#  Some unipotent Arthur packets for reductive p-adic Groups

Dan Ciubotaru, Lucas Mason-Brown, and Emile Okada

###### Abstract.

Let ${\mathsf{k}}$ be a $p$-adic field and let $\mathbf{G}({\mathsf{k}})$ be
the ${\mathsf{k}}$-points of a connected reductive group, inner to split. The
set of Aubert-Zelevinsky duals of the constituents of a tempered L-packet form
an Arthur packet for $\mathbf{G}({\mathsf{k}})$. In this paper, we give an
alternative characterization of such Arthur packets in terms of the wavefront
set, proving in some instances a conjecture of Jiang-Liu and Shahidi. Pursuing
an analogy with real and complex groups, we define some special unions of
Arthur packets which we call _weak_ Arthur packets and describe their
constituents in terms of their Langlands parameters.

###### Contents

  1. 1 Introduction
    1. 1.1 Acknowledgments
  2. 2 Preliminaries
    1. 2.1 Nilpotent orbits
    2. 2.2 Duality for nilpotent orbits
    3. 2.3 Nilpotent orbits over ${\mathsf{k}^{un}}$
    4. 2.4 Representations with unipotent cuspidal support
    5. 2.5 Aubert-Zelevinsky duality
    6. 2.6 Wavefront sets
    7. 2.7 Basic Arthur packets
  3. 3 Main results
    1. 3.1 Weak Arthur packets
  4. 4 Example (split $F_{4}$)

##  1\. Introduction

Let $\mathsf{k}$ be a nonarchimedean local field of characteristic $0$ with
ring of integers $\mathfrak{o}$, finite residue field $\mathbb{F}_{q}$ of
cardinality $q$ and valuation $\mathsf{val}_{\mathsf{k}}$. Fix an algebraic
closure $\bar{\mathsf{k}}$ of $\mathsf{k}$ and let
${\mathsf{k}^{un}}\subset\bar{\mathsf{k}}$ be the maximal unramified extension
of $\mathsf{k}$ in $\bar{\mathsf{k}}$. Let $\mathbf{G}$ be a connected
reductive algebraic group defined over ${\mathsf{k}}$ and let
$\mathbf{G}({\mathsf{k}})$ be the group of ${\mathsf{k}}$-rational points. We
assume throughout that $\mathbf{G}({\mathsf{k}})$ is inner to split.

Let $W_{{\mathsf{k}}}$ be the Weil group associated to ${\mathsf{k}}$ [Tat79,
(1.4)]. Then the Weil-Deligne group is the semidirect product [Del73, §8.3.6]

$W_{{\mathsf{k}}}^{\prime}=W_{{\mathsf{k}}}\ltimes\mathbb{C}$

where $W_{{\mathsf{k}}}$ acts on $\mathbb{C}$ via

$wxw^{-1}=\|w\|x,\qquad x\in\mathbb{C},\ w\in W_{k}.$

Let $G^{\vee}$ denote the complex Langlands dual group associated to
$\mathbf{G}$, see [Bor79, §2.1].

######  Definition 1.0.1.

A _Langlands parameter_ is a continuous homomorphism

(1.0.1) $\phi:W_{{\mathsf{k}}}^{\prime}\rightarrow G^{\vee}$

which respects the Jordan decompositions in $W_{{\mathsf{k}}}^{\prime}$ and
$G^{\vee}$ ([Bor79, §8.1]).

To each Langlands parameter $\phi$, one hopes to associate an $L$-packet
$\Pi^{\mathsf{Lan}}_{\phi}(\mathbf{G}({\mathsf{k}}))$ of irreducible
admissible $\mathbf{G}({\mathsf{k}})$-representations [Bor79, §10] (note:
these packets have not been defined in full generalality, see [Art13] or
[Kal22] for a discussion of the status of the local Langlands conjectures).

######  Definition 1.0.2.

An _Arthur parameter_ is a continuous homomorphism

$\psi:W_{{\mathsf{k}}}^{\prime}\times\mathrm{SL}(2,\mathbb{C})\stackrel{{\scriptstyle}}{{\to}}G^{\vee}$

such that

  * (i)

The restriction of $\psi$ to $W_{{\mathsf{k}}}^{\prime}$ is a tempered
Langlands parameter.

  * (ii)

The restriction of $\psi$ to $\mathrm{SL}(2,\mathbb{C})$ is algebraic.

Arthur parameters are considered modulo the $G^{\vee}$-action on the target.

For each Arthur parameter $\psi$ there is an associated Langlands parameter
$\phi_{\psi}:W_{{\mathsf{k}}}^{\prime}\stackrel{{\scriptstyle}}{{\to}}G^{\vee}$
defined by the formula

(1.0.2) $\phi_{\psi}(w)=\psi(w,\begin{pmatrix}\|w\|^{1/2}&0\\\
0&\|w\|^{-1/2}\end{pmatrix}),\qquad w\in W_{{\mathsf{k}}}^{\prime},$

where $\|\bullet\|$ is the pull-back to $W_{{\mathsf{k}}}^{\prime}$ of the
norm map on $W_{{\mathsf{k}}}$. The local version of Arthur’s conjectures can
be stated succinctly as follows.

######  Conjecture 1.0.3 (Section 6, [Art89]).

For each Arthur parameter $\psi$, there is an associated finite set
$\Pi_{\psi}^{\mathsf{Art}}(\mathbf{G}({\mathsf{k}}))$ of irreducible
admissible $\mathbf{G}({\mathsf{k}})$-representations called the _Arthur
packet_ attached to $\psi$. The set
$\Pi_{\psi}^{\mathsf{Art}}(\mathbf{G}({\mathsf{k}}))$ should satisfy various
properties, including:

  * (i)

$\Pi^{\mathsf{Lan}}_{\phi_{\psi}}(\mathbf{G}({\mathsf{k}}))\subseteq\Pi_{\psi}^{\mathsf{Art}}(\mathbf{G}({\mathsf{k}}))$,

  * (ii)

$\Pi_{\psi}^{\mathsf{Art}}(\mathbf{G}({\mathsf{k}}))$ consists of unitary
representations,

along with several other properties (e.g. stability, endoscopy), see [Art89,
Section 4] or [ABV92, Chapter 1].

Although a general definition is lacking, the packets
$\Pi_{\psi}^{\mathsf{Art}}(\mathbf{G}({\mathsf{k}}))$ have been defined in
various special cases. For quasisplit orthogonal and symplectic groups, Arthur
has defined them using endoscopic transfer and has proved that they satisfy
all of the conjectured properties [Art13, Theorems 1.5.1 and 2.2.1]. His
constructions were extended in [Mok15] to quasisplit unitary groups. Explicit
definitions, compatible with Arthur’s, were proposed by Mœglin in [Mœg06,
Mœg14], see also the exposition in [Xu17] and the references therein. For
split $G_{2}$, some Arthur packets were constructed in [GG05]. Recently,
Cunningham and his collaborators have proposed a ‘microlocal’ approach to
defining Arthur packets, see [CFMMX17].

There is one simple class of Arthur parameters for which a general (i.e. case-
free) definition of Arthur packets is available.

######  Definition 1.0.4.

An Arthur parameter $\psi$ is _basic_ if the restriction
$\psi|_{W_{{\mathsf{k}}}^{\prime}}$ is trivial.

