can appear more
than once). For each $i$, let $L_{i}$ be a finite extension of
$\mathbb{Q}_{\ell_{i}}$ and $\mathbf{G}_{i}$ a reductive group over $L_{i}$.
We make the same definitions as in Section 2.1 and write them with subindex
$i$. Thus, for example we have subgroups
$\mathbf{A}_{i}\subseteq\mathbf{S}_{i}\subseteq\bm{\mathfrak{Z}}_{i}\subseteq\mathbf{M}_{i}\subseteq\mathbf{P}_{i}\subseteq\mathbf{G}_{i}$,
and the pair $(\mathbf{G}_{i},\mathbf{S}_{i})$ has root system $\Phi_{i}$ with
simple roots $\Delta_{i}$ and Weyl group $W_{i}$. We set
$G_{i}=\mathbf{G}_{i}(L_{i})$ and $G=\prod_{i=1}^{n}G_{i}$, and similarly for
other groups. We will mostly view these simply as locally profinite groups, or
sometimes as products of locally profinite groups and $p$-adic locally
analytic groups (if some of the $\ell_{i}$ are $p$). We will also set
$\Phi=\coprod_{i=1}^{n}\Phi_{i}$, etc.

All of the results we have proven thus far remain valid for this setting after
making the appropriate modifications to the definitions if necessary. For
example, we say that a subgroup of the form $H=\prod_{i=1}^{n}H_{i}$ with
$H_{i}\subseteq G_{i}$ admits an Iwahori decomposition if the product
$(H\cap\overline{N})\times(H\cap M)\times(H\cap N)\longrightarrow H$ is a
homeomorphism (or equivalently, surjective), and that it is rooted if there
exists a convex function $f\colon\Phi\longrightarrow\mathbb{R}$ such that the
multiplication map
$\prod_{\alpha\in\Phi^{\text{ndiv},+}\smallsetminus\Phi_{M}}U_{f,\alpha}\longrightarrow
H\cap N$ is a homeomorphism (or equivalently, surjective) for any ordering of
its factors and similarly for $\overline{N}$. Here, given $\alpha\in\Phi$
belonging to $\Phi_{i}$, the group $U_{f,\alpha}$ is the group
$U_{f|_{\Phi_{i}},\alpha}\times\prod_{j\neq i}\\{1\\}$. Then, it is clear that
$H$ admits a (rooted) Iwahori decomposition if and only if each of the $H_{i}$
does.

Because all of these results generalize immediately to this setting, we will
not repeat them and instead quote directly the original results. Let us also
mention that if $H_{i,0}$ is a compact open subgroup of $G_{i}$ and
$H_{i}^{-}$ a submonoid containing $H_{i,0}$, then there is an isomorphism
$\mathcal{H}(\prod_{i=1}^{n}H_{i}^{-},\prod_{i=1}^{n}H_{i,0})_{R}\simeq\bigotimes_{i=1}^{n}\mathcal{H}(H_{i}^{-},H_{i,0})_{R}$.

##  3\. A vanishing criterion for Tor

The key technical result in [KS12] (as well as in this article) is the
vanishing of certain Tor groups with respect to the monoid algebra of
$\Lambda^{-}$. To obtain them, Kohlhaase-Schraen use that, in their setting,
this monoid algebra is isomorphic to a polynomial algebra, and they carry out
an explicit computation. In this section, we will give a more general (but
still especially tailored for our needs) criterion for the vanishing of the
relevant Tor groups, which we obtain by reducing to the case of polynomial
algebras. Let us forget the notation from previous sections.

Let $R$ be a commutative ring and $\Lambda$ a free abelian group of finite
rank, whose group law we write using multiplicative notation, and let
$\Lambda^{\prime}\subseteq\Lambda^{-}\subseteq\Lambda$ be submonoids
satisfying $\Lambda^{\prime}(\Lambda^{\prime})^{-1}=\Lambda$. If
$\chi\colon\Lambda\longrightarrow R^{\times}$ is a group homomorphism, we will
write $R(\chi)$ for the $R[\Lambda]$-module whose underlying $R$-module is $R$
and where $\Lambda$ acts via $\chi$.

######  Lemma 3.1.

Let $\mathcal{M}$ be an $R[\Lambda^{-}]$-module and $q\in\mathbb{Z}$.

  1. (i)

The natural $R[\Lambda^{-}]$-module map
$\text{Tor}^{R[\Lambda^{-}]}_{q}(R(\chi),\mathcal{M})\longrightarrow\text{Tor}^{R[\Lambda]}_{q}(R(\chi),R[\Lambda]\otimes_{R[\Lambda^{\prime}]}\mathcal{M})$
is an isomorphism.

  2. (ii)

Inclusion $\Lambda^{\prime}\subseteq\Lambda^{-}$ induces an isomorphism
$\text{Tor}^{R[\Lambda^{\prime}]}_{q}(R(\chi),\mathcal{M})\stackrel{{\scriptstyle\sim}}{{\longrightarrow}}\text{Tor}^{R[\Lambda^{-}]}_{q}(R(\chi),\mathcal{M})$.

  3. (iii)

Assume that $\mathcal{M}$ contains an $R[\Lambda^{-}]$-submodule
$\mathcal{M}^{-}$ such that for all $m\in\mathcal{M}$ there exists
$\lambda\in\Lambda^{-}$ such that $\lambda m\in\mathcal{M}^{-}$. Then,
inclusion $\mathcal{M}^{-}\subseteq\mathcal{M}$ induces an isomorphism
$\text{Tor}^{R[\Lambda^{-}]}_{q}(R(\chi),\mathcal{M}^{-})\stackrel{{\scriptstyle\sim}}{{\longrightarrow}}\text{Tor}^{R[\Lambda^{-}]}_{q}(R(\chi),\mathcal{M})$

Proof. Note that $R[\Lambda]$ is the localization of $R[\Lambda^{-}]$ at
$S=\Lambda^{-}$. Hence,
$S^{-1}\text{Tor}^{R[\Lambda^{-}]}_{q}(R(\chi),\mathcal{M})$ is isomorphic to
$\text{Tor}^{R[\Lambda]}_{q}(S^{-1}R(\chi),S^{-1}\mathcal{M})$. But $R(\chi)$
is an $R[\Lambda]$-module, and hence so is
$\text{Tor}^{R[\Lambda^{-}]}_{q}(R(\chi),\mathcal{M})$, and thus
$\text{Tor}^{R[\Lambda^{-}]}_{q}(R(\chi),\mathcal{M})\simeq\text{Tor}^{R[\Lambda]}_{q}(R(\chi),R[\Lambda]\otimes_{R[\Lambda^{-}]}\mathcal{M})$,
which proves part (1).

By (1) applied to both $\Lambda^{-}$ and $\Lambda$, statement (2) will follow
if we show that the natural map
$R[\Lambda]\otimes_{R[\Lambda^{\prime}]}\mathcal{M}\longrightarrow
R[\Lambda]\otimes_{R[\Lambda^{-}]}\mathcal{M}$ is an isomorphism. This is
clearly surjective, and its kernel is generated by elements of the form
$\lambda\otimes m-1\otimes\lambda m$ with $\lambda\in\Lambda^{-}$,
$m\in\mathcal{M}$. Writing $\lambda=\lambda^{\prime}(\mu^{\prime})^{-1}$ with
$\lambda^{\prime},\mu^{\prime}\in\Lambda^{\prime}$ we see that

$\mu^{\prime}(\lambda\otimes m)=\lambda^{\prime}\otimes
m=1\otimes\lambda^{\prime}m=\mu^{\prime}(1\otimes\lambda m).$

As $\mu^{\prime}$ is invertible in $R[\Lambda]$, this means that
$\lambda\otimes m-1\otimes\lambda m=0$, so the above homomorphism is also
injective.

