# Bruhat-Tits buildings, representations of $p$-adic groups and Langlands
correspondence

Anne-Marie Aubert  Sorbonne Université and Université Paris Cité, CNRS, IMJ-
PRG, F-75005 Paris, France<EMAIL_ADDRESS>
###### Abstract.

The Bruhat-Tits theory is a key ingredient in the construction of irreducible
smooth representations of $p$-adic reductive groups. We describe
generalizations to arbitrary such representations of several results recently
obtained in the case of supercuspidal representations, in particular regarding
the local Langlands correspondence and the internal structure of the
$L$-packets.

We prove that the enhanced $L$-parameters with semisimple cuspidal support are
those which are obtained via the (ordinary) Springer correspondence.

Let $\mathbf{G}$ be a connected reductive group over a non-archimedean field
$F$ of residual characteristic $p$. In the case where $\mathbf{G}$ splits over
a tamely ramified extension of $F$ and $p$ does not divide the order of the
Weyl group of $\mathbf{G}$, we show that the enhanced $L$-parameters with
semisimple cuspidal support correspond to the irreducible smooth
representations of $\mathbf{G}(F)$ with non-singular supercuspidal support via
the local Langlands correspondence constructed by Kaletha, under the
assumption that the latter satisfies certain expected properties. As a
consequence, we obtain that every compound $L$-packet of $\mathbf{G}(F)$
contains at least one representation with non-singular supercuspidal support.

To the memory of Jacques Tits with admiration

###### Contents

  1. 1 Introduction
  2. 2 Bruhat-Tits buildings
  3. 3 Theory of types
    1. 3.1 Construction of types
    2. 3.2 Types and Bernstein decomposition
    3. 3.3 Regular representations
    4. 3.4 Non-singular representations
  4. 4 Langlands correspondence and Bernstein Center
    1. 4.1 Generalized Springer correspondence
    2. 4.2 The ordinary Springer correspondence
    3. 4.3 Enhanced Langlands parameters
    4. 4.4 A Bernstein-type decomposition on the Galois side
    5. 4.5 The $\mathrm{LLC}$ for non-singular representations

##  1\. Introduction

Let $F$ be a non-archimedean local field, let $\mathbf{G}$ be a connected
reductive algebraic group defined over $F$, and let $G$ denote the group of
$F$-points of $\mathbf{G}$. A fundamental problem is the construction (and
classification) of all irreducible smooth complex representations of the group
$G$. A remarkable breakthrough in this direction was done by Moy and Prasad
via the use of the Bruhat-Tits theory. In [BT1, BT2], Bruhat and Tits defined
an affine building $\mathcal{B}(\mathbf{G},F)$ attached to $G$ on which the
group $G$ acts. In particular, for each point $x$ in
$\mathcal{B}(\mathbf{G},F)$, they constructed a compact subgroup $G_{x,0}$ of
$G$, called a parahoric subgroup of $G$, which generalizes the notion of
Iwahori subgroup.

In [MP1, MP2], Moy and Prasad defined a filtration of these parahoric
subgroups by smaller, normal in $G_{x,0}$, subgroups $G_{x,0}\rhd
G_{x,r_{1}}\rhd G_{x,r_{2}}\rhd G_{x,r_{3}}\rhd\cdots$, where
$0<r_{1}<r_{2}<r_{3}<\cdots$ are real numbers depending on $x$. These
subgroups play a fundamental role in the study and construction of
supercuspidal representations of $G$. Thanks to this filtration, they
introduced the notion of depth of a representation, which measures the first
occurrence of a fixed vector in a given representation.

In [Adl], Adler used the Moy-Prasad filtration to construct certain positive-
depth supercuspidal representations of $G$, in the case where $\mathbf{G}$
splits over a tamely ramified extension. His construction was generalized by
Yu in [Yu1]. When $\mathbf{G}$ splits over a tamely ramified extension and the
residual characteristic of $F$ does not divide the order of the Weyl group of
$\mathbf{G}$, Yu’s construction gives all supercuspidal irreducible
representations of $G$ (see [Fi1, Ki]). However, it was recently noticed by
Spice that the proofs of two essential results in this construction, [Yu1,
Prop. 14.1 and Th. 14.2], are not correct, due to the usage of a misstated
lemma in a reference. Although [Fi2, Th. 3.1] shows that Yu’s construction
still produces irreducible supercuspidal representations, it appeared to be
important to restore the validity of Yu’s results. This was recently done in
[FKS, Th. 4.1.13] by modifying Yu’s construction with a certain quadratic
character.

The original Yu’s method is based on the construction of very specific pairs
formed by an open compact subgroup of $G$ and an irreducible smooth
representation of the latter, called types (see [BK2]). Yu’s construction of
types has been generalized to not necessarily supercuspidal representations in
[KY] and [Fi1]. In §3, we similarly extend the “twisted Yu’s construction” of
[FKS] to arbitrary irreducible smooth representations.

If the group $\mathbf{G}$ splits over a tamely ramified extension and the
residual characteristic of $F$ does not divide the order of the Weyl group of
$\mathbf{G}$, for a certain class of supercuspidal irreducible
representations, called “non-singular” (which contains the class of “regular”
supercuspidal representations considered in [Ka3]), Kaletha obtained a
simplification of Yu’s construction and constructed a Langlands correspondence
in [Ka4]. The $L$-packets which contain non-singular supercuspidal
representations of $G$ are those attached to discrete $L$-parameters for $G$
that have trivial restriction to $\operatorname{SL}_{2}(\mathbb{C})$. These
$L$-packets contain only supercuspidal representations.

We generalize these notions to arbitrary smooth representations as follows: we
call an irreducible smooth representation $\pi$ of $G$ regular (resp. non-
singular) if its supercuspidal support is regular (resp. non-singular). We
study the local Langlands correspondence for these representations by using
the approach we developed in [AMS1, AMS2, AX1]. In particular, assuming some
properties of the local Langlands correspondence (proved to be satisfied in
the case of classical $p$-adic groups in [AMS3], in the case of the group
$\operatorname{G}_{2}$ in [AX2], and, for $G$ arbitrary, in the case where $L$
a torus in [ABPS2, So2]), we obtain in Theorem 4.5.5, for every $L$-parameter
$\varphi$, a description of the equivalence classes of the irreducible non-
singular representations in a compound $L$-packet $\Pi_{\varphi}$ via the
Springer correspondence of a (possibly disconnected) reductive complex group
attached to $\varphi$. As a consequence, we prove that, if $G$ is quasi-split,
then any $L$-packet for $G$ contains a non-singular representation.

Acknowledgements. We would like to thank the referees for their thourough
reports and their suggestions for improvements. We are also grateful to
Alexandre Afgoustidis, Alexander Bertoloni-Meli, Arnaud Mayeux, and Masao Oi
for several helpful comments and discussions.

Notations. Let $F$ be a non-archimedean local field. Let $\mathfrak{o}_{F}$
denote the ring of integers of $F$ and $\mathfrak{p}_{F}$ the maximal ideal in
$\mathfrak{o}_{F}$. There is in $\mathfrak{o}_{F}$ a prime element
$\varpi_{F}$ such that $\mathfrak{p}_{F}=\varpi_{F}\mathfrak{o}_{F}$. Let
$k_{F}:=\mathfrak{o}_{F}/\mathfrak{p}_{F}$ be the residue field of $F$. Then
$k_{F}$ is finite, and we denote by $p$ its characteristic and by $q$ its
cardinality. We fix a separable closure $F_{\mathrm{s}}$ of $F$, and denote by
$\Gamma_{F}:=\operatorname{Gal}(F_{\mathrm{s}}/F)$ the absolute Galois group
of $F$. We set $\Sigma_{F}:=\Gamma_{F}\times\\{\pm 1\\}$. We denote by $W_{F}$
the absolute Weil group of $F$ and by $I_{F}$ (resp $P_{F})$ the inertia
(resp. wild inertia) subgroup. Subextensions of $F_{\mathrm{s}}/F$ are always
endowed with the valuation extending the valuation ${\mathrm{ord}}$ on $F$. We
set $|a|_{F}:=q^{-{\mathrm{ord}}(a)}$ for any $a\in F^{\times}$ and
$|0|_{F}:=0$. We fix an additive character $\psi\colon
F\to\mathbb{C}^{\times}$ of $F$ of conductor $\mathfrak{p}_{F}$.

Let $F_{\operatorname{ur}}$ denote the maximal unramified extension of $F$
within $F_{\mathrm{s}}$. The residue field of $F_{\operatorname{ur}}$ is an
algebraic closure $\overline{k}_{F}$ of $k_{F}$. We denote by $\mathrm{Fr}$
the element of $\operatorname{Gal}(F_{\operatorname{ur}}/F)$ that induces the
automorphism $\overline{a}\mapsto\overline{a}^{q}$ on $\overline{k}_{F}$.

Let $\mathbf{G}$ a connected reductive algebraic group defined over $F$, and
let $G$ be the group of $F$-rational points of $\mathbf{G}$. We denote by
$\mathbf{G}_{\mathrm{ad}}:=\mathbf{G}/{\mathrm{Z}}_{\mathbf{G}}$ the adjoint
quotient of $\mathbf{G}$, by $\mathbf{G}_{\mathrm{der}}$ the derived subgroup
of $\mathbf{G}$, and by
$\mathbf{G}_{{\mathrm{ab}}}:=\mathbf{G}/\mathbf{G}_{{\mathrm{der}}}$ the
maximal abelian quotient of $\mathbf{G}$. Let $\mathbf{G}_{\mathrm{sc}}$
denote the simply-connected cover of $\mathbf{G}_{\mathrm{der}}$, and let
${\mathrm{Z}}_{\mathbf{G}}$ be the center of $\mathbf{G}$. Let $\mathbf{L}$ be
a Levi subgroup of $\mathbf{G}$. We write $\mathbf{L}_{\mathrm{sc}}$ for the
pre-image of $\mathbf{L}$ in $\mathbf{G}_{\mathrm{sc}}$, and
$\mathbf{L}_{\mathrm{ad}}$ for the image of $\mathbf{L}$ in
$\mathbf{G}_{\mathrm{ad}}$, and we write
$\mathbf{L}_{{\mathrm{sc}},{\mathrm{ab}}}:=\mathbf{L}_{\mathrm{sc}}/\mathbf{L}_{{\mathrm{sc}},{\mathrm{der}}}$
for the abelianization of $\mathbf{L}_{\mathrm{sc}}$.

If $\mathbf{T}$ is a maximal torus in $\mathbf{G}$, which is defined over $F$.
We denote by $X(\mathbf{T})$ the group of algebraic group homomorphisms
$\mathbf{T}\to\mathbb{G}_{m}$ and by $R(\mathbf{G},\mathbf{T})\subset
X(\mathbf{T})$ the root system of $\mathbf{G}$ with respect to $\mathbf{T}$.
Let ${\mathrm{Q}}(\mathbf{G},\mathbf{T})$ be the $\mathbb{Z}$-sublattice of
$X(\mathbf{T})$ generated by $R(\mathbf{G},\mathbf{T})$. Let
$R(\mathbf{G},\mathbf{T})^{\vee}$ the set of coroots. We write

${\mathrm{P}}(\mathbf{G},\mathbf{T}):=\left\\{\xi\in
X(\mathbf{T})\,:\,\langle\xi,\alpha^{\vee}\rangle\in\mathbb{Z},\;\;\forall\alpha^{\vee}\in
R(\mathbf{G},\mathbf{T})^{\vee}\right\\}$

for the lattice of weights. Following [Bou, VI.§1.9], we call connection index
the index of the root lattice in the weight lattice, that is the cardinality
of ${\mathrm{P}}(\mathbf{G},\mathbf{T})/{\mathrm{Q}}(\mathbf{G},\mathbf{T})$.
Recall that a prime is said to be good (or not bad) for $\mathbf{G}$ if it
does not divide the coefficients of the highest root of
$R(\mathbf{G},\mathbf{T})$ (see [SS, §4.3]). If a good prime for $\mathbf{G}$
does not divide the connection index, it is said to be very good for
$\mathbf{G}$ (see for instance [Let, §1]).

If $\mathbf{L}$ is a connected reductive $F$-subgroup of $\mathbf{G}$
containing $\mathbf{T}$, we set

(1.0.1)
$R(\mathbf{G}|\mathbf{L},\mathbf{T}):=R(\mathbf{G},\mathbf{T})\backslash
R(\mathbf{L},\mathbf{T}).$

The sets $R(\mathbf{G},\mathbf{T})$, $R(\mathbf{L},\mathbf{T})$ and
$R(\mathbf{G}|\mathbf{L},\mathbf{T})$ are equipped with a $\Gamma_{F}$-action.

For each $\alpha\in R(\mathbf{G},\mathbf{T})$, we denote by $\Gamma_{\alpha}$
(resp. $\Gamma_{\pm\alpha}$) the stabilizer of $\alpha$ (resp.
$\\{\pm\alpha\\}$) in $\Gamma_{F}$. Let $F_{\alpha}$ (resp. $F_{{\pm\alpha}}$)
be the subfield of $F_{\mathrm{s}}$ fixed by $\Gamma_{\alpha}$ (resp.
$\Gamma_{\pm\alpha}$):

(1.0.2) $F\subset F_{\pm\alpha}\subset
F_{\alpha}\quad\longleftrightarrow\quad\Gamma_{F}\supset\Gamma_{{\pm\alpha}}\supset\Gamma_{{\alpha}}.$

We recall the following terminology of [AS]:

  * •

When $F_{\alpha}=F_{\pm\alpha}$, we say that $\alpha$ is an asymmetric root.

  * •

When $F_{\alpha}\neq F_{\pm\alpha}$, we say that $\alpha$ is a symmetric root.
Note that, in this case, the extension $F_{\alpha}/F_{\pm\alpha}$ is
necessarily quadratic. Furthermore,

    * –

when $F_{\alpha}/F_{\pm\alpha}$ is unramified, we say that $\alpha$ is
symmetric unramified,

    * –

when $F_{\alpha}/F_{\pm\alpha}$ is ramified, we say $\alpha$ is symmetric
ramified.

Let $k_{\alpha}$ (resp. $k_{\pm\alpha}$) denote the residue fiel of
$F_{\alpha}$ (resp. $F_{\pm\alpha}$), let
${\mathrm{N}}_{k_{\alpha}|k_{F}}\colon k_{\alpha}^{\times}\to k_{F}^{\times}$
be the norm map, and let $k_{\alpha}^{1}$ be the kernel of
${\mathrm{N}}_{k_{\alpha}|k_{F}}$. If $p$ is odd, the groups
$k_{\alpha}^{\times}$ and $k_{\alpha}^{1}$ are finite cyclic of even order and
we denote by $\operatorname{sgn}_{k_{\alpha}}$ and
$\operatorname{sgn}_{k_{\alpha}^{1}}$ the corresponding unique non-trivial
$\\{\pm 1\\}$-valued characters.

For any maximal $F$-split torus $\mathbf{T}$, $R(\mathbf{G},\mathbf{T},F)$
denotes the corresponding set of roots in $\mathbf{G}$. For $\alpha\in
R(\mathbf{G},\mathbf{T},F)$, let $\mathbf{U}_{\alpha}$ be the root subgroup
corresponding to $\alpha$.

From now on, we suppose that the group $\mathbf{G}$ splits over a tamely
ramified extension of $F$, and that $p$ is very good for $\mathbf{G}$.

##  2\. Bruhat-Tits buildings

In [BT1, BT2], Bruhat and Tits defined a building associated to the $p$-adic
group $G=\mathbf{G}(F)$ on which the group $G$ acts. Let
$\mathcal{B}(\mathbf{G}/{\mathrm{Z}}_{\mathbf{G}},F)$ denote the Bruhat and
Tits building of $(\mathbf{G}/{\mathrm{Z}}_{\mathbf{G}})(F)$. The enlarged
building of $G$ is defined to be

(2.0.1)
$\mathcal{B}(\mathbf{G},F):=\mathcal{B}(\mathbf{G}/{\mathrm{Z}}_{\mathbf{G}},F)\times
X_{*}({\mathrm{Z}}_{G})\otimes_{\mathbb{Z}}\mathbb{R},$

where $X_{*}({\mathrm{Z}}_{G})$ is the set of $F$-algebraic cocharacters of
${\mathrm{Z}}_{G}$. For each point $x$ in $\mathcal{B}(\mathbf{G},F)$, they
constructed a compact subgroup $G_{x,0}$ of $G$, that they called a parahoric
subgroup. Let $G_{x,0^{+}}$ denote the pro-$p$ unipotent radical of $G_{x,0}$,
and $\mathbb{G}_{x,0}$ the quotient $G_{x,0}/G_{x,0+}$, (the points of) a
connected reductive group over the residue field of $F$. We denote by $[x]$
the image of $x$ in the reduced Bruhat–Tits building and we write $G_{[x]}$
for the stabilizer of $[x$] under the action $G$ on the reduced building. The
group $G_{[x]}$ coincides with the normalizer in $G$ of $G_{x,0}$. Indeed, if
$gG_{x,0}g^{-1}$ (which is equal to $G_{g\cdot x,0}$) is equal to $G_{x,0}$,
then we have $g.[x]=[x]$ by [BT2, proof of 5.2.8].