By the Jacobson-Morozov theorem, there is a natural bijection

$\displaystyle\\{\text{nilpotent adjoint }G^{\vee}\text{-orbits}\\}$
$\displaystyle\xrightarrow{\sim}\\{\text{basic Arthur parameters}\\}/G^{\vee}$
$\displaystyle\mathbb{O}^{\vee}$
$\displaystyle\mapsto\psi_{\mathbb{O}^{\vee}}.$

For each nilpotent orbit $\mathbb{O}^{\vee}\subset\mathfrak{g}^{\vee}$, we
choose an $\mathfrak{sl}(2)$-triple $(e^{\vee},f^{\vee},h^{\vee})$ and
consider the semisimple element $q^{\frac{1}{2}h^{\vee}}\in G^{\vee}$. This
element is well-defined modulo conjugation by $G^{\vee}$, and therefore
determines an ‘infinitesimal character’, see Section 2.4. There is a well-
known involution defined by Aubert and Zelevinsky on the set of irreducible
admissible $\mathbf{G}({\mathsf{k}})$-representations, see Section 2.5. We
denote this involution by $X\mapsto\mathrm{AZ}(X)$. The Arthur packets
attached to basic Arthur parameters can be defined as follows, , see for
example [Art13, §7.1].

######  Definition 1.0.5.

Let $\mathbb{O}^{\vee}$ be a nilpotent adjoint $G^{\vee}$-orbit and let
$\psi_{\mathbb{O}^{\vee}}$ be the associated basic Arthur parameter. Then
$\Pi^{\mathsf{Art}}_{\psi_{\mathbb{O}^{\vee}}}(\mathbf{G}({\mathsf{k}}))$ is
the set of irreducible $\mathbf{G}({\mathsf{k}})$-representations $X$ with
unipotent cuspidal support such that

  * (i)

The infinitesimal character of $X$ is $q^{\frac{1}{2}h^{\vee}}$.

  * (ii)

$\mathsf{AZ}(X)$ is tempered.

The _wavefront set_ of an admissible $\mathbf{G}({\mathsf{k}})$-representation
is a fundamental invariant coming from the Harish-Chandra-Howe local character
expansion. In its classical form, the wavefront set $\mathrm{WF}(X)$ of $X$ is
a collection of nilpotent ${\mathbf{G}}({\mathsf{k}})$-orbits in the Lie
algebra $\mathfrak{g}({\mathsf{k}})$, namely the _maximal_ nilpotent orbits
for which the Fourier transforms of the associated orbital integrals
contribute to the local character expansion of the distribution character of
$X$, [Har99, Theorem 16.2]. In this paper, we consider two coarser invariants
(see Section 2.6 for the precise definitions). The first of these invariants
is the _algebraic wavefront set_ , denoted
$\hphantom{}{}^{\bar{{\mathsf{k}}}}\mathrm{WF}(X)$. This is a collection of
nilpotent orbits in $\mathfrak{g}({\bar{\mathsf{k}}})$, see for example
[Wal18, p. 1108] (where it is simply referred to as the ‘wavefront set’ of
$X$). The second invariant is $\underline{\mathrm{WF}}(X)$, a natural
refinement of ${}^{{\bar{\mathsf{k}}}}\mathrm{WF}(X)$ called the _canonical
unramified wavefront set_ , defined recently in [Oka21]. This is a collection
of nilpotent orbits $\mathfrak{g}({\mathsf{k}^{un}})$ (modulo a certain
equivalence relation $\sim_{A}$). The relationship between these three
invariants is as follows: the algebraic wavefront set
$\hphantom{}{}^{\bar{{\mathsf{k}}}}\mathrm{WF}(X)$ is deducible from the usual
wavefront set $\mathrm{WF}(X)$ as well as the canonical unramified wavefront
set $\underline{\mathrm{WF}}(X)$. It is not known whether the canonical
unramified wavefront set is deducible from the usual wavefront set, but we
expect this to be the case, see [Oka21, Section 5.1] for a careful discussion
of this topic.

For real and complex groups, it is possible to define Arthur packets in terms
of the wavefront set, see [BV85] and [ABV92, Section 27]. Thus, it is natural
to hope for a similar definition in the $p$-adic setting. Our main result
(Theorem 3.0.3) gives an alternative characterization of the Arthur packets
attached to basic Arthur parameters in terms of the canonical unramified
wavefront set. The most precise version of our result requires two additional
ingredients which will be introduced in Section 2 (the classification of
nilpotent orbits in $\mathfrak{g}(\bar{{\mathsf{k}}})$ and Achar duality).
Here is a simplified version.

######  Theorem 1.0.6.

Let $\mathbb{O}^{\vee}$ be a nilpotent adjoint $G^{\vee}$-orbit and let
$\psi_{\mathbb{O}^{\vee}}$ be the associated basic Arthur parameter. Then
$\Pi^{\mathsf{Art}}_{\mathbb{O}^{\vee}}(\mathbf{G}({\mathsf{k}}))$ is the set
of irreducible representations $X$ with unipotent cuspidal support such that

  * (i)

The infinitesimal character of $X$ is $q^{\frac{1}{2}h^{\vee}}$.

  * (ii)

The canonical unramified wavefront set $\underline{\mathrm{WF}}(X)$ is minimal
subject to (i).

As a consequence of Theorem 1.0.6, we prove a conjecture of Jiang-Liu and
Shahidi in the case of basic Arthur packets (see Corollary 3.0.5 below).

If we replace $\underline{\mathrm{WF}}(X)$ in condition (ii) with the coarser
invariant ${}^{\bar{{\mathsf{k}}}}\mathrm{WF}(X)$, we get a larger set of
representations, which we call a _weak Arthur packet_. We conjecture that this
set is a union of Arthur packets (and, in particular, that its constituents
are unitary).

###  1.1. Acknowledgments

The authors would like to thank Kevin McGerty and David Vogan for many helpful
conversations. The authors would also like to thank Anne-Marie Aubert, Colette
Mœglin, David Renard, and Maarten Solleveld for their helpful comments and
corrections on an earlier draft of this paper. This research was partially
supported by the Engineering and Physical Sciences Research Council under
grant EP/V046713/1. The third author was supported by Aker Scholarship.

##  2\. Preliminaries

Let $\mathsf{k}$ be a nonarchimedean local field of characteristic $0$ with
residue field $\mathbb{F}_{q}$ of sufficiently large characteristic, ring of
integers $\mathfrak{o}\subset\mathsf{k}$, and valuation
$\mathsf{val}_{\mathsf{k}}$. Fix an algebraic closure $\bar{\mathsf{k}}$ of
$\mathsf{k}$ with Galois group $\Gamma$, and let
${\mathsf{k}^{un}}\subset\bar{\mathsf{k}}$ be the maximal unramified extension
of $\mathsf{k}$ in $\bar{\mathsf{k}}$. Let $\mathfrak{O}$ be the ring of
integers of ${\mathsf{k}^{un}}$. Let $\mathrm{Frob}$ be the geometric
Frobenius element of $\mathrm{Gal}({\mathsf{k}^{un}}/\mathsf{k})$, the
topological generator which induces the inverse of the automorphism
$x\stackrel{{\scriptstyle}}{{\to}}x^{q}$ of $\mathbb{F}_{q}$.

Let ${\mathbf{G}}$ be a connected reductive algebraic group defined over
${\mathsf{k}}$, inner to split, and let ${\mathbf{T}}\subset\mathbf{G}$ be a
maximal torus. For any field $F$ containing ${\mathsf{k}}$, we write
$\mathbf{G}(F)$, $\mathbf{T}(F)$, etc. for the groups of $F$-rational points.
Let ${\mathbf{G}}_{\operatorname{ad}}={\mathbf{G}}/Z({\mathbf{G}})$ denote the
adjoint group of ${\mathbf{G}}$.