For statement (3) it’s enough to show by (1) that the natural map

$R[\Lambda]\otimes_{R[\Lambda^{-}]}\mathcal{M}^{-}\longrightarrow
R[\Lambda]\otimes_{R[\Lambda^{-}]}\mathcal{M}$

is an isomorphism. As localization is exact, it’s enough to show that
$R[\Lambda]\otimes_{R[\Lambda^{-}]}(\mathcal{M}/\mathcal{M}^{-})$ vanishes.
This follows from our assumption on $\mathcal{M}^{-}$, since any element of
the form $\mu\otimes m$ can also be written as $\mu\lambda^{-1}\otimes\lambda
m$ for any $\lambda\in\Lambda^{-}$. ∎

######  Lemma 3.2.

Assume that $\mathcal{M}$ admits a decomposition
$\mathcal{M}=\mathcal{M}^{+}\oplus\bigoplus_{\lambda\in\Lambda^{-}}\mathcal{M}_{\lambda}$
as $R$-modules satisfying

  1. (a)

$\lambda\mathcal{M}_{\mu}\subseteq\mathcal{M}_{\lambda\mu}$ for
$\lambda,\mu\in\Lambda^{-}$,

  2. (b)

for all $m\in\mathcal{M}$ there exists $\lambda\in\Lambda^{-}$ such that
$\lambda m\in\bigoplus_{\lambda\in\Lambda^{-}}\mathcal{M}_{\lambda}$.

Then, the same conditions hold for $\Lambda^{\prime}$ instead of
$\Lambda^{-}$, and there is an isomorphism

$\text{Tor}^{R[\Lambda^{-}]}_{q}(R(\chi),\mathcal{M})\simeq\text{Tor}^{R[\Lambda^{\prime}]}_{q}(R(\chi),\bigoplus_{\lambda\in\Lambda^{\prime}}\mathcal{M}_{\lambda}).$

Proof. We first show that (a) and (b) remain valid if we replace $\Lambda^{-}$
by $\Lambda^{\prime}$ everywhere. This is immediate for (a). Given
$m\in\mathcal{M}$, by (b) there exist
$\lambda,\lambda_{1},...,\lambda_{n}\in\Lambda^{-}$ and
$m_{i}\in\mathcal{M}_{\lambda_{i}}$ for $i=1,...,n$ such that $\lambda
m=m_{1}+\cdots+m_{n}$. We may write
$\lambda=\lambda^{\prime}(\mu^{\prime})^{-1}$ with
$\lambda^{\prime},\mu^{\prime}\in\Lambda^{\prime}$ and
$\lambda_{i}=\lambda^{\prime}_{i}(\mu^{\prime}_{i})^{-1}$ similarly. Then,

$(\lambda^{\prime}\mu^{\prime}_{1}\cdots\mu^{\prime}_{n})m=\mu^{\prime}(\mu^{\prime}_{1}\cdots\mu^{\prime}_{n})\lambda
m\in\Lambda^{\prime}\cdot\bigoplus_{i=1}^{n}\mathcal{M}_{\mu^{\prime}_{i}\lambda_{i}}\subseteq\bigoplus_{\mu\in\Lambda^{\prime}}\mathcal{M}_{\mu}.$

This proves the first statement. The second statement follows from applying
Lemma 3.1 (3) to $\Lambda^{\prime}$ and
$\bigoplus_{\lambda^{\prime}\in\Lambda^{\prime}}\mathcal{M}_{\lambda}\subseteq\mathcal{M}$.
∎

######  Lemma 3.3.

Let $\beta_{1},...,\beta_{r}\colon\Lambda\longrightarrow\mathbb{Z}$ be
linearly independent $\mathbb{Z}$-linear maps.

  1. (i)

$\Lambda^{-}:=\\{\lambda\in\Lambda:\beta_{i}(\lambda)\leq 0$ for all $i\\}$
contains a free abelian monoid $\Lambda^{\prime}$ of rank $d$ such that
$\Lambda^{\prime}(\Lambda^{\prime})^{-1}=\Lambda$.

  2. (ii)

If $\mathcal{M}$ is an $R[\Lambda^{-}]$-module as in Lemma 3.2, then
$\text{Tor}_{q}^{R[\Lambda^{-}]}(R(\chi),\mathcal{M})=0$ for $q>0$.

Proof. Part (1) is a consequence of [CLS11, Theorem 11.1.9]. By Lemma 3.2,
part (2) is equivalent to
$\text{Tor}_{q}^{R[\Lambda^{\prime}]}(R(\chi),\bigoplus_{\lambda\in\Lambda^{\prime}}\mathcal{M}_{\lambda})=0$
for $q>0$, which follows from Lemma 3.4 below and [Bou06, X.9.6, Proposition
5]. ∎

######  Lemma 3.4.

Let $R$ be a ring, $R[X_{1},...,X_{d}]$ a polynomial ring in $d$ variables
with its usual $\mathbb{Z}_{\geq 0}^{d}$ grading. Let
$\mathcal{M}=\bigoplus_{m\in\mathbb{Z}_{\geq 0}^{d}}\mathcal{M}_{m}$ be a
graded $R[X_{1},...,X_{d}]$-module. Let $a_{1},...,a_{d}\in R^{\times}$, and
set $Y_{i}:=X_{i}-a_{i}$. Then, the sequence $(Y_{1},...,Y_{d})$ of
endomorphisms of $\mathcal{M}$ is regular, i.e. if $1\leq j\leq d$, then the
endomorphism of $\mathcal{M}/\sum_{i=1}^{j-1}\operatorname{Im}(Y_{i})$ induced
by $Y_{j}$ is injective.

Proof. This statement is a generalization of [KS12, Theorem 2.6], and the same
proof works in this general setting. ∎

######  Lemma 3.5.

Let $\bm{\Lambda}$ be a free abelian group of finite rank, $\bm{\Lambda}^{-}$
a submonoid of $\bm{\Lambda}$ such that
$\bm{\Lambda}^{-}(\bm{\Lambda}^{-})^{-1}=\bm{\Lambda}$. Let $\Lambda$ be a
finite index subgroup of $\bm{\Lambda}$ and
$\Lambda^{-}=\Lambda\cap\bm{\Lambda}^{-}$, and assume that
$\bm{\Lambda}=\bm{\Lambda}^{-}\Lambda$. Assume also that the index
$[\bm{\Lambda}:\Lambda]$ is invertible in $R$. Then, for any
$R[\bm{\Lambda}^{-}]$-module $\mathcal{M}$, character
$\chi\colon\bm{\Lambda}\longrightarrow R^{\times}$ and $q\in\mathbb{Z}$, the
$R[\bm{\Lambda}]$-module the natural map
$\text{Tor}^{R[\Lambda^{-}]}_{q}(R(\chi),\mathcal{M})\longrightarrow\text{Tor}^{R[\bm{\Lambda}^{-}]}_{q}(R(\chi),\mathcal{M})$
is a homomorphism of $R[\bm{\Lambda}]$-modules that identifies the target with
a direct summand of the source.

