On the group of homeomorphisms of the Basilica

Yury A.Neretin111Supported by the grant FWF, P31591.

We show that the group of all homeomorphisms of the Basilica fractal coincides
with a group of transformations of a certain non-locally finite ribbon tree.
Also, we show that Basilica Thompson group defined by Belk and Forrest is
dense in the group of all orientation preserving homeomorphisms of the
Basilica.

1\. The Basilica. The Basilica $\mathcal{B}$ is a compact topological space
obtained as a quotient of the circle $S={\mathbb{R}}/2{\mathbb{Z}}$ under the
equivalence relation

(1) $\tfrac{3k+1}{3^{n}}\sim\tfrac{3k+2}{3^{n}},\qquad\text{where $k$, $n$ are
nonnegative integers,}$

In particular $1\sim 2=0$. In other words, we write numbers in ternary system
and identify $a_{0}.a_{1}\dots a_{n-1}1\sim a_{0}.a_{1}\dots a_{n-1}2$. The
reader can find nice pictures of the Basilica in papers [7], [1]–[2] or their
preprint versions in arXiv, see also Fig. 1.

We say that a separating point of $\mathcal{B}$ is a point corresponding a
nontrivial (two-point) equivalence class. Denote by
$\operatorname{Sep}(\mathcal{B})$ the set of separating points. If
$x\in\operatorname{Sep}(\mathcal{B})$, then $\mathcal{B}\setminus x$ consists
of two components. Otherwise, $\mathcal{B}\setminus x$ is connected.

It is convenient to think that our circle $S$ is a boundary of a disk $D$, and
equivalent points are connected by chords $[a\text{---}b]$ (leafs), clearly
they are disjoint. Denote the set of all leafs by $L(\mathcal{B})$. Leafs
divide the disc into domains, say Cantorgons (cf. [4]), having a countable
number of sides, an intersection of a Cantorgon with $S$ is a totally
disconnected (Cantor) set. The set of sides of a Cantorgon has a canonical
cyclic order; as a cyclic ordered set this set is equivalent to
${\mathbb{Q}}/{\mathbb{Z}}$ (or to a countable dense subset of a circle, all
such subsets are equivalent).

Remark. The set of all leafs can be obtained as follows. Consider a circle of
length 2. Its diameter $[0\text{---}1]$ splits a circle into two arcs. We
divide each arc by a pair of points with ratios $1:1:1$ and connect elements
of each pair by a chord. We get 6 ends of chords and 6 arcs and divide arcs
with ratios $1:1:1$, etc. $\boxtimes$

Consider a Cantorgon $C$ and contract each side to a point. Then we get a
topological disc, its boundary $\eth(C)$ is a topological circle equipped with
a canonical orientation induced from $S$. Each circle $\eth(C_{j})$ is
homeomorphically embedded to $\mathcal{B}$, we call such image a component of
the Basilica. Two components are disjoint or contain one joint point, which
with necessity is a separating point. The union of all components is dense in
the Basilica.

Clearly, homeomorphisms of $\mathcal{B}$ send components to components. We say
that a homeomorphism of $\mathcal{B}$ preserves orientation if it is an
orientation preserving map on each component. Denote by
$\operatorname{Homeo}(\mathcal{B})$ the group of all homeomorphisms of
$\mathcal{B}$, by $\operatorname{Homeo}_{+}(\mathcal{B})$ the subgroup of
orientation preserving homeomorphisms.

2\. The tree of the Basilica. Consider a graph $T$, whose vertices are
enumerated by Cantorgons of $\mathcal{B}$ and edges are enumerated by leafs;
an edge contains a vertex if the corresponding leaf is a side of the
corresponding Cantorgon. Clearly, $T$ is a tree and valences of all vertices
are countable. Denote by $\mathop{\mathrm{edge}}\nolimits(v)$ the set of edges
incident to a vertex $v$. As we have seen above, each set
$\mathop{\mathrm{edge}}\nolimits(v)$ has a canonical cyclic order. Denote the
tree equipped with this structure by $T(\mathcal{B})$.

Consider an automorphism $g$ of the tree $T$ (i.e., a bijection of the set of
vertices to itself sending adjacent vertices to adjacent vertices). We say
that $g$ is orientation preserving if for each vertex $v$ the map $g$ induces
a cyclic order preserving map
$\mathop{\mathrm{edge}}\nolimits(v)\to\mathop{\mathrm{edge}}\nolimits(gv)$. We
say that $g$ respects orientation if each induced map
$\mathop{\mathrm{edge}}\nolimits(v)\to\mathop{\mathrm{edge}}\nolimits(gv)$
preserves or inverses the cyclic order. We denote by
$\operatorname{Aut}_{+}(T(\mathcal{B}))$ (resp.,
$\operatorname{Aut}(T(\mathcal{B}))$) the groups of all orientation preserving
(resp., orientation respecting) automorphisms of $T(\mathcal{B})$ (this
construction is inside the formalism of Burger–Mozes groups, see [3], Subsect.
3.2).

Figure 1. We draw a circle ${\mathbb{R}}/2{\mathbb{Z}}$ as a segment of the
horizontal line and connect equivalent points by arcs.

We define topology on these group by the assumption: stabilizers of finite
subtrees are open subgroups. It is easy to see that we get topological groups,
and, moreover, Polish groups (a group is called Polish if it is homeomorphic
to a complete metric space222Denote by $E$ the set of edges of $T$, by $V$ the
set of vertices, by $S(E)$, $S(T)$ the groups of all permutations of these
sets equipped with the unique separable topology on infinite symmetric group,
see, e.g., [5]. Let $A\subset V\times E$ consists of pairs $(v,e)$ such that
$v\in e$. Let $B\subset V\times E\times E\times E$ consists of
$(v,e_{1},e_{2},e_{3})$ such that $v\in e_{j}$ and $e_{1}$, $e_{2}$, $e_{3}$
are arranged clockwise. Then $\operatorname{Aut}_{+}(T(\mathcal{B}))$ is the
subgroup in a Polish group $S(E)\times S(V)$ consisting of elements sending
$A$ to $A$ and $B$ to $B$. It is easy to see that this condition is closed,
and therefore $\operatorname{Aut}_{+}(T(\mathcal{B}))$ is a closed subgroup in
a Polish group, so it is Polish.).

3\. The statement of the paper. Since a homeomorphism of the Basilica sends
components to components and separating points to separating points, we have a
canonical homomorphism

$\Pi:\operatorname{Homeo}(\mathcal{B})\to\operatorname{Aut}(T(\mathcal{B})),$

sending
$\operatorname{Homeo}_{+}(\mathcal{B})\to\operatorname{Aut}_{+}(T(\mathcal{B}))$.
Since separating points are dense, our homomorphism is injective.

######  Theorem 1.

a) $\Pi$ establishes an isomorphism
$\operatorname{Homeo}(\mathcal{B})\simeq\operatorname{Aut}(T(\mathcal{B}))$.

b) $\Pi$ establishes an isomorphism
$\operatorname{Homeo}_{+}(\mathcal{B})\simeq\operatorname{Aut}_{+}(T(\mathcal{B}))$.

c) The Basilica Thompson group $\mathrm{Th}(\mathcal{B})$ is dense in
$\operatorname{Aut}_{+}(T(\mathcal{B}))$.

4\. The Basilica Thompson group $\mathrm{Th}(\mathcal{B})$. We say that a
diagram is a finite collection $\Delta\subset L(\mathcal{B})$ of leafs such
that

1) for any $\eta\in L(\mathcal{B})\setminus\Delta$ all elements of $\Delta$
lie on one side of $\eta$.

2) $\Delta$ contains the leaf $[0\text{---}1]$.

Leafs of $\Delta$ divide the disk $D$ into domains, say cells. Our condition
means that for each cell all adjacent leafs are sides of one Cantorgon. We
call ends of leafs vertices of a diagram, and arcs of $S$ between vertices
intervals of a diagram.

We say that a diagram is perfect if it can be obtained by the following
procedure. We start with the diameter $[1-2]$. It splits $S$ into two
intervals. We divide one of intervals with ratios $1:1:1$ by points $a$, $b$
and draw the corresponding chord $[a-b]$. Now a circle is split into 4
intervals. We choose one interval, divide it with ratios $1:1:1$ and add the
corresponding chord. Etc. By the construction, intervals of a perfect diagram
have forms

(2)
$\bigl{[}\tfrac{3\alpha}{3^{a}},\tfrac{3\alpha+1}{3^{a}}\bigr{]},\qquad\bigl{[}\tfrac{3\beta+1}{3^{b}},\tfrac{3\beta+2}{3^{b}}\bigr{]},\qquad\bigl{[}\tfrac{3\gamma+2}{3^{c}},\tfrac{3\gamma}{3^{c}}\bigr{]}.$

We say that two perfect diagrams $\Delta_{1}$, $\Delta_{2}$ are isomorphic if
there is an orientation preserving bijective continuous map identifying
$\Delta_{1}$ and $\Delta_{2}$. An isomorphism determines a bijective map on
the set of vertices. We extend this map to each interval of $\Delta_{1}$ in a
linear way (all slopes have a form $3^{k}$). According Belk and Forrest [1],
such maps form a group—the Basilica Thompson group $\mathrm{Th}(\mathcal{B})$,
see [1] (for instance, it is generated by 4 explicit elements, Theorem 7.1),
for further properties, see [7], [15], see also [2] on extensions of this
construction. On the other hand, such a map respects the equivalence relation
(1) and therefore determines a homeomorphism of the Basilica.

5\. Correspondence with the initial model of the Basilica Thompson group. The
Basilica naturally arises as an object of complex dynamics (as a Julia set,
see, e.g., [7]), a natural equivalence relation on a circle is defined in the
following way. We consider a circle $S^{\diamond}={\mathbb{R}}/{\mathbb{Z}}$
of length 1 and the corresponding disc. We start with chords
$[\frac{1}{6}\text{---}(-\frac{1}{6})]$, $[\frac{1}{3}\text{---}\frac{2}{3}]$
and apply the procedure described in Remark in Sect.1 changing ratios $1:1:1$
to $1:2:1$. In this way, we get the same combinatorial picture and the same
topological object. Again, we consider perfect diagrams, their isomorphisms,
and linear maps on intervals (now all slopes have form $2^{k}$).

Let us describe the correspondence more precisely. First, we pass to the
circle $S^{\circ}={\mathbb{R}}/3{\mathbb{Z}}$ with chords $[0\text{---}1]$,
$[3/2\text{---}5/2]$ (we rotate the circle $S^{\diamond}$ and dilatate it).
The homeomorphism $\Theta:S\to S^{\circ}$ is defined in the following way. Let
$a$ range in the set $\\{0,1,2\\}$. We define functions $\varphi(a)$,
$\psi(a)$ by

$\varphi(0)=\tfrac{1}{4},\,\varphi(1)=\tfrac{1}{2},\,\varphi(2)=\tfrac{1}{4},\qquad\psi(0)=0,\,\psi(1)=\tfrac{1}{4},\,\psi(2)=\tfrac{3}{4}.$

Then for a ternary number $A=a_{0}.a_{1}a_{2}\dots$ we set

$\Theta(A)=\frac{\psi(a_{0})}{\varphi(0)}+\sum_{n=1}^{\infty}\Bigl{(}\prod_{p=0}^{n-1}\varphi(a_{p})\cdot\psi(a_{n})\Bigr{)}.$

This homeomorphism identifies the equivalence relations and identify the group
$\mathrm{Th}(\mathcal{B})$ defined above with the group from [1].

6\. Identification of $\operatorname{Homeo}_{+}(\mathcal{B})$ and
$\operatorname{Aut}_{+}T(\mathcal{B})$.

######  Lemma 2.

Let $\Delta$ be a diagram. Consider all cells $M_{j}$ of $\Delta$ adjacent to
$>1$ leafs and the corresponding Cantorgons $C_{j}$. Then there is a finite
collection of leafs $l_{1},l_{2},\dots$, such that each $l_{q}$ is a side of
some Cantorgon $C_{p}$ and the diagram $\Delta\cup\\{l_{1},l_{2},\dots\\}$ is
perfect.