Write $X^{*}(\mathbf{T},{\bar{\mathsf{k}}})$ (resp.
$X_{*}(\mathbf{T},{\bar{\mathsf{k}}})$) for the lattice of algebraic
characters (resp. co-characters) of $\mathbf{T}({\bar{\mathsf{k}}})$, and
write $\Phi(\mathbf{T},{\bar{\mathsf{k}}})$ (resp.
$\Phi^{\vee}(\mathbf{T},{\bar{\mathsf{k}}})$) for the set of roots (resp. co-
roots). Let

$\mathcal{R}=(X^{*}(\mathbf{T},{\bar{\mathsf{k}}}),\
\Phi(\mathbf{T},{\bar{\mathsf{k}}}),X_{*}(\mathbf{T},{\bar{\mathsf{k}}}),\
\Phi^{\vee}(\mathbf{T},{\bar{\mathsf{k}}}),\ \langle\ ,\ \rangle)$

be the root datum corresponding to $\mathbf{G}$, and let $W$ the associated
(finite) Weyl group. Let $G$ be the complex reductive group with the same
absolute root data as ${\mathbf{G}}$ and let $\mathbf{G}^{\vee}$ be the
Langlands dual group of ${\mathbf{G}}$, i.e. the connected reductive algebraic
group defined and split over $\mathbb{Z}$ corresponding to the dual root datum

$\mathcal{R}^{\vee}=(X_{*}(\mathbf{T},{\bar{\mathsf{k}}}),\
\Phi^{\vee}(\mathbf{T},{\bar{\mathsf{k}}}),X^{*}(\mathbf{T},{\bar{\mathsf{k}}}),\
\Phi(\mathbf{T},{\bar{\mathsf{k}}}),\ \langle\ ,\ \rangle).$

Set
$\Omega=X_{*}(\mathbf{T},{\bar{\mathsf{k}}})/\mathbb{Z}\Phi^{\vee}(\mathbf{T},{\bar{\mathsf{k}}})$.
The center $Z({\mathbf{G}}^{\vee})$ can be naturally identified with the
irreducible characters $\mathsf{Irr}\Omega$, and dually, $\Omega\cong
X^{*}(Z({\mathbf{G}}^{\vee}))$. For $\omega\in\Omega$, let $\zeta_{\omega}$
denote the corresponding irreducible character of $Z({\mathbf{G}}^{\vee})$.

For details regarding the parametrization of inner twists of a group
${\mathbf{G}}(\mathsf{k})$, see [Vog93, §2], [Kot84], [Kal16, §2], or [ABPS17,
§1.3] and [FOS20, §1]. We only record here that the equivalence classes of
inner twists of $\mathbf{G}$ are parameterized by the Galois cohomology group

$H^{1}(\Gamma,\mathbf{G}_{\operatorname{ad}})\cong
H^{1}(F,\mathbf{G}_{\operatorname{ad}}({\mathsf{k}^{un}}))\cong\Omega_{\operatorname{ad}}\cong\operatorname{Irr}Z({\mathbf{G}}^{\vee}_{\mathsf{sc}}),$

where ${\mathbf{G}}^{\vee}_{\mathsf{sc}}$ is the Langlands dual group of
${\mathbf{G}}_{\operatorname{ad}}$, i.e., the simply connected cover of
${\mathbf{G}}^{\vee}$, and $F$ denotes the action of $\mathrm{Frob}$ on
${\mathbf{G}}({\mathsf{k}^{un}})$. We identify $\Omega_{\operatorname{ad}}$
with the fundamental group of ${\mathbf{G}}_{\operatorname{ad}}$. The
isomorphism above is determined as follows: for a cohomology class $h$ in
$H^{1}(F,\mathbf{G}_{\operatorname{ad}}({\mathsf{k}^{un}}))$, let $z$ be a
representative cocycle. Let
$u\in{\mathbf{G}}_{\operatorname{ad}}({\mathsf{k}^{un}})$ be the image of $F$
under $z$, and let $\omega$ denote the image of $u$ in
$\Omega_{\operatorname{ad}}$. Set $F_{\omega}=\operatorname{Ad}(u)\circ F$.
The corresponding rational structure of ${\mathbf{G}}$ is given by
$F_{\omega}$. Let ${\mathbf{G}}^{\omega}$ be the connected reductive group
defined over ${\mathsf{k}}$ such that
${\mathbf{G}}({\mathsf{k}^{un}})^{F_{\omega}}={\mathbf{G}}^{\omega}(\mathsf{k})$.

If $H$ is a complex reductive group and $x$ is an element of $H$ or
$\mathfrak{h}$, we write $H(x)$ for the centralizer of $x$ in $H$, and
$A_{H}(x)$ for the group of connected components of $H(x)$. If $S$ is a subset
of $H$ or $\mathfrak{h}$ (or indeed, of $H\cup\mathfrak{h}$), we can similarly
define $H(S)$ and $A_{H}(S)$. We will sometimes write $A(x)$, $A(S)$ when the
group $H$ is implicit. The subgroups of $H$ of the form $H(x)$ where $x$ is a
semisimple element of $H$ are called _pseudo-Levi_ subgroups of $H$.

Let $\mathcal{C}({\mathbf{G}}(\mathsf{k}))$ be the category of smooth complex
${\mathbf{G}}(\mathsf{k})$-representations and let
$\Pi(\mathbf{G}(\mathsf{k}))\subset\mathcal{C}({\mathbf{G}}(\mathsf{k}))$ be
the set of irreducible objects. Let $R({\mathbf{G}}(\mathsf{k}))$ denote the
Grothendieck group of $\mathcal{C}({\mathbf{G}}(\mathsf{k}))$.

###  2.1. Nilpotent orbits

Let $\mathcal{N}$ be the functor which takes a field $F$ containing
${\mathsf{k}}$ to the set of nilpotent elements of $\mathfrak{g}(F)$. By
‘nilpotent’ in this context we mean the unstable points (in the sense of GIT)
with respect to the adjoint action of ${\mathbf{G}}(F)$, see [DeB02, Section
2]. For $F$ algebraically closed this coincides with all the usual notions of
nilpotence. Let $\mathcal{N}_{o}$ be the functor which takes $F$ to the set of
orbits in $\mathcal{N}(F)$ under the adjoint action of ${\mathbf{G}}(F)$. When
$F$ is ${\mathsf{k}}$ or ${\mathsf{k}^{un}}$, we view $\mathcal{N}_{o}(F)$ as
a partially ordered set with respect to the closure ordering in the topology
induced by the topology on $F$. When $F$ is algebraically closed, we view
$\mathcal{N}_{o}(F)$ as a partially ordered set with respect to the closure
ordering in the Zariski topology. For brevity we will write
$\mathcal{N}(F^{\prime}/F)$ (resp. $\mathcal{N}_{o}(F^{\prime}/F)$) for
$\mathcal{N}(F\stackrel{{\scriptstyle}}{{\to}}F^{\prime})$ (resp.
$\mathcal{N}_{o}(F\stackrel{{\scriptstyle}}{{\to}}F^{\prime})$) where
$F\stackrel{{\scriptstyle}}{{\to}}F^{\prime}$ is a morphism of fields. For
$(F,F^{\prime})=({\mathsf{k}},{\mathsf{k}^{un}})$ (resp.
$({\mathsf{k}},{\bar{\mathsf{k}}})$,
$({\mathsf{k}^{un}},{\bar{\mathsf{k}}})$), the map
$\mathcal{N}_{o}(F^{\prime}/F)$ is strictly increasing. We will write
$\mathcal{N}$ for the nilpotent cone of the Lie algebra of $G$ and
$\mathcal{N}_{o}$ for its $\operatorname{Ad}(G)$ orbits. In this case we also
define $\mathcal{N}_{o,c}$ (resp. $\mathcal{N}_{o,\bar{c}}$) to be the set of
all pairs $(\mathbb{O},C)$ such that $\mathbb{O}\in\mathcal{N}_{o}$ and $C$ is
a conjugacy class in the fundamental group $A(\mathbb{O})$ of $\mathbb{O}$
(resp. Lusztig’s canonical quotient $\bar{A}(\mathbb{O})$ of $A(\mathbb{O})$,
see [Som01, Section 5]). There is a natural map