Proof. We begin by observing that for any $R[\bm{\Lambda}^{-}]$-module $M$,
the natural map

$R[\Lambda]\otimes_{R[\Lambda^{-}]}M\longrightarrow
R[\bm{\Lambda}]\otimes_{R[\bm{\Lambda}^{-}]}M$

is an isomorphism. Indeed, as
$R[\Lambda]\otimes_{R[\Lambda^{-}]}M=R[\Lambda]\otimes_{R[\Lambda^{-}]}R[\bm{\Lambda}^{-}]\otimes_{R[\bm{\Lambda}^{-}]}M$,
it’s enough to prove that the natural map
$R[\Lambda]\otimes_{R[\Lambda^{-}]}R[\bm{\Lambda}^{-}]\longrightarrow
R[\bm{\Lambda}]$ is an isomorphism. This is a localization of the
$R[\Lambda^{-}]$-module inclusion
$R[\bm{\Lambda}^{-}]\xhookrightarrow{}R[\bm{\Lambda}]$, and hence is
injective, and surjectivity follows from the condition
$\bm{\Lambda}=\bm{\Lambda}^{-}\Lambda$. Apply this to $\mathcal{M}$ to see
that, if
$\mathcal{N}=R[\bm{\Lambda}]\otimes_{R[\bm{\Lambda}^{-}]}\mathcal{M}$, then by
Lemma 3.1 (1) it’s enough to show that the homomorphism
$\text{Tor}^{R[\Lambda]}_{q}(R(\chi),\mathcal{N})\longrightarrow\text{Tor}^{R[\bm{\Lambda}]}_{q}(R(\chi),\mathcal{N})$
identifies the target with an $R[\bm{\Lambda}]$-module direct summand of the
source. Let $P_{\bullet}$ be a resolution of $R(\chi)$ by free
$R[\bm{\Lambda}]$-modules. It’s enough to show that the natural map
$P_{\bullet}\otimes_{R[\Lambda]}\mathcal{N}\longrightarrow
P_{\bullet}\otimes_{R[\bm{\Lambda}]}\mathcal{N}$ identifies the target with a
direct summand of the source. This follows from the fact that
$P_{\bullet}\otimes_{R[\Lambda]}\mathcal{N}\simeq(P_{\bullet}\otimes_{R[\Lambda]}R[\bm{\Lambda}])\otimes_{R[\bm{\Lambda}]}\mathcal{N}$
and the map $p\otimes\lambda\longmapsto\lambda p\colon
P_{\bullet}\otimes_{R[\Lambda]}R[\bm{\Lambda}]\longrightarrow P_{\bullet}$ is
a section of
$\frac{1}{[\bm{\Lambda}:\Lambda]}\sum_{\lambda\in\bm{\Lambda}/\Lambda}\lambda
p\otimes\lambda^{-1}\mapsfrom p$. ∎

Putting everything together, we obtain the main result of this section.

######  Proposition 3.6.

Let $R,\bm{\Lambda},\bm{\Lambda}^{-},\Lambda,\Lambda^{-},\mathcal{M}$ and
$\chi$ be as in Lemma 3.5. Assume that the submonoid $\Lambda^{-}$ of
$\Lambda$ is of the form described in Lemma 3.3 (1) and that $\mathcal{M}$ the
conditions of Lemma 3.2 for $\bm{\Lambda}^{-}$. Namely, $\mathcal{M}$ admits a
decomposition
$\mathcal{M}=\mathcal{M}^{+}\oplus\bigoplus_{\lambda\in\bm{\Lambda}^{-}}\mathcal{M}_{\lambda}$
as $R$-modules satisfying

  1. (a)

$\lambda\mathcal{M}_{\mu}\subseteq\mathcal{M}_{\lambda\mu}$ for
$\lambda,\mu\in\bm{\Lambda}^{-}$,

  2. (b)

for all $m\in\mathcal{M}$ there exists $\lambda\in\bm{\Lambda}^{-}$ such that
$\lambda m\in\bigoplus_{\lambda\in\bm{\Lambda}^{-}}\mathcal{M}_{\lambda}$.

Then, $\text{Tor}_{q}^{R[\bm{\Lambda}^{-}]}(R(\chi),\mathcal{M})=0$ for $q>0$.

Proof. By Lemma 3.1 and Lemma 3.5 it’s enough to show that
$\text{Tor}^{R[\Lambda]}_{q}(R(\chi),R[\Lambda]\otimes_{R[\Lambda^{-}]}\mathcal{M})$
vanishes for $q>0$. Recall from the proof of Lemma 3.2 that there is a natural
isomorphism

$R[\Lambda]\otimes_{R[\Lambda^{-}]}\bigoplus_{\lambda\in\bm{\Lambda}^{-}}\mathcal{M}_{\lambda}\stackrel{{\scriptstyle\sim}}{{\longrightarrow}}R[\Lambda]\otimes_{R[\Lambda^{-}]}\mathcal{M}.$

Write $\mathcal{N}$ for the source of the isomorphism. Then, $\mathcal{N}$
admits a direct sum decomposition
$\mathcal{N}=\bigoplus_{\lambda\in\bm{\Lambda}}\mathcal{N}_{\lambda}$, where
$\mathcal{N}_{\lambda}$ is the $R$-submodule generated by $\mu\otimes m$ with
$m\in\mathcal{M}_{\mu^{-1}\lambda}$ and $\mu\in\Lambda$ such that
$\mu^{-1}\lambda\in\Lambda^{-}$. For every coset $L$ in
$\bm{\Lambda}/\Lambda$, the $R[\Lambda]$-submodule
$\mathcal{N}_{L}=\bigoplus_{\lambda\in L}\mathcal{N}_{\lambda}$ satisfies the
assumptions of Lemma 3.2, and hence
$\text{Tor}^{R[\Lambda]}_{q}(R(\chi),\mathcal{N}_{L})=0$ for $q>0$ by Lemma
3.3. In particular, $\text{Tor}^{R[\Lambda]}_{q}(R(\chi),\mathcal{N})=0$ for
$q>0$. ∎

##  4\. The locally analytic representations

In this section we will introduce the locally analytic representations whose
homology we will be studying in the rest of the article and study some of
their properties. For example, we will prove Theorem 1.6.

###  4.1. Locally $\Sigma$-analytic and compact induction

Recall that a Hausdorff topological space $X$ is called strictly paracompact
if every open cover has a refinement by pairwise disjoint open subsets. A
locally $\Sigma$-analytic manifold is a pair $X=(X_{\Sigma},X^{\Sigma})$ where
$X_{\Sigma}$ is a (strictly) paracompact $p$-adic locally analytic manifold
and $X^{\Sigma}$ is a Hausdorff topological space that can be written as a
disjoint union of compact, totally disconnected subspaces. We will usually
write $X=X_{\Sigma}\times X^{\Sigma}$. A morphism of locally $\Sigma$-analytic
manifolds $f\colon X\longrightarrow Y$ consists of a pair of a morphism
$f_{\Sigma}\colon X_{\Sigma}\longrightarrow Y_{\Sigma}$ of locally analytic
manifolds and a continuous map $f^{\Sigma}\colon X^{\Sigma}\longrightarrow
Y^{\Sigma}$. We may similarly define a locally $\Sigma$-analytic group as a
pair of a $p$-adic locally analytic group and a locally profinite group with
the property that their quotients by some (or equivalently, any) compact open
subgroups are countable (these spaces are then automatically strictly
paracompact). We will start by defining locally $\Sigma$-analytic functions on
a locally $\Sigma$-analytic manifold $X$, which stated roughly are those that
are locally analytic in $X_{\Sigma}$ and smooth in $X^{\Sigma}$. If $Y$ is an
open subset of $X_{\Sigma}$ equipped with an isomorphism
$\mathbb{Y}(\mathbb{Q}_{p})\stackrel{{\scriptstyle\sim}}{{\longrightarrow}}Y$,
where $\mathbb{Y}$ is a rigid $\mathbb{Q}_{p}$-analytic closed ball, and $U$
is an open compact subset of $X$, we will say that a function from $Y\times U$
to a ($p$-adic) Banach space $W$ is $\mathbb{Y}\times U$-analytic if it
factors through projection to $Y$ and is given by a rigid
$\mathbb{Q}_{p}$-analytic function of $\mathbb{Y}$, i.e. by an element of
$\mathcal{O}(\mathbb{Y})\operatorname{\widehat{\otimes}}_{\mathbb{Q}_{p}}W$.
If $V$ is a Hausdorff locally convex $\mathbb{Q}_{p}$-vector space, we define
the space of locally $\Sigma$-analytic functions $X\longrightarrow V$ as the
direct limit