Proof. Let a leaf $[a_{0}.a_{1}\dots a_{n}1\text{---}a_{0}.a_{1}\dots a_{n}2]$
be contained in a perfect diagram $\Sigma$. It can appear as a result of
cutting of the arc $[a_{0}.a_{1}\dots a_{n},a_{0}.a_{1}\dots a_{n}+3^{-n}]$
with ratios $1:1:1$ and therefore ends of this arc are vertices of $\Sigma$,
i.e., ends of arcs of $\Sigma$. Minimal ternary notation of these vertices has
$\leqslant n$ places, therefore the corresponding leafs are longer than our
initial leaf.

Consider the diagram $\Delta$. For each cell $M_{j}$ we find an adjacent leaf
$m_{j}$ of minimal length and add to $\Delta$ all sides of the Cantorgon
$C_{j}$ whose lengths are larger than the length of $m_{j}$. We get a perfect
diagram. $\square$

######  Lemma 3.

Let $r\in\operatorname{Aut}_{+}(T(\mathcal{B}))$, let $P$ be a finite subtree
in $T(\mathcal{B})$. Then there exists $g\in\mathrm{Th}(\mathcal{B})$ such
that $g\Bigr{|}_{P}=\Pi\circ r\Bigr{|}_{P}$.

Proof. Without loss of generality we can assume that both $P$, $rP$ contain
the edge corresponding the leaf $[0\text{---}1]$.

For each vertex of $P$ the set of adjacent vertices is equipped with a cyclic
order. So $P$ has a structure of a planar tree; $rP$ is an isomorphic planar
tree. For each edge of $P$ we draw the corresponding leaf and get a diagram,
say $\Delta$. Let $\Sigma$ be the similar diagram for $rP$. These diagrams are
combinatorially isomorphic. Next, we extend them to perfect diagrams
$\widetilde{\Delta}$, $\widetilde{\Sigma}$ as in Lemma 2. This means that we
added additional leafs between each pair of neighbouring leafs of each cell of
$\Delta$ (resp. $\Sigma$). Generally, diagrams $\widetilde{\Delta}$,
$\widetilde{\Sigma}$ are non-isomorphic since number of additional leafs for
corresponding pairs are different. However, adding more leafs to
$\widetilde{\Delta}$ and $\widetilde{\Sigma}$ we can get isomorphic perfect
diagrams. It remains to take the corresponding $g\in\mathrm{Th}(\mathcal{B})$.
$\square$

Proof of statements b), c) of Theorem 1. Fix
$r\in\operatorname{Aut}_{+}(T(\mathcal{B}))$ Consider a sequence of finite
subtrees $P_{j}\subset T(\mathcal{B})$ such that $\cup P_{j}=T(\mathcal{B})$.
For each $P_{j}$, consider an element
$g_{j}\in\operatorname{Aut}_{+}(T(\mathcal{B}))$ such that
$g_{j}\Bigr{|}_{P_{j}}=\Pi\circ r\Bigr{|}_{P_{j}}$.

Then for each $x_{\alpha}\in\operatorname{Sep}(\mathcal{B})$ the sequence
$g_{j}(x_{\alpha})$ is eventually constant, denote its limit by
$\rho(x_{\alpha})$. For each $m$ denote by $\Omega_{m}$ the set of
$x_{\alpha}\in\operatorname{Sep}(\mathcal{B})$ such that
$g_{j}x_{\alpha}=\rho(x_{\alpha})$ for $j\geqslant m$. Clearly, we have
$\cup\Omega_{m}=\operatorname{Sep}(\mathcal{B})=\cup\rho(\Omega_{m})$. For
this reason, the map
$\rho:\operatorname{Sep}(\mathcal{B})\to\operatorname{Sep}(\mathcal{B})$ is a
bijection.

Next, sets $\mathcal{B}\setminus\Omega_{m}$,
$\mathcal{B}\setminus\rho(\Omega_{m})$ split into finite collection of
connected pieces and diameters of these pieces tend to 0 as $m\to\infty$.
Elements $g_{j}$ for $j\geqslant m$ send pieces to pieces. This implies
uniform convergence of homeomorphisms $g_{j}$. We define $\rho$ as limit of
$g_{j}$, the corresponding transformation of the tree $T(\mathcal{B})$ is $r$.

Thus, we established that $\mathrm{Th}(\mathcal{B})$ is dense in
$\operatorname{Homeo}_{+}(\mathcal{B})$ and the homomorphism
$\Pi:\operatorname{Homeo}_{+}(\mathcal{B})\to\operatorname{Aut}_{+}(T(\mathcal{B}))$
is surjective. Continuity of $\Pi$ is obvious. Banach’s bounded inverse
theorem remains true for Polish groups, see e.g., [5], Corollary 2.3.4. For
this reason, $\Pi$ is an isomorphism of Polish groups.

7\. Identification of $\operatorname{Homeo}(\mathcal{B})$ and
$\operatorname{Aut}T(\mathcal{B})$. For any leaf
$[A\text{---}A^{\prime}]=[a_{0}.a_{1}\dots a_{n}1\text{---}a_{0}.a_{1}\dots
a_{n}2]$ consider the following homeomorphism $\gamma_{A,A^{\prime}}$ of
$\mathcal{B}$: it permutes $A$ and $A^{\prime}$, it is linear on the interval
$[A,A^{\prime}]$, and is identical outside this interval.

We can repeat the previous considerations for the countable group $\Gamma$
generated by $\mathrm{Th}(\mathcal{B})$ and $\gamma_{A,A^{\prime}}$. Indeed,
denote by $e_{A,A^{\prime}}$ the corresponding edge of $T(\mathcal{B})$.
Cutting it in a midpoint we splits the tree into two branches. The
transformation $\gamma_{A,A^{\prime}}$ is identical on one of branches and is
a map changing cyclic ordering in each vertex of another branch. Now we can
get the analogue of Lemma 3. Let $q\in\operatorname{Aut}(T(\mathcal{B}))$, $P$
be a finite subtree. We can choose a collection $(A_{1},A^{\prime}_{1})$, …,
$(A_{k},A^{\prime}_{k})$ such that

$r=\gamma_{A_{k},A^{\prime}_{k}}\dots\gamma_{A_{1},A^{\prime}_{1}}q\in\operatorname{Aut}_{+}(T(\mathcal{B}))$

We apply Lemma 3 and get an element $g\in\Gamma$ such that $g=\Pi\circ
q\Bigr{|}_{P}$. Further arguments are identical.

8\. Final remarks. a) I have not seen the groups
$\operatorname{Aut}_{+}(T(\mathcal{B}))$ and
$\operatorname{Aut}(T(\mathcal{B}))$ in literature. However, these objects are
not as exotic as it may seem at first glance. In 1982 Olshanski [10]
classified unitary representations of the group of all automorphisms of a non-
locally finite Bruhat–Tits tree, in a certain sense this example is toy, but
this work was an important test site for representation theory of infinite-
dimensional groups. The group of spheromorphisms of this tree is a nontrivial
object for representation theory [9].

The group of order preserving bijections of the set ${\mathbb{Q}}$ is a
standard example of Polish groups, see, e.g., [11], [14], [6]. It also admits
explicit classification of unitary representations [14] (but in this case the
list of representations is poor). On the other hand, dense cyclic orderings
participate in the construction of ’Chinese restaurant process’ by Pitman,
see, e.g., [12], this process is a starting point for harmonic analysis
related to infinite symmetric groups.

b) Belk and Forrest [2] proposed a general construction of discrete groups of
Thompson type acting on certain fractal compact spaces. Apparently, there are
statements similar to Theorem 1 for some of these spaces. Certainly, this is
so for the Doaudy rabbit known in complex dynamics. Another interesting
possible example is the Airplane fractal defined in [2], see also [13].
Apparently, in this case we must consider the Bruhat–Tits ${\mathbb{R}}$-tree
(see, e. g., [8]) of the field of formal Laurent series
${\mathbb{Q}}[[t,t^{1,2},t^{1/4},\dots]]$ equipped with a ribbon structure.
Apparently, the group of homeomorphisms of the Airplane is isomorphic to the
group of homeomorphisms of this ${\mathbb{R}}$-tree respecting the cyclic
ordering structure of germs of edges in vertices.

c) It is well-known that unitary representation of discrete groups are a
dangerous topic. However, classification [10] of unitary representations of a
non-locally finite Bruhat–Tits tree contains a counterpart of the
complementary series. Restricting it to $\mathrm{Th}(\mathcal{B})$ we get
unitary representations that are at least unusual and are spherical with
respect to the stabilizer of a Cantorgon (this stabilizer is described in [1],
Sect. 6).

I am grateful to Matteo Tarocchi for a discussion of
$\mathrm{Th}(\mathcal{B})$.

## References

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  * [2] Belk J., Forrest B. Rearrangement groups of fractals. Trans. Amer. Math. Soc. 372 (2019), no. 7, 4509-4552.
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  * [7] Lyubich M., Merenkov S. Quasisymmetries of the basilica and the Thompson group. Geom. Funct. Anal. 28 (2018), no. 3, 727-754.
  * [8] Morgan J.W., $\Lambda$-trees and their applications, Bull. Amer. Math. Soc. 26 (1992), 87-112.
  * [9] Neretin Yu. A. On the group of spheromorphisms of a homogeneous non-locally finite tree. Izv. Math. 84 (2020), no. 6, 1161-1191.
  * [10] Olshanskii G. I., “New “large” groups of type I. J. Soviet Math., 18:1 (1982), 22-39.
  * [11] Pestov V. Dynamics of infinite-dimensional groups. The Ramsey–Dvoretzky–Milman phenomenon. American Mathematical Society, Providence, RI, 2006.
  * [12] Pitman J. Combinatorial stochastic processes. Lectures from the 32nd Summer School on Probability Theory held in Saint-Flour, July 7-24, 2002. Lect. Notes Math., 1875. Springer-Verlag, Berlin, 2006.
  * [13] Tarocchi M. Generation and Simplicity in the Airplane Rearrangement Group. Preprint, https://arxiv.org/abs/2107.08744.
  * [14] Tsankov T. Unitary representations of oligomorphic groups. Geom. Funct. Anal. 22 (2012), no. 2, 528-555.
  * [15] Witzel S., Zaremsky M. C. B. The basilica Thompson group is not finitely presented. Groups Geom. Dyn. 13 (2019), no. 4, 1255-1270.

Fakultät für Mathematik, Universität Wien;

Institute for Information Transmission Problems;

Moscow State University

e-mail: yurii.neretin(frog)univie.ac.at

URL: mat.univie.ac.at/$\sim$neretin


# Typical representations, parabolic induction and the inertial local
Langlands correspondence

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

We prove a result which provides a link between the decomposition of
parabolically induced representations and the Bushnell–Kutzko theory of
typical representations. As an application, we show that there exists a well-
defined inertial Langlands correspondence which respects the monodromy action
of $L$-parameters, under some standard conjectures regarding the local
Langlands correspondence. To allow for potential applications of this inertial
Langlands correspondence, we also provide a complete construction of the set
of typical representations, giving a parametrization of these in terms of the
structure of the Bruhat–Tits building of $G$.

###### Key words and phrases:

Theory of types, Langlands correspondence, parabolic induction

###### 1991 Mathematics Subject Classification:

22E50, 11F80

##  1\. Introduction

In this paper, we show that there exists a well-defined “inertial local
Langlands correspondence”, as long as one assumes some standard conjectures
about the existence and properties of the local Langlands correspondence.
Recall that the local Langlands correspondence is a conjectural
parametrization of the irreducible representations of a $p$-adic group $G$ in
terms of _$L$ -parameters_, which are objects closely related to continuous
Galois representations (see Section 5 for a precise definition).

The inertial Langlands correspondence is a modification of the local Langlands
correspondence which tracks only the information which occurs “on the compact
level”—it replaces $L$-parameters with the restrictions to the compact inertia
subgroup, and replaces irreducible111In the main body of the paper, we
consider only _regular_ representations due to non-regular representations
introducing additional technical complications. We overlook this distinction
during the introduction. representations with (objects closely related to)
their _types_ , in the sense of [BK98].

The existence of the inertial Langlands correspondence not only provides an
affirmative answer to the natural question of whether the many available
explicit constructions translate over the local Langlands correspondence, but
is also of arithmetic significance. Most notably, the inertial Langlands
correspondence is a key pre-requisite to the formation of the Breuil–Mézard
conjecture [BM02, EG14], which is a central problem in modern algebraic number
theory, the partial solution of which has already found numerous
applications—for example, Kisin’s proof of (many cases of) the 2-dimensional
Fontaine–Mazur conjecture [Kis09] is largely based on establishing the
corresponding cases of the Breuil–Mézard conjecture.