As shown in [BT2, §5], the point $x$ specifies a smooth connected affine
$\mathfrak{o}_{F}$-group scheme ${\mathfrak{G}}^{\circ}_{x}$ with
${\mathfrak{G}}_{x}^{\circ}(F)=\mathbf{G}(F)$ and
${\mathfrak{G}}_{x}^{\circ}(\mathfrak{o}_{F})=G_{x,0}$. We denote by
${\underline{\mathbb{G}}}_{x,0}$ the reductive quotient of the special fiber
of ${\mathfrak{G}}^{\circ}_{x}$. We have
${\underline{\mathbb{G}}}_{x,0}(k_{F})=\mathbb{G}_{x,0}$111Vérifier!. We
further have an $\mathfrak{o}_{F}$-group scheme ${\mathfrak{G}}_{x}$ with
${\mathfrak{G}}_{x}(F)=G$ and
${\mathfrak{G}}_{x}(\mathfrak{o}_{F})=\operatorname{Stab}_{G_{1}}(x)$, where
$G_{1}$ denotes the intersection of the kernels of the group homomorphisms
${\mathrm{ord}}\circ\chi\colon G\to\mathbb{Z}$ for all $F$-rational characters
$\chi\colon G\to\mathbb{G}_{m}$. We denote by ${\underline{\mathbb{G}}}_{x},$
the quotient of the special fiber of this group scheme by its maximal
connected normal unipotent subgroup. Then ${\mathfrak{G}}^{\circ}_{x}$
coincides with the neutral connected component of ${\mathfrak{G}}_{x}$. In
particular, ${\underline{\mathbb{G}}}_{x}$ is a (possibly disconnected)
algebraic group over $k_{F}$ with reductive neutral connected component. We
have ${\underline{\mathbb{G}}}_{x}(k_{F})=\mathbb{G}_{x}$.

In [MP1, MP2], Moy and Prasad defined a filtration of these parahoric
subgroups by normal in $G_{x,0}$ subgroups

(2.0.2) $G_{x,0}\rhd G_{x,r_{1}}\rhd G_{x,r_{2}}\rhd G_{x,r_{3}}\rhd\cdots,$

that are normal inside each other and whose intersection is trivial, where
$0<r_{1}<r_{2}<r_{3}<\cdots$ are real numbers depending on $x$. For
$r\in\mathbb{R}_{\geq 0}$, we write

(2.0.3) $G_{x,r+}:=\bigcup_{t>r}G_{x,t}.$

When $r>0$, the quotient $G_{x,r}/G_{x,r+}$ is abelian and can be identified
with a $k_{F}$-vector space. Moy and Prasad defined also filtration submodules
${\mathfrak{g}}_{x,r}$ of ${\mathfrak{g}}:={\mathrm{Lie}}(\mathbf{G})(F)$ and
${\mathfrak{g}}^{*}_{x,r}$ of the linear dual ${\mathfrak{g}}^{*}$ of
${\mathfrak{g}}$, for $r\in\mathbb{R}$. We write
${\mathfrak{g}}_{x,r+}:=\bigcup_{t>r}{\mathfrak{g}}_{x,t}$.

The ${\mathfrak{g}}_{x,r}$ were used in [MP1] to define the notion of depth of
a representation. We recall it below, following [DB1] in which the reliance on
${\mathfrak{g}}$ has been removed from the definition. Let $(\pi,V)$ be a
smooth representation of $G$. By [DB1, Lemma 5.2.1], there exists
$0pt(\pi)\in\mathbb{Q}$ with the following properties.

  1. (1)

If $(x,r)\in\mathcal{B}(\mathbf{G},F)\times\mathbb{R}_{\geq 0}$ such that
$V^{G_{x,r+}}$ is nontrivial, then $r\geq 0pt(\pi)$.

  2. (2)

There exists a $x^{\prime}\in\mathcal{B}(\mathbf{G},F)$ such that
$V^{G_{x^{\prime},0pt(\pi)+}}$ is nontrivial.

Then the depth of $\pi$ is defined to be the least nonnegative real number
$0pt(\pi)$ for which there exists an $x\in\mathcal{B}(\mathbf{G},F)$ such that
$V^{G_{x,0pt(\pi)+}}$ is nontrivial.

Note that in [Yu2], different filtrations where introduced. It could be
possible to define a notion of depth using these filtrations instead. However,
as shown in [Yu2, §8.1] or [KP, §13], in the case of quasi-split groups that
split over a tamely ramified extension, both notions of filtrations coincide.
Related filtrations by affinoid groups, in the Berkovich analytification of a
connected reductive group, have been defined in [Ma] (see also [RTW] and the
references there).

Let $\mathbf{S}$ be a torus of $\mathbf{G}$ defined over $F$. The topological
group $S:=\mathbf{S}(F)$ has a unique maximal bounded subgroup
$S_{\mathrm{b}}$ (which is also the unique maximal compact subgroup, as $F$ is
locally compact) and this subgroup is equipped with a decreasing filtration
$S_{r}$ indexed by the non-negative real numbers, namely the Moy-Prasad
filtration corresponding to the unique point in the reduced Bruhat-Tits
building of $\mathbf{S}$. The torus $\mathbf{S}$ possesses an lft-Néron model
$\mathfrak{S}^{\mathrm{lft}}$ by [BLR, §10]. This is a smooth group scheme
over $\mathfrak{o}_{F}$ satisfying a certain universal property. It is locally
of finite type and the maximal subgroup-scheme of finite type is called the
ft-Néron model $\mathfrak{S}^{\mathrm{ft}}$. Both models share the same
neutral connected component $\mathfrak{S}^{\circ}$, called the connected Néron
model of $\mathbf{S}$. We have

(2.0.4) $S_{0}=\mathfrak{S}^{\circ}(\mathfrak{o}_{F})\quad\text{and}\quad
S_{\mathrm{b}}=\mathfrak{S}^{\mathrm{ft}}(\mathfrak{o}_{F}).$

The quotient $S/S_{\mathrm{b}}$ is a finitely generated free abelian group,
and, by [MP2, §3.2], [Yu2, §4.2], for each positive real number $r$ we have

(2.0.5) $S_{r}=\left\\{s\in S_{0}\,:\,{\mathrm{ord}}(\chi(s)-1)\geq r\text{
for all $\chi\in X^{*}(\mathbf{S})$}\right\\}.$

We set $S_{r+}:=\bigcup_{t>r}S_{t}$. The group $S_{0+}$ is the pro-$p$-Sylow
of $S_{0}$. Let $\mathbf{T}$ be the maximal unramified subtorus of
$\mathbf{S}$. By [Ka3, Lem.3.1.16], we have

(2.0.6) $\mathbb{T}:=S_{0}/S_{0+}\simeq T_{0}/T_{0+}.$

Let $E/F$ be a finite extension. By a twisted $E$-Levi subgroup of
$\mathbf{G}$, we mean an $F$-subgroup $\mathbf{G}^{\prime}$ of $\mathbf{G}$
such that $\mathbf{G}^{\prime}\otimes_{F}E$ is an $E$-rational Levi subgroup
of an $E$-rational parabolic subgroup of $\mathbf{G}\otimes_{F}E$. If $E/F$ is
tamely ramified, then $\mathbf{G}^{\prime}$ is called a tamely ramified
twisted Levi subgroup of $\mathbf{G}$. Recall that when $\mathbf{G}^{\prime}$
is a tamely ramified twisted Levi subgroup of $\mathbf{G}$, there is a family
of natural embeddings of $\mathcal{B}(\mathbf{G}^{\prime},F)$ into
$\mathcal{B}(\mathbf{G},F)$.

Let $\mathbf{G}^{\prime}$ be a tamely ramified twisted Levi subgroup of
$\mathbf{G}$, and denote by
$({\mathrm{Lie}}^{*}(\mathbf{G}^{\prime}))^{\mathbf{G}^{\prime}}$ the
subscheme of ${\mathrm{Lie}}^{*}(\mathbf{G}^{\prime})$ that is fixed by the
dual of the adjoint action of $\mathbf{G}^{\prime}$. Let
$Y\in({\mathrm{Lie}}^{*}(\mathbf{G}^{\prime}))^{\mathbf{G}^{\prime}}$. Recall
that $({\mathrm{Lie}}^{*}\mathbf{T})\otimes F_{\mathrm{s}}$ can be identified
with $X(\mathbf{T})\otimes_{\mathbb{Z}}F_{\mathrm{s}}$. Therefore, we can view
$\varpi_{r}Y\in{\mathrm{Lie}}^{*}({\mathrm{Z}}_{\mathbf{G}^{\prime}}\otimes
F_{\mathrm{s}})^{\circ}\subset{\mathrm{Lie}}^{*}(\mathbf{T}\otimes
F_{\mathrm{s}})$ as an element of
$X(\mathbf{T})\otimes_{\mathbb{Z}}\mathfrak{o}_{F_{\mathrm{s}}}$, where
$\varpi_{r}$ is an element of $F_{\mathrm{s}}^{\times}$ of valuation $r$. The
residue class $\overline{Y}$ of $\varpi_{r}Y$ is an element of
$X(\mathbf{T})\otimes_{\mathbb{Z}}k_{F_{\mathrm{s}}}$. Then $\overline{Y}$ is
well defined up to a multiplicative constant in $k_{F_{\mathrm{s}}}^{\times}$.

Following [Yu1, §8], as corrected in [FKS, Rem.4.1.3], as in [Fi3, Def. 2.1],
we say that an element $Y$ of
$({\mathrm{Lie}}^{*}(\mathbf{G}^{\prime}))^{\mathbf{G}^{\prime}}$ is
$\mathbf{G}$-generic of depth $r\in\mathbb{R}$ if the following conditions
hold:

  1. (1)

for some (equivalently, any) point $x\in\mathcal{B}(\mathbf{G}^{\prime},F)$,
we have $Y\in{\mathrm{Lie}}^{*}(\mathbf{G}^{\prime})_{x,-r}$;

  2. (2)

${\mathrm{ord}}(Y(H_{\alpha}))=-r$ for all $\alpha\in
R(\mathbf{G}|\mathbf{G}^{\prime},\mathbf{T})$ for some maximal torus
$\mathbf{T}$ of $\mathbf{G}^{\prime}$, where
$H_{\alpha}:={\mathrm{Lie}}(\alpha^{\vee})(1)$ with $\alpha^{\vee}$ the coroot
of $\alpha$.

  3. (3)

the subgroup of $W$ fixing $\overline{Y}$ is precisely the Weyl group of
$R(\mathbf{G}^{\prime},T,F_{\mathrm{s}})$.

As observed in [Yu1, Lem. 8.1], (2) implies (3) if $p$ is not a torsion prime
for the dual root datum of $\mathbf{G}$.

Let $x\in\mathcal{B}(\mathbf{G}^{\prime},F)$ and $r\in\mathbb{R}_{>0}$. A
character $\phi$ of $G^{\prime}$ is called $\mathbf{G}$-generic of depth $r$
relative to $x$ if $\phi$ is trivial on $G^{\prime}_{x,r+}$, non-trivial on
$G^{\prime}_{x,r}$, and the restriction of $\phi$ to
$G^{\prime}_{x,r}/G^{\prime}_{x,r+}\simeq{\mathfrak{g}}_{x,r}^{\prime}/{\mathfrak{g}}^{\prime}_{x,r+}$
is given by $\phi\circ Y$ for some generic element
$Y\in({\mathrm{Lie}}^{*}(\mathbf{G}^{\prime}))^{\mathbf{G}^{\prime}}$ of depth
$r$.

Let $\mathbf{L}$ be an $F$-rational Levi subgroup of an $F$-rational parabolic
subgroup of $\mathbf{G}$. Let $\mathbf{T}$ be an elliptic maximally unramified
maximal torus of $\mathbf{L}$, and let $y\in\mathcal{B}(\mathbf{L},F)$ be the
point associated to $\mathbf{S}$. It is a vertex [Ka3, Lem.3.4.3]. If
$\alpha\in R(\mathbf{L},\mathbf{T})$, we set

(2.0.7)
${\mathrm{ord}}_{y}(\alpha):=\left\\{t\in\mathbb{R}\,:\,\mathfrak{g}_{\alpha}(F_{\alpha})_{y,t+}\subsetneq\mathfrak{g}_{\alpha}(F_{\alpha})_{y,t}\right\\}.$

An element
$X\in{\mathrm{Lie}}(\mathbf{L}_{{\mathrm{sc}},{\mathrm{ab}}})^{*}(F)\subset{\mathrm{Lie}}(\mathbf{L}_{\mathrm{sc}})^{*}(F)$
is called $\mathbf{G}$-good if there exists $r\in\mathbb{R}$ such that for all
tamely ramified maximal tori $\mathbf{T}\subset\mathbf{L}$ and all $\alpha\in
R(\mathbf{G}|\mathbf{L},\mathbf{T})$ we have ${\mathrm{ord}}(\langle
X,H_{\alpha}\rangle)=-r$.

Let
$\vv\mathbf{G}=(\mathbf{G}^{0}\subset\mathbf{G}^{1}\subset\cdots\subset\mathbf{G}^{d}=\mathbf{G})$
be a tamely ramified twisted Levi sequence in $\mathbf{G}$, that is, a finite
sequence of twisted $E$-Levi subgroups of $\mathbf{G}$, with $E/F$ tamely
ramified. Let $\mathbf{L}^{0}$ be a Levi subgroup of $\mathbf{G}^{0}$. We
denote by $A_{\mathbf{L}^{0}}$ the maximal $F$-split torus of the center
${\mathrm{Z}}_{\mathbf{L}^{0}}$ of $L^{0}$. For each $i\in\\{1,\ldots,d\\}$,
we define $\mathbf{L}^{i}$ to be the centralizer of $A_{\mathbf{L}^{0}}$ in
$\mathbf{G}^{i}$, it is a Levi subgroup of $\mathbf{G}^{i}$. The sequence
$\vv\mathbf{L}(\vv\mathbf{G}):=(\mathbf{L}^{0},\cdots,\mathbf{L}^{d})$ is a
tamely ramified generalized twisted Levi sequence in $\mathbf{L}$ and
${\mathrm{Z}}_{\mathbf{L}^{0}}/{\mathrm{Z}}_{\mathbf{L}^{d}}$ is
$F$-anisotropic (see [KY, Lem. 2.4]).

Folowing [KY, Def. 3.2], we say that an embedding
$\iota\colon\mathcal{B}(\mathbf{L},F)\hookrightarrow\mathcal{B}(\mathbf{G},F)$
is $(y,r)$-generic if
$\mathbf{U}_{\alpha,\iota(y),r}=\mathbf{U}_{\alpha,\iota(y),r+}$ for all
$\alpha\in R(\mathbf{G}|\mathbf{L},\mathbf{S},F)$ where $\mathbf{S}$ is any
maximal $F$-split torus of $\mathbf{L}$ such that
$y\in\mathcal{A}(\mathbf{L},\mathbf{T},F)$. Here
$\mathcal{A}(\mathbf{L},\mathbf{T},F)$ is the apartment associated to
$\mathbf{S}$ in $\mathcal{B}(\mathbf{L},F)$ and
$(\mathbf{U}_{\alpha,\iota(y),r})_{r\in\mathbb{R}}$ is the filtration on the
root group $\mathbf{U}_{\alpha}$, $\alpha\in R(\mathbf{G},\mathbf{S},F)$ so
that
$\mathbf{U}_{\alpha,\iota(y),r}=\mathbf{U}_{\alpha}\cap\mathbf{G}_{\iota(y),r}$,
where $(\mathbf{G}_{\iota(y),r})_{r\in\mathbb{R}}$ is the Moy-Prasad
filtration.

Let $\\{\iota\\}$ be a commutative diagram of embeddings:

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We denote by $\iota$ any composite embedding in this diagram from
$\mathcal{B}(\mathbf{L}^{i},F)$ to $\mathcal{B}(\mathbf{L}^{j},F)$ or
$\mathcal{B}(\mathbf{G}^{j},F)$, and from $\mathcal{B}(\mathbf{G}^{i},F)$ to
$\mathcal{B}(\mathbf{G}^{j},F)$ for $i\leq j$.

The image of a point $y\in\mathcal{B}(\mathbf{L}^{0},F)$ in
$\mathcal{B}(\mathbf{L}^{i},F)$ is uniquely determined, modulo the translation
of $A_{\mathbf{L}^{0}}$. To specify a diagram of embeddings as above, we need
to give the image of $y$ in $\mathcal{B}(\mathbf{G}^{i},F)$ for $0\leq i\leq
d$. We have the disjoint-union decomposition

(2.0.9)
$R(\mathbf{L}^{i}|\mathbf{L}^{i-1},\mathbf{T})=R(\mathbf{L}^{i}|\mathbf{L}^{i-1},\mathbf{T})_{\mathrm{asym}}\sqcup
R(\mathbf{L}^{i}|\mathbf{L}^{i-1},\mathbf{T})_{\mathrm{sym}}$

into the subsets of asymmetric, respectively symmetric, elements. We suppose
there exists and fix a character $\phi$ of $L^{0}$ that is
$\mathbf{L}$-generic relative to $y$ of depth $r$ for some non-negative real
number $r$ in the sense of [Yu1, p.599], and put $s=r/2$. To the character
$\phi$ one can associate a generic element of the dual Lie algebra of $L^{0}$.