(4.1)
$\displaystyle\mathcal{C}^{{\Sigma-\text{la}}}(X,V)=\varinjlim_{\\{(\mathbb{X}_{i},U_{j},W_{i})_{(i,j)\in
I\times J}\\}}\prod_{(i,j)\in I\times
J}\mathcal{O}(\mathbb{X}_{i})\operatorname{\widehat{\otimes}}_{{\mathbb{Q}_{p}}}W_{i}$

running over all collections of triples
$(\mathbb{X}_{i},U_{j},W_{i})_{(i,j)\in I\times J}$, where
$(\mathbb{X}_{i})_{i\in I}$ is an analytic partition of $X_{\Sigma}$ in the
sense of [Eme17, Section 2.1], $(U_{j})_{j\in J}$ is a partition of
$X^{\Sigma}$ into pairwise disjoint open compact subspaces, and
$W_{i}\xhookrightarrow{}V$ is a BH-subspace of $V$ for each $i\in I$ (i.e. an
injective continuous map from a Banach space into $V$). We equip
$\mathcal{C}^{{\Sigma-\text{la}}}(X,V)$ with the locally convex direct limit
topology. We identify an element $f\in\mathcal{C}^{{\Sigma-\text{la}}}(X,V)$
with a continuous function $X\longrightarrow V$: if $f$ belongs to an element
of the product in 4.1, say $f=(\sum_{m}f_{i,j,m}\otimes w_{i,j,m})_{i,j}$, and
$x\in X$ belongs to $\mathbb{X}_{i}(\mathbb{Q}_{p})\times U_{j}$, then
$f(x)=\sum_{m}f_{i,j,m}(x)w_{i,j,m}$. If $X^{\Sigma}$ is compact, then there
is a topological isomorphism
$\mathcal{C}^{{\Sigma-\text{la}}}(X,V)\simeq\mathcal{C}^{\text{sm}}(X^{\Sigma},\mathcal{C}^{\text{la}}(X_{\Sigma},V))$.

Now, let $G=G_{\Sigma}\times G^{\Sigma}$ be a locally $\Sigma$-analytic group.
If $V$ is endowed with a continuous action of $G$, we will say that this
action is locally $\Sigma$-analytic if for all $v\in V$ the orbit map
$g\longmapsto gv$ is locally $\Sigma$-analytic. We do not impose any
conditions on the topology of $V$ such as being of compact type.

If $H\subseteq G$ is a closed (locally $\Sigma$-analytic) subgroup and $V$ is
a locally $\Sigma$-analytic representation of $H$ on a locally convex
${\mathbb{Q}_{p}}$-vector space, we write
$\operatorname{Ind}_{H}^{G}(V)^{\Sigma-\text{la}}$ for the locally analytic
induction

$\operatorname{Ind}_{H}^{G}(V)^{\Sigma-\text{la}}=\\{f\in\mathcal{C}^{\Sigma-\text{la}}(G,V):f(gh)=h^{-1}f(g)\text{
for all }g\in G,h\in H\\},$

and equip it with the topology it inherits as a closed subspace of
$\mathcal{C}^{\Sigma-\text{la}}(G,V)$. The group $G$ acts on
$\operatorname{Ind}_{H}^{G}(V)^{\Sigma-\text{la}}$ by left translations:
$(gf)(g^{\prime})=f(g^{-1}g^{\prime})$ for $g,g^{\prime}\in G$ and for
$f\in\operatorname{Ind}_{H}^{G}(V)^{\Sigma-\text{la}}$. This action is locally
$\Sigma$-analytic if the quotient $G/H$ is compact, which will always be the
case for us. Given a subset $\Omega\subseteq G$, we will write
$\operatorname{Ind}_{H}^{G}(V)^{{\Sigma-\text{la}}}(\Omega)$ for the subspace
of $\operatorname{Ind}_{H}^{G}(V)^{{\Sigma-\text{la}}}$ consisting of
functions $f$ whose support $\operatorname{supp}(f)$ is contained in $\Omega$.

We are also interested the compact induction

$\displaystyle\operatorname{c-Ind}_{H}^{G}(V)=\\{$ $\displaystyle f\colon
G\longrightarrow V:f(gh)=h^{-1}f(g)\text{ for all }g\in G,h\in H,$
$\displaystyle\operatorname{supp}(f)/H\text{ is compact }\\}$

with its action of $G$ by left translations. Again, for $\Omega\subseteq G$,
we will write $\operatorname{c-Ind}_{H}^{G}(V)(\Omega)$ for the subspace of
$\operatorname{c-Ind}_{H}^{G}(V)$ consisting of functions whose support is
contained in $\Omega$. We equip $\operatorname{c-Ind}_{H}^{G}(V)$ with its
locally convex direct limit topology obtained from the isomorphism

$\operatorname{c-Ind}_{H}^{G}(V)\simeq\varinjlim_{\Omega}\operatorname{c-Ind}_{H}^{G}(V)(\Omega),$

where $\Omega$ runs through finite unions of left cosets of $H$ in $G$. If $V$
has its finest locally convex topology (for example, if $V$ is finite-
dimensional and Hausdorff), then the same is true of
$\operatorname{c-Ind}_{H}^{G}(V)$. Given $g\in G$ and $v\in V$, we write
$[g,v]$ for the element of $\operatorname{c-Ind}_{H}^{G}(V)$ satisfying
$\operatorname{supp}([g,v])=gH$ and $[g,v](g)=v$.

The following lemma guarantees that our definition of $\Sigma$-analytic
induction agrees with the example in the introduction.

######  Lemma 4.2.

Assume that $G_{\Sigma}=G_{\Sigma,1}\times G_{\Sigma,2}$ and
$G^{\Sigma}=G^{\Sigma}_{1}\times G^{\Sigma}_{2}$. For $i=1,2$, let
$G_{i}=G_{\Sigma,i}\times G^{\Sigma}_{i}$ and let
$H_{\Sigma,i},H^{\Sigma}_{i}$ be closed subgroups of $G_{\Sigma,i}$,
$H^{\Sigma}_{i}$, $H_{i}=H_{\Sigma,i}\times H^{\Sigma}_{i}$, and $V_{i}$ be a
locally $\Sigma$-analytic representation of $H_{i}$ on a Hausdorff locally
convex $E$-vector space of compact type, where $E$ is some finite extension of
$\mathbb{Q}_{p}$.

  1. (i)

If $H_{i}$ is open in $G_{i}$ for $i=1,2$, then there is a topological
isomorphism

$\operatorname{c-Ind}_{H_{1}}^{G_{1}}(V_{1})\widehat{\boxtimes}_{E}\operatorname{c-Ind}_{H_{2}}^{G_{2}}(V_{2})\stackrel{{\scriptstyle\sim}}{{\longrightarrow}}\operatorname{c-Ind}_{H_{1}\times
H_{2}}^{G_{1}\times G_{2}}(V_{1}\widehat{\boxtimes}_{E}V_{2}).$

  2. (ii)

If $G_{i}/H_{i}$ is compact for $i=1,2$, then there is a topological
isomorphisms

$\operatorname{Ind}_{H_{1}}^{G_{1}}(V_{1})^{\Sigma-\text{la}}\widehat{\boxtimes}_{E}\operatorname{Ind}_{H_{2}}^{G_{2}}(V_{2})^{\Sigma-\text{la}}\stackrel{{\scriptstyle\sim}}{{\longrightarrow}}\operatorname{Ind}_{H_{1}\times
H_{2}}^{G_{1}\times
G_{2}}(V_{1}\widehat{\boxtimes}_{E}V_{2})^{\Sigma-\text{la}}.$

Proof. Let us start with the case of compact induction. The morphism we seek
sends $f_{1}\otimes f_{2}$ to the map $(g_{1},g_{2})\longmapsto
f_{1}(g_{1})\otimes f_{2}(g_{2})$. We must check that this is indeed a
topological isomorphism. As $H_{i}$ is open, a (right) $H$-equivariant
function $G_{i}\longrightarrow V_{i}$ has compact support if and only if its
support is a finite union of left cosets of $H_{i}$. Write each $V_{i}$ as a
countable direct limit $V_{i}=\varinjlim_{n}V_{i,n}$ of Banach spaces with
compact injective transition maps and choose a countable increasing system
$(\Omega_{i,n})_{n}$ of finite sets of left $H_{i}$-cosets whose union is
$G_{i}$. Then,

