# Restricting admissible representations to fixed-point subgroups

Peter Latham  Department of Mathematics and Statistics, University of Ottawa,
Ottawa, Canada<EMAIL_ADDRESS>and  Monica Nevins  Department of
Mathematics and Statistics, University of Ottawa, Ottawa, Canada
<EMAIL_ADDRESS>
###### Abstract.

Given a $p$-adic group $G=\mathbf{G}(F)$ and a finite group
$\Gamma\subset\mathrm{Aut}_{F}(\mathbf{G})$ such that the fixed-point subgroup
$\mathbf{G}^{\Gamma}$ is reductive, we establish a relationship between
constructions of types for $G$ and $G^{\Gamma}$ due to Yu, Kim–Yu and Fintzen,
which generalizes the relationship between general linear and classical groups
observed by Stevens. As an application, given an irreducible representation
$\pi$ of $G$, we explicitly identify a number of the inertial equivalence
classes occurring in the representation $\pi|_{G^{[\Gamma]}}$.

###### Key words and phrases:

$p$-adic groups, theory of types, fixed-point subgroups

###### 1991 Mathematics Subject Classification:

22E50

The second author’s research is supported by a Discovery Grant from NSERC
Canada.

Given a reductive $p$-adic group $G$, there are two distinct approaches to
constructing types for $G$ (in the sense of [BK98]), each of which relies on
differing hypotheses (and, whenever these two constructions overlap, they
produce the same types). Firstly, one has the construction of Yu, Kim–Yu and
Fintzen [Yu01, KY17, Fin21], which is valid for an arbitrary tamely ramified
group $G$, but which only produces types for every inertial equivalence class
if one imposes additional hypotheses on the prime number $p$. This
construction is notable in that it is uniform across all such $G$.

In contrast, the constructions of Bushnell–Kutzko, Stevens, and
Blasco–Blondel, each of which tackles only a specific family of groups, come
with the advantage of requiring no tameness assumptions on the prime $p$. In
each, the key is the Bushnell–Kutzko construction of _semisimple characters_
for $\mathbf{GL}_{n}(F)$ [BK93, BK99], which are a family of characters of
certain pro-$p$ subgroups of $\mathbf{GL}_{n}(F)$ satisfying a number of
remarkable properties. The constructions of Stevens [Ste05, Ste08], and
Blasco–Blondel [BB12] each rely on exploiting the relationships between $G$
and $\mathbf{GL}_{n}(F)$ in order to define a new family of semisimple
characters by restriction of those of Bushnell–Kutzko. This approach by nature
reveals a network of relationships between the theories of types for the
various groups for which such a construction has been carried out. Nowadays
such relationships fall under the umbrella term of _endo-equivalence_ , with
the case of $\mathbf{GL}_{n}(F)$ and classical groups being just one
particularly simple example of a more general phenomenon [KSS21].

Since the constructions of Yu _et al._ treat each group in abstraction, such
relationships among related groups do not immediately follow—while they do
produce a family of semisimple characters (with a more straightforward
construction than is seen in the Bushnell–Kutzko setting, due to the
hypotheses on $p$), to date the only results relating these constructions have
been between $G$ and its derived subgroup [Bou21]. Our goal in this paper is
to show that, in the setting of Yu _et al._ ’s construction, the relationship
between semisimple characters for $\mathbf{GL}_{n}(F)$ and semisimple
characters for classical groups admits a significant generalization.

Before stating our results, it is useful to quickly recap the approaches of
[Ste05] and [BB12], which provide our motivating examples. In [Ste05], Stevens
considers a classical group $G$, identified as the fixed points in
$\tilde{G}=\mathbf{GL}_{n}(F)$ of a Galois involution $\gamma$. The
Bushnell–Kutzko construction, along with a generalization due to Stevens in
_loc. cit._ , associates to each _semisimple stratum_ $(\mathfrak{a},\beta)$
for $\tilde{G}$ a finite set $\tilde{\mathcal{C}}(\mathfrak{a},\beta)$ of
semisimple characters of a pro-$p$ open subgroup
$\tilde{H}_{+}(\mathfrak{a},\beta)$ of $\tilde{G}$. Stevens shows that if
$(\mathfrak{a},\beta)$ is _skew_ , _i.e._ if $\mathfrak{a}$ is $\gamma$-stable
and $\beta$ is contained in the Lie algebra of $G$, then
$\tilde{H}_{+}(\mathfrak{a},\beta)$ is stable under the action of $\gamma$, in
which case he is able to define
$H_{+}(\mathfrak{a},\beta)=\tilde{H}_{+}(\mathfrak{a},\beta)^{\gamma}$ and
equip this group with a set $\mathcal{C}(\mathfrak{a},\beta)$ of semisimple
characters by restriction. It transpires that the semisimple characters
obtained in this way still have the desired properties, and this approach
ultimately leads to a construction of types for classical groups [Ste08].
Building upon this, in [BB12] Blondel and Blasco consider the case where
$\mathbf{G}_{2}(F)$ is the fixed points under triality in
$\mathbf{SO}_{8}(F)$; they establish the analogous relationships between
semisimple characters, with this yielding a construction of supercuspidal
representations of $\mathbf{G}_{2}(F)$.

The technical result underlying the present paper is a generalization of this
relationship to the Yu setting, and to a wider class of groups. Specifically,
we consider an arbitrary tamely ramified group $G$, equipped with a finite
group $\Gamma$ of algebraic automorphisms, the order of which is coprime to
$p$. Then (the connected component of) the fixed-point subgroup $G^{\Gamma}$
is a closed reductive subgroup of $G$ [PY02]. Here, the situation is rather
different to that considered by Stevens, in that _one already has_ a
construction of semisimple characters for both $G$ and $G^{\Gamma}$. Thus,
rather than providing a new construction of types, our goal is to show that
the existing constructions of types are related in a manner directly
generalizing the situation of [Ste05, BB12].

The construction of semisimple characters for the group $G$ proceeds by
defining a family of _truncated data_ $\Sigma$. To each such datum $\Sigma$,
one may associate a pro-$p$ open subgroup $H_{+}(\Sigma)$ of $G$, together
with a semisimple character $\theta_{\Sigma}$ of $H_{+}(\Sigma)$. We identify
a family of truncated data which we call _$\Gamma$ -stable_ that provide the
natural generalization of the notion of _skewness_ considered in [Ste05], and
prove the following result.

######  Theorem 1 (Theorem 4.4).

Suppose that $\Sigma$ is a $\Gamma$-stable truncated datum. Then the group
$H_{+}(\Sigma)$ is stable under the action of $\Gamma$, and the pair
$(H_{+}(\Sigma)^{\Gamma},\theta_{\Sigma})$ is a semisimple character for
$G^{\Gamma}$.

In fact, this is explicit: we are able to write down a datum $\Sigma^{\Gamma}$
for $G^{\Gamma}$ such that
$(H_{+}(\Sigma)^{\Gamma},\theta_{\Sigma})=(H_{+}(\Sigma^{\Gamma}),\theta_{\Sigma^{\Gamma}})$.

Given such a natural relationship between the semisimple characters of $G$ and
$G^{\Gamma}$, it is natural to ask to what extent this relationship extends to
one between the corresponding types. Specifically, given a type $(J,\lambda)$
for $G$ which contains some $\Gamma$-stable semisimple character
$(H_{+},\theta)$, we may ask for a description of the irreducible components
of $\lambda|_{J\cap G^{\Gamma}}$. The situation here is rather less elegant,
due to the fact that not every irreducible component $\xi$ of $\lambda|_{J\cap
G^{[\Gamma]}}$ will be a type. Nonetheless, we are able to adapt the arguments
of [LN21, §7] in order to associate to each such $\xi$ an inertial equivalence
class $\mathfrak{s}_{\xi}$ of representations of $G^{\Gamma}$, and prove the
following.

######  Theorem 2 (Theorem 5.4).

Let $\pi$ be a representation of $G$ which is generated by its
$\lambda$-isotypic vectors. For each irreducible component $\xi$ of
$\lambda|_{J\cap G^{\Gamma}}$, the representation $\pi|_{G^{\Gamma}}$ contains
an irreducible subquotient of inertial support $\mathfrak{s}_{\xi}$.

These two results give a uniform framework for some important questions of
branching to reductive subgroups that can otherwise only be treated case-by-
case. For example, taking $\Gamma$ to be a finite-order inner automorphism of
$\mathbf{G}$ yields information about branching to some Levi (and twisted
Levi) subgroups $\mathbf{G}^{\Gamma}$. On the other hand, working with a group
$\Gamma$ of outer automorphisms is a key tool in understanding representations
of exceptional groups.

Several interesting questions remain open. One is to classify the semisimple
characters of $G^{\Gamma}$ obtained through this construction; one obstacle to
lifting arbitrary truncated data from $G^{\Gamma}$ to $G$ relates to
satisfying the genericity condition (iv) of Definition 2.1. For another, the
two hypotheses on $\Gamma$—that it is finite, and that its order is prime to
$p$—are used in the proofs of most of the results in Section 3, through the
work of [PY02] and [Por14]. It would be particularly interesting to relax the
first hypothesis, at the expense of some generality, to be able to say more
about restrictions to more general subgroups of $G$.

This paper is organized as follows. We establish our notation in Section 1 and
define (truncated) data as well as their associated semisimple characters and
types in Section 2, making use of several results from [Por14] and [Fin21]. In
Section 3 we define the notion of $\Gamma$-stable truncated data, and use
these to prove Theorem 4.4 in Section 4. We conclude in Section 5 with an
application to restrictions of representations, in the form of Theorem 5.4.

##  1\. Notation

Let $F$ be a locally compact field, complete with respect to a non-archimedean
discrete valuation $\nu$, with ring of integers $\mathfrak{o}$, maximal ideal
$\mathfrak{p}$, and finite residue field
$\mathfrak{f}=\mathfrak{o}/\mathfrak{p}$ of cardinality $q=p^{r}$. We fix a
choice of additive character $\psi\colon F\to\mathbb{C}^{\times}$ that is
trivial on $\mathfrak{p}$ but non-trivial on $\mathfrak{o}$. Write
$\mathrm{val}$ for a valuation of $F$ and, for any finite extension $E$ of
$F$, denote also by $\mathrm{val}$ its unique $F$-normalized extension to $E$.
We let $\bar{F}$ denote a separable algebraic closure of $F$.