(2.1.1)
$\mathfrak{Q}:\mathcal{N}_{o,c}\stackrel{{\scriptstyle}}{{\to}}\mathcal{N}_{o,\bar{c}},\qquad(\mathbb{O},C)\mapsto(\mathbb{O},\bar{C})$

where $\bar{C}$ is the image of $C$ in $\bar{A}(\mathbb{O})$ under the natural
homomorphism $A(\mathbb{O})\twoheadrightarrow\bar{A}(\mathbb{O})$. There are
also projection maps
$\mathrm{pr}_{1}:\mathcal{N}_{o,c}\stackrel{{\scriptstyle}}{{\to}}\mathcal{N}_{o}$,
$\mathrm{pr}_{1}:\mathcal{N}_{o,\bar{c}}\stackrel{{\scriptstyle}}{{\to}}\mathcal{N}_{o}$.
We will typically write $\mathcal{N}^{\vee}$,
$\mathcal{N}^{\vee}_{o},\mathcal{N}^{\vee}_{o,c}$, and
$\mathcal{N}^{\vee}_{o,\bar{c}}$ for the sets $\mathcal{N}$,
$\mathcal{N}_{o},\mathcal{N}_{o,c}$, and $\mathcal{N}_{o,\bar{c}}$ associated
to the Langlands dual group $G^{\vee}$. When we wish to emphasise the group we
are working with we include it as a superscript e.g.
$\mathcal{N}_{o}^{\mathbf{G}}(k)$.

Recall the following classical result.

######  Lemma 2.1.1 (Corollary 3.5, [Pom77] and Theorem 1.5, [Pom80]).

Let $F$ be algebraically closed with good characteristic for ${\mathbf{G}}$.
Then there is canonical isomorphism of partially ordered sets
$\Theta_{F}:\mathcal{N}_{o}(F)\xrightarrow{\sim}\mathcal{N}_{o}$.

###  2.2. Duality for nilpotent orbits

Write

(2.2.1)
$d:\mathcal{N}_{0}\stackrel{{\scriptstyle}}{{\to}}\mathcal{N}_{0}^{\vee},\qquad
d:\mathcal{N}_{0}^{\vee}\stackrel{{\scriptstyle}}{{\to}}\mathcal{N}_{0}.$

for the _Barbasch-Lusztig-Spaltenstein-Vogan duality maps_ (see [Spa82,
Chapter 3] or [BV85, Appendix A]). Write

(2.2.2)
$d_{S}:\mathcal{N}_{o,c}\twoheadrightarrow\mathcal{N}^{\vee}_{o},\qquad
d_{S}:\mathcal{N}^{\vee}_{o,c}\twoheadrightarrow\mathcal{N}_{o}$

for the duality maps defined by Sommers in [Som01, Section 6] and

(2.2.3)
$D:\mathcal{N}_{o,\bar{c}}\stackrel{{\scriptstyle}}{{\to}}\mathcal{N}^{\vee}_{o,\bar{c}},\qquad
D:\mathcal{N}^{\vee}_{o,\bar{c}}\stackrel{{\scriptstyle}}{{\to}}\mathcal{N}_{o,\bar{c}}$

for the duality maps defined by Achar in ([Ach03, Section 1]). Since the
latter two maps are less well known we give a (slightly non-standard) account
of their definitions and basic properties.

The original precursor to the duality map $d$ is the involution on the set of
two sided cells of the finite Hecke algebra $\mathcal{H}$ attached to $G$ with
equal parameters apparent in [Lus79]. Upon composing this involution with the
Springer correspondence (and identifying the two Hecke algebras obtained from
$G$ and $G^{\vee}$) one obtains the duality map $d$ [Spa82, Chapter 3]. The
map $d$ is not an involution, but satisfies $d^{3}=d$. The reason is that the
2-sided cells of $\mathcal{H}$ only biject with $\operatorname{im}(d)$ (the so
called special nilpotent orbits) instead of whole of $\mathcal{N}_{o}$ and so
$d$ is only an involution when restricted to this set.

One drawback of this construction however is that the non-special nilpotent
orbits only play a peripheral role. Here the affine Hecke algebra
$\mathcal{H}_{aff}$ of $G^{\vee}$ with equal parameters hints at a remedy for
the situation. In particular, the 2-sided cells of $\mathcal{H}_{aff}$ are in
bijection with $\mathcal{N}_{o}^{\vee}$. Moreover, for any element $s\in T$,
the finite Hecke algebra $\mathcal{H}_{s}$ associated to connected centraliser
$C_{G}^{\circ}(s)$ embeds into $\mathcal{H}_{aff}$. Along with an induction
operation on 2-sided cells defined in [Lus89, Section 6.5], we are now in a
position to define $d_{S}$. For a pair $(\mathbb{O},C)\in\mathcal{N}_{o,c}$
let $x\in\mathbb{O}$ and $s$ be a semisimple element of $C_{G}(x)$ such that
$sC_{G}^{\circ}(x)$ lies in $C$. Conjugating $s$ appropriately we may assume
that it lies in $T$. Write $L_{s}=C_{G}^{\circ}(s)$ and $d^{L_{s}}$ for the
duality map $d$ for the complex reductive group $L_{s}$. Then

$d_{S}(\mathbb{O},C)=\operatorname{Ind}d^{L_{s}}(L_{s}.x).$

Here we interpret $d^{L_{s}}(L_{s}.x)$ as a 2-sided cell of $\mathcal{H}_{s}$
and the resulting induced cell of $\mathcal{H}_{aff}$ as an elements of
$\mathcal{N}_{o}^{\vee}$ using the bijections outlined in this and the
preceeding paragraph. Unravelling these constructions gives the clean
construction given in [Som01] using Springer theory, and the resulting map
$d_{S}:\mathcal{N}_{o,c}\stackrel{{\scriptstyle}}{{\to}}\mathcal{N}_{o}^{\vee}$
is well-defined and has image the whole of $\mathcal{N}_{o}^{\vee}$. Moreover,
it is clear that $d_{S}(\mathbb{O},1)=d(\mathbb{O})$ and so it extends the
usual map $d$.