$\operatorname{c-Ind}_{H_{i}}^{G_{i}}(V_{i})\simeq\varinjlim_{\Omega}\operatorname{c-Ind}_{H_{i}}^{G_{i}}(V_{i})(\Omega)\simeq\varinjlim_{n}\operatorname{c-Ind}_{H_{i}}^{G_{i}}(V_{i,n})(\Omega_{i,n})$

and the transition maps are injective and compact, and we have a similar
expression for the compact induction to $G_{1}\times G_{2}$. Thus, [Eme17,
Proposition 1.1.32 (i)] implies that

$\displaystyle\operatorname{c-Ind}_{H_{1}}^{G_{1}}(V_{1})\operatorname{\widehat{\otimes}}_{E}\operatorname{c-Ind}_{H_{2}}^{G_{2}}(V_{2})$
$\displaystyle\simeq\varinjlim_{n}\operatorname{c-Ind}_{H_{1}}^{G_{1}}(V_{1,n})(\Omega_{1,n})\operatorname{\widehat{\otimes}}_{E}\operatorname{c-Ind}_{H_{2}}^{G_{2}}(V_{2,n})(\Omega_{2,n})$
$\displaystyle\simeq\varinjlim_{n}\operatorname{c-Ind}_{H_{1}\times
H_{2}}^{G_{1}\times
G_{2}}(V_{1,n}\operatorname{\widehat{\otimes}}_{E}V_{2,n})(\Omega_{1,n}\times\Omega_{2,n})$
$\displaystyle\simeq\operatorname{c-Ind}_{H_{1}\times H_{2}}^{G_{1}\times
G_{2}}(V_{1}\operatorname{\widehat{\otimes}}_{E}V_{2}),$

as required (for the second isomorphism note that, for example,
$\operatorname{c-Ind}_{H_{1}}^{G_{1}}(V_{1,n})(\Omega_{1,n})$ is isomorphic to
a direct sum of $|\Omega_{1,n}/H_{1}|$ copies of $V_{1,n}$). This proves (1).

Next, let us deal with (2). The map above is defined by the same formula as in
the case of compact induction. Choose for $i=1,2$ a section $s_{i}\colon
G_{i}/H_{i}\longrightarrow G_{i}$ of locally $\Sigma$-analytic manifolds, i.e.
sections $G_{\Sigma,i}/H_{\Sigma,i}\longrightarrow G_{\Sigma,i}$ of locally
analytic manifolds and $G^{\Sigma}_{i}/H^{\Sigma}_{i}\longrightarrow
G^{\Sigma}_{i}$ of topological spaces. These sections induce a commutative
diagram

${\operatorname{Ind}_{H_{1}}^{G_{1}}(V_{1})^{\Sigma-\text{la}}\widehat{\boxtimes}_{E}\operatorname{Ind}_{H_{2}}^{G_{2}}(V_{2})^{\Sigma-\text{la}}}$${\operatorname{Ind}_{H_{1}\times
H_{2}}^{G_{1}\times
G_{2}}(V_{1}\widehat{\boxtimes}_{E}V_{2})^{\Sigma-\text{la}}}$${\mathcal{C}^{\Sigma-\text{la}}(G_{1}/H_{1},V_{1})\operatorname{\widehat{\otimes}}_{E}\mathcal{C}^{\Sigma-\text{la}}(G_{2}/H_{2},V_{2})}$${\mathcal{C}^{\Sigma-\text{la}}(G_{1}\times
G_{2}/H_{1}\times
H_{2},V_{1}\operatorname{\widehat{\otimes}}_{E}V_{2}),}$$\scriptstyle{\sim}$$\scriptstyle{\sim}$

and it’s enough to show that the bottom map is a topological vector space
isomorphism. Choose for each $i$ a cofinal system
$((\mathbb{X}_{i,n,k},U_{i,n,j})_{k,j})_{n}$ of pairs of analytic partitions
of $G_{\Sigma,i}/H_{\Sigma,i}$, where $(\mathbb{X}_{i,n+1,k})_{k}$ is a
relatively compact refinement of $(\mathbb{X}_{i,n,k})_{k}$, and disjoint
partitions of $G^{\Sigma}_{i}/H^{\Sigma}_{i}$ into open compact subspaces.
Then, [Eme17, Proposition 1.1.32 (i)] shows that

$\mathcal{C}^{\Sigma-\text{la}}(G_{i}/H_{i},V_{i})\simeq\varinjlim_{n}\prod_{k,j}\mathcal{O}(\mathbb{X}_{i,n,k})\operatorname{\widehat{\otimes}}_{{\mathbb{Q}_{p}}}V_{i,n},$

and the transition maps are injective and compact. In particular, applying
[Eme17, Proposition 1.1.32 (i)] again shows that

(4.3)
$\displaystyle\mathcal{C}^{\Sigma-\text{la}}(G_{1}/H_{1},V_{1})\operatorname{\widehat{\otimes}}_{E}\mathcal{C}^{\Sigma-\text{la}}(G_{2}/H_{2},V_{2})\simeq\varinjlim_{n}\prod_{k,j}\mathcal{O}(\mathbb{X}_{1,n,k}\times\mathbb{X}_{2,n,k})\operatorname{\widehat{\otimes}}_{{\mathbb{Q}_{p}}}(V_{1,n}\operatorname{\widehat{\otimes}}_{E}V_{2,n}).$

As the quotients $G_{i}/H_{i}$ are compact,
$((\mathbb{X}_{1,n,k}\times\mathbb{X}_{2,n,k})_{k})_{n}$ is a cofinal system
of analytic partitions of $G_{\Sigma,1}\times G_{\Sigma,2}/H_{\Sigma,1}\times
H_{\Sigma,2}$ and $((U_{1,n,j}\times U_{2,n,j})_{j})_{n}$ is a cofinal system
of partitions of $G^{\Sigma}_{1}\times G^{\Sigma}_{2}/H^{\Sigma}_{1}\times
H^{\Sigma}_{2}$ into open compact subspaces. In particular, the right-hand
side of 4.3 is isomorphic to $\mathcal{C}^{\Sigma-\text{la}}(G_{1}\times
G_{2}/H_{1}\times H_{2},V_{1}\operatorname{\widehat{\otimes}}_{E}V_{2})$,
which completed the proof. ∎

######  Lemma 4.4.

Assume that $H_{0}$ is an open subgroup of $G$. If the action of $H_{0}$ on a
vector space $V$ extends to an action of a submonoid $H^{+}$ of $G$ containing
$H_{0}$, then there is a natural right action of the Hecke algebra
$\mathcal{H}(H^{-},H_{0})$ on $\operatorname{c-Ind}_{H_{0}}^{G}(V)$, where
$H^{-}=(H^{+})^{-1}$, such that for $\mu\in H^{-}$,

$[g,v][H_{0}\mu H_{0}]=\sum_{x\mu\in H_{0}\mu
H_{0}/H_{0}}[gx\mu,(x\mu)^{-1}v].$

This action commutes with the action of $G$.