Let $\mathbf{G}$ denote a connected reductive group defined over $F$, and
write $G=\mathbf{G}(F)$ for its group of rational points, equipped with the
locally profinite topology. We use similar notation for closed subgroups
without comment. The Lie algebra of the algebraic group $\mathbf{G}$ will be
denoted by $\mathrm{Lie}(\mathbf{G})$, with its $F$-rational points being
$\mathfrak{g}=\mathrm{Lie}(\mathbf{G})(F)$; we denote the duals of these
algebras by $\mathrm{Lie}(\mathbf{G})^{*}$ and $\mathfrak{g}^{*}$,
respectively. For any suitable object $*$, write $\mathrm{Cent}_{G}(*)$ for
the centralizer in $G$ of $*$ and $Z(*)$ for its centre.

We write $\mathscr{B}(G)=\mathscr{B}(\mathbf{G},F)$ for the (extended)
Bruhat–Tits building of $G$. Given a point $x\in\mathscr{B}(G)$, we write
$G_{x}$ for the subgroup of $G$ which fixes $x$; this group carries the
Moy–Prasad filtration $G_{x,r}$ for $r\geq 0$; the group $G_{x,0}$ is the
parahoric subgroup associated to $x$, and for any $r>0$ the group $G_{x,r}$ is
a pro-$p$ normal subgroup of $G_{x}$. For any $0<s<r$ we abbreviate
$G_{x,s:r}=G_{x,s}/G_{x,r}$ and for any $r\geq 0$, we abbreviate
$G_{x,r+}=\bigcup_{s>r}G_{x,s}$. In fact, as established in [Yu15], for $r\geq
0$ the groups $G_{x,r}$ are schematic, in the sense that there exist canonical
smooth connected affine $\mathfrak{o}$-models $\mathbf{G}_{x,r}$ of
$\mathbf{G}$ such that $\mathbf{G}_{x,r}(\mathfrak{o})=G_{x,r}$. As a special
case, due to Bruhat and Tits, note that the special fibre of
$\mathbf{G}_{x,0}$ is a connected reductive $\mathfrak{f}$-algebraic group
whose group of $\mathfrak{f}$ -points is $G_{x,0:0+}$.

Similarly, for any $r\in\mathbb{R}$ and $x\in\mathscr{B}(G)$ we have a
filtration of $\mathfrak{g}$ by $\mathfrak{o}$-lattices $\mathfrak{g}_{x,r}$
and $\mathfrak{g}_{x,r+}$; this induces a corresponding filtration on
$\mathfrak{g}^{*}$ by setting $\mathfrak{g}_{x,r}^{*}=\\{X\in\mathfrak{g}^{*}\
|\ \forall Y\in\mathfrak{g}_{x,-r+},\ X(Y)\in\mathfrak{p}\\}$. For any
$0<s<r\leq 2s$ the Moy–Prasad isomorphism identifies
$\mathfrak{g}_{x,s:r}=\mathfrak{g}_{x,s}/\mathfrak{g}_{x,r}$ with $G_{x,s:r}$.

We fix a finite subgroup $\Gamma\subset\mathrm{Aut}_{F}(\mathbf{G})$, and
assume that $p$ is coprime to the order of $\Gamma$. Via the action of
$\Gamma$ on $\mathbf{G}$, we obtain natural actions of $\Gamma$ on $G$, on
$\mathscr{B}(G)$, on $\mathfrak{g}$, and on $\mathfrak{g}^{*}$. Given a
$\Gamma$-set $X$, we write $X^{\Gamma}$ for the set of points $x\in X$ which
are fixed by $\Gamma$.

To ease notation, if $\mathbf{H}$ is a $\Gamma$-stable subgroup of
$\mathbf{G}$, then we write
$\mathbf{H}^{[\Gamma]}=(\mathbf{H}^{\Gamma})^{\circ}$ for the connected
component of $\mathbf{H}^{\Gamma}$, and set
$H^{[\Gamma]}=\mathbf{H}^{[\Gamma]}(F)$, to be distinguished from $H\cap
G^{[\Gamma]}$ as necessary.

By [PY02], $\mathbf{G}^{[\Gamma]}$ is a closed reductive subgroup of
$\mathbf{G}$, whence $G^{[\Gamma]}$ is a closed subgroup of $G$; moreover,
there is a family of $G^{[\Gamma]}$-equivariant embeddings
$\mathscr{B}(G^{[\Gamma]})\hookrightarrow\mathscr{B}(G)$, each of which
identifies $\mathscr{B}(G^{[\Gamma]})$ with $\mathscr{B}(G)^{\Gamma}$. We fix,
once and for all, such a choice of embedding.

Finally, throughout the paper we impose the hypothesis that $\mathbf{G}$
splits over a tamely ramified extension of $F$, and that $p$ is coprime to the
orders of the (absolute) Weyl groups of both $\mathbf{G}$ and
$\mathbf{G}^{[\Gamma]}$. This hypothesis is required in order to apply the
results of [Fin21], and it ensures that each twisted Levi subgroup of
$\mathbf{G}$ that is defined over $F$ splits over a tamely ramified extension.

##  2\. Data and semisimple characters

In this section, we recall the fundamental notion of a datum for constructing
a type, following the approach of [Fin21] (although with our indices running
in the opposite direction, following the convention from the initial
definition in [Yu01]). We then introduce various groups and representations
associated to a datum, choosing notation similar to that found in [LN21, §5].

######  Definition 2.1.

A _truncated datum_ for $G$ is a tuple
$\Sigma=(\vec{G},x,\vec{r},\vec{X},\vec{\phi})$ consisting of the following
objects.

  1. (i)

A sequence
$\vec{G}=(\mathbf{G}^{0}\subsetneq\mathbf{G}^{1}\subsetneq\cdots\subsetneq\mathbf{G}^{d}\subseteq\mathbf{G}^{d+1}=\mathbf{G})$
of twisted Levi subgroups of $\mathbf{G}$, each of which is defined over $F$
(and splits over a tamely ramified extension of $F$).

  2. (ii)

A point
$x\in\mathscr{B}(G^{0})\subset\mathscr{B}(G^{1})\subset\cdots\subset\mathscr{B}(G)$,
relative to a fixed choice of embeddings of buildings.

  3. (iii)

A sequence $\vec{r}=(0<r_{0}<r_{1}<\cdots<r_{d-1}<r_{d})$ of real numbers; we
set $s_{i}=r_{i}/2$ and $s_{-1}=0$ throughout.

  4. (iv)

A sequence $\vec{X}=(X^{0},X^{1},\ldots,X^{d})$ consisting of elements
$X^{i}\in(\mathfrak{g}^{i})^{*}$ which are _almost strongly stable and
$\mathbf{G}^{i+1}$-generic of depth $-r_{i}$ at $x\in\mathscr{B}(G^{i+1})$_.
(We recall these terms from [Fin21, Def. 3.2, Def. 3.5] in the next section.)
Further, for each $0\leq i\leq d$ we require
$\mathrm{Cent}_{G^{i+1}}(X^{i})=G^{i}$.

  5. (v)

A sequence $\vec{\phi}=(\phi^{0},\phi^{1},\ldots,\phi^{d})$ of characters
$\phi^{i}$ of $G^{i}$ of depth $r_{i}$ realized by $X^{i}$, that is, such that
via the Moy-Prasad isomorphism
$G^{i}_{x,s_{i}+:r_{i}+}\cong\mathfrak{g}^{i}_{x,s_{i}+:r_{i}+}$, the
restriction of $\phi^{i}$ to $G^{i}_{x,s_{i}+}$ is given by $\psi\circ X^{i}$.

######  Remark 2.2.

Omitting (v) from the definition above gives the “truncated extended data” of
[Fin21], which parametrize strictly larger collections of representations. It
is convenient for us to include the sequence of characters $\vec{\phi}$ so
that our truncated data directly parametrize semisimple characters, as below.

######  Definition 2.3.

A truncated datum $\Sigma$ is completed to a datum
$\tilde{\Sigma}=(\Sigma;\sigma)$ if in addition to (i) through (v) above, we
have

  1. (vi)

$\sigma$ is an irreducible representation of $G^{0}_{x}$ that is trivial on
$G^{0}_{x,0+}$ and whose restriction to $G^{0}_{x,0}$ is a sum of cuspidal
representations inflated from $G^{0}_{x,0:0+}$.

######  Remark 2.4.

Given a datum $\tilde{\Sigma}$, by omitting reference to the sequence
$\vec{X}$ and letting $M^{0}$ denote the Levi subgroup of $G^{0}$ associated
to the point $x$ by [MP96, §6.3], one recovers a datum in the sense of [KY17].
A consequence of the main result of [Fin21] is that every irreducible
representation of $G$ contains a datum satisfying (i)–(vi).

Let us now define two key compact open subgroups of $G$, each equipped with an
irreducible representation, derived from these data. Note that in [Yu01], our
group $H_{+}(\Sigma)$ is denoted $K_{+}$ and our group $J(\Sigma)$ is denoted
${}^{\circ}K$.

First, to a truncated datum $\Sigma$ one may attach the pro-$p$ group

(2.1) $H_{+}(\Sigma)=G_{x,0+}^{0}G_{x,s_{0}+}^{1}\cdots G_{x,s_{d}+}^{d+1}.$

To define the _semisimple character_ $\theta_{\Sigma}$ of $H_{+}(\Sigma)$, we
proceed as follows.

For each $i$ such that $0\leq i\leq d$, let
$\mathbf{Z}=Z(\mathbf{G}^{i})^{\circ}$; then $Z=\mathbf{Z}(F)$ decomposes
$\mathfrak{g}$ into a direct sum of isotypic subspaces such that
$\mathfrak{g}^{i}$ is the $0$-isotypic subspace. Write $\mathfrak{n}^{i}$ for
the sum of the remaining isotypic components.