This finally brings us to the definition of $D$. In defining $d_{S}$ we
obtained a map which is no longer symmetric in target and source. Achar’s
duality map rectifies this. In [Som01, Proposition 15], Sommers shows that
$d_{S}$ factors through
$\mathfrak{Q}:\mathcal{N}_{o,c}\stackrel{{\scriptstyle}}{{\to}}\mathcal{N}_{o,\bar{c}}$,
and by [Ach03, Theorem 1] we have an embedding

$\mathcal{N}_{o,\bar{c}}\xhookrightarrow{i}\mathcal{N}_{o}\times\mathcal{N}_{o}^{\vee},\quad(\mathbb{O},C)\mapsto(\mathbb{O},d_{S}(\mathbb{O},C)).$

There is also of course the corresponding embedding for the dual object

$\mathcal{N}^{\vee}_{o,\bar{c}}\xhookrightarrow{i^{\vee}}\mathcal{N}_{o}^{\vee}\times\mathcal{N}_{o}$

and this suggests a very natural candidate for $D$, namely
$D=(i^{\vee})^{-1}\circ\tilde{D}\circ i$ where

$\tilde{D}:\mathcal{N}_{o}\times\mathcal{N}_{o}^{\vee}\stackrel{{\scriptstyle}}{{\to}}\mathcal{N}_{o}^{\vee}\times\mathcal{N}_{o},\quad(\mathbb{O},\mathbb{O}^{\vee})\mapsto(\mathbb{O}^{\vee},\mathbb{O}).$

Indeed, whenever $\tilde{D}\circ i(\xi)\in\operatorname{im}i^{\vee}$, this is
equivalent to the definition for $D(\xi)$ given by Achar. However in general,
not all $\xi\in\mathcal{N}_{o,\bar{c}}$ enjoy this property. Let us call
$\xi\in\mathcal{N}_{o,\bar{c}}$ _special_ if $\tilde{D}\circ
i(\xi)\in\operatorname{im}i^{\vee}$ and let us endow
$\mathcal{N}_{o}\times\mathcal{N}_{o}^{\vee}$ with the partial order

$(\mathbb{O}_{1},\mathbb{O}_{1}^{\vee})\leq(\mathbb{O}_{2},\mathbb{O}_{2}^{\vee}),\text{
if
}\mathbb{O}_{1}\leq\mathbb{O}_{2},\mathbb{O}_{1}^{\vee}\geq\mathbb{O}_{2}^{\vee}$

(which in turn endows $\mathcal{N}_{o,\bar{c}}$ with a partial order via the
embedding $i$). By [Ach03, Theorem 2.4], every $\xi\in\mathcal{N}_{o,\bar{c}}$
is dominated by a unique smallest special $\xi^{\prime}$. We can then define
$D(\xi)=(i^{\vee})^{-1}\circ\tilde{D}\circ i(\xi^{\prime})$, and it is easily
seen that this is equivalent to Achar’s definition of $D$. The resulting map
$D:\mathcal{N}_{o,\bar{c}}\stackrel{{\scriptstyle}}{{\to}}\mathcal{N}_{o,\bar{c}}^{\vee}$
is then a duality map in much that same way that $d$ is: it satisfies
$D^{3}=D$. It also extends $d_{S}$ in the sense that $\mathrm{pr}_{1}\circ
D=d_{S}$. These results are enough for our purposes, but let us just note that
in obvious analogy to the preceding situations one might wonder if
$\operatorname{im}D$ is in bijection with the 2-sided cells of some Hecke
algebra. This is to the best of out knowledge still unknown.

###  2.3. Nilpotent orbits over ${\mathsf{k}^{un}}$

Let ${\mathbf{T}}$ be a maximal ${\mathsf{k}}$-split torus of ${\mathbf{G}}$,
${\mathbf{T}}_{1}$ be a maximal ${\mathsf{k}^{un}}$-split torus of
${\mathbf{G}}$ defined over ${\mathsf{k}}$ and containing ${\mathbf{T}}$, and
$x_{0}$ be a special point in
$\mathcal{A}({\mathbf{T}}_{1},{\mathsf{k}^{un}})$. In [Oka21, Section 2.1.5]
the third-named author constructed a bijection

$\theta_{x_{0},{\mathbf{T}}_{1}}:\mathcal{N}_{o}^{{\mathbf{G}}}({\mathsf{k}^{un}})\xrightarrow{\sim}\mathcal{N}_{o,c}$

which should be viewed as an analogoue of the bijection in Lemma 2.1.1 for
nilpotent orbits over ${\mathsf{k}^{un}}$. This bijection enjoys a number of
properties summarised in the following theorem.

######  Theorem 2.3.1.

[Oka21, Theorem 2.20, Theorem 2.27, Proposition 2.29] The bijection

$\theta_{x_{0},{\mathbf{T}}_{1}}:\mathcal{N}_{o}^{{\mathbf{G}}}({\mathsf{k}^{un}})\xrightarrow{\sim}\mathcal{N}_{o,c}$

is natural in ${\mathbf{T}}_{1}$, equivariant in $x_{0}$, and makes the
following diagram commute:

(2.3.1)
${\mathcal{N}_{o}^{{\mathbf{G}}}({\mathsf{k}^{un}})}$${\mathcal{N}_{o,c}}$${\mathcal{N}_{o}({\bar{\mathsf{k}}})}$${\mathcal{N}_{o}.}$$\scriptstyle{\theta_{x_{0},{\mathbf{T}}_{1}}}$$\scriptstyle{\mathcal{N}_{o}({\bar{\mathsf{k}}}/{\mathsf{k}^{un}})}$$\scriptstyle{\mathrm{pr}_{1}}$$\scriptstyle{\Theta_{{\bar{\mathsf{k}}}}}$

One important consequence of the equivariance in $x_{0}$ is that the
composition

$d^{un}:=d_{S}\circ\theta_{x_{0},{\mathbf{T}}_{1}}:\mathcal{N}_{o}({\mathsf{k}^{un}})\stackrel{{\scriptstyle}}{{\to}}\mathcal{N}_{o}^{\vee}$

is independent of the choice of $x_{0}$ [Oka21, Proposition 2.32]. We suppress
the ${\mathbf{T}}_{1}$ from the notation since this choice is implicit from
fixing the root data and the dual group.

For $\mathbb{O}_{1},\mathbb{O}_{2}\in\mathcal{N}_{o}({\mathsf{k}^{un}})$
define $\mathbb{O}_{1}\leq_{A}\mathbb{O}_{2}$ by

$\mathbb{O}_{1}\leq_{A}\mathbb{O}_{2}\iff\mathcal{N}_{o}({\bar{\mathsf{k}}}/{\mathsf{k}^{un}})(\mathbb{O}_{1})\leq\mathcal{N}_{o}({\bar{\mathsf{k}}}/{\mathsf{k}^{un}})(\mathbb{O}_{2}),\text{
and }d^{un}(\mathbb{O}_{1})\geq d^{un}(\mathbb{O}_{2})$

and let $\sim_{A}$ denote the equivalence classes of this pre-order. Recall
from the previous section that $\mathcal{N}_{o,\bar{c}}$ can be endowed with a
partial ordering coming from the embedding
$\mathcal{N}_{o,\bar{c}}\hookrightarrow\mathcal{N}_{o}\times\mathcal{N}_{o}^{\vee}$.
The following proposition parameterises
$\mathcal{N}_{o}({\mathsf{k}^{un}})/\sim_{A}$ in terms of
$\mathcal{N}_{o,\bar{c}}$.

######  Proposition 2.3.2.

[Oka21, Theorem 2.33] The composition
$\mathfrak{Q}\circ\theta_{x_{0},{\mathbf{T}}_{1}}:\mathcal{N}_{o}({\mathsf{k}^{un}})\stackrel{{\scriptstyle}}{{\to}}\mathcal{N}_{o,\bar{c}}$
descends to an isomorphism of partial orders

$\bar{\theta}:\mathcal{N}_{o}({\mathsf{k}^{un}})/\sim_{A}\stackrel{{\scriptstyle}}{{\to}}\mathcal{N}_{o,\bar{c}}$

which does not depend on $x_{0}$.