Proof. We may view the action of $H^{+}$ on $V$ as a right action of $H^{-}$.
Consider the space $E[G]\otimes_{E}V$, it is a $(G,H^{-})$-bimodule with
$H^{-}$ acting via $h\cdot(g\otimes v)=gh\otimes h^{-1}v$ and
$\operatorname{c-Ind}_{H_{0}}^{G}(V)$ can be identified with its right
$H_{0}$-coinvariants (an element $g\otimes v$ of the latter corresponding to
$[g,v]$). The action of the Hecke algebra is that described in Lemma 2.29, and
$G$-equivariance is immediate from this formula. ∎

###  4.2. Parabolic and parahoric induction

We now retake the notation from Section 2, more specifically Section 2.9.
Thus, we let $\ell_{1},...,\ell_{n}$ be prime numbers, for each $i$ $L_{i}$ is
a finite extension of $\mathbb{Q}_{\ell_{i}}$ and $\mathbf{G}_{i}$ is a
reductive group over $L_{i}$, etc. Fix a subset $\Sigma$ of $\\{1,...,n\\}$
such that for all $i\in\Sigma$, $\ell_{i}=p$, and set
$G_{\Sigma}=\prod_{i\in\Sigma}G_{i}$ and
$G^{\Sigma}=\prod_{i\not\in\Sigma}G_{i}$, and similarly for the subgroups of
$G$ defined in Section 2.9. Hence, $G=G_{\Sigma}\times G^{\Sigma}$ is a
locally $\Sigma$-analytic manifold and similarly for its subgroups $P$, $M$,
etc. Analogously, we also have similar decompositions
$\Phi=\Phi_{\Sigma}\coprod\Phi^{\Sigma},\Delta=\Delta_{\Sigma}\coprod\Delta^{\Sigma}$,
etc. Recall from Section 2.9 that we will usually reference results from
Section 2 that were stated (and proven) for each of the $G_{i}$ but also hold
for $G$ when we want to use them in the latter case. Let $\mathcal{J}_{M}$ be
a subgroup of $\widetilde{J}_{M}$ containing $AJ_{M}$. Many of our results on
$\widetilde{J}_{M}$ and $\widetilde{J}_{M}^{-}$ from Section 2 (for example,
Lemma 2.15 and Lemma 2.16) also hold for $\mathcal{J}_{M}$ and
$\mathcal{J}_{M}^{-}$ for trivial reasons, and we will also use them without
comment in this situation.

Let $R$ be a Noetherian Banach ${\mathbb{Q}_{p}}$-algebra and $\sigma$ be a
locally $\Sigma$-analytic representation of $\mathcal{J}_{M}$ on some finitely
generated $R$-module, which we also write as $\sigma$, equipped with its
unique Banach $R$-module topology. We will write
$V=\operatorname{c-Ind}_{\mathcal{J}_{M}}^{M}(\sigma)$ for the compact
induction of $\sigma$ to $M$. As explained in the previous section, we equip
$V$ with the direct limit locally convex topology induced from the expression

(4.5)
$V\simeq\varinjlim_{\Omega}\operatorname{c-Ind}_{\mathcal{J}_{M}}^{M}(\sigma)(\Omega),$

where $\Omega$ runs over finite unions of cosets of $\mathcal{J}_{M}$ in $M$,
and the terms in the limit, which are isomorphic as $R$-modules to direct sums
of finitely many copies of $\sigma$, are given their unique Banach $R$-module
topologies. We extend the action of $M$ on $V$ to a locally $\Sigma$-analytic
action of $P$ by letting $N$ act trivially, and we extend $\sigma$ to $J\cap
P$ similarly. Restriction of functions induces a topological isomorphism
$\operatorname{Ind}_{J\cap
P}^{J}(\sigma)^{\Sigma-\text{la}}\stackrel{{\scriptstyle\sim}}{{\longrightarrow}}\mathcal{C}^{{\Sigma-\text{la}}}(J\cap\overline{N},\sigma)$
as in the proof of Lemma 4.2. Likewise, restriction of functions induces
topological isomorphisms

$\operatorname{Ind}_{P}^{G}(V)^{\Sigma-\text{la}}(JP)\stackrel{{\scriptstyle\sim}}{{\longrightarrow}}\operatorname{Ind}_{J\cap
P}^{J}(V)^{\Sigma-\text{la}}\stackrel{{\scriptstyle\sim}}{{\longrightarrow}}\mathcal{C}^{{\Sigma-\text{la}}}(J\cap\overline{N},V),$

the first of which is $J$-equivariant.

Let $s>0$. Recall from Proposition 2.20 that $J_{s}$ admits a rooted Iwahori
decomposition, $J_{s}=\overline{N}_{s}J_{M,s}N_{s}$. For each $i\in\Sigma$, in
Section 2.4 we constructed an affinoid $L_{i}$-space
$\overline{\mathbf{N}}_{i,s}$ whose set of $L_{i}$-points is
$\overline{N}_{i,s}$. In particular, we may view $\overline{N}_{\Sigma,s}$ as
the set of $\mathbb{Q}_{p}$-points of the $\mathbb{Q}_{p}$-affinoid
$\overline{\mathbf{N}}_{\Sigma,s}:=\prod_{i\in\Sigma}\operatorname{Res}_{L_{i}/\mathbb{Q}_{p}}\overline{\mathbf{N}}_{i,s}$.
Given $s$ and $s^{\prime}$, we will say that a locally analytic function
$J\cap\overline{N}\longrightarrow\sigma$ is $(s,s^{\prime})$-$\Sigma$-analytic
if its restriction to each left
$\overline{N}_{\Sigma,s}\times\overline{N}^{\Sigma}_{s^{\prime}}$-coset is
$\overline{\mathbf{N}}_{\Sigma,s}\times\overline{N}^{\Sigma}_{s^{\prime}}$-analytic.
Let us also say that a function $f$ in $\operatorname{Ind}_{J\cap
P}^{J}(\sigma)^{\text{la}}$ is $(s,s^{\prime})$-$\Sigma$-analytic if its
restriction to $J\cap\overline{N}$ is. We will write
$\mathcal{A}_{\sigma}^{(s,s^{\prime})-{\Sigma-\text{an}}}$ for the subspace of
$\operatorname{Ind}_{J\cap P}^{J}(\sigma)^{\Sigma-\text{la}}$ consisting of
such functions. Restriction to $J\cap\overline{N}$ induces an isomorphism from
$\mathcal{A}_{\sigma}^{(s,s^{\prime})-{\Sigma-\text{an}}}$ to the space of
$(s,s^{\prime})$-$\Sigma$-analytic functions
$J\cap\overline{N}\longrightarrow\sigma$, which we use to give the former the
structure of a Banach $R$-module. If $\sigma$ is free over $R$, then
$\mathcal{A}_{\sigma}^{(s,s^{\prime})-{\Sigma-\text{an}}}$ is
orthonormalizable. When $s=s^{\prime}$, we will speak of $s$-$\Sigma$-analytic
functions instead and write
$\mathcal{A}_{\sigma}^{s-{\Sigma-\text{an}}}=\mathcal{A}_{\sigma}^{(s,s)-{\Sigma-\text{an}}}$.
Note that $\operatorname{Ind}_{J\cap P}^{J}(\sigma)^{\Sigma-\text{la}}$ is
isomorphic to the colimit
$\varinjlim_{s}\mathcal{A}_{\sigma}^{s-{\Sigma-\text{an}}}$ since any rigid
analytic closed ball in $\overline{\mathbf{N}}$ containing the identity is
contained in some $\overline{\mathbf{N}}_{s}$ and vice versa (but the
$\overline{\mathbf{N}}_{s}$ will not necessarily be closed balls if the
$\mathbf{G}_{i}$ aren’t split).

######  Lemma 4.6.

If $s,s^{\prime}$ are large enough, then
$\mathcal{A}_{\sigma}^{(s,s^{\prime})-{\Sigma-\text{an}}}$ is stable under the
action of $J$.