From the pair $(X^{i},\phi^{i})$ we construct a character $\hat{\phi}^{i}$ of
$G^{i}_{x}G_{x,s_{i}+}$ following [Yu01, §4]: since $X^{i}$ is of depth
$-r_{i}$, $\psi\circ X^{i}$ defines a character of
$\mathfrak{g}^{i}_{x,s_{i}+}$ that is trivial on
$\mathfrak{g}^{i}_{x,r_{i}+}$. Define a character $\tilde{\phi}^{i}$ of
$G_{x,s_{i}+}$ by extending $\psi\circ X^{i}$ trivially across
$\mathfrak{n}^{i}\cap\mathfrak{g}_{x,s_{i}+}$, and then using the Moy–Prasad
isomorphism to lift this to the group. Since $\tilde{\phi}^{i}$ and $\phi^{i}$
agree on $G^{i}_{x}\cap G_{x,s_{i}+}$ we may define, for any $g\in G^{i}_{x}$
and $h\in G_{x,s_{i}+}$, a character $\hat{\phi}^{i}$ of
$G^{i}_{x}G_{x,s_{i}+}$ by

(2.2) $\hat{\phi}^{i}(gh)=\phi^{i}(g)\tilde{\phi^{i}}(h).$

Write again $\hat{\phi}^{i}$ for its restriction to the group
$H_{+}(\Sigma)\subset G^{i}_{x}G_{x,s_{i}+}$. The semisimple character
$\theta_{\Sigma}$ is then defined as
$\theta_{\Sigma}=\prod_{i=0}^{d}\hat{\phi}^{i}.$

Next, to $\Sigma$ one may also associate the group

$J(\Sigma)=G^{0}_{x}G_{x,s_{0}}^{1}\cdots G_{x,s_{d}}^{d+1},$

which contains $H_{+}(\Sigma)$ as an open normal subgroup. The semisimple
character $\theta_{\Sigma}$ admits a _Heisenberg–Weil lift_ to $J(\Sigma)$.
This is a certain irreducible representation $\kappa_{\Sigma}$ of $J(\Sigma)$
which restricts to $H_{+}(\Sigma)$ as a sum of copies of $\theta_{\Sigma}$; we
refer to [HM08, §2.3, 2.4] for details and will briefly recall the
construction where needed in Section 5.

If we now complete $\Sigma$ to a datum $\tilde{\Sigma}$ by choosing a
representation $\sigma$, then we may also define a representation
$\lambda_{\tilde{\Sigma}}=\sigma\otimes\kappa_{\Sigma}$ of $J(\Sigma)$. The
fundamental fact is that the pair $(J(\Sigma),\lambda_{\tilde{\Sigma}})$ is an
$\mathfrak{s}$-type, for some inertial equivalence class
$\mathfrak{s}=\mathfrak{s}_{\Sigma}$ of $G$ (in the sense of [BK98]).
Moreover, under our hypotheses on $p$, every irreducible representation of $G$
contains a type of the form $(J(\Sigma),\lambda_{\tilde{\Sigma}})$ for some
datum ${\tilde{\Sigma}}$ [Fin21].

##  3\. $\Gamma$-stable truncated data

Let $\Gamma\subset\mathrm{Aut}_{F}(\mathbf{G})$ be as in Section 1. We begin
with some straightforward lemmas.

######  Lemma 3.1.

Let $\mathbf{S}\subset\mathbf{G}^{[\Gamma]}$ be a maximal $F$-torus of
$\mathbf{G}^{[\Gamma]}$. Then
$\mathbf{T}=\mathrm{Cent}_{\mathbf{G}}(\mathbf{S})$ is a $\Gamma$-stable
maximal $F$-torus of $\mathbf{G}$.

###### Proof.

Let $\mathbf{H}=(\mathrm{Cent}_{\mathbf{G}}(\mathbf{S}))^{\circ}$. Then
$\mathbf{H}$ is a twisted Levi subgroup of $\mathbf{G}$, hence a closed
connected reductive $F$-subgroup. Since $\mathbf{S}$ is fixed pointwise by
$\Gamma$, $\mathbf{H}$ is $\Gamma$-stable. By [PY02], $\mathbf{H}^{[\Gamma]}$
is a connected reductive group containing $\mathbf{S}$ in its centre.

Suppose that the derived group $\mathbf{H}^{\prime}=[\mathbf{H},\mathbf{H}]$
were non-trivial. This group is $\Gamma$-stable, and so again
$(\mathbf{H}^{\prime})^{[\Gamma]}$ is a (non-trivial) reductive group. Choose
a non-trivial $F$-torus
$\mathbf{S}^{\prime}\subset(\mathbf{H}^{\prime})^{[\Gamma]}$. Then
$\mathbf{S}^{\prime}$ commutes with $\mathbf{S}$ and
$\mathbf{S}^{\prime}\cap\mathbf{S}$ is finite, so that the torus
$\mathbf{S}^{\prime}\mathbf{S}$ is of rank strictly larger than $\mathbf{S}$
and yet contained in $\mathbf{H}^{[\Gamma]}\subset\mathbf{G}^{[\Gamma]}$,
which contradicts the maximality of $\mathbf{S}$. Therefore
$\mathbf{H}=\mathbf{T}$ is a $\Gamma$-stable torus. ∎

######  Remark 3.2.

Note that since $p$ is coprime to the orders of the absolute Weyl groups of
both $\mathbf{G}$ and $\mathbf{G}^{[\Gamma]}$, and $\mathbf{G}$ splits over a
tamely ramified extension of $F$, by [Fin21, Lem. 2.2] this implies that every
$F$-rational twisted Levi subgroup of $\mathbf{G}$ or of
$\mathbf{G}^{[\Gamma]}$ splits over a tamely ramified extension of $F$. Thus
the tori $\mathbf{S}$ and $\mathbf{T}$ of the lemma split over a tamely
ramified extension, but as remarked by [Por14], it is not generally true that
$\mathbf{T}$ splits over $F$ even if $\mathbf{S}$ does.

Continuing in the setting of Lemma 3.1, let $\Phi(\mathbf{G},\mathbf{T})$
denote the set of (absolute) roots of $\mathbf{G}$ relative to $\mathbf{T}$
and $\Phi(\mathbf{G}^{[\Gamma]},\mathbf{S})$ the set of (absolute) roots of
$\mathbf{G}^{[\Gamma]}$ relative to $\mathbf{S}$. The elements $a$ of
$\Phi(\mathbf{G}^{[\Gamma]},\mathbf{S})$ occur as restrictions to $\mathbf{S}$
of certain roots of $\mathbf{G}$ relative to $\mathbf{T}$. To describe these,
note that the action of $\Gamma$ on $\mathbf{T}$ and $\mathbf{G}$ induces an
action on the set of roots, and in consequence for each root $\alpha$ a linear
action of the stabilizer $\Gamma_{\alpha}$ on the root space
$\mathrm{Lie}(\mathbf{G})_{\alpha}$.

######  Lemma 3.3.

The roots of $\mathbf{G}^{[\Gamma]}$ relative to $\mathbf{S}$ are the
restrictions of those roots $\alpha$ of $\mathbf{G}$ relative to $\mathbf{T}$
for which $\Gamma_{\alpha}$ admits a nonzero fixed vector on
$\mathrm{Lie}(\mathbf{G})_{\alpha}$, that is,

$\Phi(\mathbf{G}^{[\Gamma]},\mathbf{S})=\\{\alpha|_{\mathbf{S}}\ |\
\alpha\in\Phi(\mathbf{G},\mathbf{T}),\
\mathrm{Lie}(\mathbf{G})_{\alpha}^{\Gamma_{\alpha}}\neq\\{0\\}\\}.$

###### Proof.

Let $\Gamma\cdot\alpha$ denote the orbit of a root
$\alpha\in\Phi(\mathbf{G},\mathbf{T})$ and let $a=\alpha|_{\mathbf{S}}$ be the
common restriction to $\mathbf{S}$. The action of $\Gamma$ on
$\oplus_{\beta\in\Gamma\cdot\alpha}\mathrm{Lie}(\mathbf{G})_{\beta}$ is
induced from the action of $\Gamma_{\alpha}$ on
$\mathrm{Lie}(\mathbf{G})_{\alpha}$, implying the result. In particular, for
any nonzero vector $Y$ in
$\mathrm{Lie}(\mathbf{G})_{\alpha}^{\Gamma_{\alpha}}$, taking the average over
its $\Gamma$-orbit
$\overline{Y}=\frac{1}{[\Gamma:\Gamma_{\alpha}]}\sum_{\gamma\in\Gamma/\Gamma_{\alpha}}\gamma\cdot
Y$ yields a $\Gamma$-fixed sum of linearly independent vectors, so a nonzero
vector in $\mathrm{Lie}(\mathbf{G}^{[\Gamma]})_{a}$. ∎

Our next lemmas relate to $\mathrm{Lie}(G)^{*}$.

######  Lemma 3.4.

There is a canonical identification
$\mathrm{Lie}(\mathbf{G}^{[\Gamma]})^{*}=(\mathrm{Lie}(\mathbf{G})^{*})^{\Gamma}$.

###### Proof.

We have
$\mathrm{Lie}(\mathbf{G})^{\Gamma}=\mathrm{Lie}(\mathbf{G}^{[\Gamma]})$. Since
$p$ does not divide $|\Gamma|$, Maschke’s theorem applies, so that both
$\mathrm{Lie}(\mathbf{G})$ and $\mathrm{Lie}(\mathbf{G})^{*}$ are completely
decomposable as $\Gamma$-modules. Thus there is a natural $F$-linear
isomorphism
$\mathrm{Lie}(\mathbf{G}^{[\Gamma]})^{*}\rightarrow(\mathrm{Lie}(\mathbf{G})^{*})^{\Gamma}$
defined by extending any functional $f$ on $\mathrm{Lie}(G^{[\Gamma]})$ by
zero on the unique $\Gamma$-invariant complement in
$\mathrm{Lie}(\mathbf{G})$; its inverse is given by restriction. ∎

We write $\mathfrak{g}^{\Gamma}$ for
$\mathrm{Lie}(\mathbf{G})^{\Gamma}(F)=\mathrm{Lie}(\mathbf{G}^{[\Gamma]})(F)$,
and similarly for the dual.

Given a nonzero element $X\in\mathfrak{g}^{*}$ and a point
$x\in\mathscr{B}(G)$, the _depth of $X$ at $x$_ is defined to be
$d_{G}(x,X)=\mathrm{max}\\{r\ |\ X\in\mathfrak{g}_{x,r}^{*}\\}$. The _depth_
of $X$ is $d_{G}(X)=\mathrm{sup}\\{d_{G}(x,X)\ |\ x\in\mathscr{B}(G)\\}$, when
the supremum exists.

######  Lemma 3.5.

Suppose that $X\in\mathfrak{g}^{*}\setminus\\{0\\}$ and $x\in\mathscr{B}(G)$
are both $\Gamma$-fixed. Then, viewing $X$ as an element of
$(\mathfrak{g}^{\Gamma})^{*}$ and $x$ as a point in
$\mathscr{B}(G^{[\Gamma]})$, we have $d_{G}(x,X)=d_{G^{[\Gamma]}}(x,X)$.