For $\alpha\in R(\mathbf{L}^{i}|\mathbf{L}^{i-1},\mathbf{T})$, denote by
$\alpha_{0}$ the restriction of $\alpha$ to ${\mathrm{Z}}_{\mathbf{L}^{0}}$,
and by $e_{\alpha}$ (resp. $e_{\alpha_{0}}$) the ramification degree of the
extensions $F_{\alpha}/F$ (resp. $F_{\alpha_{0}}/F$). We set
$e(\alpha/\alpha_{0}):=e_{\alpha}/e_{\alpha_{0}}$. Let
$\ell_{p^{\prime}}(\alpha^{\vee})$ denote the prime-to-$p$ part of the
normalized square length of $\alpha^{\vee}$, and let
$f_{(\mathbf{L},\mathbf{T})}(\alpha)$ be the toral invariant from [Ka1, §4.1].
Let $\Sigma_{F}$ act on $F_{\mathrm{s}}$ with $\\{\pm 1\\}$ acting trivially.
For $t\in\mathbb{T}$, we write

$\epsilon_{\natural,y}^{\mathbf{L}^{i}|\mathbf{L}^{i-1}}(t):=\prod_{\begin{smallmatrix}\alpha\in
R(\mathbf{L}^{i}|\mathbf{L}^{i-1},\mathbf{T})_{\mathrm{asym}}/\Sigma_{F}\cr
s\in{\mathrm{ord}}_{y}(\alpha)\end{smallmatrix}}\operatorname{sgn}_{k_{\alpha}}(\alpha(t))\,\cdot\,\prod_{\begin{smallmatrix}\alpha\in
R(\mathbf{L}^{i}|\mathbf{L}^{i-1},\mathbf{T})_{\mathrm{sym},\operatorname{ur}}/\Gamma_{F}\cr
s\in{\mathrm{ord}}_{y}(\alpha)\end{smallmatrix}}\operatorname{sgn}_{k_{\alpha}^{1}}(\alpha(t));$
$\epsilon_{\flat,0}^{\mathbf{L}^{i}|\mathbf{L}^{i-1}}(t):=\prod_{\begin{smallmatrix}\alpha\in
R(\mathbf{L}^{i}|\mathbf{L}^{i-1},\mathbf{T})_{\mathrm{asym}}/\Sigma_{F}\cr\alpha_{0}\in
R(\mathbf{L}^{i}|\mathbf{L}^{i-1},{\mathrm{Z}}_{\mathbf{L}^{0}})_{\mathrm{sym},\operatorname{ur}}\cr
2\nmid
e(\alpha/\alpha_{0})\end{smallmatrix}}\operatorname{sgn}_{k_{\alpha}}\alpha(t)\,\cdot\,\prod_{\begin{smallmatrix}\alpha\in
R(\mathbf{L}^{i}|\mathbf{L}^{i-1},\mathbf{T})_{\mathrm{sym},\operatorname{ram}}/\Gamma_{F}\cr\alpha_{0}\in
R(\mathbf{L}^{i}|\mathbf{L}^{i-1},{\mathrm{Z}}_{\mathbf{L}^{0}})_{\mathrm{sym},\operatorname{ur}}\cr
2\nmid
e(\alpha|\alpha_{0})\end{smallmatrix}}\operatorname{sgn}_{k_{\alpha}^{1}}\alpha(t);$
$\epsilon_{\flat,1}^{\mathbf{L}^{i}|\mathbf{L}^{i-1}}(t):=\prod_{\begin{smallmatrix}\alpha\in
R(\mathbf{L}^{i}|\mathbf{L}^{i-1},\mathbf{T})_{\mathrm{sym},\operatorname{ram}}/\Gamma_{F}\cr\alpha(t)\in-1+\mathfrak{p}_{\alpha}\end{smallmatrix}}(-1)^{[k_{\alpha}:k_{F}]+1}\operatorname{sgn}_{k_{\alpha}}(e_{\alpha}\ell_{p^{\prime}}(\alpha^{\vee}));$
$\epsilon_{\flat,2}^{\mathbf{L}^{i}|\mathbf{L}^{i-1}}(t):=\prod_{\begin{smallmatrix}\alpha\in
R(\mathbf{L}^{i}|\mathbf{L}^{i-1},\mathbf{T})_{\mathrm{sym},\operatorname{ram}}/\Gamma_{F}\cr\alpha(t)\in-1+\mathfrak{p}_{\alpha}\end{smallmatrix}}\operatorname{sgn}_{k_{\alpha}}((-1)^{\frac{(e(\alpha/\alpha_{0})-1)}{2}});$
(2.0.10)
$\epsilon_{\flat,0}^{\mathbf{L}^{i}|\mathbf{L}^{i-1}}:=\epsilon_{\flat,0}^{\mathbf{L}^{i}|\mathbf{L}^{i-1}}\cdot\epsilon_{\flat,1}^{\mathbf{L}^{i}|\mathbf{L}^{i-1}}\cdot\epsilon_{\flat,2}^{\mathbf{L}^{i}|\mathbf{L}^{i-1}};$
(2.0.11)
$\epsilon_{f}^{\mathbf{L}^{i}|\mathbf{L}^{i-1}}(t):=\prod_{\begin{smallmatrix}\alpha\in
R(\mathbf{L}^{i}|\mathbf{L}^{i-1},\mathbf{T})_{\mathrm{sym},\operatorname{ram}}/\Gamma_{F}\cr\alpha(t)\in-1+\mathfrak{p}_{\alpha}\end{smallmatrix}}f_{(\mathbf{L},\mathbf{T})}(\alpha).$

By [FKS, Th. 3.4 and Cor. 3.6], there exists a unique character

(2.0.12)
$\epsilon_{y}^{\mathbf{L}^{i}|\mathbf{L}^{i-1}}\colon\mathbb{L}^{i-1}_{y}\to\\{\pm
1\\}$

satisfying the following property: for every tamely ramified maximal torus
$\mathbf{T}$ of $\mathbf{L}^{i-1}$ such that $y\in\mathcal{B}(\mathbf{T},F)$
the restriction of $\epsilon_{y}^{\mathbf{L}^{i}|\mathbf{L}^{i-1}}$ to
$\mathbb{T}$ equals
$\epsilon_{\natural,y}^{\mathbf{L}^{i}|\mathbf{L}^{i-1}}\cdot\epsilon_{\flat}^{\mathbf{L}^{i}|\mathbf{L}^{i-1}}\cdot\epsilon_{f}^{\mathbf{L}^{i}|\mathbf{L}^{i-1}}$.
By inflation we consider $\epsilon_{y}^{\mathbf{L}^{i}|\mathbf{L}^{i-1}}$ as a
character of $L_{y}^{i-1}$ when convenient. For every $i\in\\{1,\ldots,d\\}$,
we define

(2.0.13)
$\epsilon_{y}^{i}:=\prod_{j=1}^{i}\epsilon_{y}^{\mathbf{L}^{j}|\mathbf{L}^{j-1}},\quad\text{and
we write}\quad\epsilon_{y}:=\epsilon_{y}^{d}.$

##  3\. Theory of types

###  3.1. Construction of types

######  Definition 3.1.1.

A depth-zero $\mathbf{G}$-datum is a triple
$((\mathbf{G},\mathbf{L}),(y,\iota),{\boldsymbol{\tau}}_{L})$ satisfying the
following

  * (1)

$\mathbf{G}$ is a connected reductive group over $F$, and $\mathbf{L}$ is a
Levi subgroup of $\mathbf{G}$;

  * (2)

$y$ is a point in $\mathcal{B}(\mathbf{L},F)$ such that $L_{y,0}$ is a maximal
parahoric subgroup of $L$, and
$\iota\colon\mathcal{B}(\mathbf{L},F)\hookrightarrow\mathcal{B}(\mathbf{G},F)$
is a $(y,0)$-generic embedding;

  * (3)

${\boldsymbol{\tau}}_{L}$ is an irreducible smooth representation of $L_{y}$
such that ${\boldsymbol{\tau}}_{L}|L_{y,0}$ contains the inflation to
$L_{y,0}$ of a cuspidal representation of $\mathbb{L}_{y,0}$.

######  Definition 3.1.2.

A $\mathbf{G}$-datum is a $5$-tuple

(3.1.3) $\mathcal{D}=((\vv\mathbf{G},\mathbf{L}^{0}),(y,\\{\iota\\}),\vv
r,{\boldsymbol{\tau}}_{L^{0}},\vv\phi)$

satisfying the following:

  * D1.

$\vv\mathbf{G}=(\mathbf{G}^{0},\mathbf{G}^{1},\cdots,\mathbf{G}^{d})$ is a
tamely ramified twisted Levi sequence in $\mathbf{G}$, and $\mathbf{L}^{0}$ a
Levi subgroup of $\mathbf{G}^{0}$. Let $\vv\mathbf{L}$ be associated to
$\vv\mathbf{G}$ as above;

  * D2.

$y$ is a point in $\mathcal{B}(\mathbf{L}^{0},F)$, and $\\{\iota\\}$ is a
commutative diagram of $\vv s$-generic embeddings of buildings relative to
$y$, where $\vv s=(0,r_{0}/2,\cdots,r_{d-1}/2)$;

  * D3.

$\vv r=(r_{0},r_{1},\cdots,r_{d})$ is a sequence of real numbers satisfying
$0<r_{0}<r_{1}<\cdots<r_{d-1}\leq r_{d}$ if $d>0$, and $0\leq r_{0}$ if $d=0$;

  * D4.

$\mathcal{D}^{0}:=((\mathbf{G}^{0},\mathbf{L}^{0}),(y,\iota),{\boldsymbol{\tau}}_{L^{0}})$
is a depth zero $\mathbf{G}^{0}$-datum;

  * D5.

$\vv\phi=(\phi_{0},\phi_{1},\cdots,\phi_{d})$ is a sequence of quasi-
characters, where $\phi_{i}$ is a quasi-character of $G^{i}$ such that
$\phi_{i}$ is $G^{i+1}$-generic of depth $r_{i}$ relative to $y$ for all
$y\in\mathcal{B}(\mathbf{G}^{i},F)$.

For $i\in\\{0,\ldots,d\\}$, we write $s_{i}:=r_{i}/2$. We set $s_{-1}:=0$. For
$i\in\\{1,\ldots,d\\}$ and $t=s_{i-1}$ or $t=s_{i-1}+$, we denote by
$(\mathbf{G}^{i-1},\mathbf{G}^{i})(F)_{\iota(y),(r_{i-1},t)}$ the group

$G^{i-1}\cap\langle\mathbf{T}(E)_{r_{i-1}},\mathbf{U}_{\alpha}(E)_{\iota(y),r_{i-1}},\mathbf{U}_{\beta}(E)_{\iota(y),t}\;:\;\alpha\in
R(\mathbf{G}^{i},\mathbf{T}),\beta\in
R(\mathbf{G}^{i-1}|\mathbf{G}^{i},\mathbf{T})\rangle,$

where $\mathbf{T}$ is a maximal torus of $\mathbf{G}^{i}$ that splits over a
tamely ramified extension $E/F$ with
$\iota(y)\in\mathcal{B}(\mathbf{T},F)\subset\mathcal{B}(\mathbf{G}^{i},F)$,
and $\mathbf{U}_{\alpha}(E)_{\iota(y),r_{i-1}}$ denotes the Moy-Prasad
filtration subgroup of depth $r_{i-1}$ at $y$ of the root group
$\mathbf{U}_{\alpha}(E)\subset\mathbf{G}^{i-1}(E)$ corresponding to the root
$\alpha$, and similarly for $\mathbf{U}_{\beta}(E)_{\iota(y),t}$. We set

$J^{i}:=(\mathbf{G}^{i-1},\mathbf{G}^{i})(F)_{\iota(y),(r_{i-1},s_{i-1})}\quad\text{and}\quad
J^{i}_{+}:=(\mathbf{G}^{i-1},\mathbf{G}^{i})(F)_{\iota(y),(r_{i-1},s_{i-1}+)}.$

The construction. Let
$\mathcal{D}=((\vv\mathbf{G},\mathbf{L}^{0}),(y,\\{\iota\\}),\vv
r,{\boldsymbol{\tau}}_{L^{0}},\vv\phi)$ be a $\mathbf{G}$-datum. Let
$K_{G^{0}}$ denote the group generated by $L^{0}_{y}$ and
$G_{\iota(y),0}^{0}$. For $0\leq i\leq d$, as in [KY, §7.4], we define

(3.1.4) $\begin{cases}K^{i}:=K_{G^{0}}G^{1}_{\iota(y),s_{0}}\cdots
G^{i}_{\iota(y),s_{i-1}}\cr
K^{i}_{+}:=G_{\iota(y),0+}^{0}G^{1}_{\iota(y),s_{0}+}\cdots
G^{i}_{\iota(y),s_{i-1}+}.\end{cases}$

Inductively, we show that $K^{i}$ and $K^{i}_{+}$ are groups. For $i=0$, we
have $K^{0}:=K_{G^{0}}$ and $K^{0}_{+}:=G_{\iota(y),0+}^{0}$. For $i>0$,
$K_{G^{0}}G^{1}_{\iota(y),s_{0}}\cdots G^{i-1}_{\iota(y),s_{i-2}}$ is a group
by our induction hypothesis. It is clearly a subgroup of $G^{i}_{\iota(y),0}$.
Since $G^{i}_{\iota(y),0}$ normalizes $K_{G^{0}}$, $G^{1}_{\iota(y),s_{0}}$,
$\ldots$, $G^{i-1}_{\iota(y),s_{i-2}}$, we see that
$K_{G^{0}}G^{1}_{\iota(y),s_{0}}\cdots G^{i}_{\iota(y),s_{i-1}}$ is a group.
The proof that $K^{i}_{+}$ is a group is analogous.

By [KY, Prop. 4.3(b)], we have

(3.1.5)
$K_{G^{0}}/G^{0}_{\iota(y),0+}\overset{\mathrm{q}}{\simeq}L^{0}_{y}/L_{y,0+}^{0}.$

We define ${\boldsymbol{\tau}}_{\mathcal{D}^{0}}$ to be the representation of
$K_{G^{0}}$ obtained by composing the isomorphism (3.1.5) with
${\boldsymbol{\tau}}_{L^{0}}$.

Following similar lines as in [Yu1, KY], we construct, for $0\leq i\leq d$,
irreducible representations ${\boldsymbol{\tau}}_{i}$ and
${\boldsymbol{\tau}}_{i}^{\prime}$ of $K^{i}$ inductively as follows. First we
set ${\boldsymbol{\tau}}_{0}^{\prime}:={\boldsymbol{\tau}}_{\mathcal{D}^{0}}$
and
${\boldsymbol{\tau}}_{0}:={\boldsymbol{\tau}}_{0}^{\prime}\otimes\phi_{0}$.
Let $i\in\\{1,\ldots,d\\}$. We suppose that ${\boldsymbol{\tau}}_{i-1}$ and
${\boldsymbol{\tau}}_{i-1}^{\prime}$ are already constructed, and that
${\boldsymbol{\tau}}_{i-1}^{\prime}|_{G^{i-1}_{\iota(y),r_{i-1}}}$ is
$\mathbf{1}$-isotypic. Let $\hat{\phi}_{i-1}$ be the extension of
$\phi_{i-1}|_{K_{G^{0}}G^{i-1}_{\iota(y),0}}$ to
$K_{G^{0}}G^{i-1}_{\iota(y),0}G_{y,s_{i-1}+}$ defined as in [Yu1, §4].

Since every element of $J^{i}/J^{i}_{+}$ has order dividing $p$, we can view
$J^{i}/J^{i}_{+}$ as an ${\mathbb{F}_{p}}$-vector space. We define a pairing
$\langle\,,\,\rangle$ on $J^{i}/J^{i}_{+}$ by

(3.1.6) $\langle
j_{1},j_{2}\rangle:=\hat{\phi}_{i-1}(j_{1}j_{2}j_{1}^{-1}j_{2}^{-1})=[j_{1},j_{2}].$

By [BT1, Prop. 6.4.44], it is well-defined, since $[J^{i},J^{i}]$ is contained
in $J^{i}_{+}$, and by [Yu1, Lem. 11.1], it is non-degenerate. For any $j\in
J_{+}^{i}$, the kernel of $\hat{\phi}_{i-1}$ contains $j^{p}$. Hence the order
of every element of $\hat{\phi}_{i-1}(J^{i}_{+})$ divides $p$. Since
$\hat{\phi}_{i-1}(J^{i}_{+})$ is a nontrivial subgroup of
$\mathbb{C}^{\times}$, this shows that $\hat{\phi}_{i-1}(J^{i}_{+})$ is
isomorphic to ${\mathbb{F}_{p}}$. Thus we can view $\langle\,,\,\rangle$ as a
non-degenerate symplectic form on the ${\mathbb{F}_{p}}$-vector space
$V_{i}:=J^{i}/J^{i}_{+}$. Let $J^{i}_{+}(\hat{\phi}_{i-1})$ denote the
intersection of $J^{i}_{+}$ with the kernel of $\hat{\phi}_{i-1}$. From [Yu1,
Prop. 11.4] there is a canonical isomorphism

(3.1.7) $J^{i}/J^{i}_{+}(\hat{\phi}_{i-1})\longrightarrow
V_{i}\times(J^{i}_{+}/J^{i}_{+}(\hat{\phi}_{i-1}))\simeq H(V_{i}),$

where $H(V_{i})$ is the Heisenberg group defined to be the set
$V_{i}\times{\mathbb{F}_{p}}$ with the multiplication
$(v,t)(v^{\prime},t^{\prime}):=(v+v^{\prime},t+t^{\prime}+\frac{1}{2}\langle
v,v^{\prime}\rangle)$. By combining (3.1.7) with the map
$K^{i-1}\to\operatorname{Sp}(V_{i})$ induced by the conjugation, we define a
map

(3.1.8) $K^{i-1}\ltimes J^{i}\to(J^{i}/J^{i}_{+}(\hat{\phi}_{i-1}))\rtimes
K^{i-1}\to H(V_{i})\rtimes\operatorname{Sp}(V_{i}).$

By combining (3.1.8) with the Weil representation of
$H(V_{i})\rtimes\operatorname{Sp}(V_{i})$ with central character
$\hat{\phi}_{i-1}$, we obtain a representation $\widetilde{\phi}_{i-1}$ of
$K^{i-1}\ltimes J^{i}$. Let $\mathrm{infl}(\phi_{i-1})$ denote the inflation
of $\phi_{i-1}$ via the map $K^{i-1}\ltimes J^{i}\to K^{i-1}$. Then
$\mathrm{infl}(\phi_{i-1})\otimes\widetilde{\phi}_{i-1}$ factors through the
map $K^{i-1}\ltimes J^{i}\to K^{i-1}J^{i}=K^{i}$. Let $\phi^{\prime}_{i-1}$
denote the representation of $K^{i}$ whose inflation to $K^{i-1}\ltimes J^{i}$
is $\mathrm{infl}(\phi_{i-1})\otimes\widetilde{\phi}_{i-1}$. If
$r_{i-1}<r_{i}$, the restriction of $\phi^{\prime}_{i-1}$ to
$G^{i}_{\iota(y),r_{i}}$ is $\mathbf{1}$-isotypic.