Proof. As $\sigma$ is finitely generated, we may choose $s$ and $s^{\prime}$
large enough so that the orbits of $J_{i,M,s}$ in $\sigma$ are analytic for
$i\in\Sigma$ (i.e. they come from rigid-analytic morphisms of
$\mathbf{J}_{i,M,s}$, as defined in Section 2.4) and the orbits of
$J_{i,M,s^{\prime}}$ are constant for $i\not\in\Sigma$. Let $y\in J$ and
$f\in\mathcal{A}_{\sigma}^{(s,s^{\prime})-{\Sigma-\text{an}}}$. We will show
that $(y^{-1}f)|_{J\cap\overline{N}}$ is $(s,s^{\prime})$-$\Sigma$-analytic.
By writing $f$ as a sum of functions with support in different cosets of
$\overline{N}_{\Sigma,s}\times\overline{N}^{\Sigma}_{s^{\prime}}$, it’s enough
to show that
$(y^{-1}f)|_{\overline{N}_{\Sigma,s}\times\overline{N}^{\Sigma}_{s^{\prime}}}$
is
$\overline{\mathbf{N}}_{\Sigma,s}\times\overline{N}^{\Sigma}_{s^{\prime}}$-analytic.
Write $y=\overline{n}_{0}m_{0}n_{0}$ according to the Iwahori decomposition of
$J$ (Lemma 2.10), set $p_{0}=m_{0}n_{0}$, and let $\overline{n}\colon
J_{\Sigma,s}\times
J^{\Sigma}_{s^{\prime}}\longrightarrow\overline{N}_{\Sigma,s}\times\overline{N}^{\Sigma}_{s^{\prime}}$
and $m\colon J_{\Sigma,s}\times J^{\Sigma}_{s^{\prime}}\longrightarrow
J_{\Sigma,M,s}\times J^{\Sigma}_{M,s^{\prime}}$ be the projections given by
the Iwahori decomposition of $J_{s}$ (Proposition 2.20). Then, for all
$x\in\overline{N}_{\Sigma,s}\times\overline{N}^{\Sigma}_{s^{\prime}}$ one has

(4.7)
$\displaystyle(y^{-1}f)(x)=m(p_{0}xp_{0}^{-1})m_{0}^{-1}f(\overline{n}_{0}\cdot\overline{n}(p_{0}xp_{0}^{-1})).$

By Lemma 2.24 (1) and (2), the maps
$x\longmapsto\overline{n}(p_{0}xp_{0}^{-1})\colon\overline{N}_{\Sigma,s}\longrightarrow\overline{N}_{\Sigma,s}$
and $x\longmapsto m(p_{0}xp_{0}^{-1})\colon$
$\overline{N}_{\Sigma,s}\longrightarrow J_{\Sigma,M,s}$ are induced from
rigid-analytic morphisms. We have similar continuous maps for
$\overline{N}^{\Sigma}_{s^{\prime}}$. By our choice of $s$ and $s^{\prime}$,
this implies that 4.7 is
$\overline{\mathbf{N}}_{\Sigma,s}\times\overline{N}^{\Sigma}_{s^{\prime}}$-analytic
as a function of
$x\in\overline{N}_{\Sigma,s}\times\overline{N}^{\Sigma}_{s^{\prime}}$. ∎

We will write $s_{\sigma}$ for the smallest $s$ such that
$\mathcal{A}_{\sigma}^{s-{\Sigma-\text{an}}}$ is stable under the action of
$J$, and when writing $\mathcal{A}_{\sigma}^{s-{\Sigma-\text{an}}}$ we will
always implicitly assume $s\geq s_{\sigma}$. Similarly, when we consider
$\mathcal{A}_{\sigma}^{(s,s^{\prime})-{\Sigma-\text{an}}}$ we will always
assume that $s$ and $s^{\prime}$ are large enough that this is stable under
the action of $J$.

######  Lemma 4.8.

If $\widetilde{s}\geq s$ and $\widetilde{s}^{\prime}\geq s^{\prime}$, then
$\mathcal{A}_{\sigma}^{(s,s^{\prime})-{\Sigma-\text{an}}}\subseteq\mathcal{A}_{\sigma}^{(\widetilde{s},\widetilde{s}^{\prime})-{\Sigma-\text{an}}}$.
If $\widetilde{s}>s$, then this inclusion is a compact map of Banach
$R$-modules.

Proof. The first statement is immediate. Note that there is an isomorphism of
$R$-Banach modules

(4.9)
$\displaystyle\left(\mathcal{C}^{\overline{\mathbf{N}}_{s}-\text{an}}(J_{\Sigma}\cap\overline{N}_{\Sigma},\mathbb{Q}_{p})\otimes_{\mathbb{Q}_{p}}\mathbb{Q}_{p}^{|(J^{\Sigma}\cap\overline{N}^{\Sigma})/\overline{N}^{\Sigma}_{s^{\prime}}|}\right)\operatorname{\widehat{\otimes}}_{\mathbb{Q}_{p}}\sigma\stackrel{{\scriptstyle\sim}}{{\longrightarrow}}\mathcal{A}_{\sigma}^{(s,s^{\prime})-{\Sigma-\text{an}}},$

where
$\mathcal{C}^{\overline{\mathbf{N}}_{s}-\text{an}}(J_{\Sigma}\cap\overline{N}_{\Sigma},\mathbb{Q}_{p})$
denotes the space of functions
$J_{\Sigma}\cap\overline{N}_{\Sigma}\longrightarrow\mathbb{Q}_{p}$ whose
restriction to each $\overline{N}_{s}$ coset is
$\overline{\mathbf{N}}_{s}$-analytic. The second statement follows follows
from this isomorphism by Lemma 2.9 and [Buz07, Corollary 2.9]. ∎

The $J_{M}$-equivariant injection $\sigma\xhookrightarrow{}V$ given by
$v\longmapsto[1,v]$ induces $J$-equivariant maps $\operatorname{Ind}_{J\cap
P}^{J}(\sigma)^{\Sigma-\text{la}}\longrightarrow\operatorname{Ind}_{J\cap
P}^{J}(V)^{\Sigma-\text{la}}$. Composing this with $\operatorname{Ind}_{J\cap
P}^{J}(V)^{\Sigma-\text{la}}\stackrel{{\scriptstyle\sim}}{{\longrightarrow}}\operatorname{Ind}_{P}^{G}(V)^{\Sigma-\text{la}}(JP)$
we obtain a $J$-equivariant injection $\operatorname{Ind}_{J\cap
P}^{J}(\sigma)^{\Sigma-\text{la}}\longrightarrow\operatorname{Ind}_{P}^{G}(V)^{\Sigma-\text{la}}(JP)$.

######  Lemma 4.10.

The image of this map is stable under the action of
$\mathcal{J}_{M}^{+}=(\mathcal{J}_{M}^{-})^{-1}$.

Proof. A function $f\in\operatorname{Ind}_{J\cap
P}^{J}(\sigma)^{\Sigma-\text{la}}$ is in the image of this map if and only if
there exists some $F\in\operatorname{Ind}_{J\cap
P}^{J}(\sigma)^{\Sigma-\text{la}}$ such that $f(y)=[1,F(y)]$ for all $y\in J$,
or equivalently, for all $y\in J\cap\overline{N}$. Consider some such $f$ and
$F$, and let $\mu\in\mathcal{J}_{M}^{+}$ and $y\in J\cap\overline{N}$. Then,
$(\mu f)(y)=\mu f(\mu^{-1}y\mu)=[1,\mu F(\mu^{-1}y\mu)]$, so $\mu f$ lies in
the image of this map. ∎

It follows from Lemma 2.15 that $J\mathcal{J}_{M}^{+}J$ is a submonoid of $G$,
and Lemma 4.10 implies that we may extend the action of $J$ on
$\operatorname{Ind}_{J\cap P}^{J}(\sigma)^{\Sigma-\text{la}}$ to an action of
$J\mathcal{J}_{M}^{+}J$ making the injection $\operatorname{Ind}_{J\cap
P}^{J}(\sigma)^{\Sigma-\text{la}}\longrightarrow\operatorname{Ind}_{P}^{G}(V)^{\Sigma-\text{la}}(JP)$
a $J\mathcal{J}_{M}^{+}J$-equivariant map. Explicitly, if $y$ is an element of
$J$ with Iwahori decomposition $y=\overline{n}mn$ and
$F\in\operatorname{Ind}_{J\cap P}^{J}(\sigma)^{\Sigma-\text{la}}$, then the
action of $\mathcal{J}_{M}^{+}$ is given by $(\mu F)(y)=m^{-1}\mu
F(\mu^{-1}\overline{n}\mu)$. If $\mu\in\mathcal{J}_{M}^{+}$, then $\mu$
preserves $\mathcal{A}_{\sigma}^{s-{\Sigma-\text{an}}}$ by Lemma 2.24 (3), and
for a fixed norm on $\sigma$, the norm of $\mu$ as an operator on
$\mathcal{A}_{\sigma}^{s-{\Sigma-\text{an}}}$ (with its norm induced by 4.9
and that of $\sigma$) is bounded above by the norm of $\mu$ on $\sigma$. If,
moreover,
$\mu^{-1}\overline{\mathbf{N}}_{\Sigma,s}\overline{N}^{\Sigma}_{s^{\prime}}\mu\subseteq\overline{\mathbf{N}}_{\Sigma,\widetilde{s}}\overline{N}^{\Sigma}_{s^{\prime}}$
for some $\widetilde{s}<s$ (for example, if $\mu=a^{-1}$ with $a\in
A^{--}_{\Sigma}$), then the action of $\mu$ on
$\mathcal{A}_{\sigma}^{(\widetilde{s},s^{\prime})-{\Sigma-\text{an}}}$ factors
through the inclusion
$\mathcal{A}_{\sigma}^{(s,s^{\prime})-\text{an}}\subseteq\mathcal{A}_{\sigma}^{(\widetilde{s},s^{\prime})-{\Sigma-\text{an}}}$,
and therefore induces an $R$-compact endomorphism of
$\mathcal{A}_{\sigma}^{(\widetilde{s},s^{\prime})-{\Sigma-\text{an}}}$ by
Lemma 4.8. Moreover, applying Lemma 4.4 and the results of Section 2.6, we
obtain the following.