Each of these applications relies on the inertial Langlands correspondence for
$\mathbf{GL}_{N}(F)$, the existence and fine properties of which have been
understood for quite some time. However, if one wishes to investigate similar
questions within the more general context of the Langlands programme for an
arbitrary $p$-adic group, then one requires a similar understanding of the
inertial Langlands correspondence for such groups. This is something of a
problematic topic within the literature—the term “inertial Langlands
correspondence” actually refers to _two_ (closely related) folklore
conjectures. The author is not aware of an exposition of either anywhere in
the literature, and in fact one of the two is not provably well-defined, given
the current state of the art.

To explain this distinction, recall that an $L$-parameter for $G$ consists of
a continuous representation of the Weil–Deligne group
$W_{F}\ltimes\mathbb{C}$, taking image in the Langlands dual group ${}^{L}G$.
The action of $\mathbb{C}$ contained within such a representation is the
_monodromy action_ of the $L$-parameter; the role of the monodromy action is
to distinguish between distinct irreducible representations with equivalent
supercuspidal supports.

  * •

The most basic form of the inertial Langlands correspondence is a canonical
map which assigns to each Bushnell–Kutzko type a map
$I_{F}\rightarrow{}^{L}G$, where $I_{F}\subset W_{F}$ is the inertia group.
The existence of this map is not a problem; given the existence of the local
Langlands correspondence, together with a construction of types, deducing the
existence of this inertial correspondence is an exercise in the definitions.
The issue here is that one forgets the monodromy action, which leads to a
somewhat coarse correspondence—for applications such as the Breul–Mézard
conjecture, one must keep track of the monodromy action. For this reason, we
refer to this construction as the _weak inertial Langlands correspondence_.

  * •

One then expects the existence of the “strong” inertial Langlands
correspondence. This is entirely conjectural aside from the cases of
$\mathbf{GL}_{N}(F)$ and $\mathbf{SL}_{N}(F)$, with even these being poorly
represented in the literature222At the end of Section 9, we provide an
extended discussion of exactly what is currently known in these cases, and to
what extent it is available in published form.. It predicts that by replacing
Bushnell–Kutzko types with a well-chosen set of _typical representations_ (in
the sense of [BK98]), one may refine the weak correspondence to a natural map
which associates to each typical representation a representation
$I_{F}\rightarrow{}^{L}G$, together with a monodromy action. This is something
of a widely-held folklore belief within the community, but cannot be clearly
stated given our current understanding of parabolically induced
representations—there is no hope of precisely defining _which_ typical
representations one should consider.

In this paper, we present what appears to be an extremely satisfactory remedy
to the situation. We make a minor concession, in allowing the domain of the
inertial Langlands correspondence to consist not of typical representations,
but of certain _explicitly defined_ finite sets of typical representations. In
doing so, one obtains a canonical map $\varphi$ which associates to each
irreducible representation $\pi$ of $G$ the (finite-modulo-conjugacy) set of
typical representations which are contained in $\pi|_{K}$, as $K$ ranges over
the maximal compact subgroups of $G$. Call a set $X$ of conjugacy classes of
typical representations _admissible_ if it lies in the image of $\varphi$. Our
main result is the following (see Theorem 9.2 for a precise statement).

######  Theorem 1.

Let $X$ be an admissible set of typical representations. For each irreducible
representation $\pi$ of $G$ such that $\varphi(\pi)=X$, let
$\mathrm{iner}(X,\pi)$ denote the restriction to $I_{F}\ltimes\mathbb{C}$ of
the $L$-parameters corresponding to $\pi$. Then
$\mathrm{iner}(X)=\mathrm{iner}(X,\pi)$ is independent of the choice of $\pi$.
In particular, one obtains a canonical map $X\mapsto\mathrm{iner}(X)$ which
associates to each admissible set $X$ of typical representations the
restriction to $I_{F}\ltimes\mathbb{C}$ of an $L$-parameter.

Our proof takes place entirely on the $p$-adic side of the correspondence,
with the difficulty lying in showing that the map $\mathrm{iner}$ is well-
defined. This requires us to prove the following result, which may be of
interest independent of any applications to the Langlands programme.

######  Theorem 2.

Suppose that $\pi_{1}$ and $\pi_{2}$ are two irreducible representations of
$G$ which have the same supercuspidal supports, and suppose furthermore that
$\varphi(\pi_{1})=\varphi(\pi_{2})$. Then $\pi_{1}\simeq\pi_{2}$.

This is intriguing in that, while not allowing for a complete description of
the Jordan–Hölder series of a parabolically induced representation, it does
allow for a combinatorial description of the irreducible components of such a
representation, in terms of sets of typical representations.

In order to apply either this description of (semisimplified) parabolically
induced representations, or to make a useful application of the above inertial
local Langlands correspondence, one requires a workable description of the
various typical representations. Our final result provides just this, showing
that the isomorphism classes of typical representations admit a combinatorial-
geometric description via Bruhat–Tits theory. Specifically, for each inertial
equivalence class $\mathfrak{s}$ and each maximal compact subgroup $K$ of $G$,
the $\mathfrak{s}$-typical representations of $K$ occur as components of
$\mathrm{Ind}_{J}^{K}\ \lambda$, where $(J,\lambda)$ is an
$\mathfrak{s}$-type. In Section 10, we construct an injective system of
subrepresentations $V_{i}$ of $\mathrm{Ind}_{J}^{K}\ \lambda$ such that
$\mathrm{Ind}_{J}^{K}\ \lambda=\varinjlim V_{i}$, and show that the maximal
elements of $\\{V_{i}\\}$ (which are largely determined by a choice of vertex
in a certain sub-building of the Bruhat–Tits building of $G$) are exactly the
irreducible components of $\mathrm{Ind}_{J}^{K}\ \lambda$.

We now summarize the layout of the paper. Section 2 briefly introduces the key
notation which will be used throughout the paper. Section 3 recalls some key
definitions from [BK98], and Section 4 sketches the construction of types due
to Kim–Yu [KY17]. From Section 5 onwards, we turn towards the Langlands
correspondence, introducing the notion of $L$-parameters, and setting up some
useful functors. Section 6 recalls Kaletha’s construction [Kal19] of the local
Langlands correspondence for regular supercuspidal representations; Section 7
then extends this to regular non-cuspidal representations, under some standard
conjectures. Section 8 extends the notions of Section 3 in order to introduce
typical representations, which are then used in Section 9 to construct our
inertial Langlands correspondence. Finally, Section 10 provides an explicit
description of the set of typical representations.

Acknowledgements. This work was partially supported by an NSERC grant via the
University of Ottawa. I thank Monica Nevins for asking the questions which
eventually led to these results, and Maarten Solleveld for a number of helpful
comments on an earlier version of the paper.

##  2\. Notation

Let $F$ be a locally compact field, complete with respect to a non-archimedean
discrete valuation, and with finite residue field $\mathfrak{f}$ of odd
characteristic $p$. We fix, once and for all, a separable algebraic closure
$\bar{F}/F$.

Let $\mathbf{G}$ be a quasi-split connected reductive algebraic group defined
over $F$, and suppose that there exists a finite tamely ramified extension
$E/F$ such that $\mathbf{G}\times_{\mathrm{Spec}\ F}\mathrm{Spec}\ E$ is
split. Moreover, we assume that $p$ is coprime to the order of the Weyl group
of $\mathbf{G}\times_{\mathrm{Spec}\ F}\mathrm{Spec}\ E$333In particular, this
hypothesis ensures that all representations and $L$-parameters are
_essentially tame_ (see [Fin21])., as well as to the orders of the algebraic
fundamental groups of both $\mathbf{G}$ and its reductive dual. We set
$G=\mathbf{G}(F)$.

Our approach will be grounded in Bruhat–Tits theory. We write $\mathscr{B}(G)$
for the Bruhat–Tits building of $G$; it should be understood that this means
the _enlarged_ Bruhat–Tits building. On the rare occasion where we wish to
work with the reduced building $\mathscr{B}^{\mathrm{red}}(G)$, we will make
this clear. In particular, we denote the canonical projection
$\mathscr{B}(G)\rightarrow\mathscr{B}^{\mathrm{red}}(G)$ by $x\mapsto[x]$, and
always enclose points in square brackets when we wish to consider them in the
reduced building.

Given a point $x\in\mathscr{B}(G)$, we write $G_{x}$ for the stabilizer of
$G$, and $G_{x,0}$ for the parahoric subgroup contained in $G_{x}$ with finite
index. The Moy–Prasad filtration (see [MP94]) of $G_{x,0}$ will be denoted by
$G_{x,r}$, $r\geq 0$, with the convention that $G_{x,r+}$ denotes
$\bigcup_{s>r}G_{x,s}$. We write $G_{x,r:s}$ for the quotient
$G_{x,s}/G_{x,s}$ whenever $s>r$; this quotient is abelian if $r\geq s/2$.

We denote smooth induction and compact induction by $\mathrm{Ind}_{A}^{B}\ -$
and $\textrm{c-}\mathrm{Ind}_{A}^{B}\ -$, respectively. In the cases where
these functors coincide, we prefer to denote the functor by
$\mathrm{Ind}_{A}^{B}\ -$. Moreover, in the case where $B=G$ and $A=P$ is a
parabolic subgroup of $G$, then $\mathrm{Ind}_{P}^{G}\
-=\textrm{c-}\mathrm{Ind}_{P}^{G}\ -$; in this special case we commit a slight
abuse of notation and use $\mathrm{Ind}_{P}^{G}\ -$ to denote _normalized_
parabolic induction, i.e. we pre-compose with the functor
$-\otimes\delta_{P}^{-1/2}$, where $\delta_{P}$ is the modular function of
$P$.

##  3\. Bushnell–Kutzko types

In this section, we briefly recap the key definitions which are required in
order to work with Bushnell–Kutzko types.

Given an irreducible representation $\pi$ of $G$, one may always find a Levi
subgroup $M$ of $G$, together with a supercuspidal representation $\pi_{M}$ of
$M$, such that $\pi$ is isomorphic to a subquotient of $\mathrm{Ind}_{P}^{G}\
\pi_{M}$, for some parabolic subgroup $P$ of $G$ with Levi factor $M$. The
pair $(M,\pi_{M})$ is uniquely determined up to $G$-conjugacy; one therefore
obtains a _supercuspidal support map_ $\pi\mapsto\mathrm{sc}(\pi)$, which
associates to $\pi$ the $G$-conjugacy class of $(M,\pi_{M})$.

Given two pairs $(M,\rho)$ and $(M^{\prime},\rho^{\prime})$ consisting of
supercuspidal representations of Levi subgroups of $G$, we say that $(M,\rho)$
is _$G$ -inertially equivalent_ to $(M^{\prime},\rho^{\prime})$ if there
exists a $g\in G$ and an unramified character $\omega$ of $M^{\prime}$ such
that ${}^{g}M=M^{\prime}$ and ${}^{g}\rho\simeq\rho^{\prime}\otimes\omega$. We
write $[M,\rho]_{G}$ for the $G$-inertial equivalence class of $(M,\rho)$, and
$\mathfrak{B}(G)$ for the set of equivalence classes of such pairs. There is
an _inertial support_ map
$\mathfrak{I}:\mathrm{Irr}(G)\rightarrow\mathfrak{B}(G)$ which maps $\pi$ to
the inertial equivalence class of $\mathrm{sc}(\pi)$.

For each class $\mathfrak{s}\in\mathfrak{B}(G)$, consider the full subcategory
$\mathrm{Rep}^{\mathfrak{s}}(G)$ consisting of those representations every
irreducible subquotient of which has inertial support $\mathfrak{s}$.

######  Theorem 3.1 ([Ber84]).

Each category $\mathrm{Rep}^{\mathfrak{s}}(G)$ is indecomposable, and one has
a block decomposition

$\mathrm{Rep}(G)=\prod_{\mathfrak{s}\in\mathfrak{B}(G)}\mathrm{Rep}^{\mathfrak{s}}(G).$

######  Definition 3.2 ([BK98]).

Let $(J,\lambda)$ be a pair consisting of an irreducible representation
$\lambda$ of a compact open subgroup $J$ of $G$. Consider the full subcategory
$\mathrm{Rep}_{\lambda}(G)$ of $\mathrm{Rep}(G)$ consisting of those
representations which are generated by their $\lambda$-isotypic subspaces.