Let $\mathrm{infl}({\boldsymbol{\tau}}^{\prime}_{i-1})$ be the inflation of
${\boldsymbol{\tau}}^{\prime}_{i-1}$ via the map

(3.1.9) $K^{i}=K^{i-1}J^{i}\longrightarrow K^{i-1}J^{i}/J^{i}\simeq
K^{i-1}/G^{i-1}_{\iota(y),r_{i-1}}.$

Then we define
${\boldsymbol{\tau}}^{\prime}_{i}:=\mathrm{infl}({\boldsymbol{\tau}}_{i-1}^{\prime})\otimes\phi^{\prime}_{i-1}$,
which is trivial on $G^{i}_{\iota(y),r_{i}}$ if $r_{i-1}<r_{i}$, and
${\boldsymbol{\tau}}_{i}:={\boldsymbol{\tau}}_{i}^{\prime}\otimes\phi_{i}$.

Recall the tamely ramified twisted Levi sequence
$\vv\mathbf{L}:=\vv\mathbf{L}(\vv\mathbf{G})=(\mathbf{L}^{0},\ldots,\mathbf{L}^{d})$
of $\mathbf{L}$. We attach to $\vv\mathbf{L}$ the cuspidal datum for
$\mathbf{L}$:

(3.1.10) $\mathcal{D}_{\mathbf{L}}:=(\vv\mathbf{L},y,\vv
r,{\boldsymbol{\tau}}_{L^{0}},\vv\phi).$

For $i\in\\{1,\ldots,d\\}$, we define

(3.1.11) $J_{L}^{i}:=L^{i}\cap J^{i}\quad\text{and}\quad
J_{L,+}^{i}:=L^{i}\cap J_{+}^{i}.$

Let ${\boldsymbol{\epsilon}}_{y}^{\mathbf{L}^{i}|\mathbf{L}^{i-1}}$ denote the
character of $L^{i-1}_{[y]}J_{L}^{i}$ that is trivial on $J_{L}^{i}$, and
whose restriction to $L^{i-1}_{[y]}$ is
$\epsilon_{y}^{\mathbf{L}^{i}_{\mathrm{ad}}|\mathbf{L}^{i-1}_{\mathrm{ad}}}\circ\operatorname{pr}^{i-1}_{\mathrm{ad}}$,
where
$\epsilon_{y}^{\mathbf{L}^{i}_{\mathrm{ad}}|\mathbf{L}^{i-1}_{\mathrm{ad}}}$
is the character defined in (2.0.12) and
$\operatorname{pr}^{i-1}_{\mathrm{ad}}\colon(L_{\mathrm{ad}}^{i-1})_{[y]}\to(\mathbf{L}_{\mathrm{ad}}^{i-1})_{[y]}/(\mathbf{L}_{\mathrm{ad}}^{i-1})_{y,0+}$
is the natural projection. We set

(3.1.12) $K^{i}_{L}:=L_{y}^{0}L^{1}_{y,s_{0}}\cdots
L^{i}_{y,s_{i-1}}\quad\text{and}\quad
K^{i}_{L,+}:=L_{y,0+}^{0}L^{1}_{y,s_{0}+}\cdots L^{i}_{y,s_{i-1}+}.$

For $1\leq j\leq i$, we define $\epsilon_{y}^{K_{L}^{i}}$ to be the quadratic
character of $K^{i}_{L}$ whose restriction to $L^{0}_{\iota(y)}$ is given by
${\boldsymbol{\epsilon}}^{\mathbf{L}^{i}|\mathbf{L}^{i-1}}_{y}|_{L^{0}_{y}}$
and that is trivial on $L^{1}_{y,s_{0}}\cdots L^{i}_{y,s_{i-1}}$. The datum
$\mathcal{D}_{\mathbf{L}}$ gives a supercuspidal type in $L^{i}$ as follows.
Let $\widetilde{K}_{L}^{i}$ denote the normalizer in $L^{i}$ of $K_{L}^{i}$.
The group $\widetilde{K}_{L}^{i}$ is a compact modulo center subgroup of $L$.
We set

(3.1.13)
${\boldsymbol{\tau}}_{L}^{i}:={\boldsymbol{\tau}}^{i}|_{K_{L}}\quad\text{and}\quad\sigma_{\mathcal{D}_{L}}^{i}:=\operatorname{ind}_{\widetilde{K}_{L}^{i}}^{L}(\epsilon_{y}^{K_{L}^{i}}{\boldsymbol{\tau}}^{i}_{L}).$

Then $\sigma_{\mathcal{D}_{L}}^{i}$ is an irreducible supercuspidal
representation of $L^{i}$ (see [FKS, proof of Th. 4.1.13]).

###  3.2. Types and Bernstein decomposition

Let $\mathbf{L}$ be an $F$-rational Levi subgroup of an $F$-rational parabolic
of $\mathbf{G}$. Let ${\mathfrak{X}}_{\mathrm{nr}}(L)$ denote the group of
unramified characters of $L$. Let $\sigma$ be an irreducible supercuspidal
smooth representation of $L$. We write:

  * •

$(L,\sigma)_{G}$ for the $G$-conjugacy class of the pair $(L,\sigma)$;

  * •

$\mathfrak{s}:=[L,\sigma]_{G}$ for the $G$-conjugacy class of the pair
$(L,{\mathfrak{X}}_{\mathrm{nr}}(L)\cdot\sigma)$.

We denote by $\mathfrak{B}(G)$ the set of such classes $\mathfrak{s}$. We set
$\mathfrak{s}_{L}:=[L,\sigma]_{L}$.

Let $\mathfrak{R}(G)$ denote the category of smooth representations of $G$,
and let $\mathfrak{R}^{\mathfrak{s}}(G)$ be the full subcategory of
$\mathfrak{R}(G)$ whose objects are the representations $(\pi,V)$ such that
every $G$-subquotient of $\pi$ is equivalent to a subquotient of a
parabolically induced representation
$\operatorname{i}_{P}^{G}(\sigma^{\prime})$, where $\operatorname{i}_{P}^{G}$
is the functor of normalized parabolic induction and
$\sigma^{\prime}\in{\mathfrak{X}}_{\mathrm{nr}}(M)\cdot\sigma$. By [Ber], the
categories $\mathfrak{R}^{\mathfrak{s}}(G)$ are indecomposable and split the
category $\mathfrak{R}(G)$ into a direct product:

(3.2.1)
$\mathfrak{R}(G)=\prod_{\mathfrak{s}\in\mathfrak{B}(G)}\mathfrak{R}^{\mathfrak{s}}(G).$

Every object of $\mathfrak{R}^{\mathfrak{s}}(G)$ has depth $0pt(\sigma)$ (see
[DB1, Lem.5.2.4 and Lem.5.2.6] for instance). Let $\operatorname{Irr}(G)$
(resp. $\operatorname{Irr}_{\mathrm{sc}}(G)$) denote the set of classes of
irreducible (resp. irreducible supercuspidal) objects in $\mathfrak{R}(G)$,
and by $\operatorname{Irr}^{\mathfrak{s}}(G)$ the classes of irreducible
objects in $\mathfrak{R}^{\mathfrak{s}}(G)$, i.e. the equivalence classes of
irreducible representations whose supercuspidal support lies in
$\mathfrak{s}$, and call $\operatorname{Irr}^{\mathfrak{s}}(G)$ the Bernstein
series attached to $\mathfrak{s}$. By (3.2.1), Bernstein series give a
partition of the set $\operatorname{Irr}(G)$ of isomorphism classes of
irreducible smooth irreducible representations of $G$:

(3.2.2)
$\operatorname{Irr}(G)=\bigsqcup_{\mathfrak{s}\in\mathfrak{B}(G)}\operatorname{Irr}^{\mathfrak{s}}(G).$

Let $\mathbf{P}=\mathbf{L}\mathbf{U}$ be a parabolic subgroup of $\mathbf{G}$
defined over $F$ with Levi factor $\mathbf{L}$, and let
$\overline{\mathbf{P}}=\mathbf{L}\overline{\mathbf{U}}$ be the opposite
parabolic subgroup. A compact open subgroup $K$ of $G$ is said to decompose
with respect to $(U,L,\overline{U})$ if $K=(K\cap U)\cdot(K\cap
L)\cdot(K\cap\overline{U}$ Let $K$ (resp. $K_{L}$) be a compact open subgroup
of $G$ (resp. $L$), and $\lambda$ (resp. $\lambda_{L}$) an irreducible smooth
representation of $K$ (resp. $K_{L}$). The pair $(K,\lambda)$ is called a
$G$-cover of the pair $(K_{L},\lambda_{L})$ (see [BK2, Blo]) if for any
opposite pair of parabolic subgroups $\mathbf{P}=\mathbf{L}\mathbf{U}$ and
$\overline{\mathbf{P}}=\mathbf{L}\overline{\mathbf{U}}$ with with Levi factor
$\mathbf{L}$, we have

  1. (1)

$K$ decomposes with respect to $(U,L,\overline{U})$;

  2. (2)

$\lambda|_{K_{L}}=\lambda_{L}$ and $K\cap
U,K\cap\overline{U}\subset\ker(\lambda)$;

  3. (3)

for any smooth representation $V$ of $G$, the natural map from $V$ to its
Jacquet module $V_{U}$ induces an injection on $V^{(K,\lambda)}$ the
$(K,\lambda)$-isotypic subspace of $V$.

A pair $(K,\lambda)$ is called an $\mathfrak{s}$-type for $G$ if the following
property is satisfied

(3.2.3) $\lambda$ occurs in the restriction of $\pi\in\operatorname{Irr}(G)$
if and only if $\pi\in\mathfrak{s}$.

We recall that if $(K_{L},\lambda_{L})$ is an $\mathfrak{s}_{L}$-type for $L$,
then any $G$-cover of $(K_{L},\lambda_{L})$ is an $\mathfrak{s}$-type for $G$
by [BK2, Th. 8.3].

We define $\epsilon_{y}^{K^{i}}$ to be the quadratic character of $K^{i}$
defined as

(3.2.4) $\epsilon^{K^{i}}:=\epsilon_{y}^{i}\,\circ\,\mathrm{q},$

where $\mathrm{q}$ is the isomorphism (3.1.5).

######  Theorem 3.2.5.

For each $i\in\\{0,\ldots,d\\}$, the pair
$(K_{L}^{i},\epsilon^{K_{L}^{i}}{\boldsymbol{\tau}}_{L}^{i})$ is an
$\mathfrak{s}_{L}$-type for $L^{i}$, and
$(K^{i},\epsilon^{K^{i}}{\boldsymbol{\tau}}^{i})$ is a $G^{i}$-cover of
$(K_{L}^{i},\epsilon^{K_{L}^{i}}{\boldsymbol{\tau}}_{L}^{i})$, thus it is an
$\mathfrak{s}^{i}$-type for
$\mathfrak{s}^{i}=[L^{i},\sigma_{\mathcal{D}_{L}}^{i}]_{G^{i}}$.

###### Proof.

The proof is analogous to that of [KY, Th. 7.5]. ∎

###  3.3. Regular representations

Let ${\underline{\mathbb{L}}}$ be a connected reductive group defined over the
finite field $k_{F}$, let ${\underline{\mathbb{T}}}$ be a maximal torus of
${\underline{\mathbb{L}}}$. In [DL, Def. 5.15], Deligne and Lusztig defined
two regularity conditions for a character $\theta$ of
$\mathbb{T}:={\underline{\mathbb{T}}}(k_{F})$, which we now recall:

  * •

$\theta$ is said to be in general position if its stabilizer in
$(\operatorname{N}_{\underline{\mathbb{G}}}({\underline{\mathbb{T}}})/{\underline{\mathbb{T}}})(k_{F})$
is trivial.

  * •

$\theta$ is said to be non-singular if it is not orthogonal to any coroot.

By [DL, Prop. 5.16], if the centre of $\mathbf{G}$ is connected, $\theta$ is
non-singular if and only if it is in general position.

Let $\mathbf{S}$ be a maximal torus of $\mathbf{L}$ which defined over $F$,
and let $\mathbf{T}$ denote the maximal unramified subtorus of $\mathbf{S}$.
Let $F^{\prime}/F$ be an unramified extension splitting $\mathbf{T}$ and
$R_{\mathrm{res}}(\mathbf{L},\mathbf{T})$ denote the set of restrictions to
$\mathbf{T}$ of the absolute roots in $R(\mathbf{L},\mathbf{S})$. Let $y$ be a
vertex in $\mathcal{B}(\mathbf{L}/{\mathrm{Z}}_{\mathbf{L}},F)$, and let
${\underline{\mathbb{T}}}$ be an elliptic maximal torus of
${\underline{\mathbb{L}}}_{y}$. By [DB2, Lem. 2.3.1] there exists a maximally
unramified elliptic maximal torus $\mathbf{T}$ of $\mathbf{G}$ such that the
reductive quotient of the special fiber of the connected Néron model of
$\mathbf{T}$ is ${\underline{\mathbb{T}}}$. In [Ka3, Def.3.4.16], Kaletha
introduced the following definition: a character $\theta$ of
${\underline{\mathbb{T}}}$ is called regular if its stabilizer in
$(\operatorname{N}_{\mathbf{L}}(\mathbf{T})/\mathbf{T})(F)$ is trivial. By
[Ka3, Fact 3.4.18], if $\theta$ is regular, then it is in general position.

Let $\vartheta$ be a depth-zero character of $S$. Let $S_{0}$ and $\mathbb{T}$
be defined as in (2.0.4) and (2.0.6). The character $\vartheta$ is said to be
regular if its restriction to $S_{0}$ equals the inflation of a regular
character of $\mathbb{T}$ (see [Ka3, Def. 3.4.16]).

Let $\sigma$ be any irreducible depth-zero supercuspidal representation of
$L$. There exists a vertex $y\in\mathcal{B}_{\mathrm{red}}(\mathbf{L},F)$ and
an irreducible cuspidal representation $\tau$ of
$\mathbb{L}_{y}:={\underline{\mathbb{L}}}_{y}(k_{F})$, such that the
restriction of $\sigma$ to $L_{y,0}$ contains the inflation
$\mathrm{infl}(\tau)$ of $\tau$ (see [Mor, §1-2] or [MP2, Prop. 6.6]). The
normalizer ${\mathrm{N}}_{L}(L_{y,0})$ of $L_{y,0}$ in $L$ is a totally
disconnected group that is compact mod center, which, by [BT2, proof of
(5.2.8)], coincides with the fixator $L_{[y]}$ of $[y]$ under the action of
$L$ on the reduced building of $\mathbf{L}$. Let ${\boldsymbol{\tau}}$ denote
an extension of $\mathrm{infl}(\tau)$ to $L_{[y]}$. Then $\sigma$ is compactly
induced from a representation of ${\mathrm{N}}_{L}(L_{y,0})$:

(3.3.1) $\sigma=\operatorname{c-Ind}_{L_{[y]}}^{L}({\boldsymbol{\tau}}).$

The representation $\sigma$ is called regular if $\tau=\pm
R_{\underline{\mathbb{T}}}^{{\underline{\mathbb{L}}}_{y,0}}(\theta)$ for some
pair $({\underline{\mathbb{T}}},\theta)$, where ${\underline{\mathbb{T}}}$ is
an elliptic maximal torus of ${\underline{\mathbb{L}}}_{y,0}$ and $\theta$ is
a regular character of $\mathbb{T}$ (see [Ka3, Def. 3.4.19]).

A supercuspidal representation of $L$ is called regular if it arises via the
construction described in §3.1 from a cuspidal $L$-datum $\mathcal{D}^{L}$
such that the representation $\sigma^{0}$ of $L^{0}$ is regular (see [Ka3,
Def. 3.7.9]).

######  Definition 3.3.2.

An irreducible smooth representation $\pi$ of $G$ is regular if its
supercuspidal support is regular, i.e. if
$\pi\in\operatorname{Irr}^{\mathfrak{s}}(G)$ with
$\mathfrak{s}=[L,\sigma]_{L}$ such that $\sigma$ is regular.

######  Proposition 3.3.3.

Any irreducible smooth representation of $\mathrm{GL}_{N}(F)$ is regular.

###### Proof.

Let $G:=\mathrm{GL}_{N}(F)$, let $\mathfrak{s}\in\mathfrak{B}(G)$ and let
$\pi\in\operatorname{Irr}^{\mathfrak{s}}(G)$. There is a partition
$(N_{1},N_{2},\ldots,N_{r})$ of $N$ such that $\mathfrak{s}=[L,\sigma]_{G}$
with

$L=\mathrm{GL}_{N_{1}}(F)\times\mathrm{GL}_{N_{2}}(F)\times\cdots\times\mathrm{GL}_{N_{r}}(F)\quad\text{and}\quad\sigma=\sigma_{1}\otimes\sigma_{2}\otimes\cdots\otimes\sigma_{r},$

with
$\mathfrak{s}_{i}:=[\mathrm{GL}_{N_{i}}(F),\sigma_{i}]_{\mathrm{GL}_{N_{i}(F)}}\in\mathfrak{B}(\mathrm{GL}_{N_{i}(F)})$.
The representation $\sigma$ is regular if and only if the representations
$\sigma_{1}$, $\ldots$, $\sigma_{r}$ are all regular. Thus, by Definition
3.3.2, the representation $\pi$ is regular if and only if the representations
$\sigma_{1}$, $\ldots$, $\sigma_{r}$ are all regular. Thus we are reduced to
show that any supercuspidal irreducible smooth representation $\pi$ of
$\mathrm{GL}_{N}(F)$ is regular. Then the result follows from [Ka3,
Lem.3.7.7]. We can also recover it as follows. By applying the construction
above with $L=G$, we obtain that $G^{0}\simeq\mathrm{GL}_{N_{E}}(E)$, where
$E/F$ is finite tamely ramified extension (here $N_{E}=N/[E:F]$) and
$\mathbb{G}^{0}_{y}$ is isomorphic to $\mathrm{GL}_{N_{E}}(k_{E})$. Any
cuspidal irreducible representation of $\mathrm{GL}_{N_{E}}(k_{E})$ is up to
sign a Deligne-Lusztig character induced from a character $\theta_{E}$ in
general position of the $k_{E}$-points of an elliptic maximal torus, but
equivalently $\theta_{E}$ is regular. ∎

######  Remark 3.3.4.

When $G=\mathrm{GL}_{N}(F)$ we have also at our disposal the Bushnell-Kutzko
construction of supercuspidal irreducible smooth representation in [BK1],
which works for any prime number $p$. By [MY], this construction coincides
with the one in [Yu1] when $p$ does not divide the order of the Weyl group of
$G$.