###  2.4. Representations with unipotent cuspidal support

We briefly recall the classification of irreducible representations with
unipotent cuspidal support defined in [Lus95, §1.6,§1.21]. If $\mathcal{S}$ is
a subset of $G^{\vee}$, let

$Z^{1}_{G^{\vee}_{\mathsf{sc}}}(\mathcal{S})=\text{ preimage of
}G^{\vee}(\mathcal{S})/Z(G^{\vee})\text{ under the projection
}G^{\vee}_{\mathsf{sc}}\stackrel{{\scriptstyle}}{{\to}}G^{\vee}_{\operatorname{ad}},$

and let $A^{1}(\mathcal{S})$ denote the component group of
$Z^{1}_{G^{\vee}_{\mathsf{sc}}}(\mathcal{S})$.

Write $\Phi(G^{\vee})$ for the set of $G^{\vee}$-orbits (under conjugation) of
triples $(s,n,\rho)$ where

  * •

$s\in G^{\vee}$ is semisimple,

  * •

$n\in\mathfrak{g}^{\vee}$ such that $\operatorname{Ad}(s)n=qn$,

  * •

$\rho\in\mathrm{Irr}(A^{1}_{G^{\vee}}(s,n))$.

Without loss of generality, we may assume that $s\in T^{\vee}$. Note that
$n\in\mathfrak{g}^{\vee}$ is necessarily nilpotent. The group $G^{\vee}(s)$
acts with finitely many orbits on the $q$-eigenspace of $\operatorname{Ad}(s)$

$\mathfrak{g}_{q}^{\vee}=\\{x\in\mathfrak{g}^{\vee}\mid\operatorname{Ad}(s)x=qx\\}$

In particular, there is a unique open $G^{\vee}(s)$-orbit in
$\mathfrak{g}_{q}^{\vee}$.

Fix an $\mathfrak{sl}(2)$-triple
$\\{n^{-},h^{\vee},n\\}\subset\mathfrak{g}^{\vee}$ with
$h^{\vee}\in\mathfrak{t}^{\vee}_{\mathbb{R}}$ and set

$s_{0}:=sq^{-\frac{h^{\vee}}{2}}.$

Then $\operatorname{Ad}(s_{0})n=n$.

A parameter $(s,n,\rho)\in\Phi(G^{\vee})$ is called _discrete_ if
$Z_{G^{\vee}}(s,n)$ does not contain a nontrivial torus. A discrete parameter
is called _cuspidal_ if $(u,\rho)$ is a cuspidal pair in $Z_{G^{\vee}}(s)$ in
the sense of Lusztig.

Let $\Pi^{\mathsf{Lus}}({\mathbf{G}}^{\omega}(\mathsf{k}))$ denote the
equivalence classes of irreducible
${\mathbf{G}}^{\omega}(\mathsf{k})$-representations with unipotent cuspidal
support and

$\Pi^{\mathsf{Lus}}({\mathbf{G}})=\bigsqcup_{\omega\in\Omega_{\mathsf{ad}}}\Pi^{\mathsf{Lus}}({\mathbf{G}}^{\omega}(\mathsf{k})).$

(In this subsection, ${\mathbf{G}}^{1}(\mathsf{k})$ is the split form.) The
following theorem is a combination of several results, namely [KL87, Theorems
7.12, 8.2, 8.3] for ${\mathbf{G}}$ adjoint and Iwahori-spherical
representations, [Lus95, Corollary 6.5] and [Lus02, Theorem 10.5] for
${\mathbf{G}}$ adjoint and representations with unipotent cuspidal support,
[Ree02, Theorem 3.5.4] for ${\mathbf{G}}$ arbitrary and Iwahori-spherical
representations, and [FOS20, FO20, Sol18] for ${\mathbf{G}}$ arbitrary and
representations with unipotent cuspidal support. See [AMS21, §2.3] for a
discussion of the compatibility between these classifications.

######  Theorem 2.4.1 (Deligne-Langlands-Lusztig correspondence).

There is a bijection

$\Phi(G^{\vee})\xrightarrow{\sim}\Pi^{\mathsf{Lus}}({\mathbf{G}}),\qquad(s,n,\rho)\mapsto
X(s,n,\rho),$

such that

  1. (1)

$X(s,n,\rho)$ is tempered if and only if $s_{0}\in T_{c}^{\vee}$ (in
particular, $\overline{G^{\vee}(s)n}=\mathfrak{g}_{q}^{\vee}$),

  2. (2)

$X(s,n,\rho)$ is square integrable (modulo the center) if and only if it is
tempered and $(s,n,\rho)$ is discrete.

  3. (3)

$X(s,n,\rho)$ is supercuspidal if and only if $(s,n,\rho)$ is a cuspidal
parameter.

  4. (4)

$X(s,n,\rho)\in\Pi^{\mathsf{Lus}}({\mathbf{G}}^{\omega}(\mathsf{k}))$ if and
only if $\rho|_{Z(G^{\vee})}$ is a multiple of $\zeta_{\omega}$.

This bijection satisfies several natural desiderata (including formal degrees,
equivariance with respect to tensoring by weakly unramified characters), see
[Sol18, Theorem 1] and [FOS20, Theorem 2].

We denote by $\Pi^{\mathsf{Lus}}_{s}({\mathbf{G}}(\mathsf{k}))$ the set of
irreducible ${\mathbf{G}}(\mathsf{k})$-representations $X(s,n,\rho)$ for a
fixed $s\in T^{\vee}$.

###  2.5. Aubert-Zelevinsky duality

There is an involution $\mathsf{AZ}$ on the Grothendieck group
$R(\mathbf{G}({\mathsf{k}}))$ of smooth
$\mathbf{G}({\mathsf{k}})$-representation, called _Aubert-Zelevinsky duality_
[Aub95, §1]. This involution can defined in the following manner. Let
$\mathcal{Q}$ denote the set of parabolic subgroups of ${\mathbf{G}}$ defined
over $\mathsf{k}$ and containing $\mathbf{B}$. For every
$\mathbf{Q}\in\mathcal{Q}$, let
$i_{\mathbf{Q}(\mathsf{k})}^{\mathbf{G}(\mathsf{k})}$ and
$\mathsf{r}_{\mathbf{Q}(\mathsf{k})}^{\mathbf{G}(\mathsf{k})}$ denote the
normalized parabolic induction and normalized Jacquet functors, respectively.
Then $\mathsf{AZ}$ is defined by

(2.5.1)
$\widetilde{\mathsf{AZ}}:R(\mathbf{G}(\mathsf{k}))\stackrel{{\scriptstyle}}{{\to}}R(\mathbf{G}(\mathsf{k})),\qquad\widetilde{\mathsf{AZ}}(X)=\sum_{\mathbf{Q}\in\mathcal{Q}}(-1)^{r_{\mathbf{Q}}}~{}i_{\mathbf{Q}(\mathsf{k})}^{{\mathbf{G}}(\mathsf{k})}(\mathsf{r}_{\mathbf{Q}(\mathsf{k})}^{{\mathbf{G}}(\mathsf{k})}(X)),$

where $r_{\mathbf{Q}}$ is the semisimple rank of the reductive quotient of
$\mathbf{Q}$. If a class $X\in R(\mathbf{G}({\mathsf{k}}))$ is irreducible and
$X$ occurs as a composition factor of a parabolically induced module
$i_{\mathbf{Q}(\mathsf{k})}^{{\mathbf{G}}(\mathsf{k})}(\sigma)$, where
$\sigma$ is a supercuspidal representation, then, by [Aub95, Corollaire 3.9]
$(-1)^{r_{\mathbf{Q}}}\widetilde{\mathsf{AZ}}(X)$ is the class of an
irreducible representation, which we denote $\mathsf{AZ}(X)$.