We say that $(J,\lambda)$ is an $\mathfrak{s}$_-type_ if
$\mathrm{Rep}_{\lambda}(G)=\mathrm{Rep}^{\mathfrak{s}}(G)$.

Next, we recall the notion of $G$-covers from [BK98], which will be crucial to
our approach.

Fix a Levi subgroup $M\subset G$ and a supercuspidal representation $\pi_{M}$
of $M$; write $\mathfrak{s}_{M}=[M,\pi_{M}]_{M}$, and suppose that
$(J_{M},\lambda_{M})$ is an $\mathfrak{s}_{M}$-type. Let
$\mathfrak{s}=[M,\pi_{M}]_{G}$.

######  Definition 3.3.

A _$G$ -cover_ of $(J_{M},\lambda_{M})$ is a pair $(J,\lambda)$ consisting of
an irreducible representation $\lambda$ of a compact open subgroup $J$ of $G$,
satisfying the following properties.

  1. (i)

$J\cap M=J_{M}$.

  2. (ii)

The restriction to $J_{M}$ of $\lambda$ is equal to $\lambda_{M}$.

  3. (iii)

For any parabolic subgroup $P$ of $G$ with Levi factor $M$, unipotent radical
$N$, if one write $N^{\mathrm{op}}$ for the opposite of $N$ then one has that

$J=(N\cap J)J_{M}(N^{\mathrm{op}}\cap J).$

  4. (iv)

Both $N\cap J$ and $N^{\mathrm{op}}\cap J$ are contained in $\ker\lambda$.

  5. (v)

For any smooth representation $\Pi$ of $G$, the natural map from $V$ to its
Jacquet module $\Pi_{N}$ induces an injection on the $\lambda$-isotypic
subspace of $\Pi$.

The key fact is the following.

######  Theorem 3.4 ([BK98]).

Suppose that $(J,\lambda)$ is a $G$-cover of the $\mathfrak{s}_{M}$-type
$(J_{M},\lambda_{M})$. Then $(J,\lambda)$ is an $\mathfrak{s}$-type.

##  4\. The Kim–Yu construction

In this section, we briefly recall the key points of the Kim–Yu construction
of types [Yu01, KY17]. We adhere to the notational conventions of [LN20, §5],
and refer to _ibid._ for many of the more technical details—in particular, we
do not specify how the construction of Heisenberg–Weil representations works,
and we largely ignore the subtlety of generic diagrams of embeddings of
buildings; whenever these issues become relevant, we will refer to the
corresponding passages in _ibid._

The reader who is only interested in reaching the statement of our inertial
correspondence in Section 9 will be able to get away without knowing the
precise details of the construction discussed during this section. However,
these become essential for the results of Section 10.

######  Definition 4.1.

A _datum_ is a tuple $\Sigma=(\vec{G},M^{0},u,\sigma,\vec{r},\vec{\phi})$,
consisting of the following objects.

  1. (i)

A sequence $\vec{G}=(G^{0}\subsetneq G^{1}\subsetneq\cdots\subsetneq G^{d}=G)$
of tame twisted Levi subgroups of $G$, i.e. subgroups $G^{i}$ of $G$ which are
of the form $G^{i}=\mathbf{G}^{i}(F)$, where $\mathbf{G}^{i}\subset\mathbf{G}$
is a connected reductive subgroup for which there exists a finite tamely
ramified extension $E/F$ such that $\mathbf{G}^{i}\times_{F}E$ is a Levi
subgroup of $\mathbf{G}\times_{F}E$.

  2. (ii)

A (genuine) Levi subgroup $M^{0}$ of $G^{0}$.

  3. (iii)

A vertex $u$ contained in $\mathscr{B}(M^{0})$.

  4. (iv)

An irreducible representation $\sigma$ of $G_{u}^{0}$ such that
$(G_{u}^{0},\sigma)$ is an $\mathfrak{s}^{0}$-type for some inertial
equivalence class $\mathfrak{s}^{0}\in\mathfrak{B}(G^{0})$ which consists of
depth-zero supercuspidal representations.

  5. (v)

A sequence $\vec{r}=(0<r_{0}<r_{1}<\cdots<r_{d-1}=r_{d})$ of real numbers. We
set $s_{i}=r_{i}/2$ for each $i$.

  6. (vi)

A sequence $\vec{\phi}=(\phi^{0},\phi^{1},\dots,\phi^{d})$ of characters
$\phi^{i}$ of $G^{i}$, where $\phi^{i}$ is $G^{i+1}$-generic of depth $r_{i}$,
in the sense of [HM08, 3.9]. Moreover, we take $\phi^{d}$ to be the trivial
character.

A _truncated datum_ is a tuple of the form
$(\vec{G},M^{0},u,\vec{r},\vec{\phi})$, i.e. a datum without any specified
representation $\sigma$ of $G_{x}^{0}$.

To illuminate the requirement of $\sigma$ set out in point (iv), note that
$G_{u}^{0}/G_{u,0+}^{0}$ identifies naturally with the points of a
disconnected algebraic group over the residue field of $F$, the connected
component of which is $G_{u}^{0}/G_{u,0+}^{0}$. The requirements of (iv) are
equivalent to asking that $\ker\sigma\supset G_{u,0+}^{0}$ and that, when one
views $\sigma$ as a representation of $G_{u}^{0}/G_{u,0+}^{0}$ and restricts
it to $G_{u,0:0+}^{0}$, it becomes isomorphic to a sum of pairwise
$G_{u}^{0}$-conjugate cuspidal representations of $G_{u,0:0+}^{0}$.

######  Remark 4.2.

As discussed at length in [KY17], in order for this definition to work out
entirely as desired, one must also specify a _generic diagram of embeddings of
buildings_. This is a highly technical notion which will merely play a
background role in results we cite. However, it is not difficult to see that
there exist _many_ such generic diagrams (in fact, almost all choices of
embeddings lead to such diagrams). Due to this, we omit it from our notation
and refer the reader to _ibid._ for a discussion; it is to be understood
throughout that we have fixed a generic diagram of embeddings.

To such a diagram one associates a great many groups and representations, as
detailed in [LN20, §5]. For the purposes of this paper, we require the
following three groups:

$\displaystyle J(\Sigma)$ $\displaystyle=G_{x}^{0}G_{x,s_{0}}^{1}\cdots
G_{x,s_{d-1}}^{d};$ $\displaystyle J_{+}(\Sigma)$
$\displaystyle=G_{x:0+}^{0}G_{x,s_{0}}^{1}\cdots
G_{x,s_{d-1}}^{d};\quad\text{and}$ $\displaystyle H_{+}(\Sigma)$
$\displaystyle=G_{x,0+}^{0}G_{x,s_{0}+}^{1}\cdots G_{x,s_{d-1}+}^{d}.$

Then one has a chain of inclusions $H_{+}(\Sigma)\subset J_{+}(\Sigma)\subset
J(\Sigma)$, and $H_{+}(\Sigma)$ and $J_{+}(\Sigma)$ are both pro-$p$ normal
subgroups of $J(\Sigma)$. Moreover, there is a canonical isomorphism
$J(\Sigma)/J_{+}(\Sigma)=G_{u}^{0}/G_{u,0+}^{0}$, i.e. we may identify
$\sigma$ with an irreducible representation of $J(\Sigma)$ whose kernel
contains $J_{+}(\Sigma)$.

After restricting each character $\phi^{i}$ to $G_{u,s_{i}+}^{i}$, one may
extend these restricted characters to obtain a sequence $\hat{\phi}^{i}$ of
characters, each of which is defined on $H_{+}(\Sigma)$; we set
$\theta_{\Sigma}=\bigotimes_{i=0}^{d}\ \hat{\phi}_{i}$. We refer to
$\theta_{\Sigma}$ as the _semisimple character_ associated to the datum
$\Sigma$.

For each character $\hat{\phi^{i}}$ there exists a canonical (and essentially
unique; see [HM08, §3.3]) irreducible representation $\kappa_{\Sigma}^{i}$ of
$J(\Sigma)$ which is isomorphic to a sum of copies of $\hat{\phi}^{i}$ upon
restriction to $H_{+}(\Sigma)$. The representation $\kappa_{\Sigma}^{i}$ is
called the _Heisenberg–Weil lift_ of $\hat{\phi}_{i}$. Moreover, if we define
$\kappa_{\Sigma}=\bigotimes_{i=0}^{d}\kappa_{\Sigma}^{i}$, then
$\kappa_{\Sigma}$ is an irreducible representation of $J(\Sigma)$, the
restriction to $H_{+}(\Sigma)$ of which is $\theta_{\Sigma}$-isotypic.

######  Definition 4.3.

We set $\lambda_{\Sigma}=\sigma\otimes\kappa_{\Sigma}$.

######  Theorem 4.4 ([KY17, Fin21]).

The representation $\lambda_{\Sigma}$ of $J(\Sigma)$ is irreducible, and there
exists an inertial equivalence class $\mathfrak{s}\in\mathfrak{B}(G)$ such
that $(J(\Sigma),\lambda_{\Sigma})$ is an $\mathfrak{s}$-type.

Moreover, under our hypothesis that $p$ is coprime to the order of the
absolute Weyl group of $\mathbf{G}$, every inertial equivalence class
$\mathfrak{s}\in\mathfrak{B}(G)$ admits a type of the form
$(J(\Sigma),\lambda_{\Sigma})$ for some datum $\Sigma$.

The construction of [KY17] actually produces covers, in the sense of the
previous section. Given a datum
$\Sigma=(\vec{G},M^{0},u,\sigma,\vec{r},\vec{\phi})$, one can write $M$ for
the $G$-centralizer of the centre of $M^{0}$; this is a Levi subgroup of $G$.
The restriction of $\lambda_{\Sigma}$ to $J(\Sigma)_{M}:=J(\Sigma)\cap M$ is
irreducible, and $(J(\Sigma),\lambda_{\Sigma})$ is a $G$-cover of
$(J(\Sigma)_{M},\lambda_{\Sigma})$.

For convenience, we will equip the set of all types of the form
$(J(\Sigma),\lambda_{\Sigma})$ with a very simple equivalence relation—we say
that $(J(\Sigma),\lambda_{\Sigma})$ and
$(J(\Sigma^{\prime}),\lambda_{\Sigma^{\prime}})$ are equivalent if they are
both types for the same inertial equivalence class, i.e. if one has that
$\mathrm{Rep}_{\lambda_{\Sigma}}(G)=\mathrm{Rep}_{\lambda_{\Sigma^{\prime}}}(G)$.
It is worth remarking that, via the results of [HM08], we know rather more
about what the “correct” equivalence relation to impose on the set of data is
, but the above equivalence relation is the natural one for our intended
application.

##  5\. $L$-parameters

We fix, once and for all, a separable algebraic closure $\bar{F}/F$, and write
$W_{F}\subset\mathrm{Gal}(\bar{F}/F)$ for the Weil group and $I_{F}\subset
W_{F}$ for the inertia subgroup. Write $\mathbf{G}^{\vee}$ for the complex
reductive dual of $\mathbf{G}$, and $G^{\vee}=\mathbf{G}^{\vee}(\mathbb{C})$;
the Langlands dual group is then the semidirect product ${}^{L}G=W_{F}\ltimes
G^{\vee}$, containing $G^{\vee}$ as its identity component. In particular,
since $G^{\vee}$ is normal in ${}^{L}G$, one obtains a natural action of
$W_{F}$ on $G^{\vee}$.

We will also require the _Weil–Deligne group_. This is the semidirect product
$W_{F}^{\prime}:=W_{F}\ltimes\mathbb{C}$, where $w\in W_{F}$ acts on
$x\in\mathbb{C}$ as $wxw^{-1}=\|w\|x$. Observe that we have a diagram of
inclusions:

$\textstyle{W_{F}^{\prime}}$$\textstyle{W_{F}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{I_{F}^{\prime}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{I_{F}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$

######  Definition 5.1.

An _$L$ -parameter_ for $G$ is an element of the continuous cohomology set
$\mathrm{H}^{1}(W_{F}^{\prime},G^{\vee})$.444More properly, such an element is
an _equivalence class_ of $L$-parameters; since everything we will consider
will be invariant under change of representative of the equivalence class,
this viewpoint is most convenient for our purposes.

Note that when we say _continuous_ cohomology, we mean the set of cocycles
which are smooth when restricted to $W_{F}$ and Zariski-continuous when
restricted to $\mathbb{C}$, modulo coboundaries.