Proposition 3.3.3 extends to this more general setting as follows. Let $\pi$
be a supercuspidal irreducible smooth representation of $\mathrm{GL}_{N}(F)$.
By [BK1, Chap.6], for $p$ arbitrary, the representation $\pi$ is compactly
induced from a representation $\Lambda$ of
$E^{\times}J(\beta,{\mathfrak{A}})$, where $J(\beta,{\mathfrak{A}})$ is a
compact open subgroup of $\mathrm{GL}_{N}(F)$ and $\Lambda$ is an extension of
an irreducible smooth representation $\lambda$ of $J(\beta,{\mathfrak{A}})$.
Here $E/F$ is a (possibly wildly ramified) finite extension, $\beta$ an
element of $\operatorname{M}_{n}(F)$ and ${\mathfrak{A}}$ an hereditary
$\mathfrak{o}_{F}$-order such that $[{\mathfrak{A}},n,0,\beta]$ is a simple
stratum. There is a subgroup $J^{1}(\beta,{\mathfrak{A}})$ such that

(3.3.5) $J(\beta,{\mathfrak{A}})/J^{1}(\beta,{\mathfrak{A}})\simeq
G^{\beta}_{y_{\mathfrak{A}},0}/G^{\beta}_{y_{\mathfrak{A}},0+},$

where $G^{\beta}\simeq\mathrm{GL}_{N_{E}}(E)$ is the centralizer of $E$ in
$\mathrm{GL}_{N}(F)$ and $G^{\beta}_{y_{\mathfrak{A}},0}$ denotes the
parahoric subgroup of $G^{\beta}$ associated to a vertex $y_{\mathfrak{A}}$ in
$\mathcal{B}(G^{\beta},E)$ which corresponds to ${\mathfrak{A}}\cap
G^{\beta}$. Moreover, $\lambda=\mathrm{infl}(\tau_{E})\otimes\kappa$, where
$\mathrm{infl}(\tau_{E})$ is the inflation to $G^{\beta}_{y_{\mathfrak{A}},0}$
of an irreducible cuspidal representation of
$G^{\beta}_{y_{\mathfrak{A}},0}/G^{\beta}_{y_{\mathfrak{A}},0+}\simeq\mathrm{GL}_{N_{E}}(k_{E})$,
and $\kappa$ is certain representation of $J(\beta,{\mathfrak{A}})$ that
extends a Heisenberg representation. The normalizer of
$G^{\beta}_{y_{\mathfrak{A}},0}$ coincides with
$E^{\times}G^{\beta}_{y_{\mathfrak{A}},0}$. Since $\tau_{E}$ is regular, the
depth-zero representation
$\operatorname{c-Ind}_{E^{\times}G^{\beta}_{y_{\mathfrak{A}},0}}^{G^{\beta}}({\boldsymbol{\tau}}_{E})$
is a regular supercuspidal irreducible representation of $G^{\beta}$.

###  3.4. Non-singular representations

As in [Ka4, Def. 3.1.1], we call a depth-zero character $\vartheta\colon
S\to\mathbb{C}^{\times}$ $F$-non-singular if for every $\alpha\in
R_{\mathrm{res}}(\mathbf{L},\mathbf{T})$ the character
$\vartheta\circ\operatorname{N}_{F^{\prime}|F}\circ\alpha^{\vee}\colon
F^{\prime\times}\to\mathbb{C}^{\times}$ has non-trivial restriction to
$\mathfrak{o}_{F^{\prime}}$. The character $\vartheta$ factors through a
character $\theta$ of $\mathbb{T}$. We denote the restriction of $\theta$ to
${\underline{\mathbb{S}}}^{\circ}(k_{F})$ by $\theta^{\circ}$. The character
$\vartheta$ is called $k$-non-singular, if for every $\overline{\alpha}\in
R({\underline{\mathbb{S}}}^{\circ},{\underline{\mathbb{L}}}_{y})$ the
character
$\theta^{\circ}\circ\operatorname{N}_{k_{F^{\prime}}|k_{F}}\circ\overline{\alpha}^{\vee}\colon
k_{F^{\prime}}^{\times}\to\mathbb{C}^{\times}$ is non-trivial. As noticed in
[Ka4, Rem. 3.1.2], these notions do not depend on the choice of $F^{\prime}$.

Let $\vartheta\colon S\to\mathbb{C}^{\times}$ be a character. The pair
$(\mathbf{S},\vartheta)$ is said to be tame $F$-non-singular elliptic in
$\mathbf{L}$ (see [Ka4, Def. 3.4.1]) if

  * •

$\mathbf{S}$ is elliptic (i.e. $\mathbf{S}$ is anisotropic modulo
${\mathrm{Z}}_{\mathbf{L}}$) and its splitting extension $E/F$ is tame;

  * •

Inside the connected reductive subgroup $\mathbf{L}^{0}$ of $\mathbf{L}$ with
maximal torus $\mathbf{S}$ and root system

(3.4.1) $R_{0+}:=\\{\alpha\in
R(\mathbf{L},\mathbf{S})|\vartheta(\operatorname{N}_{E|F}(\alpha^{\vee}(E_{0+}^{\times})))=1\\},$

the torus $\mathbf{S}$ is maximally unramified.

  * •

The character $\vartheta$ is $F$-non-singular with respect to
$\mathbf{L}^{0}$.

In [Ka3, Ka4], Kaletha describes how to construct supercuspidal
representations $\sigma_{\mathbf{S},\vartheta}$ of $L$ from tame $F$-non-
singular elliptic pairs $(\mathbf{S},\vartheta)$ in $\mathbf{L}$. The
representation $\sigma_{\mathbf{S},\vartheta}$ is obtained in two steps. One
starts by unfolding the pair $(\mathbf{S},\vartheta)$ into a cuspidal
$\mathbf{L}$-datum $\mathcal{D}_{\mathbf{S},\vartheta}:=(\vv\mathbf{L},y,\vv
r,\sigma^{0},\vv\phi)$. The properties of $\mathbf{S}$ and $\vartheta$ allow
to go to the reductive quotient and use the theory of Deligne-Lusztig cuspidal
representations in order to construct $\sigma^{0}$, the so-called depth-zero
part of the datum $\mathcal{D}_{\mathbf{S},\vartheta}$. The second step
involves the construction described in §3.1 to the obtained
$\mathbf{L}$-datum.

Let ${\underline{\mathbb{U}}}$ be the unipotent radical of a Borel subgroup of
${\underline{\mathbb{L}}}_{y}$. The corresponding Deligne-Lusztig variety is

(3.4.2)
$Y_{\underline{\mathbb{U}}}^{{\underline{\mathbb{L}}}_{y}}:=\left\\{l{\underline{\mathbb{U}}}\in{\underline{\mathbb{L}}}_{y}/{\underline{\mathbb{U}}}\,:\,l^{-1}\mathrm{Fr}(l)\in{\underline{\mathbb{U}}}\cdot\mathrm{Fr}({\underline{\mathbb{U}}})\right\\}.$

The group $\mathbb{L}_{y}={\underline{\mathbb{L}}}_{y}(k_{F})$ acts on
$Y_{\underline{\mathbb{U}}}^{{\underline{\mathbb{L}}}_{y}}$ by the left
multiplication and $\mathbb{T}={\underline{\mathbb{T}}}(k_{F})$ acts on
$Y_{\underline{\mathbb{U}}}^{{\underline{\mathbb{L}}}_{y}}$ by right
multiplication. So these groups act on the $\ell$-adic cohomology group
$H^{i}_{c}(Y_{\underline{\mathbb{U}}}^{{\underline{\mathbb{L}}}_{y}},\overline{\mathbb{Q}}_{\ell})$.
We define

(3.4.3)
$H_{c}^{d_{\underline{\mathbb{U}}}}(Y_{\underline{\mathbb{U}}}^{\underline{\mathbb{L}}},\overline{\mathbb{Q}}_{\ell})_{\theta}:=\left\\{h\in
H^{i}_{c}(Y_{\underline{\mathbb{U}}}^{{\underline{\mathbb{L}}}_{y}},\overline{\mathbb{Q}}_{\ell})\,:\,h\cdot
s=\theta(s)h\text{ for all $s\in\mathbb{T}$}\right\\}.$

Let
$\kappa_{({\underline{\mathbb{T}}},\theta)}^{{\underline{\mathbb{L}}}_{y}}$ be
the isomorphism class of the representation
$H_{c}^{d_{\underline{\mathbb{U}}}}(Y_{\underline{\mathbb{U}}}^{\underline{\mathbb{L}}},\overline{\mathbb{Q}}_{\ell})_{\theta}$.
The representation
$\sigma_{({\underline{\mathbb{T}}},\theta)}:=\operatorname{c-Ind}_{L_{y}}^{L}\mathrm{infl}(\kappa_{({\underline{\mathbb{T}}},\theta)}^{\mathbb{L}_{y}})$
is supercuspidal but not necessarily irreducible.

Let ${\mathrm{N}}_{\mathbb{L}_{[y]}}({\underline{\mathbb{T}}},\theta)$ denote
the stabilizer of the pair $({\underline{\mathbb{S}}},\theta)$ in
$\mathbb{L}_{[y]}$ by the conjugate action, and let
$\operatorname{Irr}_{\theta}({\mathrm{N}}_{\mathbb{L}_{[y]}}({\underline{\mathbb{S}}},\theta))$
be the set of irreducible representations of
${\mathrm{N}}_{\mathbb{L}_{[y]}}({\underline{\mathbb{T}}},\theta)$ whose
restriction to $\mathbb{S}$ is $\theta$-isotypic. When
$\sigma_{({\underline{\mathbb{T}}},\theta)}$ is reducible, it decomposes as
[Ka4, 3.3.3]

(3.4.4)
$\sigma_{({\underline{\mathbb{T}}},\theta)}=\sum\limits_{\rho\in\operatorname{Irr}_{\theta}({\mathrm{N}}_{\mathbb{L}_{[y]}}({\underline{\mathbb{T}}},\theta))}\dim(\rho)\sigma_{({\underline{\mathbb{T}}},\theta,\rho)}^{\varepsilon},$

where the constituents
$\sigma_{({\underline{\mathbb{T}}},\theta,\rho)}^{\varepsilon}:=\operatorname{c-Ind}_{L_{y}}^{L}\mathrm{infl}(\kappa_{({\underline{\mathbb{T}}},\theta,\rho)}^{\mathbb{L}_{y},\varepsilon})$,
constructed from
$\kappa_{({\underline{\mathbb{T}}},\theta,\rho)}^{\varepsilon}$ as in [Ka4,
Def. 2.7.6], are irreducible non-singular supercuspidal representations. Here
$\varepsilon$ is a fixed coherent splitting of the family of $2$-cocycles
$\\{\eta_{\Psi,{\underline{\mathbb{U}}}}\\}$ as in [Ka4, §2.4]. The positive-
depth supercuspidals can be described similarly, by applying the construction
of §3.1 to the representation
$\sigma_{(L^{0},{\underline{\mathbb{T}}},\phi_{-1})}$ of $L^{0}$ associated to
the pair $({\underline{\mathbb{T}}},\phi_{-1})$ in [Ka4, (3.2)], where
$\phi_{-1}$ is a character of $S$ which is trivial if
$\mathbf{L}^{0}=\mathbf{S}$ and otherwise satisfies $\phi_{-1}|_{S_{0+}}=1$.

Let $\sigma\in\operatorname{Irr}_{\mathrm{sc}}(L)$ be non-singular, and let
$\chi\in{\mathfrak{X}}_{{\mathrm{nr}}}(L)$. We observe that
$\sigma\otimes\chi$ is also non-singular. Then we say that
$\mathfrak{s}=[L,\sigma]_{G}\in\mathfrak{B}(G)$ is non-singular. In the
following definition, we extend the notions of regularity and non-singularity
to arbitrary irreducible smooth representations of $G$:

######  Definition 3.4.5.

An irreducible smooth representation $\pi$ of $G$ is non-singular if its
supercuspidal support is non-singular, i.e. if
$\pi\in\operatorname{Irr}^{\mathfrak{s}}(G)$ with
$\mathfrak{s}=[L,\sigma]_{L}$ such that $\sigma$ is non-singular.

######  Example 3.4.6.

We assume that the cardinality $q$ of $k_{F}$ is odd. Let $W=\\{1,w\\}$ denote
the Weyl group of $\mathrm{GL}_{2}$ with respect to a split maximal torus
$\widetilde{\mathbf{T}}$. We have
$\widetilde{\mathbf{T}}(k_{F})\simeq\mathbb{F}_{q}^{\times}\times\mathbb{F}_{q}^{\times}$.
Let $\widetilde{\mathbf{T}}_{w}$ be a torus of type $w$ with respect to
$\widetilde{\mathbf{T}}$. We have

$\widetilde{\mathbf{T}}_{w}(k_{F})\simeq\left\\{\left(\begin{smallmatrix}\lambda&0\cr
0&\lambda^{q}\end{smallmatrix}\right)\,:\,\lambda\in\mathbb{F}_{q^{2}}^{\times}\right\\}\simeq\mathbb{F}_{q^{2}}^{\times}.$

The irreducible cuspidal representations of $\mathrm{GL}_{2}(k_{F})$ are the
$-R_{\widetilde{\mathbf{T}}_{w}}^{\mathrm{GL}_{2}}(\widetilde{\omega})$, where
$\widetilde{\omega}$ denote a character of $\mathbb{F}_{q^{2}}^{\times}$ such
that $\widetilde{\omega}\neq\widetilde{\omega}^{q}$. Let $\mathbf{T}$ be the
split maximal torus of $\operatorname{SL}_{2}$. We have
$\mathbf{T}(k_{F})\simeq\mu_{q-1}$. Let $\mathbf{T}_{w}$ be a torus of type
$w$ with respect to $\mathbf{T}$. We have

$\mathbf{T}_{w}({\mathbb{F}_{q}})\simeq\left\\{\left(\begin{smallmatrix}\lambda&0\cr
0&\lambda^{q}\end{smallmatrix}\right)\,:\,\lambda^{q+1}=1\right\\}\simeq\mu_{q+1}.$

Let $\omega$ be a character of $\mu_{q+1}$. Then we have

(3.4.7)
$-R_{T_{w}}^{\operatorname{SL}_{2}}(\omega)=-\operatorname{Res}^{\mathrm{GL}_{2}(k_{F})}_{\operatorname{SL}_{2}(k_{F})}(R_{\widetilde{\mathbf{T}}_{w}}^{\mathrm{GL}_{2}}(\widetilde{\omega})).$

If $\omega^{2}\neq 1$, then the representation
$\tau:=-R_{T_{w}}^{\operatorname{SL}_{2}}(\omega)$ is an irreducible cuspidal
representation of $\operatorname{SL}_{2}({\mathbb{F}_{q}})$. If
$\omega=\omega_{0}$ is the quadratic character of $\mu_{q+1}$, then

(3.4.8) $-R_{T_{w}}^{\operatorname{SL}_{2}}(\omega)=\tau_{+}+\tau_{,}$

where $\tau_{+}$ and $\tau_{-}$ are irreducible cuspidal representations of
$\operatorname{SL}_{2}(k_{F})$ of dimension $\frac{q-1}{2}$ (see [DM, Table
2]). Let $K:=\operatorname{SL}_{2}(\mathfrak{o}_{F})$ and
$K^{\prime}:=uKu^{-1}$ where $u=\left(\begin{smallmatrix}1&0\cr
0&\varpi_{F}\end{smallmatrix}\right)$.Then $\\{K,K^{\prime}\\}$ is a set of
representatives of the maximal parahoric subgroups of
$\operatorname{SL}_{2}(F)$, and the reductive quotients of $K$ and
$K^{\prime}$ are both isomorphic to $\operatorname{SL}_{2}(k_{F})$. We denote
by $\mathrm{infl}(\tau)$ and $\mathrm{infl}(\tau)^{\prime}$ the inflations of
$\tau$ to $K$ and $K^{\prime}$ respectively, and by
$\mathrm{infl}(\tau_{\pm})$ and $\mathrm{infl}(\tau_{\pm})^{\prime}$ the
inflations of $\tau_{\pm}$ to $K$ and $K^{\prime}$ respectively. Thus, the
depth-zero supercuspidal representations of $\operatorname{SL}_{2}(F)$ are the
representations $\operatorname{c-Ind}_{K}^{G}(\mathrm{infl}(\tau))$,
$\operatorname{c-Ind}_{K^{\prime}}^{G}(\mathrm{infl}(\tau)^{\prime})$, which
are regular, and the four representations
$\operatorname{c-Ind}_{K}^{G}(\mathrm{infl}(\tau_{\pm}))$,
$\operatorname{c-Ind}_{K^{\prime}}^{G}(\mathrm{infl}(\tau_{\pm})^{\prime})$,
which are non-regular non-singular.

##  4\. Langlands correspondence and Bernstein Center

###  4.1. Generalized Springer correspondence

Let $\mathcal{G}$ be a complex (possibly disconnected) reductive group, and
let $\mathcal{G}^{\circ}$ be the identity component of $\mathcal{G}$. If
$g\in\mathcal{G}$, we denote by $(g)_{\mathcal{G}}$ its
$\mathcal{G}$-conjugacy class. We denote by $\mathfrak{U}(\mathcal{G})$ the
unipotent variety of $\mathcal{G}$, and by
$\mathfrak{U}_{\mathrm{e}}(\mathcal{G})$ the set of $\mathcal{G}$-conjugacy
classes of pairs $(u,\rho)$, with $u\in\mathcal{G}$ unipotent and
$\rho\in\operatorname{Irr}(A_{\mathcal{G}}(u))$, where
$A_{\mathcal{G}}(u):={\mathrm{Z}}_{\mathcal{G}}(u)/{\mathrm{Z}}_{\mathcal{G}}(u)^{\circ}$
denotes the component group of the centralizer of $u$ in $\mathcal{G}$.