###### Proof.

Taking $\Gamma$-fixed points preserves Moy–Prasad filtrations on the Lie
algebra, by [Por14, Lem. 2.8]. That is, for all $r\in\mathbb{R}$ and
$x\in\mathscr{B}(G^{[\Gamma]})$, we have
$(\mathfrak{g}^{\Gamma})_{x,r}=\mathfrak{g}^{\Gamma}\cap\mathfrak{g}_{x,r}$.
Since $\mathfrak{g}_{x,r}^{*}$ is defined as $\\{X\in\mathfrak{g}^{*}\ |\
\forall Y\in\mathfrak{g}_{x,-r+},\ X(Y)\in\mathfrak{p}\\}$, taking
$\Gamma$-fixed points preserves the filtration on the dual as well. ∎

We now define almost stable and almost strongly stable elements, following
[Fin21, Def. 3.1].

We say an element $X\in\mathfrak{g}^{*}$ is _almost stable_ if its
$\mathbf{G}$-orbit in $\mathrm{Lie}(\mathbf{G})^{*}$ is closed. Note that if
we identify $\mathfrak{g}^{*}$ and $\mathfrak{g}$ via a non-degenerate
$G$-equivariant bilinear form, then almost stable elements correspond to
semisimple elements of the Lie algebra. In particular, by [AD02, Lemma 3.3.4],
such elements have finite depth.

Now suppose $X$ is almost stable and let $r=d(x,X)$; then the corresponding
quotient element $\overline{X}\in\mathfrak{g}^{*}_{x,r:r+}$ is nonzero. We say
that $X$ is _almost strongly stable_ at $x$ if the orbit of $\overline{X}$ in
$\mathfrak{g}^{*}_{x,r:r+}$ under the special fibre of the
$\mathfrak{o}$-scheme $\mathbf{G}_{x,0}$ does not contain $0$ in its Zariski
closure. This is the condition that the coset $X+\mathfrak{g}^{*}_{x,r+}$ is
nondegenerate, in the sense of [AD02]; for $p$ sufficiently large (and again
identifying $\mathfrak{g}^{*}$ and $\mathfrak{g}$), it is equivalent to the
statement that this coset does not contain any nilpotent elements.

######  Lemma 3.6.

Suppose that $X\in\mathfrak{g}^{*}$ is almost stable under the coadjoint
action of $\mathbf{G}$ and that $\Gamma\cdot X=X$. Then, viewed as an element
of $(\mathfrak{g}^{\Gamma})^{*}$, $X$ is almost stable under the coadjoint
action of $\mathbf{G}^{[\Gamma]}$.

###### Proof.

Since $X$ is $\Gamma$-fixed, it lies in $(\mathfrak{g}^{*})^{\Gamma}$. Since
$\mathbf{G}^{[\Gamma]}$ is a closed subgroup of $\mathbf{G}$, the orbit of $X$
under $\mathbf{G}^{[\Gamma]}$ is also closed, and is contained in the
$\mathbf{G}^{[\Gamma]}$-invariant subspace
$(\mathrm{Lie}(\mathbf{G})^{*})^{\Gamma}\simeq\mathrm{Lie}(\mathbf{G}^{[\Gamma]})^{*}$.
∎

######  Lemma 3.7.

Suppose that $X\in\mathfrak{g}^{*}$ is almost stable under the coadjoint
action of $\mathbf{G}$, and that $\Gamma\cdot X=X$. Let
$\mathbf{G}^{\prime}=\mathrm{Cent}_{\mathbf{G}}(X)$. Then
$(\mathbf{G}^{\prime})^{[\Gamma]}=\mathrm{Cent}_{\mathbf{G}^{[\Gamma]}}(X)$.
In particular, $(\mathbf{G}^{\prime})^{[\Gamma]}$ is a twisted Levi subgroup
of $\mathbf{G}^{[\Gamma]}$.

###### Proof.

From the identification
$(\mathfrak{g}^{\Gamma})^{*}=(\mathfrak{g}^{*})^{\Gamma}$, one sees that
$\mathbf{G}^{\prime}\cap\mathbf{G}^{[\Gamma]}=\mathrm{Cent}_{\mathbf{G}^{[\Gamma]}}(X)$.
Since by Lemma 3.6, $X$ is almost stable relative to $\mathbf{G}^{[\Gamma]}$,
we have by [Fin21, Lem. 3.4] that this intersection is a twisted Levi subgroup
of $\mathbf{G}^{[\Gamma]}$. Moreover, as recapped in _loc. cit._ , the
hypotheses on $p$ ensure it is therefore connected and thus equal to
$(\mathbf{G}^{\prime})^{[\Gamma]}$. ∎

We have the inclusion $Z(\mathbf{G}^{\prime})^{[\Gamma]}\subseteq
Z({\mathbf{G}^{\prime}}^{[\Gamma]})^{\circ}$, but note that equality should
fail in general (_cf._ Lemma 3.3).

We now recall the notion of _genericity_ , as defined in [Fin21, Def. 3.5].
Let $X\in\mathfrak{g}^{*}$ be an almost stable element and let
$\mathbf{G}^{\prime}=\mathrm{Cent}_{\mathbf{G}}(X)$. Let $\mathbf{T}$ be a
maximal $F$-torus contained in $\mathbf{G}^{\prime}$ that splits over a tamely
ramified extension $E$ of $F$. Let $\mathscr{A}(\mathbf{G},\mathbf{T},E)$,
respectively $\mathscr{A}(\mathbf{G}^{\prime},\mathbf{T},E)$, denote the
apartment of $\mathscr{B}(\mathbf{G},E)$, respectively
$\mathscr{B}(\mathbf{G}^{\prime},E)$, corresponding to $\mathbf{T}$. Let
$\Phi=\Phi(\mathbf{G},\mathbf{T})$ and
$\Phi^{\prime}=\Phi(\mathbf{G}^{\prime},\mathbf{T})$ be the corresponding root
systems. For each $\alpha\in\Phi$, let
$H_{\alpha}=d\alpha(1)\in\mathrm{Lie}(\mathbf{T})$ denote the corresponding
coroot. By construction, we have $X(H_{\alpha})=0$ for each
$\alpha\in\Phi^{\prime}$.

######  Definition 3.8.

Let $X\in\mathfrak{g}^{*}$ be an almost stable element. Choose a maximal
$F$-torus $\mathbf{T}$ in $\mathbf{G}^{\prime}=\mathrm{Cent}_{\mathbf{G}}(X)$
and
$x\in\mathscr{A}(\mathbf{G}^{\prime},\mathbf{T},E)\cap\mathscr{B}(\mathbf{G}^{\prime},F)\subset\mathscr{B}(\mathbf{G},F)$.
We say $X$ is _$\mathbf{G}$ -generic of depth $r$ relative to $x$_ if
$d(x,X)=r$ and furthermore $\mathrm{val}(X(H_{\alpha}))=r$ for each
$\alpha\in\Phi(\mathbf{G},\mathbf{T})\setminus\Phi(\mathbf{G}^{\prime},\mathbf{T})$.

Note that if $\mathbf{G}=\mathbf{G}^{\prime}$ then the condition in the
definition is only that $d(x,X)=r$.

######  Lemma 3.9.

Suppose two almost stable elements $X$ and $Y$ are $\mathbf{G}$-generic of
depth $r$ and $s$, respectively, relative to a point $x$, and $r<s$. If
$\mathrm{Cent}_{\mathbf{G}}(X)=\mathrm{Cent}_{\mathbf{G}}(Y)$ then $X+Y$ is
almost stable and $\mathbf{G}$-generic of depth $r$ relative to $x$.

###### Proof.

Let
$\mathbf{G}^{\prime}=\mathrm{Cent}_{\mathbf{G}}(X)\subset\mathrm{Cent}_{\mathbf{G}}(X+Y)$.
Under the $\mathbf{G}$-equivariant identification of
$\mathrm{Lie}(\mathbf{G})$ with $\mathrm{Lie}(\mathbf{G})^{*}$, the almost
stable elements $X$ and $Y$ correspond to semisimple elements of the centre of
$\mathrm{Lie}(\mathbf{G}^{\prime})$, so their sum is again almost stable. Thus
by [Fin21, Lemma 3.4] the centralizer of $X+Y$ is again connected. Since
$d(x,X+Y)=\min\\{d(x,X),d(x,Y)\\}=r$, we have
$\mathrm{val}((X+Y)(H_{\alpha}))=\mathrm{val}(X(H_{\alpha}))=r$ for all
$\alpha\in\Phi(\mathbf{G},\mathbf{T})\setminus\Phi(\mathbf{G}^{\prime},\mathbf{T})$.
Thus in particular $\mathrm{Cent}_{\mathbf{G}}(X+Y)\subset\mathbf{G}^{\prime}$
and genericity follows. ∎

######  Lemma 3.10.

Suppose that $X\in\mathfrak{g}^{*}$ is almost strongly stable and
$\mathbf{G}$-generic of depth $r$ at $x\in\mathscr{B}(G)$, and that both $x$
and $X$ are $\Gamma$-fixed. Then $X$ is almost strongly stable and
$\mathbf{G}^{[\Gamma]}$-generic of depth $r$ at
$x\in\mathscr{B}(G^{[\Gamma]})$.

###### Proof.

Set $\mathbf{G}^{\prime}=\mathrm{Cent}_{\mathbf{G}}(X)$; this is
$\Gamma$-stable. Since $X$ is almost stable and generic of depth $r$ at $x$,
by [Fin21, Lemma 3.11] we have $x\in\mathscr{B}(G^{\prime})$. In fact
$x\in\mathscr{B}(G^{\prime})^{\Gamma}=\mathscr{B}({G^{\prime}}^{[\Gamma]})$
and $X$, viewed as an element of $(\mathfrak{g}^{\Gamma})^{*}$, also has depth
$r$ relative to $x$, by Lemma 3.5. To carry out the proof we now make a choice
of torus and move to a more suitable point $y$, as follows.

Let $\mathbf{S}$ be a maximal $\Gamma$-fixed torus in $\mathbf{G}^{\prime}$
and $\mathbf{T}$ its centralizer, which by Lemma 3.1 is a $\Gamma$-stable
maximal torus. Let $E$ be a tamely ramified extension of $F$ over which
$\mathbf{T}$ splits, and choose a point
$y\in\mathscr{A}({\mathbf{G}^{\prime}}^{[\Gamma]},\mathbf{S},E)\cap\mathscr{B}({G^{\prime}}^{[\Gamma]})\subset\mathscr{A}(\mathbf{G}^{\prime},\mathbf{T},E)\cap\mathscr{B}(G^{\prime})$.
By [Fin21, Lemma 4.4], $X$ is also almost strongly stable and generic of depth
$r$ at $y$, so in particular $X\in(\mathfrak{g}^{*}_{y,r})^{\Gamma}$.