Dual to the above diagram, we obtain a diagram of canonical restriction maps:

$\textstyle{\mathrm{H}^{1}(W_{F}^{\prime},G^{\vee})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathrm{H}^{1}(W_{F},G^{\vee})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathrm{H}^{1}(I_{F}^{\prime},G^{\vee})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathrm{H}^{1}(I_{F},G^{\vee})}$

Much of the content of this paper will be focused on imposing various
equivalence relations on the set of irreducible representations of $G$ via
this diagram, and understanding the relationships between these diagrams. In
particular, the local Langlands correspondence is expected to give a finite-
to-one parametrization
$\mathrm{rec}:\mathrm{Irr}(G)\rightarrow\mathrm{H}^{1}(W_{F}^{\prime},G^{\vee})$;
the finite fibres of this map give a partition of $\mathrm{Irr}(G)$ into _$L$
-packets_. One can alternatively consider the fibres of
$\varphi\circ\mathrm{rec}$, where $\varphi$ is any of the maps in the above
diagram. Many of our questions will be aimed towards giving a precise
description of the fibres of the resulting equivalence relation; our ultimate
goal is the construction of a set of objects which naturally parametrize the
fibres of $\mathrm{Res}_{I_{F}^{\prime}}^{W_{F}^{\prime}}\circ\mathrm{rec}$,
which is essentially achieved by chasing around the diagram in the opposite
direction.

The precise role of the action of $\mathbb{C}$ associated to an $L$-parameter
will play a particularly key role throughout the paper; we refer to this
action as the _monodromy_ of an $L$-parameter.

There are another family of useful maps on these cohomology sets which will be
of use to us. Given a closed subgroup $\mathbf{H}$ of $\mathbf{G}$, there is
often a dual embedding $\iota:H^{\vee}\hookrightarrow G^{\vee}$, which extends
to an embedding $\iota:{}^{L}H\hookrightarrow{}^{L}G$. Via this embedding, one
obtains a pushforward
$\iota_{*}:\mathrm{H}^{1}(\Gamma,H^{\vee})\rightarrow\mathrm{H}^{1}(\Gamma,G^{\vee})$
whenever $\Gamma$ is one of the above groups. While one requires some
hypotheses on $\mathbf{H}$ in order to guarantee that the dual embedding works
as desired, it is known that one can construct such a dual embedding when
$\mathbf{H}$ is either a torus or a Levi subgroup of $\mathbf{G}$; this will
be sufficient for our purposes. A description of the construction of these
dual embeddings is given in [Kal19, 5.1].

##  6\. Kaletha’s correspondence

In [Kal19], Kaletha directly constructs an extremely strong candidate for the
image under the local Langlands correspondence of the set of _regular
supercuspidal representations_ of $G$. While it is _not_ known that Kaletha’s
construction _is_ the local Langlands correspondence (due to the fact that it
is currently unclear exactly which properties should characterize the
correspondence for arbitrary groups), it is widely expected that this is the
case. For the remainder of the paper, we assume that this is so.

######  Definition 6.1.

Let $\pi$ be an irreducible representation of $G$ of inertial support
$\mathfrak{s}$, and let $(J(\Sigma),\lambda_{\Sigma})$ be an
$\mathfrak{s}$-type arising from a datum
$\Sigma=(\vec{G},M^{0},u,\sigma,\vec{r},\vec{\phi})$ as in Section 4. We say
that $\pi$ is _regular_ if the depth-zero component $\sigma$ is regular in the
following sense: the restriction of $\sigma$ to $G_{u,0}^{0}$ is a sum of
pairwise $G_{[u],0}^{0}$-conjugate representations; fix one such component
$\sigma_{0}$. Then $\sigma_{0}$ is inflated from a cuspidal irreducible
representation of $G_{u,0:0+}^{0}$, a finite group of Lie type; we say that
$\sigma$ is regular if $\sigma_{0}$ is a regular Deligne–Lusztig
representation, i.e. there exists a minisotropic maximal torus
$\mathsf{T}\subset G_{u,0:0+}^{0}$, and a character $\vartheta$ of
$\mathsf{T}$ in general position, such that
$\sigma_{0}=\pm\mathrm{R}_{\mathsf{T}}\theta$.

Kaletha shows that, when $\pi$ is a regular _supercuspidal_ representation of
$G$, the information in the datum $\Sigma$ may be encoded in a _regular
elliptic pair_ $(S,\Theta)$; this is a pair consisting of an elliptic torus
$S$, together with a character $\Theta$ of $S$ which satisfies a number of
properties. In doing so, Kaletha obtains a canonical bijection between the set
of regular supercuspidal representations of $G$ and the set of $G$-conjugacy
classes of regular elliptic pairs $(S,\Theta)$, which we denote by
$\pi_{S,\Theta}\leftrightarrow(S,\Theta)$. For the most part, we are able to
treat this construction as a black box, although we do require one basic
property: by following Kaletha’s construction of the bijection
$(S,\Theta)\rightarrow\pi_{S,\Theta}$, which proceeds by first associating a
datum $\Sigma_{S,\Theta}$ to $(S,\Theta)$, it is apparent that if $\chi$ is a
character of $G$ such that $(S,\Theta\otimes\chi)$ is also a regular elliptic
pair, then one has $\pi_{S,\Theta\otimes\chi}=\pi_{S,\Theta}\otimes\chi$. In
particular, this is always the case when $\chi$ is an unramified character of
$G$.

We write $\mathrm{Cusp}(G)$ for the set of isomorphism classes of regular
supercuspidal representations, i.e. the set of isomorphism classes of
representations of the form $\pi_{S,\Theta}$ for some regular elliptic pair
$(S,\Theta)$. More generally, we will make use of the following definition
throughout the remainder of the paper.

######  Definition 6.2.

An irreducible representation $\pi$ of $G$ is _regular_ if its supercuspidal
support $\mathrm{sc}(\pi)$ is a regular representation of some Levi subgroup
of $G$.

We write $\mathrm{Reg}(G)$ for the set of isomorphism classes of regular
representations of $G$.

######  Remark 6.3.

This is a slightly non-standard notion of regularity within the context of the
Langlands programme. More typically, there is a notion of a regular
$L$-parameter, and one says that $\pi$ is regular if the local Langlands
correspondence assigns to it a regular $L$-parameter. It is widely expected
that representations which are regular in our sense should also be regular in
this sense, and our notion of regularity is much more natural from the
$p$-adic group side of the correspondence, which is where much of our work
will take place.

The major result of [Kal19] is the construction of a local Langlands
correspondence for the regular supercuspidal representations.

######  Theorem 6.4 ([Kal19]).

There exists a natural map
$\mathrm{rec}_{G}:\mathrm{Cusp}(G)\rightarrow\mathrm{H}^{1}(W_{F},G^{\vee})$.

As discussed above, the remainder of the paper assumes that this map
$\mathrm{rec}_{G}$ _is_ the local Langlands correspondence.

######  Remark 6.5.

Notice that the image of this map is contained in
$\mathrm{H}^{1}(W_{F},G^{\vee})$, _not_ in
$\mathrm{H}^{1}(W_{F}^{\prime},G^{\vee})$—in other words, every regular
supercuspidal representation of $G$ corresponds to a parameter with a trivial
monodromy action.

As a consequence of this construction, one easily obtains an inertial
Langlands correspondence. We write $\mathrm{Type}(G)$ for the set of
equivalence classes of cuspidal Kim–Yu types which are contained in regular
supercuspidal representations, i.e. the set of pairs of the form
$(J(\Sigma),\lambda_{\Sigma})$ where $\Sigma$ is some datum with regular
Deligne–Lusztig depth-zero component. There is therefore a canonical
surjective map $\mathrm{Cusp}(G)\rightarrow\mathrm{Type}(G)$, the image of
which we denote by $\mathrm{Type}_{\mathrm{sc}}(G)$.

######  Corollary 6.6.

There exists a natural map
$\mathrm{iner}_{G}:\mathrm{Type}_{\mathrm{sc}}(G)\rightarrow\mathrm{H}^{1}(I_{F},G^{\vee})$
such that the following diagram commutes.

$\textstyle{\mathrm{Cusp}(G)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\mathrm{rec}_{G}}$$\textstyle{\mathrm{H}^{1}(W_{F},G^{\vee})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\mathrm{Res}_{I_{F}}^{W_{F}}}$$\textstyle{\mathrm{Type}_{\mathrm{sc}}(G)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\mathrm{iner}_{G}}$$\textstyle{\mathrm{H}^{1}(I_{F},G^{\vee})}$

###### Proof.

Given some $(J,\lambda)$ representing a class in $\mathrm{Type}(G)$, since
$(J,\lambda)$ is an $\mathfrak{s}$-type for some inertial equivalence class
$\mathfrak{s}=[G,\pi]_{G}$, any two irreducible representations of $G$ which
contain $\lambda$ are related by a twist by some unramified character of $G$.
In particular, the composition $\mathrm{Res}_{I_{F}}^{W_{F}}\mathrm{rec}(\pi)$
is independent of the choice of $\pi$. This gives the desired map
$\mathrm{iner}_{G}:\mathrm{Type}_{\mathrm{sc}}(G)\rightarrow\mathrm{H}^{1}(I_{F},G^{\vee})$.
∎

##  7\. Parabolic induction

Next, we explain how Kaletha’s construction is _expected_ to extend to a local
Langlands correspondence for $\mathrm{Reg}(G)$. We caution the reader that, in
the large majority of cases, this has not been proved and is entirely
conjectural.

######  Hypothesis 7.1.

Kaletha’s map
$\mathrm{rec}_{G}:\mathrm{Cusp}(G)\rightarrow\mathrm{H}^{1}(W_{F},G^{\vee})$
extends to a map
$\mathrm{rec}_{G}:\mathrm{Reg}(G)\rightarrow\mathrm{H}^{1}(W_{F}^{\prime},G^{\vee})$
such that, for any Levi subgroup $M$ of $G$, and any dual embedding
$\iota:{}^{L}M\hookrightarrow{}^{L}G$, if one denotes by $\mathrm{Reg}_{M}(G)$
the set of isomorphism classes of irreducible representations with
supercuspidal support defined on $M$, then the following diagram is
commutative.

$\textstyle{\mathrm{Reg}_{M}(G)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\mathrm{sc}}$$\scriptstyle{\mathrm{rec}_{G}}$$\textstyle{\mathrm{H}^{1}(W_{F}^{\prime},G^{\vee})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\mathrm{Res}_{W_{F}}^{W_{F}^{\prime}}}$$\textstyle{\mathrm{Cusp}(M)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\mathrm{rec}_{M}}$$\textstyle{\mathrm{H}^{1}(W_{F},M^{\vee})}$

In particular, note that from this implies that if $\pi$ has supercuspidal
support $(M,\rho)$, then $\mathrm{Res}_{W_{F}}^{W_{F}^{\prime}}\
\mathrm{rec}_{G}(\pi)\in\iota_{*}\mathrm{H}^{1}(W_{F},M^{\vee})$ .

######  Proposition 7.2 (The weak inertial Langlands correspondence).

There exists a map
$\overline{\mathrm{iner}}_{G}:\mathrm{Type}(G)\rightarrow\mathrm{H}^{1}(I_{F},G^{\vee})$
which makes the following diagram commute.

$\textstyle{\mathrm{Reg}(G)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\mathrm{rec}_{G}}$$\scriptstyle{\mathrm{sc}}$$\textstyle{\mathrm{H}^{1}(W_{F}^{\prime},G^{\vee})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\mathrm{Res}_{I_{F}}^{W_{F}^{\prime}}}$$\scriptstyle{\mathrm{Res}_{W_{F}}^{W_{F}^{\prime}}}$$\textstyle{\coprod_{M\subset
G}\mathrm{Cusp}(M)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\iota_{*}\mathrm{rec}_{M}}$$\textstyle{\mathrm{H}^{1}(W_{F},G^{\vee})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\mathrm{Res}_{I_{F}}^{W_{F}}}$$\textstyle{\coprod_{M\subset
G}\mathrm{Type}_{\mathrm{sc}}(M)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\mathrm{iner}_{M}}$$\textstyle{\mathrm{H}^{1}(I_{F},G^{\vee})}$$\textstyle{\mathrm{Type}(G)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\overline{\mathrm{iner}}_{G}}$$\textstyle{\mathrm{H}^{1}(I_{F},G^{\vee})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$

Here, the map $\mathrm{Type}(G)\rightarrow\coprod\mathrm{Type}(M)$ is the
canonical one obtained by recognizing that each $(J,\lambda)$ is a $G$-cover
of $(J_{M},\lambda_{M})$ for some $M$; one can therefore map $(J,\lambda)$ to
$(J\cap M,\lambda)$.