Let $\mathcal{P}_{0}$ be a parabolic subgroup of $\mathcal{G}^{\circ}$ with
unipotent radical $\mathcal{U}$, let $\mathcal{M}_{0}$ be a Levi factor of
$\mathcal{P}^{\circ}$, and let $v$ be a unipotent element in
$\mathcal{M}_{0}$. The group
${\mathrm{Z}}_{\mathcal{G}^{\circ}}(u)\times{\mathrm{Z}}_{\mathcal{M}_{0}}(v)\mathcal{U}$
acts on the variety

(4.1.1) $\mathcal{Y}_{u,v}:=\left\\{y\in\mathcal{G}^{\circ}\;:\;y^{-1}uy\in
v\mathcal{U}\right\\}$

by $(g,p)\cdot y:=gyp^{-1}$, with $g\in{\mathrm{Z}}_{\mathcal{G}^{\circ}}(u)$,
$p\in{\mathrm{Z}}_{\mathcal{M}_{0}}(v)\mathcal{U}$, and
$y\in\mathcal{Y}_{u,v}$. The group $A_{\mathcal{G}^{\circ}}(u)\times
A_{\mathcal{M}_{0}}(v)$ acts on the set of irreducible components of
$\mathcal{Y}_{u,v}$ of maximal dimension, and we denote by $\sigma_{u,v}$ the
corresponding permutation representation.

Let $\langle\;,\;\rangle_{A_{\mathcal{G}^{\circ}}(u)}$ be the usual scalar
product of the space of class functions on the finite group
$A_{\mathcal{G}^{\circ}}(u)$ with values in $\overline{\mathbb{Q}}_{\ell}$.
Let $\rho^{\circ}\in\operatorname{Irr}(A_{\mathcal{G}^{\circ}}(u))$. The
representation $\rho^{\circ}$ is called cuspidal if
$\langle\rho^{\circ},\sigma_{u,v}\rangle_{A_{\mathcal{G}^{\circ}}(u)}\neq 0$
for any choice of $v$ implies that $\mathcal{P}_{0}=\mathcal{G}^{\circ}$. We
denote by $\mathfrak{B}(\mathfrak{U}_{\mathrm{e}}(\mathcal{G}^{\circ}))$ the
set of $\mathcal{G}^{\circ}$-conjugacy classes of triples
$\mathfrak{t}^{\circ}:=(\mathcal{M}_{0},v,\rho^{\circ}_{\mathrm{cusp}})$ such
that $\mathcal{M}_{0}$ is a Levi subgroup of (a parabolic subgroup of)
$\mathcal{G}^{\circ}$, and $(v,\rho^{\circ}_{\mathrm{cusp}})$ is a cuspidal
unipotent pair in $\mathcal{M}_{0}$, i.e. $v$ is a unipotent element in
$\mathcal{M}_{0}$, and $\rho^{\circ}_{\mathrm{cusp}}$ is a cuspidal
irreducible representation of $A_{\mathcal{M}_{0}}(v)$. Let
$\mathfrak{t}^{\circ}\in\mathfrak{B}(\mathfrak{U}_{\mathrm{e}}(\mathcal{G}^{\circ}))$
and
$W_{\mathfrak{t}^{\circ}}:=\operatorname{N}_{\mathcal{G}^{\circ}}(\mathfrak{t}^{\circ})/\mathcal{M}_{0}$.
By [Lu2, §9.2], the group
$W_{\mathcal{M}_{0}}:=\operatorname{N}_{\mathcal{G}^{\circ}}(\mathcal{M}_{0})/\mathcal{M}_{0}$
is a Weyl group and $W_{\mathfrak{t}^{\circ}}\simeq W_{\mathcal{M}_{0}}$.

We consider now the case of the (possibly disconnected) group $\mathcal{G}$.
Let $u$ be a unipotent element in $\mathcal{G}^{\circ}$. We first observe that
${\mathrm{Z}}_{\mathcal{G}}(u)^{\circ}={\mathrm{Z}}_{\mathcal{G}^{\circ}}(u)^{\circ}$.
(The inclusion
${\mathrm{Z}}_{\mathcal{G}^{\circ}}(u)^{\circ}\subset{\mathrm{Z}}_{\mathcal{G}}(u)^{\circ}$
is obvious. On the other side, since ${\mathrm{Z}}_{\mathcal{G}}(u)^{\circ}$
is connected, it is contained in $\mathcal{G}^{\circ}$. Hence,
${\mathrm{Z}}_{\mathcal{G}}(u)^{\circ}$ is contained in
${\mathrm{Z}}_{\mathcal{G}^{\circ}}(u)$. But, since it is connected, it is
contained in ${\mathrm{Z}}_{\mathcal{G}^{\circ}}(u)$). It follows that the
natural inclusion
${\mathrm{Z}}_{\mathcal{G}^{\circ}}(u)\subset{\mathrm{Z}}_{\mathcal{G}}(u)$
induces an injection of $A_{\mathcal{G}^{\circ}}(u)$ into
$A_{\mathcal{G}}(u)$. A unipotent pair $(u,\rho)$ with
$u\in\mathcal{G}^{\circ}$ and $\rho\in\operatorname{Irr}(A_{\mathcal{G}}(u))$
is called cuspidal if (one of, or equivalently all) the irreducible
constituents of the restriction of $\rho$ to $A_{\mathcal{G}^{\circ}}(u)$ are
cuspidal.

We call quasi-Levi subgroup of $\mathcal{G}$ a subgroup $\mathcal{M}$ of the
form
$\mathcal{M}={\mathrm{Z}}_{\mathcal{G}}({\mathrm{Z}}_{\mathcal{L}}^{\circ})$,
with $\mathcal{L}$ a Levi subgroup of $\mathcal{G}^{\circ}$. We observe that
$\mathcal{M}^{\circ}=\mathcal{L}$. Note that a Levi subgroup of $\mathcal{G}$
is a quasi-Levi subgroup, and in the case when $\mathcal{G}$ connected, both
notions coincide. The group $\mathcal{M}$ is said to be cuspidal if there
exists a cuspidal unipotent pair in $\mathcal{M}$.

Let $\mathfrak{L}(\mathcal{G})$ be a set of representatives for the conjugacy
classes of quasi-Levi subgroups of $\mathcal{G}$, and let
$\mathfrak{L}_{\mathrm{cusp}}(\mathcal{G})$ denote the subset of the cuspidal
ones. Let $\mathfrak{B}(\mathfrak{U}_{\mathrm{e}}(\mathcal{G}))$ be the set of
$\mathcal{G}$-conjugacy classes of triples
$\mathfrak{t}:=(\mathcal{M},v,\rho_{\mathrm{cusp}})$, where $\mathcal{M}$ is a
cuspidal quasi-Levi subgroup of $\mathcal{G}$, and $(v,\rho_{\mathrm{cusp}})$
is a cuspidal unipotent pair in $\mathcal{M}$. Let
$\mathfrak{t}\in\mathfrak{B}(\mathfrak{U}_{\mathrm{e}}(\mathcal{G}))$. We set
$W_{\mathfrak{t}}:=\operatorname{N}_{\mathcal{G}}(\mathfrak{t})/\mathcal{M}$.
In [AMS1, §4] we constructed a bijection

(4.1.2)
$\nu_{\mathcal{G}}\colon\bigsqcup_{\mathfrak{t}\in\mathfrak{B}(\mathfrak{U}_{\mathrm{e}}(\mathcal{G}))}\operatorname{Irr}(\overline{\mathbb{Q}}_{\ell}[W_{\mathfrak{t}},\kappa_{\mathfrak{t}}])\,\overset{1-1}{\longrightarrow}\,\mathfrak{U}_{\mathrm{e}}(\mathcal{G}),$

where $\kappa_{\mathfrak{t}}\colon
W_{\mathfrak{t}}/W_{\mathfrak{t}}^{\circ}\times
W_{\mathfrak{t}}/W_{\mathfrak{t}}^{\circ}\to\overline{\mathbb{Q}}_{\ell}^{\times}$
is a $2$-cocycle and
$\overline{\mathbb{Q}}_{\ell}[W_{\mathfrak{t}},\kappa_{\mathfrak{t}}]$ is the
$\kappa_{\mathfrak{t}}$-twisted group algebra of $W_{\mathfrak{t}}$, which is
defined to be the $\overline{\mathbb{Q}}_{\ell}$-vector space
$\overline{\mathbb{Q}}_{\ell}[W_{\mathfrak{t}},\kappa_{\mathfrak{t}}]$ with
basis $\\{T_{w}\,:\,w\in W_{\mathfrak{t}}\\}$ and multiplication rules

(4.1.3)
$T_{w}T_{w^{\prime}}=\kappa_{\mathfrak{t}}(w,w^{\prime})T_{ww^{\prime}}\quad\text{for
any $w,w^{\prime}\in W_{\mathfrak{t}}$.}$

We will define a canonical map, called the cuspidal support map,

(4.1.4)
$\mathrm{Sc}_{\mathcal{G}}\colon\mathfrak{U}_{\mathrm{e}}(\mathcal{G})\longrightarrow\mathfrak{B}(\mathfrak{U}_{\mathrm{e}}(\mathcal{G})),$

and analyse its fibers. We first consider the $\mathcal{G}^{\circ}$-case.
Given a pair $(u,\rho^{\circ})$, by [Lu2, § 6.2], there exists a triple
$(\mathcal{P},\mathcal{L},v)$ as above and a cuspidal irreducible
representation $\rho_{\mathrm{cusp}}^{\circ}$ of $A_{\mathcal{L}}(v)$ such
that
$\langle\rho^{\circ}\otimes\widehat{\rho}_{\mathrm{cusp}}^{\circ},\sigma_{u,v}\rangle_{A_{\mathcal{G}}(u)\times
A_{\mathcal{L}}(v)}\neq 0$, where $\widehat{\rho}_{\mathrm{cusp}}^{\circ}$ is
the dual of $\rho_{\mathrm{cusp}}^{\circ}$. Moreover
$\mathfrak{t}:=(\mathcal{P},\mathcal{L},v,\rho_{\mathrm{cusp}}^{\circ})$ is
unique up to $\mathcal{G}^{\circ}$-conjugation (see [Lu2, Prop. 6.3]), and is
called the cuspidal support of $(u,\rho^{\circ})$. We define
$\mathrm{Sc}_{\mathcal{G}^{\circ}}$ as

(4.1.5)
$\begin{matrix}\mathrm{Sc}_{\mathcal{G}^{\circ}}\colon&\mathfrak{U}_{\mathrm{e}}(\mathcal{G}^{\circ})&\longrightarrow&\mathfrak{B}(\mathfrak{U}_{\mathrm{e}}(\mathcal{G}^{\circ}))\cr&(u,\rho^{\circ})&\mapsto&(u,\rho^{\circ})_{\mathcal{G}^{\circ}}.\end{matrix}$

We consider now the $\mathcal{G}$-case. We set
$\mathcal{M}:={\mathrm{Z}}_{\mathcal{G}}({\mathrm{Z}}_{\mathcal{L}}^{\circ})$.
Let $\mathcal{C}_{\mathcal{L}}(v)$ denote the $\mathcal{L}$-conjugacy class of
$v$ and $\mathcal{E}_{\rho_{\mathrm{cusp}}^{\circ}}$ the cuspidal
$\mathcal{L}$-equivariant local system on $\mathcal{C}_{\mathcal{L}}(v)$
corresponding to $\rho_{\mathrm{cusp}}^{\circ}$.

We denote by $\mathcal{C}_{\mathcal{M}}(v)$ the $\mathcal{M}$-conjugacy class
of $v$ and write
$\mathcal{X}_{v}:=\mathcal{C}_{\mathcal{M}}(v){\mathrm{Z}}_{\mathcal{L}}^{\circ}$.
It is a union of $\mathcal{M}$-conjugacy classes in $\mathcal{L}$. Tensoring
$\mathcal{E}_{\rho_{\mathrm{cusp}}^{\circ}}$ with the constant sheaf on
${\mathrm{Z}}_{\mathcal{L}}^{\circ}$, we obtain a $\mathcal{L}$-equivariant
local system on $\mathcal{X}_{v}$ that we still denote by
$\mathcal{E}_{\rho_{\mathrm{cusp}}^{\circ}}$. Next we build a
$\mathcal{M}$-equivariant local system on $\mathcal{X}_{v}$ as follows. Via
the map

(4.1.6) $\mathcal{X}_{v}\times\mathcal{M}\to\mathcal{X}_{v},\quad(x,m)\mapsto
m^{-1}xm,$

we pull $\mathcal{E}_{\rho_{\mathrm{cusp}}^{\circ}}$ back to a local system
$\widehat{\mathcal{E}}_{\rho_{\mathrm{cusp}}^{\circ}}$ on
$\mathcal{X}_{v}\times\mathcal{M}$. It is
$\mathcal{M}\times\mathcal{L}$-equivariant for the action

(4.1.7) $(m_{1},l)\cdot(x,m):=(m_{1}xm_{1}^{-1},m_{1}ml^{-1}).$

Since the $\mathcal{M}^{\circ}$-action is free, we can divide it out and
obtain a $\mathcal{M}$-equivariant local system
$\widetilde{\mathcal{E}}_{\rho_{\mathrm{cusp}}^{\circ}}$ on
$\mathcal{X}_{v}\times\mathcal{M}/\mathcal{M}^{\circ}$ the pull-back of which
under the natural quotient map is isomorphic to
$\widehat{\mathcal{E}}_{\rho_{\mathrm{cusp}}^{\circ}}$. Let
$\operatorname{pr}\colon\mathcal{X}_{v}\times\mathcal{M}/\mathcal{M}^{\circ}\to\mathcal{X}_{v}$
be the projection on the first coordinate. It is an $\mathcal{M}$-equivariant
fibration, so the direct image
$\operatorname{pr}_{*}\widetilde{\mathcal{E}}_{\rho_{\mathrm{cusp}}^{\circ}}$
is a $\mathcal{M}$-equivariant local system on $\mathcal{X}_{v}$. By [AMS1,
Prop. 4.5], there exists a canonical isomorphism

(4.1.8)
$f\colon\operatorname{End}_{\mathcal{G}}(\operatorname{pr}_{*}\widetilde{\mathcal{E}}_{\rho_{\mathrm{cusp}}^{\circ}})\overset{\sim}{\to}\mathbb{C}[W_{\mathfrak{t}},\kappa_{\mathfrak{t}}],$

and
$\operatorname{End}_{\mathcal{G}}(\operatorname{pr}_{*}\widetilde{\mathcal{E}}_{\rho_{\mathrm{cusp}}^{\circ}})$
contains naturally
$\operatorname{End}_{\mathcal{M}}(\operatorname{pr}_{*}\widetilde{\mathcal{E}}_{\rho_{\mathrm{cusp}}^{\circ}})$
as a subalgebra. As $\mathcal{M}/\mathcal{M}^{\circ}$ is normal in
$W_{\mathfrak{t}}$, the latter group acts on
$\operatorname{End}_{\mathcal{M}}(\operatorname{pr}_{*}\widetilde{\mathcal{E}}_{\rho_{\mathrm{cusp}}^{\circ}})$
by conjugation in $\mathbb{C}[W_{\mathfrak{t}},\kappa_{\mathfrak{t}}]$. Let
$E_{u,\rho}\in\operatorname{Irr}(\operatorname{End}_{\mathcal{G}}(\operatorname{pr}_{*}\widetilde{\mathcal{E}}_{\rho_{\mathrm{cusp}}^{\circ}}))$
which corresponds to $(u,\rho)$ by the generalized Springer representation,
i.e. $E_{u,\rho}$ is the fiber of $(u,\rho)$ under $\nu_{\mathcal{G}}\circ f$,
where $\nu_{\mathcal{G}}$ is the bijection defined in (4.1.2). Let
$E^{\prime}$ be a constituent of $E_{u,\rho}$ as an
$\operatorname{End}_{\mathcal{M}}(\operatorname{pr}_{*}\widetilde{\mathcal{E}}_{\rho_{\mathrm{cusp}}^{\circ}})$-representation.
We write
$\rho_{\mathrm{cusp}}:=(\operatorname{pr}_{*}\widetilde{\mathcal{E}}_{\rho_{\mathrm{cusp}}^{\circ}})_{E^{\prime}}$.
It is a cuspidal irreducible representation of the group $A_{\mathcal{M}}(v)$.
The cuspidal support of $(u,\rho)$ is defined to the triple
$(\mathcal{M},v,\rho_{\mathrm{cusp}})$ (see [AMS1, (65)]). It is unique up to
$\mathcal{G}$-conjugation.

###  4.2. The ordinary Springer correspondence

The bijection $\nu_{\mathcal{G}}$ extends to $\mathcal{G}$ the generalized
Springer correspondence for $\mathcal{G}^{\circ}$ defined by Lusztig in [Lu2].
The latter itself extends the Springer correspondence
$\nu_{\mathcal{G}^{\circ}}^{\mathrm{ord}}$ (sometimes called the ordinary
Springer correspondence) defined in [Sp, §6] and [Lu1].