Since $X$ is $\mathbf{G}$-generic of depth $r$ relative to $y$, we have that
$X(H_{\alpha})=0$ for all $\alpha\in\Phi(\mathbf{G}^{\prime},\mathbf{T})$ and
that $\mathrm{val}(X(H_{\alpha}))=r$ for all
$\alpha\in\Phi(\mathbf{G},\mathbf{T})\setminus\Phi(\mathbf{G}^{\prime},\mathbf{T})$.
We next show these conditions hold for the corresponding roots in
$\mathbf{G}^{[\Gamma]}$. Let $a\in\Phi(\mathbf{G}^{[\Gamma]},\mathbf{S})$ and
choose $\alpha\in\Phi(\mathbf{G},\mathbf{T})$ as in Lemma 3.3 such that
$\alpha|_{\mathbf{S}}=a$. Then the coroot $H_{a}\in\mathrm{Lie}(\mathbf{S})$
is a $\Gamma$-fixed vector in the span of
$\\{H_{\gamma\cdot\alpha}\mid\gamma\in\Gamma/\Gamma_{\alpha}\\}$. It follows
that it must be a (nonzero) multiple of
$\tilde{H}=\sum_{\gamma\in\Gamma/\Gamma_{\alpha}}H_{\gamma\cdot\alpha}$. In
particular, $\alpha(\tilde{H})\in\mathbb{Z}$ is nonzero and so
$H_{a}=2(\alpha(\tilde{H}))^{-1}\tilde{H}$. Thus, since $X$ is $\Gamma$-fixed,
we may evaluate

$X(H_{a})=2(\alpha(\tilde{H}))^{-1}\sum_{\gamma\in\Gamma/\Gamma_{\alpha}}X(H_{\gamma\cdot\alpha})=\frac{2|\Gamma/\Gamma_{\alpha}|}{\alpha(\tilde{H})}X(H_{\alpha}).$

By Lemma 3.5, $X$ has depth $r$ at $y$. If
$a\in\Phi({\mathbf{G}^{\prime}}^{[\Gamma]},\mathbf{S})$, we have $X(H_{a})=0$.
If instead
$a\in\Phi({\mathbf{G}}^{[\Gamma]},\mathbf{S})\setminus\Phi({\mathbf{G}^{\prime}}^{[\Gamma]},\mathbf{S})$
then since $H_{a}\in\mathfrak{g}^{\Gamma}_{y,0}$ we have the inequality
$\mathrm{val}(X(H_{a}))\geq r$. Since $\mathrm{val}(X(H_{\alpha}))=r$ and $p$
does not divide $2|\Gamma|$ by hypothesis, we conclude from the preceding that
$\mathrm{val}(X(H_{a}))=r$. Thus $X$ is $\mathbf{G}^{[\Gamma]}$-generic of
depth $r$ relative to $y$.

Since by Lemma 3.6 $X$ is almost stable relative to $\mathbf{G}^{[\Gamma]}$,
applying [Fin21, Lemma 3.8] yields that $X$ is almost strongly stable at $y$.
Finally, since both $x$ and $y$ lie in $\mathscr{B}({G^{\prime}}^{[\Gamma]})$,
we conclude that $X$ is almost strongly stable and
$\mathbf{G}^{[\Gamma]}$-generic of depth $r$ relative to $x$. ∎

With this preparation, we are now able to give a straightforward formulation
of the class of truncated data of $G$ that we are interested in.

######  Definition 3.11.

Let $\Sigma=(\vec{G},x,\vec{r},\vec{X},\vec{\phi})$ be a truncated datum for
$G$. We say that $\Sigma$ is _$\Gamma$ -stable_ if $\Gamma$ fixes both the
point $x$ and each of the $X^{i}\in\mathfrak{g}^{*}$.

######  Proposition 3.12.

Let $\Sigma=(\vec{G},x,\vec{r},\vec{X},\vec{\phi})$ be a $\Gamma$-stable
truncated datum for $G$. Define a tuple
$\Sigma^{\Gamma}_{\mathrm{ext}}=(\vec{G}^{[\Gamma]},x,\vec{r},\vec{X},\vec{\phi})$,
where:

  1. (i)

$\vec{G}^{[\Gamma]}$ is the twisted Levi sequence in $\mathbf{G}^{[\Gamma]}$
consisting of the groups
$\mathbf{G}^{i,[\Gamma]}=(\mathbf{G}^{i})^{[\Gamma]}$;

  2. (ii)

each element $X^{i}\in((\mathfrak{g}^{i})^{*})^{\Gamma}$ is viewed as a
element of $(\mathfrak{g}^{i,[\Gamma]})^{*}$ by restriction; and

  3. (iii)

each character $\phi^{i}$ is viewed as a character of $G^{i,[\Gamma]}$ by
restriction.

For each $i$ such that there exists $k>0$ for which
$\mathbf{G}^{i,[\Gamma]}=\mathbf{G}^{i+k,[\Gamma]}$, replace the tuple of
subsequences
$((\mathbf{G}^{i,[\Gamma]},\ldots,\mathbf{G}^{i+k,[\Gamma]}),(r_{i},\ldots,r_{i+k}),(X^{i},\ldots,X^{i+k}),(\phi^{i},\ldots,\phi^{i+k}))$
with the tuple
$(\mathbf{G}^{i,[\Gamma]},r_{i+k},\sum_{j=i}^{i+k}X^{j},\prod_{j=i}^{i+k}\phi^{j})$.
Then the resulting tuple $\Sigma^{\Gamma}$ is a truncated datum for
$G^{[\Gamma]}$.

###### Proof.

We verify the conditions of Definition 2.1. By Lemma 3.7 and Remark 3.2, (i)
holds for $\Sigma^{\Gamma}_{\mathrm{ext}}$ and thus for $\Sigma^{\Gamma}$,
with the exception that the inclusions may not be strict for the former tuple.
Conditions (ii) and (iii) are given and (v) is immediate. By Lemma 3.10, (iv)
holds for $\Sigma^{\Gamma}_{\mathrm{ext}}$ and by Lemma 3.9, recalling that
each $X^{i}$ is of depth $-r_{i}$, it thus holds for $\Sigma^{\Gamma}$. ∎

The difference between $\Sigma^{\Gamma}_{\mathrm{ext}}$ and $\Sigma^{\Gamma}$
is effectively only cosmetic. For the purposes of defining the pairs
$(H(\Sigma^{\Gamma}),\theta_{\Sigma^{\Gamma}})$ and
$(J(\Sigma^{\Gamma}),\kappa_{\Sigma^{\Gamma}})$, we may use
$\Sigma^{\Gamma}_{\mathrm{ext}}$ and the truncated datum $\Sigma^{\Gamma}$
interchangeably, as we make precise in the following sections. In all cases we
retain and refer only to the indices $i$ of the original datum.

##  4\. Semisimple characters associated to $\Gamma$-stable truncated data

We now show that the restriction of truncated data corresponds to a
restriction of semisimple characters. We begin with some results about the
compatibility of Moy–Prasad filtration subgroups.

######  Lemma 4.1.

Let $x\in\mathscr{B}(G^{[\Gamma]})=\mathscr{B}(G)^{\Gamma}$. For all $r>0$,
one has

$G_{x,r}\cap G^{[\Gamma]}=(G^{[\Gamma]})_{x,r}.$

###### Proof.

Since $\Gamma\subset\mathrm{Aut}_{F}(\mathbf{G})$ and $\Gamma$ fixes $x$, we
have that $G_{x,r}$ is $\Gamma$-stable (see, for example, [AD02, Lem. 2.2.1]).
Since $x\in\mathscr{B}(G)^{\Gamma}$ and $p$ does not divide $|\Gamma|$, it
follows that in fact
$\Gamma\subset\mathrm{Aut}_{\mathfrak{o}}(\mathbf{G}_{x,r})$. Thus

$\mathbf{G}_{x,r}\cap\mathbf{G}^{[\Gamma]}=(\mathbf{G}_{x,r})^{[\Gamma]}.$

By [Por14, Lemma 2.8] we have
$\mathfrak{g}_{x,r}\cap\mathfrak{g}^{\Gamma}=(\mathfrak{g}^{\Gamma})_{x,r}=(\mathfrak{g}^{[\Gamma]})_{x,r}$,
whence the $\mathfrak{o}$-Lie algebra of $(\mathbf{G}_{x,r})^{[\Gamma]}$
coincides with that of $(\mathbf{G}^{[\Gamma]})_{x,r}$; the result follows by
connectedness. ∎

######  Remark 4.2.

During the proof of [Por14, Lem. 2.7], Portilla shows that in the setting of
the above lemma we have

$(G^{[\Gamma]})_{x,0}=G_{x,0}\cap G^{[\Gamma]};$

he includes the proof that $(G^{[\Gamma]})_{x,0+}=G_{x,0+}\cap G^{[\Gamma]}$
as [Por14, Prop. 5.1].

Given a $\Gamma$-stable datum $\Sigma$, each twisted Levi subgroup
$\mathbf{G}^{i}$ occuring in $\Sigma$ is $\Gamma$-stable, so by Lemma 4.1 and
the remark following, for each $0\leq i\leq d+1$ and any $s\geq 0$ we have

$((G^{i})^{[\Gamma]})_{x,s}=(G_{x,s}^{i})^{[\Gamma]};$

we write $G_{x,s}^{i,[\Gamma]}$ for this group.

######  Lemma 4.3.

Let $G^{\prime}$ be a $\Gamma$-stable twisted Levi subgroup of $G$ and
$x\in\mathscr{B}(G^{\prime})^{\Gamma}\subset\mathscr{B}(G)$. For any $0<t<s$
let $K$ be a $\Gamma$-stable subgroup of $G_{x,s}$ normalized by
$G^{\prime}_{x,t}$ and containing $G^{\prime}_{x,s}$. Then we have

$(G_{x,t}^{\prime}K)^{\Gamma}={G^{\prime}}^{[\Gamma]}_{x,t}K^{\Gamma}.$

###### Proof.

Evidently
${G^{\prime}}^{[\Gamma]}_{x,t}K^{\Gamma}\subseteq(G_{x,t}^{\prime}K)^{\Gamma}$.
If $t=s$ then as $G^{\prime}_{x,s}\subseteq K$ the result is immediate. Fix
$t\in(0,s)$ and suppose the result has been shown for all $r$ satisfying
$t<r\leq s$; this amounts to a finite set of distinct equalities since the
filtrations are discrete.