Moreover, it is known that $\mathsf{AZ}$ is an involution on the set of
irreducible representations in any Bernstein component, see [Aub95, §3], also
[BBK18, §3.2]. In particular, it preserves the irreducible representations
with unipotent cuspidal support, and in fact, it induces an involution on
$\Pi^{\mathsf{Lus}}_{s}({\mathbf{G}}(\mathsf{k}))$, for each $s\in W\backslash
T^{\vee}$. In addition, by [BM89, Theorem 1.1] for Iwahori-spherical
representations and [BC13, Theorem 1.2.1] in general, $\mathsf{AZ}$ maps
unitary representations to unitary representations in the category of
representations with unipotent supercuspidal support.

###  2.6. Wavefront sets

Let $X$ be an admissible smooth representation of ${\mathbf{G}}({\mathsf{k}})$
and let $\Theta_{X}$ be the character of $X$. Recall that for each nilpotent
orbit $\mathbb{O}\in\mathcal{N}_{o}^{{\mathbf{G}}}({\mathsf{k}})$ there is an
associated distribution $\mu_{\mathbb{O}}$ on
$C_{c}^{\infty}(\mathfrak{g}({\mathsf{k}}))$ called the _nilpotent orbital
integral_ of $\mathbb{O}$ [Ran72]. Write $\hat{\mu}_{\mathbb{O}}$ for the
Fourier transform of this distribution. By a result of Harish-Chandra
([Har99]), there are complex numbers $c_{\mathbb{O}}(X)\in\mathbb{C}$ such
that

(2.6.1)
$\Theta_{X}(\mathrm{exp}(\xi))=\sum_{\mathbb{O}}c_{\mathbb{O}}(X)\hat{\mu}_{\mathbb{O}}(\xi)$

for $\xi\in\mathfrak{g}({\mathsf{k}})$ a regular element in a small
neighborhood of $0$. The formula (2.6.1) is called the _local character
expansion_ of $\pi$. There are several important invariants which can be
extracted from the local character expansion. The ($p$-adic) wavefront set of
$X$ is defined to be

$\mathrm{WF}(X):=\max\\{\mathbb{O}\mid c_{\mathbb{O}}(X)\neq
0\\}\subseteq\mathcal{N}_{o}({\mathsf{k}}).$

The _geometric wavefront set_ of $X$ is defined to be

${}^{{\bar{\mathsf{k}}}}\mathrm{WF}(X):=\max\\{\mathcal{N}_{o}({\bar{\mathsf{k}}}/{\mathsf{k}})(\mathbb{O})\mid
c_{\mathbb{O}}(X)\neq 0\\}\subseteq\mathcal{N}_{o}({\bar{\mathsf{k}}}),$

see [Wal18, p. 1108] (warning: in [Wal18], this invariant is called simply the
‘wavefront set’ of $X$). In [Oka21, Section 2.2.3] the third author has
introduced a third type of wavefront set called the _canonical unramified
wavefront set_. This is defined to be

$\underline{\mathrm{WF}}(X):=\max\\{[\mathcal{N}_{o}({\mathsf{k}^{un}}/{\mathsf{k}})(\mathbb{O})]\mid
c_{\mathbb{O}}(X)\neq
0\\}\subseteq\mathcal{N}_{o}({\mathsf{k}^{un}})/\sim_{A}.$

Recall the parameterisations
$\Theta_{{\bar{\mathsf{k}}}}:\mathcal{N}_{o}({\bar{\mathsf{k}}})\stackrel{{\scriptstyle}}{{\to}}\mathcal{N}_{o}$
and
$\bar{\theta}:\mathcal{N}_{o}({\mathsf{k}^{un}})/\sim_{A}\stackrel{{\scriptstyle}}{{\to}}\mathcal{N}_{o,\bar{c}}$.
We define

$\hphantom{}{}^{\bar{{\mathsf{k}}}}\mathrm{WF}(X,\mathbb{C}):=\Theta_{\bar{\mathsf{k}}}(\hphantom{}^{\bar{{\mathsf{k}}}}\mathrm{WF}(X)),\quad\underline{\mathrm{WF}}(X,\mathbb{C}):=\bar{\theta}(\hphantom{}\underline{\mathrm{WF}}(X)).$

We have the following basic relation between the geometric and canonical
unramified wavefront sets.

######  Proposition 2.6.1.

[Oka21, Theorem 2.37] If $\underline{\mathrm{WF}}(X)$ is a singleton then

${}^{{\bar{\mathsf{k}}}}\mathrm{WF}(X,\mathbb{C})=\mathrm{pr}_{1}(\underline{\mathrm{WF}}(X,\mathbb{C})).$

For our main result below, we will need a description of
$\underline{\mathrm{WF}}(X,\mathbb{C})$ in the case when $X$ is an irreducible
representation with unipotent cuspidal support and real infinitesimal
character. Such a description is contained in the main result of [CMO] (see
also [CMO22, Theorem 1.2.1] for the Iwahori-spherical case).

######  Theorem 2.6.2.

Let $X=X(s,n,\rho)$ be an irreducible
${\mathbf{G}}(\mathsf{k})$-representation with real infinitesimal character
and let $\mathsf{AZ}(X)=X(s,n^{\prime},\rho^{\prime})$. Then

  1. (1)

$\underline{\mathrm{WF}}(X,\mathbb{C})$ is a singleton, and

$\underline{\mathrm{WF}}(X,\mathbb{C})=D(\mathbb{O}^{\vee}_{\mathsf{AZ}(X)},1),$

where $\mathbb{O}^{\vee}_{\mathsf{AZ}(X)}$ is the $G^{\vee}$-orbit of
$n^{\prime}$.

  2. (2)

Suppose $X=X(q^{\frac{1}{2}h^{\vee}},n,\rho)$ where $h^{\vee}$ is the neutral
element of a Lie triple attached to a nilpotent orbit
$\mathbb{O}^{\vee}\subset\mathfrak{g}^{\vee}$. Then

$D(\mathbb{O}^{\vee},1)\leq\hphantom{}\underline{\mathrm{WF}}(X,\mathbb{C}).$

###  2.7. Basic Arthur packets

To formulate the relation between $\mathsf{AZ}$ and Arthur packets, it will be
convenient to recall an alternative formulation of the Langlands
classification. A _simplified Langlands parameter_ is a continuous
homomorphism
$\widetilde{\varphi}:W_{{\mathsf{k}}}\times\mathrm{SL}(2,\mathbb{C})\stackrel{{\scriptstyle}}{{\to}}G^{\vee}$
such that $\widetilde{\varphi}(W_{k})$ consists of semisimple elements and
$\widetilde{\varphi}|_{\mathrm{SL}(2,\mathbb{C})}$ is algebraic. A simplified
Langlands parameter is tempered if the image of $W_{k}$ is compact (the idea
of replacing the Weil-Deligne group $W_{k}^{\prime}$ with
$W_{k}\times\mathrm{SL}(2,\mathbb{C})$ in the non-Archimedean case was first
proposed by Langlands in [Lan79, p. 209]). Although there are good reasons for
preferring the Weil-Deligne group formulation, there is a bijection between
simplified and honest Langlands parameters, see [Kna97, p. 278]. Similarly, a
_simplified Arthur parameter_ is a continuous homomorphism
$\widetilde{\psi}:W_{k}\times\mathrm{SL}(2,\mathbb{C})_{\mathsf{Lan}}\times\mathrm{SL}(2,\mathbb{C})_{\mathsf{Art}}\stackrel{{\scriptstyle}}{{\to}}G^{\vee}$
such that the restriction of $\widetilde{\psi}$ to
$W_{k}\times\mathrm{SL}(2,\mathbb{C})_{\mathsf{Lan}}$ is a tempered simplified
Langlands parameter and the restriction of $\widetilde{\psi}$ to
$\mathrm{SL}(2,\mathbb{C})_{\mathsf{Art}}$ is algebraic. The bijection between
simplified and honest Langlands parameters induces a bijection between
simplified and honest Arthur parameters.