We recall briefly the construction of the map
$\nu_{\mathcal{G}^{\circ}}^{\mathrm{ord}}$. Let $\mathcal{B}_{u}$ be the
variety of Borel subgroups of $\mathcal{G}^{\circ}$ which contain $u$.
According to Spaltenstein, all irreducible components of $\mathcal{B}_{u}$
have same dimension, say $e(u)$. The Weyl group $W_{\mathcal{G}^{\circ}}$ of
$\mathcal{G}^{\circ}$ acts on the top cohomology group
$H^{2e(u)}(\mathcal{B}_{u},\overline{\mathbb{Q}}_{\ell})$ by permuting the
irreducible components of $\mathcal{B}_{u}$. The actions of
$W_{\mathcal{G}^{\circ}}$ and $A_{\mathcal{G}^{\circ}}(u)$ commute, and we
obtain the irreducible representations of $W_{\mathcal{G}^{\circ}}$ by taking
the $A_{\mathcal{G}^{\circ}}(u)$-isotypic components of
$H^{2e(u)}(\mathcal{B}_{u},\overline{\mathbb{Q}}_{\ell})$. We define
$\mathfrak{U}_{\mathrm{e}}^{\mathrm{ord}}(\mathcal{G}^{\circ})$ to be the set
of pairs $(u,\rho^{\circ})$, where $u$ is a unipotent element of
$\mathcal{G}^{\circ}$ (up to conjugacy) and $\rho^{\circ}$ is an irreducible
representation of $A_{\mathcal{G}^{\circ}}(u)$, such that
$\operatorname{Hom}_{A_{\mathcal{G}^{\circ}}(u)}(\rho^{\circ},H^{2e(u)}(\mathcal{B}_{u},\overline{\mathbb{Q}}_{\ell}))\neq
0$. We obtain a bijection, say $\nu_{\mathcal{G}^{\circ}}^{\mathrm{ord}}$,
from the set of all irreducible representations of $W_{\mathcal{G}^{\circ}}$
to $\mathfrak{U}_{\mathrm{e}}^{\mathrm{ord}}(G)$. In particular, we have

(4.2.1)
$(u,1)_{\mathcal{G}^{\circ}}\in\nu_{\mathcal{G}^{\circ}}^{\mathrm{ord}}(\operatorname{Irr}(W_{\mathcal{G}^{\circ}}))$,
for any $u\in\mathfrak{U}(\mathcal{G}^{\circ})$.

Let $(u,\rho)\in\mathfrak{U}_{\mathrm{e}}(\mathcal{G})$. By [AMS1, Theorem
1.2], we can write $\rho=\rho^{\circ}\rtimes\eta$, with
$\rho^{\circ}\in\operatorname{Irr}(A_{\mathcal{G}^{\circ}}(u))$. Moreover,
where $\rho^{\circ}$ is uniquely determined by $\rho$ up to
$A_{\mathcal{G}}(u)$-conjugacy. We extend the definition of
$\mathfrak{U}_{\mathrm{e}}^{\mathrm{ord}}(\mathcal{G}^{\circ})$ to the case of
disconnected groups by setting

(4.2.2)
$\mathfrak{U}_{\mathrm{e}}^{\mathrm{ord}}(\mathcal{G}):=\left\\{(u,\rho)\,:\,\text{$(u,\rho^{\circ})\in\mathfrak{U}_{\mathrm{e}}^{\mathrm{ord}}(\mathcal{G}^{\circ})$}\right\\}/\mathcal{G}.$

######  Definition 4.2.3.

A triple
$\mathfrak{t}=(\mathcal{M},v,\rho_{\mathrm{cusp}})\in\mathfrak{B}(\mathfrak{U}_{\mathrm{e}}(\mathcal{G}))$
is called semisimple if the unipotent element $v$ is trivial.

######  Lemma 4.2.4.

The semisimple triples in
$\mathfrak{B}(\mathfrak{U}_{\mathrm{e}}(\mathcal{G}))$ are those of the form
$(\mathcal{M},1,\eta)$, where
$\mathcal{M}={\mathrm{Z}}_{\mathcal{G}}(\mathcal{T})$ with $\mathcal{T}$ a
maximal torus of $\mathcal{G}^{\circ}$, and
$\eta\in\operatorname{Irr}(\mathcal{M}/\mathcal{M}^{\circ})$.

###### Proof.

Let $\mathcal{T}$ be a maximal torus of $\mathcal{G}^{\circ}$ and let
$\eta\in\operatorname{Irr}(\mathcal{M}/\mathcal{M}^{\circ})$, where
$\mathcal{M}:={\mathrm{Z}}_{\mathcal{G}}(\mathcal{T})$. By definition the
restriction of $\eta$ to $\mathcal{M}^{\circ}=\mathcal{T}$ is the trivial
representation of $\mathcal{M}^{\circ}$, which is obviously cuspidal. Hence
the representation $\eta$ is cuspidal. Thus, the triple $(\mathcal{M},1,\eta)$
belongs to $\mathfrak{B}(\mathfrak{U}_{\mathrm{e}}(\mathcal{G}))$. It is
semisimple by definition.

Conversely, let
$\mathfrak{t}=(\mathcal{M},v,\rho_{\mathrm{cusp}})\in\mathfrak{B}(\mathfrak{U}_{\mathrm{e}}(\mathcal{G}))$,
where
$\mathcal{M}={\mathrm{Z}}_{\mathcal{G}}({\mathrm{Z}}_{\mathcal{L}}^{\circ})$
with $\mathcal{L}$ a Levi subgroup of $\mathcal{G}^{\circ}$ and $v$ the
trivial element of $\mathcal{L}$. By [Lu2, Prop. 2.8], the fact that
$(v,\rho_{\mathrm{cusp}})$ is cuspidal implies that $v$ is a distinguished
unipotent element in $\mathcal{L}$, that is, $v$ does not meet the unipotent
variety of any proper Levi subgroup of $\mathcal{L}$. Since $v$ is trivial, it
is only possible if $\mathcal{L}$ does not have proper Levi subgroups, that
is, $\mathcal{L}$ is a torus, say $\mathcal{T}$. Thus we have
${\mathrm{Z}}_{\mathcal{L}}=\mathcal{T}$, which gives
$\mathcal{M}={\mathrm{Z}}_{\mathcal{G}}(\mathcal{T})$. Since
${\mathrm{Z}}_{\mathcal{M}}(v)=\mathcal{M}$, we get
$A_{\mathcal{M}}(v)=\mathcal{M}/\mathcal{M}^{\circ}$. Let
$\rho\in\operatorname{Irr}(A_{\mathcal{G}}(u))$. By [AMS1, (51)], we can write

(4.2.5)
$\rho_{\mathrm{cusp}}=\rho_{\mathrm{cusp}}^{\circ}\rtimes\eta:=\operatorname{ind}_{A_{\mathcal{M}}(v)_{\rho^{\circ}_{\mathrm{cusp}}}}^{A_{\mathcal{M}}(v)}(\rho^{\circ}\otimes\eta),$

with
$\rho^{\circ}_{\mathrm{cusp}}\in\operatorname{Irr}(A_{\mathcal{M}^{\circ}}(v))$
and
$\eta\in\operatorname{Irr}(\mathbb{C}[A_{\mathcal{M}}(v)_{\rho_{\mathrm{cusp}}^{\circ}}/A_{\mathcal{M}^{\circ}}(v),\kappa^{\prime}])$,
where $A_{\mathcal{M}}(v)_{\rho^{\circ}_{\mathrm{cusp}}}$ denotes the
stabilizer of $\rho^{\circ}_{\mathrm{cusp}}$ in $A_{\mathcal{G}}(u)$, and
$\kappa^{\prime}\colon
A_{\mathcal{M}}(v)_{\rho^{\circ}_{\mathrm{cusp}}}/A_{\mathcal{M}}(v)^{\circ}\to\overline{\mathbb{Q}}_{\ell}^{\times}$
is a certain $2$-cocycle. Since $v$ is trivial, we have
$A_{\mathcal{M}^{\circ}}(v)=\\{1\\}$. Hence $\rho^{\circ}_{\mathrm{cusp}}$ is
the trivial representation (which is cuspidal) of the trivial group
$A_{\mathcal{M}^{\circ}}(v)$, and we have
$A_{\mathcal{M}}(v)_{\rho^{\circ}_{\mathrm{cusp}}}=A_{\mathcal{M}}(v)=\mathcal{M}/\mathcal{M}^{\circ}$.
It gives $\rho_{\mathrm{cusp}}=\eta$ with
$\eta\in\operatorname{Irr}(\overline{\mathbb{Q}}_{\ell}[\mathcal{M}/\mathcal{M}^{\circ},\kappa^{\prime}])$,
where
$\kappa^{\prime}\colon\mathcal{M}/\mathcal{M}^{\circ}\to\overline{\mathbb{Q}}_{\ell}^{\times}$
is a $2$-cocycle. But, by [AMS1, (25)], the cocycle $\kappa^{\prime}$ is
trivial, since $\rho^{\circ}_{\mathrm{cusp}}$ is the trivial representation. ∎

######  Proposition 4.2.6.

The set $\mathfrak{U}_{\mathrm{e}}^{\mathrm{ord}}(\mathcal{G})$ is the set of
$\mathcal{G}$-conjugacy classes of pairs $(u,\rho)$ with semisimple cuspidal
support.

###### Proof.

Let $(u,\rho)_{\mathcal{G}}\in\mathfrak{U}_{\mathrm{e}}(\mathcal{G})$. Let
$\mathfrak{t}=(\mathcal{M},v,\rho_{\mathrm{cusp}})$ be the cuspidal support of
$(u,\rho)_{\mathcal{G}}$. The generalized Springer correspondence constructed
in [Lu2] reduces to the ordinary Springer correspondence precisely when
$\mathcal{L}$ is a maximal torus of $\mathcal{G}^{\circ}$. Thus
$(u,\rho)_{\mathcal{G}}$ belongs to
$\mathfrak{U}_{\mathrm{e}}^{\mathrm{ord}}(\mathcal{G})$ if and only if
$\mathcal{M}={\mathrm{Z}}_{\mathcal{G}}(\mathcal{T})$ with $\mathcal{T}$ a
maximal torus of $\mathcal{G}$. But then, since $v$ is a unipotent element of
$\mathcal{T}$, it has to be trivial. It follows that
$A_{\mathcal{T}}(v)=\\{1\\}$, and hence $\rho_{\mathrm{cusp}}=1$. So
$\mathfrak{t}=(\mathcal{M},1,\eta)$, where
$\eta\in\operatorname{Irr}(\mathcal{M}/\mathcal{M}^{\circ})$. Then the result
follows from Lemma 4.2.4. ∎

###  4.3. Enhanced Langlands parameters

Let $G^{\vee}$ denote the Langlands dual group of $G$, i.e. the complex Lie
group with root datum dual to that of $G$. The $L$-group of $G$ is defined to
be ${}^{L}G:=G^{\vee}\rtimes W_{F}$. Let
$W_{F}^{\prime}:=W_{F}\times\operatorname{SL}_{2}(\mathbb{C})$ be the Weil-
Deligne group of $F$. A Langlands parameter (or $L$-parameter) for $G$ and its
inner twists is a continuous morphism $\varphi\colon W_{F}^{\prime}\to{}^{L}G$
such that $\varphi(w)$ is semisimple for each $w\in W_{F}$, the restriction
$\varphi|_{\operatorname{SL}_{2}(\mathbb{C})}$ is a morphism of complex
algebraic groups, and $\varphi$ commutes to the projections to $W_{F}$. The
group $G^{\vee}$ acts on the set of $L$-parameters. We denote by $\Phi(G)$ the
set of $G^{\vee}$-classes of $G$-relevant $L$-parameters as defined in [Bor,
§3] (see also [SZ, §2.5]).

We define an action of $G^{\vee}_{\mathrm{sc}}$ on $G^{\vee}$ by setting

$h\cdot g:=h^{\prime}gh^{\prime-1}\quad\text{for $h\in G^{\vee}_{\mathrm{sc}}$
and $g\in G^{\vee}$, where $p(h)=h^{\prime}{\mathrm{Z}}_{G^{\vee}}$.}$

It induces an action of $G^{\vee}_{\mathrm{sc}}$ on ${}^{L}G$ and we denote by
${\mathrm{Z}}_{G^{\vee}_{\mathrm{sc}}}(\varphi)$ the stabilizer in
$G^{\vee}_{\mathrm{sc}}$ of $\varphi(W_{F}^{\prime})$ for this action.

We attach to each $L$-parameter $\varphi$ several (possibly disconnected)
complex reductive groups as follows. We set
${\mathrm{Z}}_{G^{\vee}}(\varphi):={\mathrm{Z}}_{G^{\vee}}(\varphi(W^{\prime}_{F}))$
and denote by ${\mathrm{Z}}_{G^{\vee}_{\mathrm{sc}}}^{1}(\varphi)$ the inverse
image of
${\mathrm{Z}}_{G^{\vee}}(\varphi)/({\mathrm{Z}}_{G^{\vee}}(\varphi)\cap{\mathrm{Z}}_{G^{\vee}})$
(viewed as a subgroup of $G^{\vee}_{\mathrm{ad}}$) under the quotient map
$G^{\vee}_{\mathrm{sc}}\to G^{\vee}_{\mathrm{ad}}$. We define
$\varphi|_{W_{F}}$ to be the restriction along the embedding of $W_{F}$ as the
first factor of $W^{\prime}_{F}$:

(4.3.1)
$\begin{matrix}\varphi|_{W_{F}}\colon&W_{F}&\to&{}^{L}G\cr&w&\mapsto&\varphi(w,1)\end{matrix}.$

We set
${\mathrm{Z}}_{G^{\vee}}(\varphi|_{W_{F}}):={\mathrm{Z}}_{G^{\vee}}(\varphi(W_{F}))$
and denote by ${\mathrm{Z}}_{G^{\vee}}^{1}(\varphi|_{W_{F}})$ the inverse
image of
${\mathrm{Z}}_{G^{\vee}}(\varphi|_{W_{F}})/({\mathrm{Z}}_{G^{\vee}}(\varphi|_{W_{F}})\cap{\mathrm{Z}}_{G^{\vee}})$
in $G^{\vee}_{\mathrm{sc}}$. We set

(4.3.2)
$\mathcal{G}_{\varphi}:={\mathrm{Z}}_{G^{\vee}}^{1}(\varphi|_{W_{F}})\quad\text{and}\quad
S_{\varphi}:={\mathrm{Z}}_{G^{\vee}}^{1}(\varphi)/({\mathrm{Z}}_{G^{\vee}}^{1}(\varphi))^{\circ}.$

We denote by $u_{\varphi}$ the image of
$\left(1,\left(\begin{smallmatrix}1&1\cr 0&1\end{smallmatrix}\right)\right)$
under $\varphi$. By [AMS1, (92)], we have
$u_{\varphi}\in\mathcal{G}_{\varphi}^{\circ}$ and $S_{\varphi}\simeq
A_{\mathcal{G}_{\varphi}}(u_{\varphi})$, where
$u_{\varphi}:=\varphi\left(1,\left(\begin{smallmatrix}1&1\cr
0&1\end{smallmatrix}\right)\right)$.

An enhancement of $\varphi$ is an irreducible representation $\rho$ of
$S_{\varphi}$. Pairs $(\varphi,\rho)$ are called enhanced $L$-parameters (for
$G$ and its inner twists). Let $G^{\vee}$ act on the set of enhanced
$L$-parameters via

(4.3.3) $g\cdot(\varphi,\rho)=(g\varphi g^{-1},g\cdot\rho).$

We denote by $\Phi_{\mathrm{e}}(G)$ the set of $G^{\vee}$-conjugacy classes of
enhanced $L$-parameters.

An $L$-parameter $\varphi$ is said to be discrete if $\varphi(W_{F}^{\prime})$
is not contained in any proper Levi subgroup of ${}^{L}G$. An enhanced
$L$-parameter $(\varphi,\rho)\in\Phi_{\mathrm{e}}(G)$ is called cuspidal if
$\varphi$ is discrete and $(u_{\varphi},\rho)$ is a cuspidal unipotent pair in
$\mathcal{G}_{\varphi}$, as defined in §4.1. We denote by
$\Phi_{{\mathrm{e}},\mathrm{cusp}}(G)$ the subset of $\Phi_{{\mathrm{e}}}(G)$
consisting of $G^{\vee}$-conjugacy classes of cuspidal enhanced
$L$-parameters.

In [AMS1, Section 7], we attached to every enhanced $L$-parameter
$(\varphi,\rho)$ its cuspidal support $\mathrm{Sc}(\varphi,\rho)$. We recall
below how it is defined. Let
$\mathfrak{t}:=(\mathcal{M},v,\rho_{\mathrm{cusp}})$ denote the cuspidal
support of $(u_{\varphi},\rho)$, as defined in 4.1. In particular,
$(v,\rho_{\mathrm{cusp}})$ is a cuspidal unipotent pair in $\mathcal{M}$. Upon
conjugating $\varphi$ with a suitable element of
${\mathrm{Z}}_{\mathcal{G}_{\varphi}^{\circ}}(u_{\varphi})$, we may assume
that the identity component $\mathcal{L}$ of $\mathcal{M}$ contains
$\varphi\left(\left(1,\left(\begin{smallmatrix}z&0\cr
0&z^{-1}\end{smallmatrix}\right)\right)\right)$ for all
$z\in\mathbb{C}^{\times}$. Recall that by the Jacobson–Morozov theorem (see
for example [CG, Theorem 3.7.1]), any unipotent element $v$ of $\mathcal{M}$
can be extended to a homomorphism of algebraic groups

(4.3.4) $j_{v}\colon\operatorname{SL}_{2}(\mathbb{C})\to\mathcal{L}\text{
satisfying }j_{v}\left(\begin{smallmatrix}1&1\\\
0&1\end{smallmatrix}\right)=v.$

By [CG, Prop. 3.7.3], the homomorphism $j_{v}$ is uniquely determined up to
conjugation by an element of the unipotent radical of
${\mathrm{Z}}_{\mathcal{L}}(v)$. In particular, $j_{1}$ is the identity map. A
homomorphism $j_{v}$ satisfying (4.3.4) will be said to be adapted to
$\varphi$.