Let $h=gk\in(G_{x,t}^{\prime}K)^{\Gamma}$ with $g\in G_{x,t}^{\prime}$ and
$k\in K$. Then for each $\gamma\in\Gamma$ we have that $\gamma(gk)=gk$, and
therefore

$g^{-1}\gamma(g)=k\gamma(k^{-1})\in G_{x,t}^{\prime}\cap K=G_{x,s}^{\prime}.$

Consequently, for all $\gamma\in\Gamma$ we have

$g\equiv\gamma(g)\quad\text{mod }G_{x,s}^{\prime}.$

Now let $\bar{g}$ denote the image of $g$ in $G_{x,t:t+}^{\prime}$; since
$s>t$ the preceding implies that $\bar{g}\in(G_{x,t:t+}^{\prime})^{\Gamma}$.
By the proof of [Por14, Lem. 2.28], we have a natural isomorphism

$(\mathfrak{g}_{x,t:t+}^{\prime})^{\Gamma}\simeq(\mathfrak{g}^{\prime[\Gamma]})_{x,t:t+};$

whence via the Moy–Prasad isomorphism we may choose a representative
$g^{\prime}$ of $\bar{g}$ satisfying
$g^{\prime}\in(G^{\prime[\Gamma]})_{x,t}$. By Lemma 4.1 we have
$g^{\prime}\in(G^{\prime}_{x,t})^{\Gamma}$. Thus
$(g^{\prime})^{-1}gk\in(G_{x,t+}^{\prime}K)^{\Gamma}$, which by induction is
equal to $G_{x,t+}^{\prime[\Gamma]}K^{\Gamma}$. We conclude that $gk\in
g^{\prime}G_{x,t+}^{\prime[\Gamma]}K^{\Gamma}\subset
G_{x,t}^{\prime[\Gamma]}K^{\Gamma}$, as required. ∎

######  Theorem 4.4.

Suppose that $\Sigma$ is a $\Gamma$-stable truncated datum for $G$ and let
$\Sigma^{\Gamma}$ be the corresponding truncated datum for $G^{[\Gamma]}$.
Then one has an equality of semisimple characters

$(H_{+}(\Sigma^{\Gamma}),\theta_{\Sigma^{\Gamma}})=(H_{+}(\Sigma)^{\Gamma},\theta_{\Sigma}|_{H_{+}(\Sigma)^{\Gamma}}).$

###### Proof.

Let $\Sigma=(\vec{G},x,\vec{r},\vec{X},\vec{\phi})$. Recall that
$H_{+}(\Sigma)=G^{0}_{x,0+}G^{1}_{x,s_{0}+}\cdots G^{d+1}_{x,s_{d}+}$.
Inductively applying Lemma 4.3 yields the equality

$H_{+}(\Sigma)^{\Gamma}=G^{0,[\Gamma]}_{x,0+}G^{1,[\Gamma]}_{x,s_{0}+}\cdots
G^{d+1,[\Gamma]}_{x,s_{d}+},$

which is the group $H_{+}(\Sigma^{\Gamma}_{\mathrm{ext}})$ as defined in
(2.1). Since $0<s_{0}<\cdots<s_{d}$, if
$G^{i,[\Gamma]}=G^{i+1,[\Gamma]}=\cdots=G^{i+k,[\Gamma]}$, then

$G^{i,[\Gamma]}_{x,s_{i-1+}}G^{i+1,[\Gamma]}_{x,s_{i}+}\cdots
G^{i+k,[\Gamma]}_{x,s_{i+k-1}+}=G^{i,[\Gamma]}_{x,s_{i-1}+}.$

Thus we have
$H(\Sigma^{\Gamma})=H(\Sigma^{\Gamma}_{\mathrm{ext}})=H_{+}(\Sigma)^{\Gamma}$.

For each $0\leq i\leq d$, we now compare the characters $\hat{\phi}^{i}$
defined in (2.2) produced from these various tuples.

Let $\mathbf{Z}^{i}=Z(\mathbf{G}^{i})^{\circ}$ and
$\mathbf{Z}^{i}_{\Gamma}=Z(\mathbf{G}^{i,[\Gamma]})^{\circ}$ and as usual let
$Z^{i}=\mathbf{Z}^{i}(F)$ and $Z^{i}_{\Gamma}=\mathbf{Z}^{i}_{\Gamma}(F)$.
Then under $Z^{i}$ we have the isotypic decomposition
$\mathfrak{g}=\mathfrak{g}^{i}\oplus\mathfrak{n}$ where $\mathfrak{g}^{i}$ is
the $1$-isotypic subspace. Since $Z^{i}$ is $\Gamma$-stable, so is
$\mathfrak{n}$, and taking $\Gamma$-fixed points yields a decomposition
$\mathfrak{g}^{[\Gamma]}=\mathfrak{g}^{i,[\Gamma]}\oplus\mathfrak{n}^{\Gamma}$.
Since $\mathfrak{g}^{i,[\Gamma]}$ is the $1$-isotypic component of
$\mathfrak{g}^{[\Gamma]}$ under $Z^{i}_{\Gamma}$, this
$Z^{i}_{\Gamma}$-invariant decomposition is isotypic. It follows that
$\tilde{\phi^{i}}$, the character of $G_{x,s_{i}+}$ arising from the trivial
extension across $\mathfrak{n}$ of $\psi\circ X^{i}$, restricts to the
extension $\tilde{\phi}^{i,\Gamma}$ of $\psi\circ X^{i}$ to
$G^{[\Gamma]}_{x,s_{i}+}$. Consequently, the character $\hat{\phi}^{i}$ of
$G^{i}_{x}G_{x,s_{i}+}$ restricts to the corresponding character
$\hat{\phi}^{i,[\Gamma]}$ of $G^{i,[\Gamma]}_{x}G^{[\Gamma]}_{x,s_{i}+}$.

Now suppose that in the above setting we have
$G^{i,[\Gamma]}=G^{i+1,[\Gamma]}=\cdots=G^{i+k,[\Gamma]}$ and have therefore
in $\Sigma^{\Gamma}$ replaced $(X^{i},\cdots,X^{i+k})$ with
$\sum_{j=i}^{i+k}X^{j}$. It suffices to note that the restriction to
$\mathfrak{g}^{[\Gamma]}_{x,s_{i+k}+}$ of
$\psi\circ\sum_{j=i}^{i+k}X^{j}=\prod_{j=i}^{i+k}\psi\circ X^{j}$ lifts to a
well-defined character of this smallest subgroup $G^{[\Gamma]}_{x,s_{i+k}+}$,
and that it moreover agrees with $\prod_{j=i}^{i+k}\phi^{j}$ on the
intersection $G^{i,[\Gamma]}_{x}\cap G^{[\Gamma]}_{x,s_{i+k}+}$. Thus in the
notation of (2.2), we have

$\prod_{j=i}^{i+k}\hat{\phi}^{j,[\Gamma]}=\widehat{\prod_{j=i}^{i+k}\phi^{j,[\Gamma]}}.$

From these two paragraphs it follows that on the subgroup
$H_{+}(\Sigma^{\Gamma})$ we have the desired equality of semisimple characters
$\theta_{\Sigma}|_{H_{+}(\Sigma)^{\Gamma}}=\theta_{\Sigma^{\Gamma}_{\mathrm{ext}}}=\theta_{\Sigma^{\Gamma}}$.
∎

##  5\. Restricting representations to $G^{[\Gamma]}$

We now fix a $\Gamma$-stable truncated datum
$\Sigma=(\vec{G},x,\vec{r},\vec{X},\vec{\phi})$ and complete it to a datum
$(\Sigma;\sigma)$ as in Definition 2.3; in order to simplify the notation in
this section, we also denote this completed datum by $\Sigma$, understanding
$\sigma$ to be fixed. Via [MP96, §6.3], the point $x\in\mathscr{B}(G^{0})$
defines a Levi subgroup $M^{0}$ of $G^{0}$; let $M$ denote the $G$-centralizer
of the centre of $M^{0}$, so that $M$ is a Levi subgroup of $G$. Then
$(J(\Sigma),\lambda_{\Sigma})$ is a $G$-cover of $(J(\Sigma)\cap
M),\lambda_{\Sigma})$, and every irreducible subquotient of
$\textrm{c-}\mathrm{Ind}_{J(\Sigma)\cap M}^{M}\ \lambda_{\Sigma}$ is a
supercuspidal representation of $M$, with any two such subquotients differing
by a twist by some unramified character of $M$. Let $\pi_{M}$ denote any such
subquotient.

Given a parabolic subgroup $P$ of $G$ with Levi factor $M$, one has a finite-
length parabolically induced representation $\mathrm{Ind}_{P}^{G}\
\pi_{M}\otimes 1$. Throughout this section, $\pi$ will denote an arbitrary
(not necessarily irreducible) subquotient of $\mathrm{Ind}_{P}^{G}\ \pi_{M}$.
The representation $\pi|_{G^{[\Gamma]}}$ is very often a non-semisimple
representation of infinite length. Nonetheless, in this section we prove a
result which allows one to explicitly identify a number of irreducible
subquotients of $\pi|_{G^{[\Gamma]}}$, up to $G^{[\Gamma]}$-inertial
equivalence. We do this by studying the decomposition of
$\lambda_{\Sigma}|_{J(\Sigma)\cap G^{[\Gamma]}}$ into irreducible components.

Recall that one has a decomposition
$\lambda_{\Sigma}=\sigma\otimes\kappa_{\Sigma}$, with $\kappa_{\Sigma}$ being
a Heisenberg–Weil lift of $\theta_{\Sigma}$. Since $\Sigma$ is
$\Gamma$-stable, one has by Theorem 4.4 that $H_{+}(\Sigma)\cap
G^{[\Gamma]}=H_{+}(\Sigma^{\Gamma})$ and
$\theta_{\Sigma}|_{H_{+}(\Sigma^{\Gamma})}=\theta_{\Sigma^{\Gamma}}$. Our
first goal is to establish an analogous relationship between the
Heisenberg–Weil representations.

######  Lemma 5.1.

The group $J(\Sigma^{\Gamma})$ is contained in $J(\Sigma)\cap G^{[\Gamma]}$
with finite index.