###### Proof.

This follows easily from the supercuspidal case, given the properties listed
in Hypothesis 7.1. ∎

######  Remark 7.3.

We refer to $\overline{\mathrm{iner}}_{G}$ as the _weak_ inertial Langlands
correspondence, since it is not capable of distinguishing between
representations with equivalent supercuspidal supports; on the Galois side of
the correspondence, this is the same as noting that its image consists of
classes with trivial monodromy. Our goal for the remainder of the paper will
be to remedy this, realizing $\overline{\mathrm{iner}}_{G}$ as a quotient of a
“strong” inertial Langlands correspondence
$X\rightarrow\mathrm{H}^{1}(I_{F}^{\prime},G^{\vee})$, where $X$ is a set of
objects which is closely related to $\mathrm{Type}(G)$.

##  8\. Typical representations

In order to construct our strong form of the inertial Langlands
correspondence, we will need to work with a slightly more general class of
objects than $\mathfrak{s}$-types.

######  Definition 8.1.

Let $(J,\lambda)$ be a pair consisting of an irreducible representation
$\lambda$ of a compact open subgroup $J$ of $G$. We say that $(J,\lambda)$ is
$\mathfrak{s}$-typical if, whenever $\pi\in\mathrm{Irr}(G)$ contains
$\lambda$, one has $\pi\in\mathrm{Rep}^{\mathfrak{s}}(G)$.

######  Remark 8.2.

In other words, this is close to asking that $\mathrm{Rep}_{\lambda}(G)$ is a
subcategory of $\mathrm{Rep}^{\mathfrak{s}}(G)$ (as opposed to being equal to
$\mathrm{Rep}^{\mathfrak{s}}(G)$, as would be the case if $(J,\lambda)$ were
an $\mathfrak{s}$-type). However, showing that this is equivalent to the above
definition encounters some difficulties when one considers inadmissible
representations of $G$, since if $(J,\lambda)$ is $\mathfrak{s}$-typical but
not an $\mathfrak{s}$-type, then $\mathrm{Rep}_{\lambda}(G)$ is not a Serre
subcategory of $\mathrm{Rep}(G)$—in this case $\mathrm{Rep}_{\lambda}(G)$ is
not closed under formation of subquotients.

As long as one has a construction of types, it is easy to see that many
typical representations exist.

######  Lemma 8.3.

Suppose that $(J,\lambda)$ is an $\mathfrak{s}$-type and that $K\supset J$ is
a maximal compact subgroup of $G$.

  1. (i)

For any irreducible component $\tau$ of $\mathrm{Ind}_{J}^{K}\ \lambda$, the
pair $(K,\tau)$ is $\mathfrak{s}$-typical.

  2. (ii)

If $\pi\in\mathrm{Irr}(G)$ contains $\lambda$, then there exists a pair
$(K,\tau)$ as above such that $\pi|K$ contains $\tau$.

###### Proof.

Each of these is a straightforward consequence of Frobenius reciprocity. ∎

In particular, one obtains a collection of typical representations which are
of the form $(K,\tau)$ where $K\subset G$ is a maximal compact subgroup and
$\tau|_{J(\Sigma)}$ contains $\lambda_{\Sigma}$ for some datum $\Sigma$ such
that $J(\Sigma)\subset K$. We denote the set of $G$-conjugacy classes of such
typical representations by $\mathrm{Typ}(G)$. Notice that there is a canonical
map $\mathrm{Typ}(G)\rightarrow\mathrm{Type}(G)$, given by mapping a typical
representation to its underlying equivalence class of types.

While there is not a canonical map
$\mathrm{Irr}(G)\rightarrow\mathrm{Typ}(G)$, there _is_ a canonical map from
$\mathrm{Irr}(G)$ to the power set $\mathcal{P}\mathrm{Typ}(G)$, namely the
map which sends $\pi$ to the set $X_{\pi}$ of $G$-conjugacy classes of
$\mathfrak{s}$-typical representations $(K,\tau)$ such that
$\mathrm{Hom}_{K}(\tau,\pi)\neq 0$, as $K$ ranges over a set of
representatives of the conjugacy classes of maximal compact subgroups of $G$.
Moreover, the canonical map $\mathrm{Typ}(G)\rightarrow\mathrm{Type}(G)$
clearly extends to a partial map
$\mathcal{P}\mathrm{Typ}(G)\rightarrow\mathrm{Type}(G)$, defined on the subset
of $\mathcal{P}\mathrm{Typ}(G)$ consisting of sets of the form $X_{\pi}$.

##  9\. The inertial Langlands correspondence

With the above discussion in mind, we write $\varphi$ for the canonical map
$\mathrm{Reg}(G)\rightarrow\mathcal{P}\mathrm{Typ}(G)$, defined by
$\varphi(\pi)=X_{\pi}$. We write $\mathrm{Typ}_{\varphi}(G)$ for the image of
$\varphi$.

The key to constructing a “strong” form of the inertial Langlands
correspondence is the following result.

######  Theorem 9.1.

Suppose that $\pi_{1},\pi_{2}\in\mathrm{Reg}(G)$ are such that
$\mathrm{sc}(\pi_{1})=\mathrm{sc}(\pi_{2})$ and
$\varphi(\pi_{1})=\varphi(\pi_{2})$. Then $\pi_{1}\simeq\pi_{2}$.

###### Proof.

Let $(M,\rho)$ denote a representative of
$\mathrm{sc}(\pi_{1})=\mathrm{sc}(\pi_{2})$, and choose a parabolic subgroup
$P$ of $G$ with Levi factor $M$, so that both $\pi_{1}$ and $\pi_{2}$ occur as
irreducible subquotients of the representation $\mathrm{Ind}_{P}^{G}\ \rho$.
It is clear that $\pi_{1}\simeq\pi_{2}$ when $M=G$, so we may assume that $M$
is a proper Levi subgroup of $G$.

Fix a datum $\Sigma=(\vec{G},M^{0},u,\sigma,\vec{r},\vec{\phi})$ such that
$(J(\Sigma),\lambda_{\Sigma})$ is an $[M,\rho]_{M}$-type; then the fact that
$M\neq G$ implies that $M^{0}\neq G^{0}$, and so $u$ is contained in the
interior of a facet of positive dimension in $\mathscr{B}(G^{0})$, and hence
also in the interior of some positive-dimensional facet
$\mathscr{F}\subset\mathscr{B}(G)$. By [MP96, 6.3], we may associate to
$\mathscr{F}$ a Levi subgroup $M_{\mathscr{F}}$ of $G$ lying in the $G$-orbit
of $M$; without loss of generality we may take $M=M_{\mathscr{F}}$. Fix an
apartment $\mathscr{A}\subset\mathscr{B}(G)$ containing $\mathscr{F}$ ,
corresponding to a maximal split torus $S\subset G$.

Let $y_{1},\dots,y_{n}$ denote the vertices of $\mathscr{F}$, so that one has
$J(\Sigma)\subset G_{y_{i}}$ for each $i$. Since $M$ contains $U_{\psi}$
whenever $\psi\in\Psi(G,S)$ is an affine root which takes constant value on
$\mathscr{F}$, one easily sees that $G_{y_{i}}P=G_{y_{j}}P$ for all $i,j$,
from which it follows that $G_{y_{i}}P=G_{\mathscr{F}}P$ for all $i$.

For each index $i$, let $V_{i}$ denote the maximal multiplicity-free
$\lambda_{\Sigma}$-isotypic subrepresentation of $\pi_{1}|_{G_{y_{i}}}$. Then
by assumption we have injective $G_{y_{i}}$-equivariant maps
$V_{i}\rightarrow\pi_{2}$. We may therefore define subrepresentations $W_{i}$
of $\mathrm{Ind}_{P}^{G}\ \rho$ generated by $V_{i}$ for each $i$ and observe
that both $\pi_{1}$ and $\pi_{2}$ are contained in $W:=\bigcap_{i}W_{i}$. We
claim that this representation $W$ must be irreducible.

Since the type $(J(\Sigma),\lambda_{\Sigma})$ is a $G$-cover of the
$[M,\rho]_{M}$-type $(J(\Sigma)\cap M,\lambda_{\Sigma})$, there is an Iwahori
decomposition $J(\Sigma)=(J\cap\bar{N})(J\cap P)$, where $\bar{N}$ denotes the
unipotent radical of the parabolic subgroup $\bar{P}$ such that $\bar{P}\cap
P=M$. Furthermore, the representation $\mathrm{Ind}_{P}^{G}\ \rho$, being
contained in the Bernstein block $\mathrm{Rep}^{[M_{\rho}]_{G}}(G)$, is
generated by its $\lambda_{\Sigma}$-isotypic vectors. We may therefore extend
$\rho$ to a representation of the group generated by $J(\Sigma)P$ by declaring
that $J(\Sigma)\cap\bar{N}$ acts trivially. As $J(\Sigma)\subset G_{y_{i}}$
for each $i$ and each representation $V_{i}$ is $\lambda_{\Sigma}$-isotypic,
we find that $W$ is contained in the representation $\mathrm{Ind}_{\langle
J(\Sigma)P\rangle}^{G}\ \rho$, which canonically occurs as a subrepresentation
of $\mathrm{Ind}_{P}^{G}\ \rho$.

Again using the fact that $J(\Sigma)$ is a $G$-cover of $(J(\Sigma)\cap
M,\lambda_{\Sigma})$, as $M$ is the Levi subgroup associated to $\mathscr{F}$
it follows from the fact that $\mathrm{Ind}_{J(\Sigma)\cap
M}^{M_{\mathscr{F}}}\ \lambda_{\Sigma}$ is irreducible that
$\mathrm{Ind}_{J(\Sigma)}^{G_{\mathscr{F}}}\ \lambda_{\Sigma}$ is irreducible,
and hence so is $\tilde{\rho}:=\mathrm{Ind}_{\langle
J(\Sigma)P\rangle}^{\langle G_{\mathscr{F}}P\rangle}\ \rho$. In particular, we
find that $W$ is contained in $\mathrm{Ind}_{\langle
G_{\mathscr{F}}P\rangle}^{G}\ \tilde{\rho}$, and that this representation is
identical to $\mathrm{Ind}_{\langle G_{y_{i}}P\rangle}^{G}\ \tilde{\rho}$. In
particular, applying Frobenius reciprocity we conclude that the
representations $\pi_{1}$ and $\pi_{2}$ of $G$ must intertwine when viewed as
representations of $\langle G_{y_{i}}P\rangle$. It remains for us to show that
$\langle G_{y_{i}}P\rangle=G$.

Fix a maximal split torus $S\subset G$ which contains the centre of $M$, and
which is such that the corresponding apartment
$\mathscr{A}\subset\mathscr{B}(G)$ contains the facet $\mathscr{F}$, and hence
each of the points $y_{i}$. Let $W_{S}^{\mathrm{aff}}=N_{G}(S)/S_{\mathrm{b}}$
denote the affine Weyl group of $S$; here, $S_{\mathrm{b}}\subset S$ denotes
the maximal bounded subgroup. For each point $v\in\mathscr{A}$, we may define
a subgroup $W_{v}$ of $W_{S}^{\mathrm{aff}}$ generated by the reflections
$s_{\psi}$ for those affine roots $\psi\in\Psi(S)$ such that $v$ lies on the
hyperplane $H_{\psi}=\\{v^{\prime}\in\mathscr{A}\ |\ \psi(v^{\prime})=0\\}$.
Each group $W_{v}$ is a finite group which, by pinning the root system
$\Phi(G,S)$ at a special vertex of $\mathscr{C}$, may be identified with a
finite subgroup of the Weyl group $W_{S}=N_{G}(S)/S$. Clearly $W_{v}$ fixes
the simplicial closure of $v$ pointwise, and so $W_{v}\subset G_{v}$.