######  Lemma 4.11.

The action of $\mathcal{J}_{M}^{+}$ on $\operatorname{Ind}_{J\cap
P}^{J}(\sigma)^{\Sigma-\text{la}}$ induces a (left) action of the Hecke
algebra $\mathcal{H}(J\mathcal{J}_{M}^{-}J,J)_{R}$ on
$\operatorname{c-Ind}_{J}^{G}(\operatorname{Ind}_{J\cap
P}^{J}(\sigma)^{\Sigma-\text{la}})$ commuting with the action of $G$, where
the characteristic functions $U_{\mu}=[J\mu J]$ for
$\mu\in\mathcal{J}_{M}^{-}$ act as

$U_{\mu}([g,F])=\sum_{x\mu\in J\mu J/J}[gx\mu,(x\mu)^{-1}F].$

Moreover, for each $s\geq s_{\sigma}$ and $\mu\in\mathcal{J}_{M}^{-}$, the
operator $U_{\mu}$ preserves the subspace
$\operatorname{c-Ind}_{J}^{G}(\mathcal{A}_{\sigma}^{s-{\Sigma-\text{an}}})$.
If $a\in A_{\Sigma}^{--}$, then the operator $U_{a}$ on this subspace is
$R$-compact.

###  4.3. A quasi-isomorphism

The $J$-equivariant embedding $\operatorname{Ind}_{J\cap
P}^{J}(\sigma)^{\Sigma-\text{la}}\longrightarrow\operatorname{Ind}_{P}^{G}(V)^{\Sigma-\text{la}}$
induces a $G$-equivariant $R$-linear map

$\varphi\colon\operatorname{c-Ind}_{J}^{G}(\operatorname{Ind}_{J\cap
P}^{J}(\sigma)^{\Sigma-\text{la}})\longrightarrow\operatorname{Ind}_{P}^{G}(V)^{\Sigma-\text{la}}.$

Explicitly, if $F\in\operatorname{Ind}_{J\cap
P}^{J}(\sigma)^{\Sigma-\text{la}}$, then $\varphi([1,F])(y)=[1,F(y)]$ for
$y\in J$. The source is a module for
$\mathcal{H}(J\mathcal{J}_{M}^{-}J,J)_{R}$, and we may regard
$\operatorname{Ind}_{P}^{G}(V)^{\Sigma-\text{la}}$ as a
$\mathcal{H}(J\mathcal{J}_{M}^{-}J,J)_{R}$-module by letting $U_{\mu}=[U\mu
U]$ act trivially. Write $R(1)$ for the
$\mathcal{H}(J\mathcal{J}_{M}^{-}J,J)_{R}$-module whose underlying $R$-module
is $R$ and where $U_{\mu}$ acts trivially. The goal of this section is to show
that $\varphi$ induces an isomorphism in the derived category
$D(\mathcal{H}(J\mathcal{J}_{M}^{-}J,J)_{R})$ of
$\mathcal{H}(J\mathcal{J}_{M}^{-}J,J)_{R}$-modules

$R(1)\otimes^{\mathbb{L}}_{\mathcal{H}(J\mathcal{J}_{M}^{-}J,J)_{R}}\operatorname{c-Ind}_{J}^{G}(\operatorname{Ind}_{J\cap
P}^{J}(\sigma)^{\Sigma-\text{la}})\stackrel{{\scriptstyle\sim}}{{\longrightarrow}}\operatorname{Ind}_{P}^{G}(V)^{\Sigma-\text{la}}.$

We will do this by following the ideas in [KS12, §2].

######  Proposition 4.12.

The map $\varphi$ is surjective.

Proof. We can write $G$ as a union of finitely many translates of $JP$,
$G=\bigcup_{i=1}^{m}g_{i}JP$. It is easy to see that we may write any element
of $\operatorname{Ind}_{P}^{G}(V)^{\Sigma-\text{la}}$ as sum of a functions in
$\operatorname{Ind}_{P}^{G}(V)^{\Sigma-\text{la}}$ whose support intersects
only one of the sets $g_{i}JP$. In particular,

$\operatorname{Ind}_{P}^{G}(V)^{\Sigma-\text{la}}=\sum_{i=1}^{m}g_{i}\cdot\operatorname{Ind}_{P}^{G}(V)^{\Sigma-\text{la}}(JP),$

so it’s enough to show that the image of $\varphi$ contains
$\operatorname{Ind}_{P}^{G}(V)^{\Sigma-\text{la}}(JP)$. Let $f$ be an element
of $\operatorname{Ind}_{P}^{G}(V)^{\Sigma-\text{la}}(JP)$. Since $f$ is
locally $\Sigma$-analytic, $f|_{J\cap\overline{N}}$ factors through some BH-
space $U\xhookrightarrow{}V$. It follows from the expression 4.5 of $V$ as an
LB-space and the closed graph theorem that any BH-space $U\xhookrightarrow{}V$
factors through $\operatorname{c-Ind}_{\mathcal{J}_{M}}^{M}(\sigma)(\Omega)$
for some finite union $\Omega$ of left cosets of $\mathcal{J}_{M}$ in $M$ (see
for example the proof of [Sch02, Proposition 8. (i)]). Hence,
$f(J\cap\overline{N})$ is contained in
$\operatorname{c-Ind}_{\mathcal{J}_{M}}^{M}(\sigma)(\Omega)$, where
$\Omega=\bigcup_{i=1}^{s}m_{i}\mathcal{J}_{M}$ for some integer $s\geq 0$ and
$m_{1},...,m_{s}\in M$. For any $\overline{n}\in J\cap\overline{N}$ we have
$f(\overline{n})=\sum_{i=1}^{s}[m_{i},F_{i}(\overline{n})]$, where we have set
$F_{i}(\overline{n}):=f(\overline{n})(m_{i})$. Thus, we may assume that $s=1$
and $f|_{J\cap\overline{N}}=[m,F]$ for some
$F\in\mathcal{C}^{\Sigma-\text{la}}(J\cap\overline{N},\sigma)$. Choose $a\in
A$ such that $am^{-1}(J\cap\overline{N})ma^{-1}\subseteq J\cap\overline{N}$,
so that by changing $F$ by $\overline{n}\longmapsto aF(\overline{n})$ we may
assume that $m^{-1}(J\cap\overline{N})m\subseteq J\cap\overline{N}$. As
$(m^{-1}f)(m^{-1}\overline{n}m)=[1,F(\overline{n})]$, we see by replacing $f$
with $m^{-1}f$ and $F$ with $y\longmapsto F(mym^{-1})$ for $y\in
m^{-1}(J\cap\overline{N})m$ that it suffices to consider the case when