###### Proof.

Let $J_{+}(\Sigma)=G_{x,0+}^{0}G_{x,s_{0}}^{1}\cdots G_{x,s_{d}}^{d+1}$. This
is a $\Gamma$-invariant normal subgroup of $J(\Sigma)$ of finite index and by
inductively applying Lemma 4.3 we deduce that
$J_{+}(\Sigma)^{\Gamma}=J_{+}(\Sigma^{\Gamma}_{\mathrm{ext}})$, which in turn
is simply equal to $J_{+}(\Sigma^{\Gamma})$. Taking $\Gamma$-fixed points of
the short exact sequence

$1\to J_{+}(\Sigma)\to J(\Sigma)\to G^{0}_{x}/G^{0}_{x,0+}\to 1$

gives an exact sequence

$1\to J_{+}(\Sigma)^{\Gamma}\to
J(\Sigma)^{\Gamma}\to(G^{0}_{x}/G^{0}_{x,0+})^{\Gamma}.$

Since $J_{+}(\Sigma)^{\Gamma}=J_{+}(\Sigma^{\Gamma})\subseteq
J(\Sigma^{\Gamma})$, this yields a coarse estimate

$[J(\Sigma)\cap
G^{[\Gamma]}:J(\Sigma^{\Gamma})]\leq[J(\Sigma)^{\Gamma}:J_{+}(\Sigma)^{\Gamma}]\leq|(G^{0}_{x}/G^{0}_{x,0+})^{\Gamma}|.$

∎

Note that the representation $\kappa_{\Sigma}$ of $J(\Sigma)$ restricts to
$H_{+}(\Sigma)$ as a sum of copies of $\theta_{\Sigma}$, and so must restrict
further to $H_{+}(\Sigma^{\Gamma})$ as a sum of copies of
$\theta_{\Sigma^{\Gamma}}$. In particular, any irreducible component of
$\kappa_{\Sigma}|_{J(\Sigma^{\Gamma})}$ restricts to $H_{+}(\Sigma^{\Gamma})$
as a sum of copies of $\theta_{\Sigma^{\Gamma}}$.

######  Proposition 5.2.

The restriction to $J(\Sigma^{\Gamma})$ of $\kappa_{\Sigma}$ is
$\kappa_{\Sigma^{\Gamma}}$-isotypic.

###### Proof.

The representation $\kappa_{\Sigma}$ is defined as a tensor product
$\bigotimes_{i=0}^{d}\kappa_{\Sigma}^{i}$ of certain representations
$\kappa_{\Sigma}^{i}$. To prove the proposition, we recall the construction of
these representations following [HM08] and show at the same time that for each
$i$, $\kappa_{\Sigma}^{i}|_{J(\Sigma^{\Gamma})}$ is
$\kappa_{\Sigma^{\Gamma}}^{i}$-isotypic. We choose notation compatible with
[LN21, Proposition 7.5]; in particularly note that the groups denoted with
script $\mathcal{J}$ here are those denoted with $J$ in [Yu01].

For each $0\leq i\leq d$, let $J^{i}(\Sigma)=J(\Sigma)\cap G^{i}$. Our first
step is to define the groups

$\mathcal{J}^{i+1}(\Sigma)=(G^{i},G^{i+1})_{x,(r_{i},s_{i})}\quad\text{and}\quad\mathcal{J}^{i+1}_{+}(\Sigma)=(G^{i},G^{i+1})_{x,(r_{i},s_{i}+)},$

that are “almost complements” of $J^{i}(\Sigma)$ in $J^{i+1}(\Sigma)$ in the
sense that for each $i$, they give the equality
$J^{i+1}(\Sigma)=J^{i}(\Sigma)\mathcal{J}^{i+1}(\Sigma)$ and their
intersection is merely $G^{i}_{x,r_{i}}$.

Choose a maximal torus $\mathbf{S}$ in $\mathbf{G}^{0,[\Gamma]}$ with tamely
ramified splitting field $E$ such that
$x\in\mathscr{A}(\mathbf{G}^{0,[\Gamma]},\mathbf{S},E)\cap\mathscr{B}(G^{0})$.
Let $\mathbf{T}$ be its centralizer, which is a $\Gamma$-stable maximal torus
by Lemma 3.1. Let $\Phi^{i}=\Phi(\mathbf{G}^{i},\mathbf{T},E)$ and
$\Phi^{i}_{\Gamma}=\Phi(\mathbf{G}^{i,[\Gamma]},\mathbf{S},E)$ be the
corresponding root systems. Writing $U_{\alpha}$ for the root subgroup
corresponding to $\alpha$ over $E$, with filtration subgroups
$U_{\alpha,x,s}$, we define

$\mathcal{J}^{i+1}(\Sigma)(E)=\langle\mathbf{T}(E)_{r_{i}},U_{\alpha,x,r_{i}},U_{\beta,x,s_{i}}\
|\ \alpha\in\Phi^{i},\beta\in\Phi^{i+1}\setminus\Phi^{i}\rangle,$

and set $\mathcal{J}^{i+1}(\Sigma)=\mathcal{J}^{i+1}(\Sigma)(E)\cap G^{i+1}$.
The group
$\mathcal{J}^{i+1}(\Sigma^{\Gamma})=\mathcal{J}^{i+1}(\Sigma^{\Gamma}_{\mathrm{ext}})$
is defined analogously. The compatibility of the root systems shown in Lemma
3.1 gives us the inclusion
$\mathcal{J}^{i+1}(\Sigma^{\Gamma})\subseteq\mathcal{J}^{i+1}(\Sigma)$.
Replacing $s_{i}$ with $s_{i}+$ in the above definition gives also
$\mathcal{J}_{+}^{i+1}(\Sigma^{\Gamma})\subseteq\mathcal{J}_{+}^{i+1}(\Sigma)$.

The character $\hat{\phi}^{i}$, restricted to
$\mathcal{J}_{+}^{i+1}(\Sigma)\subseteq G^{i+1}_{x,s_{i}+}$, has kernel
denoted $\mathcal{N}^{i+1}(\Sigma)$. Note that this kernel is uniquely
determined by the image of $X^{i}$ in $\mathfrak{g}^{*}_{x,-r_{i}:-r_{i}+}$.
Since the character $\hat{\phi}^{i,[\Gamma]}$ associated to
$\Sigma^{\Gamma}_{\mathrm{ext}}$ is the restriction of the character
$\hat{\phi}^{i}$ associated to $\Sigma$, we have
$\mathcal{N}^{i+1}(\Sigma^{\Gamma})\subset\mathcal{N}^{i+1}(\Sigma)$.

We next define

$\mathcal{H}_{\Sigma}=\mathcal{J}^{i+1}(\Sigma)/\mathcal{N}^{i+1}(\Sigma),\text{and}\;\mathcal{Z}_{\Sigma}=\mathcal{J}^{i+1}_{+}(\Sigma)/\mathcal{N}^{i+1}(\Sigma),$

as well as the analogous groups defined relative to $\Sigma^{\Gamma}$. The
group $\mathcal{H}_{\Sigma}$ is a $p$-Heisenberg group with centre
$\mathcal{Z}_{\Sigma}$. By construction, the inclusion
$\mathcal{J}^{i+1}(\Sigma^{\Gamma})\subseteq\mathcal{J}^{i+1}(\Sigma)$ induces
an embedding $\mathcal{H}_{\Sigma^{\Gamma}}\to\mathcal{H}_{\Sigma}$. Since the
centres are both isomorphic to $\mathbb{F}_{p}$, the map
$\mathcal{Z}_{\Sigma^{\Gamma}}\to\mathcal{Z}_{\Sigma}$ induced by the
embedding is an isomorphism.

Since $\hat{\phi}^{i}$ gives a (non-trivial) character of
$\mathcal{Z}_{\Sigma}\cong\mathcal{Z}_{\Sigma^{\Gamma}}$, the Stone–von
Neumann theorem associates to $\hat{\phi}^{i}$ both a unique irreducible
representation $\eta_{\Sigma}^{i}$ of $\mathcal{H}_{\Sigma}$ and a unique
irreducible representation $\eta_{\Sigma^{\Gamma}}^{i}$ of
$\mathcal{H}_{\Sigma^{\Gamma}}$. In particular, any irreducible component of
$\eta_{\Sigma}^{i}|_{\mathcal{H}_{\Sigma^{\Gamma}}}$ is an irreducible
representation of $\mathcal{H}_{\Sigma^{\Gamma}}$ with central character
$\hat{\phi}^{i}$; by uniqueness, the restriction
$\eta_{\Sigma}^{i}|_{\mathcal{H}_{\Sigma^{\Gamma}}}$ must be
$\eta_{\Sigma^{\Gamma}}^{i}$-isotypic.

Next, we define the group
$\mathcal{W}_{\Sigma}=\mathcal{J}^{i+1}(\Sigma)/\mathcal{J}_{+}^{i+1}(\Sigma)$.
When nontrivial, this group naturally carries the structure of an
$\mathfrak{f}$-vector space, on which the character $\hat{\phi}^{i}$ defines a
symplectic form by composing with the commutator. Carrying out the analogous
construction with $\Sigma^{\Gamma}$ yields an embedding
$\mathcal{W}_{\Sigma^{\Gamma}}\to\mathcal{W}_{\Sigma}$ that respects these
symplectic structures. As shown carefully in [Yu01, §10], via the conjugation
action of $J^{i}(\Sigma)$ on $\mathcal{J}^{i+1}(\Sigma)$ we may by the theory
of the Heisenberg–Weil lift extend $\eta_{\Sigma}^{i}$ to a representation
$\hat{\eta}_{\Sigma}^{i}$ of $J^{i}(\Sigma)\ltimes\mathcal{J}^{i+1}(\Sigma)$,
and similarly for $\Sigma^{\Gamma}$. Moreover, up to a minor choice (that can
be made consistently) when $p=3$ and $\dim(\mathcal{W}_{\Sigma})=2$ [HM08,
§2.3], this extension is unique. It therefore follows that the restriction of
$\hat{\eta}_{\Sigma}^{i}$ to
$J^{i}(\Sigma^{\Gamma})\ltimes\mathcal{J}^{i+1}(\Sigma^{\Gamma})$ is
$\hat{\eta}_{\Sigma^{\Gamma}}^{i}$-isotypic.

Note that when $G^{i,[\Gamma]}=G^{i+1,[\Gamma]}$ we simply have
$\mathcal{J}^{i+1}(\Sigma^{\Gamma})=\mathcal{J}^{i+1}_{+}(\Sigma^{\Gamma})=G^{i,[\Gamma]}_{x,r_{i}}$,
so that $\mathcal{H}_{\Sigma^{\Gamma}}=\mathcal{Z}_{\Sigma^{\Gamma}}$,
$\eta^{i}_{\Sigma^{\Gamma}}=\hat{\phi}^{i}$ and
$\mathcal{W}_{\Sigma^{\Gamma}}$ is trivial. In this case,
$\eta^{i}_{\Sigma^{\Gamma}}=1\ltimes\phi^{i,[\Gamma]}|_{G^{i,[\Gamma]}_{x,r_{i}}}$
and $\mathcal{J}^{i+1}(\Sigma^{\Gamma})\subset
J^{i}(\Sigma^{\Gamma})=J^{i+1}(\Sigma^{\Gamma})$.