Let $H=\langle G_{z}\ |\ z\in\mathscr{F}\rangle=\langle G_{y_{i}}\ |\
i\rangle$. Fix a chamber $\mathscr{C}$ in $\mathscr{A}$ containing
$\mathscr{F}$. The chamber $\mathscr{C}$ is a polysimplex, and therefore
admits a decomposition
$\mathscr{C}=\mathscr{C}_{1}\times\cdots\times\mathscr{C}_{k}$ as a product of
simplices. Since $M$ is a proper Levi subgroup of $G$, the facet $\mathscr{F}$
contains at least two vertices of $\mathscr{C}$, and so in the case that $k=1$
we find that every wall of $\mathscr{C}$ must contain a vertex of
$\mathscr{F}$. Then $H$ contains $s_{\psi}$ whenever $H_{\psi}$ is a wall of
$\mathscr{C}$. From this, a straightforward inductive argument shows that the
same conclusion holds true regardless of the value of $k$.

So $H$ contains $G_{y_{i}}wG_{y_{i}}$ whenever $y_{i}$ is a vertex of
$\mathscr{C}$ and $w\in W_{S}^{\mathrm{aff}}$, hence it also contains
$\bigcup_{w\in W_{S}^{\mathrm{aff}}}G_{\mathscr{C}}wG_{\mathscr{C}}$; by the
Bruhat–Tits decomposition this union contains every maximal compact subgroup
of $G$. In particular, it contains $G_{z}$ for some special vertex
$z\in\mathscr{C}$. From this, we see that $\langle G_{y_{i}}P\rangle=\langle
G_{z}P\rangle=G_{z}P=G$, by the Iwasawa decomposition. It follows that both
$\pi_{1}$ and $\pi_{2}$ both admit injective $G$-equivariant maps into the
irreducible representation $\mathrm{Ind}_{G_{\mathscr{F}}P}^{G}\ \rho$, and
are therefore isomorphic, as desired. ∎

With this, we come to our main result.

######  Theorem 9.2 (The inertial Langlands correspondence).

There exists a map
$\mathrm{iner}_{G}:\mathrm{Typ}_{\varphi}(G)\rightarrow\mathrm{H}^{1}(I_{F}^{\prime},G^{\vee})$
such that the following diagram commutes.

$\textstyle{\mathrm{Reg}(G)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\mathrm{rec}_{G}}$$\scriptstyle{\varphi}$$\textstyle{\mathrm{H}^{1}(W_{F}^{\prime},G^{\vee})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\mathrm{Res}_{I_{F}^{\prime}}^{W_{F}^{\prime}}}$$\textstyle{\mathrm{Typ}_{\varphi}(G)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\mathrm{iner}_{G}}$$\textstyle{\mathrm{H}^{1}(I_{F}^{\prime},G^{\vee})}$

Moreover, the map $\mathrm{iner}_{G}$ is compatible with
$\overline{\mathrm{iner}}_{G}$, in the sense that one has a commutative
diagram

$\textstyle{\mathrm{Typ}_{\varphi}(G)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\mathrm{iner}_{G}}$$\textstyle{\mathrm{H}^{1}(I_{F}^{\prime},G^{\vee})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\mathrm{Res}_{I_{F}}^{I_{F}^{\prime}}}$$\textstyle{\mathrm{Type}(G)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\overline{\mathrm{iner}}_{G}}$$\textstyle{\mathrm{H}^{1}(I_{F},G^{\vee})}$

###### Proof.

Given $X\in\mathrm{Typ}_{\varphi}(G)$, choose a representation
$\pi\in\mathrm{Reg}(G)$ such that $\varphi(\pi)=X$. Then we set
$\mathrm{iner}_{G}(X)=\mathrm{Res}_{I_{F}^{\prime}}^{W_{F}^{\prime}}\mathrm{rec}_{G}(\pi)$.

Suppose that $\pi^{\prime}\in\mathrm{Reg}(G)$ also satisfies
$\varphi(\pi^{\prime})=X$. We claim that
$\mathrm{Res}_{I_{F}^{\prime}}^{W_{F}^{\prime}}\mathrm{rec}_{G}(\pi^{\prime})=\mathrm{Res}_{I_{F}^{\prime}}^{W_{F}^{\prime}}\mathrm{rec}_{G}(\pi)$.
Since $\varphi(\pi)=\varphi(\pi^{\prime})$, projecting along the canonical map
$\mathrm{Typ}_{\varphi}(G)\rightarrow\mathrm{Type}(G)$ shows that $\pi$ and
$\pi^{\prime}$ are inertially equivalent. In other words, if $(M,\rho)$ is a
representative of $\mathrm{sc}(\pi)$, then there exists an unramified
character $\omega$ of $M$ such that $(M,\rho\otimes\omega)$ is a
representative of $\mathrm{sc}(\pi^{\prime})$. Since
$\pi,\pi^{\prime}\in\mathrm{Reg}(G)$, by definition we may find regular
elliptic pairs $(S,\Theta)$ and $(S^{\prime},\Theta^{\prime})$ in $M$ such
that $\rho=\pi_{S,\Theta}$ and
$\rho^{\prime}=\pi_{S^{\prime},\Theta^{\prime}}$; since
$\rho^{\prime}=\rho\otimes\omega=\pi_{S,\Theta}\otimes\omega=\pi_{S,\Theta\omega}$,
it follows that we may without loss of generality take $S=S^{\prime}$ and
$\Theta^{\prime}=\Theta\omega$.

Now consider the $L$-parameter $\mathrm{rec}_{G}(\pi^{\prime})$. By
construction, the semisimplification of this is precisely the pushforward
$\iota_{*}\mathrm{rec}_{M}(\rho^{\prime})=\iota_{*}\mathrm{rec}_{M}(\rho\otimes\omega)$
along an appropriate embedding $\iota:{}^{L}M\hookrightarrow{}^{L}G$. Since
both $\rho$ and $\rho^{\prime}=\rho\otimes\omega$ are constructed from regular
elliptic pairs defined on the same torus $S\subset M$, one finds that both
$\iota_{*}\mathrm{rec}_{M}(\rho)$ and
$\iota_{*}\mathrm{rec}_{M}(\rho^{\prime})$ have image in
${}^{L}S\hookrightarrow{}^{L}M\hookrightarrow{}^{L}G$.

Given an element $a$ of the image of $\mathrm{rec}_{G}(\pi^{\prime})$, one may
take the Jordan decomposition $a=su$ with $s$ semisimple and $u$ unipotent, so
that $s$ is contained in the image of
$\iota_{*}\mathrm{rec}_{M}(\rho^{\prime})$, which is an abelian group. If
$a=\mathrm{rec}_{G}(\pi^{\prime})(w)$, then $a$ must commute with the
semisimple element $b=\iota_{*}\mathrm{rec}_{T}(\omega)(w)$, and one finds
that $(sb)u=(bs)u=u(bs)=u(sb)$ is a Jordan decomposition of $ab$. In
particular, this implies that $\mathrm{red}_{T}(\omega)$ extends to a
1-cocycle $c_{\omega}:W_{F}^{\prime}\rightarrow G^{\vee}$ which is trivial on
both $I_{F}$ (since $\omega$ is unramified) and $\mathbb{C}$. Given a
1-cocycle $f$ representing the class $\mathrm{rec}_{G}(\pi)$, the pointwise
product $fc_{\omega}$ is a 1-cocycle whose cohomology class is independent of
the choice of $f$; in this way we obtain a class $\mathrm{rec}_{G}(\pi)\cdot
c_{\omega}\in\mathrm{H}^{1}(W_{F}^{\prime},G^{\vee})$; by the above
observations on the Jordan decomposition, we see that one must have
$\mathrm{rec}_{G}(\pi)\cdot c_{\omega}=\mathrm{rec}_{G}(\pi^{\prime})$.

We may therefore assume without loss of generality that
$\mathrm{sc}(\pi)=\mathrm{sc}(\pi^{\prime})$, in which case the assumption
that $\varphi(\pi)=\varphi(\pi^{\prime})$ allows us to apply Theorem 9.1 to
deduce that $\pi\simeq\pi^{\prime}$. As argued above, changing the
supercuspidal support of $\pi^{\prime}$ by an unramified twist does not affect
$\mathrm{Res}_{I_{F}^{\prime}}^{W_{F}^{\prime}}\mathrm{rec}_{G}(\pi^{\prime})$,
and so we conclude that
$\mathrm{Res}_{I_{F}^{\prime}}^{W_{F}^{\prime}}\mathrm{rec}_{G}(\pi)$ is
independent of the choice of $\pi$. In particular, this gives us our
definition of
$\mathrm{iner}_{G}:\mathrm{Typ}_{\varphi}(G)\rightarrow\mathrm{H}^{1}(I_{F}^{\prime},G^{\vee})$,
from which the commutativity of the first diagram is immediate.

The commutativity of the second diagram follows easily from Hypothesis 7.1,
along with the fact that the canonical map
$\mathrm{Typ}_{\varphi}(G)\rightarrow\mathrm{Type}(G)$ simply maps each
typical representation to its underlying type. ∎

######  Remarks 9.3.

One should note that, due to the hypotheses on $p$ which are in force
throughout the paper, we only claim to establish the existence of this
inertial correspondence for the so-called _essentially tame representations_
(which should be those corresponding to $L$-parameters for which the image of
the wild inertia subgroup is abelian). It is worth pointing out the precise
extent to which these hypotheses are required for the above proof to work.
Most significantly, they guarantee the existence of $\mathfrak{s}$-types for
each $\mathfrak{s}$; however, the existence of $\mathfrak{s}$-types is known
even for wildly ramified representations in a number of cases, such as for
general or special linear groups [BK93, BK94], inner forms of general linear
groups [SS08], and classical groups when $p\neq 2$ [Ste08]. Once one has a
construction of types, the major thing that one needs to know in order to
follow the template of the above proofs is that the type is associated to a
point in the building, and that the Levi subgroup associated to this point
should be equal to the Levi subgroup defining the corresponding inertial
equivalence class; this is the case in the known constructions. The other
point at which we make use of our hypotheses is when defining regular
representations as representations with regular supercuspidal support in the
sense of [Kal19]. Such an approach cannot be taken in the non-tame case,
meaning that a more complicated approach to regularity must be taken; it seems
unlikely that this would be the source of any major technical issues, however.
In short, this appears to be an approach which should go through _mutatis
mutandis_ in any situation where one has a satisfactory construction of
$\mathfrak{s}$-types for each inertial equivalence class $\mathfrak{s}$. At
the time of writing, however, the only such available constructions which are
uniform in appearance as the group $G$ varies are those leading to tame
representations.

######  Remarks 9.4.

As mentioned during the introduction, this strong form of the inertial
Langlands correspondence has been something of a problematic topic within the
literature, with varying statements and a rather complex trail of references
to be followed if one wishes to see a proof (which, ultimately, boils down to
a lengthy but largely straightforward exercise in the Bernstein–Zelevinsky
classification [BZ76]). The case of $\mathbf{GL}_{n}(G)$ is essentially
treated as [SZ99, Prop. 6.2], although this is presented in a somewhat
different language; [BC09, Prop. 6.5.3] describes how one may translate it
into a statement closer to the viewpoint taken in this paper, and such an
approach is also discussed in [EG14, §4.1].

However, while it appears to be a common expectation that such results should
generalize beyond $\mathbf{GL}_{n}(F)$, to the author’s knowledge the only
other situation in which this has been fully worked out is that of
$\mathbf{SL}_{n}(F)$ in an unpublished note by Will Conley. At best, there is
a widely believed folklore conjecture which predicts that there should be a
canonical map
$\mathrm{Typ}(G)\rightarrow\mathrm{H}^{1}(I_{F}^{\prime},G^{\vee})$ (or, more
likely, a map defined on some well-chosen subset of $\mathrm{Typ}(G)$). Such a
result would be a considerable refinement of the results of this section, but
this appears to be considerably beyond the reach of our current knowledge. In
particular, any construction of such a map appears to have several serious
pre-requisites:

  1. (i)

one must have a detailed understanding of how parabolically induced
representations decompose into irreducible components;

  2. (ii)

one must have a detailed understanding of the set of typical representations;
and

  3. (iii)

one must know that there exists a very precise correspondence between the set
of typical representations and the set of representations with a given
inertial support.

The first of these, in particular, is a serious obstacle, with very little
progress having been made beyond the Bernstein–Zelevinsky classification,
apart from for certain families of representations of classical groups.

Our approach largely consists of the observation that, by being willing to
work with sets of typical representations rather than singletons, it becomes
much easier to show that the map $\mathrm{iner}_{G}$ is well-defined, with
this problem reducing to a clear problem about the representation theory of
$G$, which we resolve as Theorem 9.1.