By [AMS1, Lem. 7.6], up to $G^{\vee}$-conjugacy, there exists a unique
homomorphism $j_{v}\colon\operatorname{SL}_{2}(\mathbb{C})\to\mathcal{L}$
which is adapted to $\varphi$, and moreover, the cocharacter

(4.3.5) $\chi_{\varphi,v}\colon
z\mapsto\varphi\left(1,\left(\begin{smallmatrix}z&0\cr
0&z^{-1}\end{smallmatrix}\right)\right)\cdot
j_{v}\left(\begin{smallmatrix}z^{-1}&0\cr 0&z\end{smallmatrix}\right)$

has image in ${\mathrm{Z}}_{\mathcal{L}}^{\circ}$. We define an $L$-parameter
$\varphi_{v}\colon
W_{F}\times\operatorname{SL}_{2}(\mathbb{C})\to{\mathrm{Z}}_{G^{\vee}}({\mathrm{Z}}_{\mathcal{L}}^{\circ})$
by

$\varphi_{v}(w,x):=\varphi(w,1)\cdot\chi_{\varphi,v}(|\\!|w|\\!|^{1/2})\cdot
j_{v}(x)\quad\text{for any $w\in W_{F}$ and any
$x\in\operatorname{SL}_{2}(\mathbb{C})$.}$

The cuspidal support of $(\varphi,\rho)$ is defined to be

(4.3.6)
$\mathrm{Sc}(\varphi,\rho):=({\mathrm{Z}}_{{}^{L}G}({\mathrm{Z}}_{\mathcal{L}}^{\circ}),(\varphi_{v},\rho_{\mathrm{cusp}}))_{G^{\vee}}.$

By [AMS1, Prop. 7.3], we can always represent
$({\mathrm{Z}}_{{}^{L}G}({\mathrm{Z}}_{\mathcal{L}}^{\circ}),\varphi_{L},\rho_{\mathrm{cusp}})_{G^{\vee}}$
by a triple of the form $(L^{\vee}\rtimes W_{F},\varphi_{L},\rho_{L})$, where
$L$ is a Levi subgroup of $G$ and $(\varphi_{L},\rho_{L})$ is a cuspidal
enhanced $L$-parameter for $L$. The cuspidal support of
$(\varphi,\rho)\in\Phi_{\mathrm{e}}(G)$ is said to be semisimple if $v=1$.

######  Definition 4.3.7.

An enhanced $L$-parameter $(\varphi,\rho)\in\Phi_{\mathrm{e}}(G)$ is called
non-singular if its cuspidal support is semisimple.

We recall that an $L$-parameter $\varphi_{L}\colon W^{\prime}_{F}\to{}^{L}L$
is said to be supercuspidal if it is discrete and has trivial restriction to
$\operatorname{SL}_{2}(\mathbb{C})$. Thus, an enhanced $L$-parameter
$(\varphi,\rho)\in\Phi_{\mathrm{e}}(G)$ is non-singular if and only if its
cuspidal support is of the form $(L^{\vee}\rtimes
W_{F},\varphi_{L},\rho_{L})_{G^{\vee}}$, with $\varphi_{L}$ a supercuspidal
$L$-parameter for a Levi subgroup $L$ of $G$.

######  Theorem 1.

An enhanced $L$-parameter $(\varphi,\rho)\in\Phi_{\mathrm{e}}(G)$ is non-
singular if and only if
$(u_{\varphi},\rho)_{\mathcal{G}_{\varphi}}\in\mathfrak{U}^{\mathrm{ord}}_{\mathrm{e}}(\mathcal{G}_{\varphi})$.

###### Proof.

It follows immediately from Proposition 4.2.6. ∎

###  4.4. A Bernstein-type decomposition on the Galois side

Let $\mathbf{L}$ be a Levi subgroup defined over $F$ of a parabolic subgroup
of $\mathbf{G}$ defined over $F$ and let $L^{\vee}$ denote the
$\mathbb{C}$-points of the Langlands dual group of $\mathbf{L}$, We define

(4.4.1)
${\mathfrak{X}}_{\mathrm{nr}}({}^{L}L):=({\mathrm{Z}}_{L^{\vee}}^{I_{F}})_{W_{F}}^{\circ}.$

The group ${\mathfrak{X}}_{\mathrm{nr}}({}^{L}L)$ is naturally isomorphic to
the group ${\mathfrak{X}}_{\mathrm{nr}}(L)$ (see [Hai, p.16]). We denote the
isomorphism
${\mathfrak{X}}_{\mathrm{nr}}(L)\xrightarrow{\sim}{\mathfrak{X}}_{\mathrm{nr}}({}^{L}L)$
by $\chi\mapsto\chi^{\vee}$. We define an action of
${\mathfrak{X}}_{\mathrm{nr}}({}^{L}L)$ on $\Phi_{\mathrm{e}}(L)$ as follows.
Given $(\varphi,\rho)\in\Phi_{\mathrm{e}}(L)$ and
$\xi\in{\mathfrak{X}}_{\mathrm{nr}}({}^{L}L)$, we define
$(\xi\varphi,\rho)\in\Phi_{\mathrm{e}}(L)$ by $\xi\varphi:=\varphi$ on
$I_{F}\times\operatorname{SL}_{2}(\mathbb{C})$ and
$(\xi\varphi)(\mathrm{Fr}_{F}):=\widetilde{\xi}\varphi(\mathrm{Fr}_{F})$. Here
$\widetilde{\xi}\in{\mathrm{Z}}_{L^{\vee}\rtimes I_{F}}^{\circ}$ represents
$z$. Let
$\mathfrak{s}^{\vee}:=[L^{\vee},\varphi_{\mathrm{c}},\rho_{\mathrm{c}}]_{G^{\vee}}$
be the $G^{\vee}$-conjugacy class of the orbit of
$(\varphi_{\mathrm{c}},\rho_{\mathrm{c}})\in\Phi_{\mathrm{e}}^{\mathrm{cusp}}(L)$
under the action of ${\mathfrak{X}}_{\mathrm{nr}}(L^{\vee})$. Let
$\mathfrak{B}^{\vee}(G)$ be the set of such $\mathfrak{s}^{\vee}$.

By [AMS1, (115)], the set $\Phi_{\mathrm{e}}(G)$ is partitioned into series à
la Bernstein as

(4.4.2)
$\Phi_{\mathrm{e}}(G)=\prod_{\mathfrak{s}^{\vee}\in\mathfrak{B}(G^{\vee})}\Phi(G)^{\mathfrak{s}^{\vee}},$

where for each $\mathfrak{s}^{\vee}$, the subset
$\Phi_{\mathrm{e}}(G)^{\mathfrak{s}^{\vee}}$ is defined to be the fiber of
$\mathfrak{s}^{\vee}$ under the supercuspidal map $\mathrm{Sc}$ defined in
(4.3.6).

For any $\mathfrak{s}=[L,\sigma]_{G}\in\mathfrak{B}(G)$, it is expected (see
[AMS1, Conj. 2]), and known for $p$-adic general linear groups and for pure
inner forms of quasi-split $p$-adic classical groups (by [Mou, AMS3]), when
$p\neq 2,3$, for the exceptional group of type $\operatorname{G}_{2}$ (by
[AX1, AX2]), and for $G$ arbitrary and $L$ a torus (by [ABPS2, So2]), that
some suitable notion of $\mathrm{LLC}$ for supercuspidal representations of
$L$, denoted by
$\sigma^{\prime}\mapsto(\varphi_{\sigma^{\prime}},\rho_{\sigma^{\prime}})$,
induces a bijection:

(4.4.3)
$\begin{matrix}\mathcal{L}^{\mathfrak{s}}_{G}\colon&\operatorname{Irr}^{\mathfrak{s}}(G)&\rightarrow&\Phi_{\mathrm{e}}^{\mathfrak{s}^{\vee}}(G)\cr&\pi&\mapsto&(\varphi_{\pi},\rho_{\pi})\end{matrix}$

where
$\mathfrak{s}^{\vee}=[L^{\vee},\varphi_{\sigma},\rho_{\sigma}]_{G^{\vee}}$,
and hence, by using (3.2.2) and (4.4.2), a bijection

(4.4.4)
$\begin{matrix}\mathcal{L}_{G}\colon&\operatorname{Irr}(G)&\rightarrow&\Phi_{\mathrm{e}}(G)\cr&\pi&\mapsto&(\varphi_{\pi},\rho_{\pi})\end{matrix}.$

Let ${\mathfrak{X}}_{{\mathrm{nr}}}(L,\sigma)$ denote the finite subgroup of
${\mathfrak{X}}_{{\mathrm{nr}}}(L)$ defined as

(4.4.5)
${\mathfrak{X}}_{{\mathrm{nr}}}(L,\sigma):=\left\\{\chi\in{\mathfrak{X}}_{{\mathrm{nr}}}(L)\;:\;\chi\otimes\sigma=\sigma\right\\}.$

Let $\natural$ and ${}^{L}\natural$ denote the collections of $2$-cocycles,
occurring respectively in [So1] and [AMS1]. In many cases these cocycles are
known to be both trivial (for instance for classical groups, as proved in
[AMS3]).

For every Levi subgroup $L$ of $G$, there is a canonical isomorphism $w\mapsto
w^{\vee}$ between $W_{G}(L):=\operatorname{N}_{G}(L)/L$ and
$W_{G^{\vee}}(L^{\vee}):=\operatorname{N}_{G^{\vee}}(L^{\vee})/L^{\vee}$, see
[ABPS1, Prop. 3.1]. For $w\in W_{G}(L)$ and
$\sigma\in\operatorname{Irr}_{\mathrm{sc}}(L)$, let ${}^{w}\sigma$ denote the
representation of $L$ defined by ${}^{w}\sigma(\ell):=\sigma(w^{-1}\ell w)$.
If the representation $\sigma$ is non-singular, then the representation
${}^{w}\sigma$ is also non-singular.

######  Remark 4.4.6.

If the following properties are satisfied for every
$\sigma^{\prime}\in\mathfrak{s}=[L,\sigma]_{G}$:

(4.4.7)
${}^{w^{\vee}}(\varphi_{\sigma^{\prime}},\rho_{\sigma^{\prime}})=(\varphi_{{}^{w}\sigma^{\prime}},\rho_{{}^{w}\sigma^{\prime}})\quad\text{for
any $w\in W(L)$.}$ (4.4.8)
${}^{L}\natural_{\chi^{\vee}}=\natural_{\chi}\quad\text{for any
$\chi\in{\mathfrak{X}}_{\mathrm{nr}}(L)/{\mathfrak{X}}_{{\mathrm{nr}}}(L,\sigma)$,}$

then
$\sigma^{\prime}\mapsto(\varphi_{\sigma^{\prime}},\rho_{\sigma^{\prime}})$
induces a bijection $\mathcal{L}^{\mathfrak{s}}_{G}$ from
$\operatorname{Irr}^{\mathfrak{s}}(G)$ to
$\Phi_{\mathrm{e}}^{\mathfrak{s}^{\vee}}(G)$, where
$\mathfrak{s}^{\vee}=[L^{\vee},\varphi_{\sigma},\rho_{\sigma}]_{G^{\vee}}$
(see [AX1]).

Moreover, as formulated in [AMS1, Conjecture-7.8], the following diagram is
expected to be commutative:

(4.4.9) $\leavevmode\hbox to244.63pt{\vbox
to60.16pt{\pgfpicture\makeatletter\hbox{\hskip
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}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-122.31485pt}{-26.81944pt}\pgfsys@invoke{
}\hbox{\vbox{\halign{\pgf@matrix@init@row\pgf@matrix@step@column{\pgf@matrix@startcell#\pgf@matrix@endcell}&#\pgf@matrix@padding&&\pgf@matrix@step@column{\pgf@matrix@startcell#\pgf@matrix@endcell}&#\pgf@matrix@padding\cr\hfil\hskip
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where $\mathfrak{L}(G)$ is a set of representatives for the conjugacy classes
of Levi subgroups of $G$. The commutativity of the diagram (4.4.9) was
established for general linear groups and for classical groups in [Mou, AMS3],
for $\operatorname{G}_{2}$ in [AX2], and for $G$ arbitrary and $L$ a torus in
[ABPS2, So2].

###  4.5. The $\mathrm{LLC}$ for non-singular representations

Let $L$ be Levi subgroup of $G$. In [Ka4, Def. 4.1.2], a supercuspidal
$L$-parameter $\varphi_{L}$ for $L$ is said to be torally wild if
$\varphi(P_{F})$ is contained in a maximal torus of $L^{\vee}$. In particular,
if $\varphi_{L}$ is torally wild then
$({\mathrm{Z}}_{L^{\vee}}(\varphi(I_{F})))^{\circ}$ is abelian.

In [Ka4], when the group $\mathbf{L}$ splits over a tamely ramified extension
and $p$ is very good for $\mathbf{L}$, Kaletha has constructed the $L$-packets
associated to torally wild supercuspidal $L$-parameters and has attached to
each non-singular irreducible supercuspidal representation $\sigma$ of $L$ a
cuspidal enhanced $L$-parameter $(\varphi_{\sigma},\rho_{\sigma})$ such that
$\varphi_{\sigma}$ is a torally wild supercuspidal $L$-parameter for $L$. When
$G=\mathrm{GL}_{N}(F)$ and $p$ is odd, as proved in [OT], the map
$\sigma\mapsto(\varphi_{\sigma},\rho_{\sigma})$ coincides with the local
Langlands correspondence defined by Harris and Taylor in [HT] and Bushnell et
Henniart in [BH1, BH2].

The connection index for $\mathbf{L}$ dividing the order of Weyl group of
$\mathbf{L}$ (see for instance [Bou, VI.2, Prop.7]), if $p$ does not divide
the order of the Weyl group of $\mathbf{L}$, then $p$ is very good for
$\mathbf{L}$. When $\mathbf{L}$ splits over a tamely ramified extension and
$p$ does not divide the order of the Weyl group of $\mathbf{L}$, all
supercuspidal $L$-parameters are torally wild, and hence, in this case, the
$L$-packets associated to all supercuspidal $L$-parameters for $L$ are
constructed in [Ka4]: these are exactly the $L$-packets that contain non-
singular supercuspidal representations of $L$.

Let ${\mathrm{Inn}}(\mathbf{G})$ be the group of inner automorphisms of
$\mathbf{G}$. Recall that given an algebraic group $\mathbf{G}^{\prime}$ over
$F$, an isomorphism $\xi\colon\mathbf{G}^{\prime}\to\mathbf{G}$ defined over
$F_{s}$ determines a $1$-cocycle

(4.5.1)
$z_{\xi}\colon\begin{array}[]{ccc}\Gamma_{F}&\to&\operatorname{Aut}(\mathbf{G})\\\
\gamma&\mapsto&\xi\gamma\xi^{-1}\gamma^{-1}.\end{array}$

An inner twist of $G$ consists of a pair $G^{\prime}_{\xi}:=(G^{\prime},\xi)$,
where $G^{\prime}=\mathbf{G}^{\prime}(F)$ for some connected reductive
$F$-group $\mathbf{G}^{\prime}$, and
$\xi\colon\mathbf{G}^{\prime}\xrightarrow{\;\sim\;}\mathbf{G}$ is an
isomorphism of algebraic groups defined over $F_{\mathrm{s}}$ such that
$\text{im }(\gamma_{\xi})\subset{\mathrm{Inn}}(\mathbf{G})$. Two inner twists
$(G^{\prime},\xi),(G^{\prime}_{1},\xi_{1})$ of $G$ are equivalent if there
exists $f\in{\mathrm{Inn}}(\mathbf{G})$ such that

(4.5.2) $z_{\xi}(\gamma)=f^{-1}z_{\xi_{1}}(\gamma)\;\gamma
f\gamma^{-1}\quad\forall\gamma\in\Gamma_{F}.$

We denote the set of equivalence classes of inner twists of $G$ by
${\mathrm{InnT}}(G)$.

An inner twist of $G$ is the same thing as an inner twist of the unique quasi-
split inner form $G^{*}=\mathbf{G}^{*}(F)$ of $G$. Thus the equivalence
classes of inner twists of $G$ are parametrized by the Galois cohomology group
$H^{1}(F,{\mathrm{Inn}}(\mathbf{G}^{*}))$.

Kottwitz [Ko, Theorem 6.4] found a natural group isomorphism

(4.5.3)
$H^{1}(F,{\mathrm{Inn}}(\mathbf{G}^{*}))\xrightarrow{\;\sim\;}\operatorname{Irr}(({\mathrm{Z}}_{G^{\vee}_{\mathrm{sc}}})^{W_{F}}).$

(When F has positive characteristic, see [Th, Theorem 2.1].) In this way every
inner twist of $G$ is associated to a unique character of
$({\mathrm{Z}}_{G^{\vee}_{\mathrm{sc}}})^{W_{F}}$.

The basic version of the local Langlands conjecture states that there exists a
surjective finite-to-one map from the set of equivalence classes of
irreducible admissible representations of $G$ to the set of
$G^{\vee}$-conjugacy classes of relevant $L$-parameters $\varphi\colon
W^{\prime}_{F}\to{}^{L}G$. The fiber over $\varphi$ is called an $L$-packet.
we denote it by $\Pi_{\varphi}(G)$. Assuming the validity of the basic local
Langlands conjecture, we can define for each Langlands parameter $\varphi$ the
“compound $L$-packet” $\Pi_{\varphi}$ as

(4.5.4)
$\Pi_{\varphi}:=\bigcup_{G^{\prime}_{\xi}\in{\mathrm{InnT}}(G)}\Pi_{\varphi}(G^{\prime}_{\xi}).$

The elements of $\Pi_{\varphi}$ are expected to be in bijection with
$\operatorname{Irr}(S_{\varphi})$, where $S_{\varphi}$ is the group introduced
in (4.3.2) (see [Ar] and [Ka2, §4.6]).

######  Theorem 4.5.5.

We suppose that $\mathbf{G}$ splits over a tamely ramified extension, that $p$
does not divide the order of the Weyl group of $\mathbf{G}$, and that, for
every Levi subgroup $L$ of $G$, the Langlands correspondence
$\mathcal{L}^{\mathrm{Ka}}_{L}\colon\sigma\mapsto(\varphi_{\sigma},\rho_{\sigma})$