Finally, the representation $\kappa_{\Sigma}^{i}$ is defined by making the
identification $J^{i+1}(\Sigma)=J^{i}(\Sigma)\mathcal{J}^{i+1}(\Sigma)$ in
order to define

$\kappa_{\Sigma}^{i}(gj)=\phi^{i}(g)\hat{\eta}^{i}_{\Sigma}(g,j),$

for $g\in J^{i}(\Sigma)$ and $j\in\mathcal{J}^{i+1}(\Sigma)$. This
representation may then be inflated to the group $J(\Sigma)$. Following the
same process for $\Sigma^{\Gamma}$ yields the representation
$\kappa^{i}_{\Sigma^{\Gamma}}$, and we deduce that the restriction of
$\kappa_{\Sigma}^{i}$ is $\kappa_{\Sigma^{\Gamma}}^{i}$-isotypic, as required.
∎

From this, one immediately deduces the following.

######  Corollary 5.3.

Any irreducible component of
$(\sigma\otimes\kappa_{\Sigma})|_{J(\Sigma^{\Gamma})}$ is isomorphic to
$\xi\otimes\kappa_{\Sigma^{\Gamma}}$ for some irreducible component $\xi$ of
$\sigma|_{J(\Sigma^{\Gamma})}$.

We therefore have achieved a decomposition into irreducible components of
$\lambda_{\Sigma}|_{J(\Sigma^{\Gamma})}$. Let us now apply this to the study
of $\pi|_{G^{[\Gamma]}}$.

If the representation $\xi$ is an irreducible component of
$\sigma|_{J(\Sigma^{\Gamma})}$ that is cuspidal when viewed as a
representation of $G_{x,0:0+}^{0,[\Gamma]}$, then
$\xi\otimes\kappa_{\Sigma^{\Gamma}}$ is an $\mathfrak{s}$-type, for
$\mathfrak{s}=\mathfrak{s}_{\xi}$ the inertial equivalence class in
$G^{[\Gamma]}$ of a supercuspidal representation of $M^{[\Gamma]}$, the Levi
subgroup corresponding to $\Sigma^{\Gamma}$ [KY17, Thm. 7.5].

If not, then it is not initially clear which inertial equivalence classes
$\xi\otimes\kappa_{\Sigma^{\Gamma}}$ may intertwine with. In order to deal
with this issue, we carry out a process established in [LN21, 7.2–7.5] in
order to construct, for each component $\xi$, an open subgroup $J_{\xi}$ of
$J(\Sigma^{\Gamma})$ and an irreducible component $\lambda_{\xi}$ of
$\xi\otimes\kappa_{\Sigma^{\Gamma}}|_{J_{\xi}}$ such that
$(J_{\xi},\lambda_{\xi})$ is an $\mathfrak{s}_{\xi}$-type for some
$G^{[\Gamma]}$-inertial equivalence class $\mathfrak{s}_{\xi}$. Rather than
repeat the arguments of [LN21, 7.2–7.5] _mutatis mutandis_ , we simply outline
the key steps of the construction:

  * •

The representation $\xi$ is an irreducible representation of
$J(\Sigma^{\Gamma})$ that is trivial on $J_{+}(\Sigma^{\Gamma})$, and
therefore descends to an irreducible representation of
$J(\Sigma^{\Gamma})/J_{+}(\Sigma^{\Gamma})\simeq
G_{x}^{0,[\Gamma]}/G_{x,0+}^{0,[\Gamma]}$. By Clifford theory, further
restriction gives a sum of pairwise $G_{x}^{0,[\Gamma]}$-conjugate irreducible
representations of $G_{x,0:0+}^{0,[\Gamma]}$, which is a finite group of Lie
type. Fix an element $\xi^{\prime}$ of this $G_{x}^{0,[\Gamma]}$-orbit, and
let $(\mathsf{L}_{\xi}^{\prime},\zeta_{\xi}^{\prime})$ denote a representative
of its cuspidal support.

  * •

Using [LN21, Lem. 7.2], we associate to $\mathsf{L}_{\xi}^{\prime}$ a point
$u_{\xi}\in\mathscr{B}(M^{0,[\Gamma]})$. The point $u_{\xi}$ in turn defines a
Levi subgroup of $G^{0,[\Gamma]}$, which we denote by $L_{\xi}^{0}$.

  * •

Via the inclusion $G_{u_{\xi}}^{0,[\Gamma]}\hookrightarrow
G_{x}^{0,[\Gamma]}$, one may choose an irreducible component $\zeta_{\xi}$ of
$\xi|_{G_{u_{\xi}}^{0,[\Gamma]}}$ such that the restriction of $\zeta_{\xi}$
to $G_{u_{\xi},0}^{0,[\Gamma]}$ is a sum of
$G_{u_{\xi}}^{0,[\Gamma]}$-conjugates of $\zeta_{\xi}^{\prime}$.

  * •

Set
$\lambda_{\Sigma_{\xi}^{\Gamma}}=\zeta_{\xi}\otimes\kappa_{\Sigma^{\Gamma}}$.
It then follows from [LN21, Lem. 7.3] that the tuple
$(\Sigma_{\xi}^{\Gamma};\zeta_{\xi})$ derived from $(\Sigma^{\Gamma};\sigma)$
by replacing $x$ with $u_{\xi}$ and $\sigma$ with $\zeta_{\xi}$ is a datum
with corresponding type
$(J(\Sigma_{\xi}^{\Gamma}),\lambda_{\Sigma_{\xi}^{\Gamma}})$. Moreover, [LN21,
Lem. 7.2] shows that $J(\Sigma_{\xi}^{\Gamma})\subset J(\Sigma^{\Gamma})$, and
it follows from [LN21, Prop. 7.5] that

$\mathrm{Hom}_{J(\Sigma_{\xi}^{\Gamma})}(\lambda_{\Sigma_{\xi}^{\Gamma}},\
\xi\otimes\kappa_{\Sigma^{\Gamma}})\neq 0.$

Let $\mathfrak{s}_{\xi}$ denote the $G^{[\Gamma]}$-inertial equivalence class
such that $(J(\Sigma_{\xi}^{\Gamma}),\lambda_{\Sigma_{\xi}^{\Gamma}})$ is an
$\mathfrak{s}_{\xi}$-type.

######  Theorem 5.4.

Let $\Sigma$ and $\pi$ be as above. For each irreducible component $\xi$ of
$\sigma|_{J(\Sigma^{\Gamma})}$, there exists an irreducible subquotient
$\pi_{\xi}$ of $\pi|_{G^{[\Gamma]}}$ that has inertial support
$\mathfrak{s}_{\xi}$.

###### Proof.

Since $\pi$ contains $\lambda_{\Sigma}$, it contains a copy of
$\lambda_{\Sigma_{\xi}^{\Gamma}}$ for each representation $\xi$, which implies
that $\pi|_{G^{[\Gamma]}}$ has a non-trivial projection onto
$\mathrm{Rep}_{\lambda_{\Sigma_{\xi}^{\Gamma}}}(G^{[\Gamma]})$, the category
of smooth representations generated by their
$\lambda_{\Sigma_{\xi}^{\Gamma}}$-isotypic vectors, for each $\xi$. As the
pair $(J(\Sigma_{\xi}^{\Gamma}),\lambda_{\Sigma_{\xi}^{\Gamma}})$ is an
$\mathfrak{s}_{\xi}$-type, one has
$\mathrm{Rep}_{\lambda_{\Sigma_{\xi}^{\Gamma}}}(G^{[\Gamma]})=\mathrm{Rep}^{\mathfrak{s}_{\xi}}(G^{[\Gamma]})$,
the corresponding Bernstein block. The representation $\pi$ therefore has non-
trivial projection onto this block, and the image of $\pi$ under this
projection, being finitely generated, admits an irreducible quotient. By
construction, any such irreducible quotient has inertial support
$\mathfrak{s}_{\xi}$. ∎

######  Remark 5.5.

In general, this does not provide an exhaustive list of the inertial supports
of representations occuring in $\pi|_{G^{[\Gamma]}}$. The finite-length
representation $\pi$ is generated by its $\lambda_{\Sigma}$-isotypic vectors,
hence by its vectors on which $J(\Sigma^{\Gamma})$ acts as
$\xi\otimes\kappa_{\Sigma^{\Gamma}}$ for some $\xi$, but only as a
representation of $G$. In general, one may apply Mackey’s theorem to say that

$\pi|_{G^{[\Gamma]}}\hookrightarrow\mathrm{Res}_{G^{[\Gamma]}}^{G}\textrm{c-}\mathrm{Ind}_{J(\Sigma)}^{G}\
\lambda_{\Sigma}=\bigoplus_{G^{[\Gamma]}\backslash
G/J(\Sigma)}\textrm{c-}\mathrm{Ind}_{{}^{g}J(\Sigma)\cap
G^{[\Gamma]}}^{G^{[\Gamma]}}\ {}^{g}\lambda_{\Sigma}.$

What we have done is describe (exhaustively) the inertial equivalence classes
of those subrepresentations of $\pi|_{G^{[\Gamma]}}$ which are contained
within the trivial Mackey summand

$\textrm{c-}\mathrm{Ind}_{J(\Sigma)\cap G^{[\Gamma]}}^{G^{[\Gamma]}}\
\lambda_{\Sigma}.$

However, the other summands are far less amenable to description. For example,
while it is true that for any $g\in G$, conjugation of the pair
$(J(\Sigma),\lambda_{\Sigma})$ gives the pair
$(J({}^{g}\Sigma),\lambda_{{}^{g}\tilde{\Sigma}})$, where ${}^{g}\Sigma$ is
still a datum for $G$, the datum ${}^{g}\Sigma$ need not be $\Gamma$-stable.
In any situations where ${}^{g}\Sigma$ _is_ $\Gamma$-stable, then we may
similarly describe the inertial equivalence classes of representations
occurring within the Mackey summand

$\textrm{c-}\mathrm{Ind}_{{}^{g}J(\Sigma)\cap G^{[\Gamma]}}^{G^{[\Gamma]}}\
{}^{g}\lambda_{\Sigma}.$

One expects, however, that many of these summands do not correspond to
$\Gamma$-stable data.

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