Nonetheless, one would hope that, at least in a great many cases, it is
actually possible to refine the map
$\mathrm{Typ}_{\varphi}(G)\rightarrow\mathrm{H}^{1}(I_{F}^{\prime},G^{\vee})$
to a (partially defined) map
$\mathrm{Typ}(G)\rightarrow\mathrm{H}^{1}(I_{F}^{\prime},G^{\vee})$ and work
with typical representations one-by-one. It would be extremely interesting to
investigate this in some explicit situations, such as for classical groups
where many parabolically induced representations have known decompositions.

##  10\. An explicit construction of typical representations

With the results of the previous section, we have a satisfactory construction
of an inertial Langlands correspondence. However, if one wishes to apply this
correspondence, then one must understand precisely the set $\mathrm{Typ}(G)$
of typical representations. In this final section, we provide a geometric
construction of these typical representations, using Bruhat–Tits theory.

We fix some notation for the remainder of the paper. Let
$\mathfrak{s}\in\mathfrak{B}(G)$ be an inertial equivalence class consisting
of regular representations, and fix a datum
$\Sigma=(\vec{G},M^{0},u,\sigma,\vec{r},\vec{\phi})$ such that
$(J(\Sigma),\lambda_{\Sigma})$ is an $\mathfrak{s}$-type. If $K$ is a maximal
compact subgroup of $G$ such that $J(\Sigma)\subset K$, then the
$\mathfrak{s}$-typical representations of $K$ are (by definition) precisely
the irreducible components of $\mathrm{Ind}_{J(\Sigma)}^{K}\
\lambda_{\Sigma}$. We will therefore describe each of these components. We
make frequent use of results established during [LN20, §7], which will provide
the technical tools necessary to carry out our construction.

From now on, let $x\in\mathscr{B}(G)$ be a point contained in the simplicial
closure of $u$, i.e. such that $G_{u}\subset G_{x}$. To state the next lemma,
we recall the notion of a _generalized $s_{i}$-facet_ from [DeB02]. We say
that two points $y,z\in\mathscr{B}(G)$ lie in the same generalized
$s_{i}$-facet of $\mathscr{B}(G)$ if one has that $G_{y,s_{i}}=G_{z,s_{i}}$
and $G_{y,s_{i}+}=G_{z,s_{i}+}$. Note that, by replacing $u$ with a point with
the same simplicial closure if necessary, it is always possible to arrange
things so that $x$ lies in the same $s_{i}$-facet as $u$.

######  Lemma 10.1.

Suppose that $x$ is contained in the same $s_{i}$-facet as $u$ for each $i$.
Write $M^{x,0}$ for the Levi subgroup of $G^{0}$ associated to the point $x$
via [MP94, 6.3–4]. Then $\Sigma_{x}:=(\vec{G},M^{x,0},x,\vec{r},\vec{\phi})$
is a truncated datum.

###### Proof.

The subtle point underlying this fact is that one may still construct a
suitable generic diagram of embeddings; this is addressed in detail during
[LN20, 7.3], where our point $u$ plays the same role as the point denoted by
$u_{\xi}$ in _loc. cit._ Our result is slightly simpler than that of _loc.
cit._ , in that we wish only to construct a truncated datum, meaning that the
details underlying the construction of $\zeta_{\xi}$ are irrelevant for our
purposes. Otherwise, the proof goes through _mutatis mutandis_. ∎

Since $u$ is contained in $\mathscr{B}(M^{x,0})$, the group
$M_{u,0:0+}^{x,0}=M_{u,0:0+}^{0}$ identifies with a Levi subgroup of
$M_{x,0:0+}^{x,0}$. Moreover, $M_{u,0}^{x,0}/M_{x,0+}^{x,0}$ is a parabolic
subgroup of $M_{x,0:0+}^{x,0}$ with $M_{u,0:0+}^{x,0}$ as its Levi factor. In
particular, we may consider the parabolically induced representation
$V_{x,\sigma}:=\mathrm{Ind}_{M_{u,0}^{x,0}/M_{x,0+}^{x,0}}^{M_{x,0:0+}^{x,0}}\
\sigma$. Let $\zeta$ be an irreducible component of $V_{x,\sigma}$.

Then we may consider the tuple
$\Sigma_{x,\zeta}:=(\vec{G},M^{0},x,\zeta,\vec{r},\vec{\phi})$. Note that,
while $\Sigma_{x}$ is a truncated datum, $\Sigma_{x,\zeta}$ is _not_ a datum
unless $x$ is contained in the interior of the same facet as $u$. Nonetheless,
we abuse notation slightly and associate groups, semisimple characters and
Heisenberg–Weil representations to $\Sigma_{x,\zeta}$ via $\Sigma_{x}$. That
is, we set $J(\Sigma_{x,\zeta})=J(\Sigma_{x})$,
$H_{+}(\Sigma_{x,\zeta})=H_{+}(\Sigma_{x})$,
$\theta_{\Sigma_{x,\zeta}}=\theta_{\Sigma_{x}}$, and
$\kappa_{\Sigma_{x,\zeta}}=\kappa_{\Sigma_{x}}$. We also set
$\lambda_{\Sigma_{x,\zeta}}=\zeta\otimes\kappa_{\Sigma_{x,\zeta}}$. The pair
$(J(\Sigma_{x,\zeta}),\lambda_{\Sigma_{x,\zeta}})$ is always
$\mathfrak{s}$-typical, but it need not be an $\mathfrak{s}$-type.

######  Lemma 10.2.

One has $J(\Sigma_{x,\zeta})\supset J(\Sigma)$ and
$H_{+}(\Sigma_{x,\zeta})\subset H_{+}(\Sigma)$. Moreover,
$\theta|_{H_{+}(\Sigma_{x,\zeta})}=\theta_{\Sigma_{x,\zeta}}$, and the
restriction to $J(\Sigma)$ of $\kappa_{\Sigma_{x,\zeta}}$ is
$\kappa_{\Sigma}$-isotypic.

###### Proof.

These claims are established during the course of the proof of [LN20, 7.5]; as
before, our point plays the role of the point denoted by $u_{\xi}$ in _loc.
cit._ ∎

In particular, from this we are able to immediately deduce the following.

######  Lemma 10.3.

For each pair $(x,\zeta)$ as above, one has that

$\mathrm{Hom}_{J(\Sigma)}(\lambda_{\Sigma},\lambda_{\Sigma_{x,\zeta}})\neq 0.$

Equivalently, via Frobenius reciprocity, we may identify
$\mathrm{Ind}_{J(\Sigma_{x,\zeta})}^{K}\lambda_{\Sigma_{x,\zeta}}$ with a
subrepresentation of $\mathrm{Ind}_{J(\Sigma)}^{K}\ \lambda_{\Sigma}$ whenever
$K\supset J(\Sigma_{x,\zeta})\supset J(\Sigma)$.

We equip the set of pairs $(x,\zeta)$ as above with a partial order $\preceq$,
where we declare that $(x,\zeta)\preceq(x^{\prime},\zeta^{\prime})$ if and
only if $x^{\prime}$ is contained in the simplicial closure of $x$ and
$\zeta^{\prime}$ is contained in $V_{x,\zeta}$. As a consequence of Lemma
10.3, we obtain the following geometric structure on
$\mathrm{Ind}_{J(\Sigma)}^{K}\ \lambda_{\Sigma}$.

######  Lemma 10.4.

For any maximal compact subgroup $K$ of $G$ which contains $J(\Sigma)$, the
set of representations $\mathrm{Ind}_{J(\Sigma_{x,\zeta})}^{K}\
\lambda_{\Sigma_{x,\zeta}}$ for which $G_{x}\subset K$ forms an injective
system via the canonical inclusion maps

$\mathrm{Ind}_{J(\Sigma_{x,\zeta})}^{K}\
\lambda_{\Sigma_{x,\zeta}}\hookrightarrow\mathrm{Ind}_{J(\Sigma_{x^{\prime},\zeta^{\prime}})}^{K}\
\lambda_{\Sigma_{x^{\prime},\zeta^{\prime}}},$

which exist whenever $(x,\zeta)\preceq(x^{\prime},\zeta^{\prime})$. In
particular, one has the identity

$\mathrm{Ind}_{J(\Sigma)}^{K}\
\lambda_{\Sigma}=\varinjlim\mathrm{Ind}_{J(\Sigma_{x,\zeta})}^{K}\
\lambda_{\Sigma_{x,\zeta}}.$

In fact, via this system we are able to exhaustively describe the isomorphism
classes of irreducible components of $\mathrm{Ind}_{J(\Sigma)}^{K}\
\lambda_{\Sigma}$ (although not necessarily the multiplicity of each such
isomorphism class).

######  Proposition 10.5.

Suppose that $(x,\zeta)$ is maximal with respect to $\preceq$ and
$G_{x}\subset K$. Then the representation $\lambda_{\Sigma_{x,\zeta}}$ of
$J(\Sigma_{x,\zeta})$ is irreducible.

Moreover, every irreducible component of $\mathrm{Ind}_{J(\Sigma)}^{K}\
\lambda_{\Sigma}$ is isomorphic to a representation of the form
$\mathrm{Ind}_{J(\Sigma_{x,\zeta})}\ \lambda_{\Sigma_{x,\zeta}}$ for some pair
$(x,\zeta)$ which is maximal with respect to $\preceq$.

###### Proof.

We freely make use of the results of [KY17, §8] and [Yu01, §15]. By carrying
out the standard inductive argument from _loc. cit._ , one finds that in order
to show that $I_{K}(\lambda_{\Sigma_{x,\zeta}})$, the set of $k\in K$ which
intertwine $\lambda_{\Sigma_{x,\zeta}}$ with itself, is equal to
$J(\Sigma_{x,\zeta})$, from which the irreducibility of
$\mathrm{Ind}_{J(\Sigma_{x,\zeta})}^{K}\ \lambda_{\Sigma_{x,\zeta}}$ follows,
it suffices to show that $I_{K\cap G^{0}}(\zeta)=K\cap G_{[x]}^{0}$. Since $K$
is a maximal compact subgroup of $G$ which contains $G_{x}$, the group $K\cap
G^{0}$ is a compact subgroup of $G^{0}$ which contains $G_{x}^{0}$. However,
we know that $(x,\zeta)$ is maximal with respect to $\preceq$, which implies
that $x$ is a vertex when viewed as a point in $\mathscr{B}(G^{0})$, i.e. that
$G_{x}^{0}$ is a maximal compact subgroup of $G^{0}$, which implies that
$K\cap G^{0}=G_{x}^{0}$. So, indeed, it must be the case that $I_{K\cap
G^{0}}(\zeta)=K\cap G_{[x]}^{0}$, from which it follows that
$\mathrm{Ind}_{J(\Sigma_{x,\zeta})}^{K}\ \lambda_{\Sigma_{x,\zeta}}$ is
irreducible.

In order to see that every irreducible component of
$\mathrm{Ind}_{J(\Sigma)}^{K}\ \lambda_{\Sigma}$ arises in such a way, it
remains for us to observe that it follows from the construction of the various
representations $\kappa_{\Sigma_{x,\zeta}}$ as Heisenberg–Weil lifts of
$\theta_{\Sigma_{x,\zeta}}$ that $\mathrm{Ind}_{J(\Sigma)}^{K}\
\lambda_{\Sigma}$ decomposes as a sum of representations of the form
$\mathrm{Ind}_{J(\Sigma_{x,\zeta})}^{K}\ \lambda_{\Sigma_{x,\zeta}}$, for
$(x,\zeta)$ maximal. In particular, the set of representations of the form
$\mathrm{Ind}_{J(\Sigma_{x,\zeta})}^{K}\ \lambda_{\Sigma_{x,\zeta}}$ exhausts
the set of isomorphism classes of irreducible components of the representation

$(\mathrm{Ind}_{J(\Sigma)}^{K}\ \sigma)\otimes(\mathrm{Ind}_{J(\Sigma)}^{K}\
\kappa_{\Sigma});$

since there is a canonical injective map

$\mathrm{Ind}_{J(\Sigma)}^{K}\
\lambda_{\Sigma}\rightarrow(\mathrm{Ind}_{J(\Sigma)}^{K}\
\sigma)\otimes(\mathrm{Ind}_{J(\Sigma)}^{K}\ \kappa_{\Sigma}),$

the claim follows. ∎

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