If $\widetilde{\psi}$ is a simplified Arthur parameter, we get a second
simplified Arthur parameter $\widetilde{\psi}^{t}$ by ‘flipping’ the
$\mathrm{SL}(2,\mathbb{C})$ factors, [Art13, p. 390], i.e.

$\widetilde{\psi}^{t}(w,x,y):=\widetilde{\psi}(w,y,x),\qquad w\in W_{k},\
x,y\in\mathrm{SL}(2,\mathbb{C}).$

Via the bijection between simplified and honest Arthur parameters, the map
$\widetilde{\psi}\mapsto\widetilde{\psi}^{t}$ induces an involution on honest
Arthur parameters, which we also denote by $\psi\mapsto\psi^{t}$. It is
expected that

(2.7.1)
$\mathsf{AZ}(\Pi_{\psi}^{\mathsf{Art}}(\mathbf{G}(\mathsf{k}))=\Pi_{\psi^{t}}^{\mathsf{Art}}(\mathbf{G}(\mathsf{k})).$

In the case when $\mathbf{G}$ is orthogonal or symplectic, this duality is
discussed in [Art13, §7.1]. If the Arthur parameter $\psi$ is trivial on the
inertia subgroup $I_{k}$, then the associated Langlands parameter
$\phi_{\psi}$ is unramified. So the representations in the Langlands packet
$\Pi_{\phi_{\psi}}^{\mathsf{Lan}}(\mathbf{G}(\mathsf{k}))$, and hence also in
the Arthur packet $\Pi_{\psi}^{\mathsf{Art}}(\mathbf{G}(\mathsf{k}))$, are of
unipotent cuspidal support.

If $\widetilde{\psi}$ is trivial on
$W_{k}\times\operatorname{SL}(2,\mathbb{C})_{\mathsf{Art}}$, then
$\Pi_{\psi}^{\mathsf{Art}}(\mathbf{G}(\mathsf{k}))=\Pi_{\phi_{\psi}}^{\mathsf{Lan}}(\mathbf{G}(\mathsf{k}))$
consists of _tempered_ representations with real infinitesimal character. So
if $\psi$ is basic (cf. Definition 1.0.4), the desideratum (2.7.1) suggests
the following definition for the associated Arthur packet
$\Pi_{\psi}^{\mathsf{Art}}(\mathbf{G}({\mathsf{k}}))$, see [Art13, (7.1.10)].

######  Definition 2.7.1.

Let $\mathbb{O}^{\vee}$ be a nilpotent adjoint $G^{\vee}$-orbit and let
$\psi_{\mathbb{O}^{\vee}}$ be the associated basic Arthur parameter (cf.
Definition 1.0.4). Then

$\Pi^{\mathsf{Art}}_{\psi_{\mathbb{O}^{\vee}}}(\mathbf{G}(\mathsf{k}))=\\{X\in\Pi^{\mathsf{Lus}}_{q^{\frac{1}{2}h^{\vee}}}(\mathbf{G}(\mathsf{k}))\mid\mathsf{AZ}(X)\text{
is tempered}\\}.$

##  3\. Main results

Fix a nilpotent orbit $\mathbb{O}^{\vee}\subset\mathcal{N}^{\vee}$ and choose
an $\mathfrak{sl}(2)$-triple $(e^{\vee},f^{\vee},h^{\vee})$ with
$e^{\vee}\in\mathbb{O}^{\vee}$. Recall that if $X=X(s,n,\rho)$, we write
$\mathbb{O}^{\vee}_{X}$ for the nilpotent adjoint $G^{\vee}$-orbit of $n$. The
semisimple operator $\operatorname{ad}(h^{\vee})$ induces a Lie algebra
grading

$\mathfrak{g}^{\vee}=\bigoplus_{n\in\mathbb{Z}}\mathfrak{g}^{\vee}[n],\qquad\mathfrak{g}^{\vee}[n]:=\\{x\in\mathfrak{g}^{\vee}\mid[h^{\vee},x]=nx\\}.$

Write $L^{\vee}$ for the connected (Levi) subgroup corresponding to the
centralizer $\mathfrak{g}^{\vee}_{0}$ of $h^{\vee}$. If we set
$s:=q^{\frac{1}{2}h^{\vee}}$, then

$L^{\vee}=G^{\vee}(s),\qquad\mathfrak{g}^{\vee}[2]=\mathfrak{g}^{\vee}_{q}$

where $G^{\vee}(s)$ and $\mathfrak{g}^{\vee}_{q}$ are as defined in Section
2.4. Note that $L^{\vee}$ acts by conjugation on each
$\mathfrak{g}^{\vee}[n]$, and in particular on $\mathfrak{g}^{\vee}[2]$.We
will need the following well-known facts, see [Kos59, Section 4] or [Lus95a,
Prop 4.2].

######  Lemma 3.0.1.

The following are true:

  * (i)

$L^{\vee}e^{\vee}$ is an open subset of $\mathfrak{g}^{\vee}[2]$ (and hence
the unique open $L^{\vee}$-orbit therein).

  * (ii)

$L^{\vee}e=\mathbb{O}^{\vee}\cap\mathfrak{g}^{\vee}[2]$.

  * (iii)

$G^{\vee}\mathfrak{g}^{\vee}[2]\subseteq\overline{\mathbb{O}^{\vee}}$.

######  Lemma 3.0.2.

Let
$X\in\Pi^{\mathsf{Lus}}_{q^{\frac{1}{2}h^{\vee}}}(\mathbf{G}(\mathsf{k}))$.
Then $X$ is tempered if and only if $\mathbb{O}^{\vee}_{X}=\mathbb{O}^{\vee}$.

###### Proof.

Let $X=X(q^{\frac{1}{2}h^{\vee}},n,\rho)$ for $n\in\mathfrak{g}^{\vee}[2]$ and
$\rho\in\mathrm{Irr}(A(q^{\frac{1}{2}h^{\vee}},n))$. Note that $s_{0}=1\in
T_{c}^{\vee}$. So

$\displaystyle X\text{ is tempered}$
$\displaystyle\iff\overline{L^{\vee}n}=g^{\vee}[2]$ (Theorem 2.4.1(i))
$\displaystyle\iff n\in L^{\vee}e^{\vee}$ (Lemma 3.0.1(i)) $\displaystyle\iff
n\in\mathbb{O}^{\vee}$ (Lemma 3.0.1(ii))
$\displaystyle\iff\mathbb{O}^{\vee}_{X}=\mathbb{O}^{\vee}.$

∎

######  Theorem 3.0.3.

Let $\mathbb{O}^{\vee}$ be a nilpotent adjoint $G^{\vee}$-orbit and let
$\psi_{\mathbb{O}^{\vee}}$ be the associated basic Arthur parameter (cf.
Definition 1.0.4). Then

(3.0.1)
$\Pi^{\mathsf{Art}}_{\psi_{\mathbb{O}^{\vee}}}(\mathbf{G}({\mathsf{k}}))=\\{X\in\Pi^{\mathsf{Lus}}_{q^{\frac{1}{2}h^{\vee}}}(\mathbf{G}(\mathsf{k}))\mid\
\underline{\mathrm{WF}}(X,\mathbb{C})\leq D(\mathbb{O}^{\vee},1)\\}.$

###### Proof.

Let
$X\in\Pi^{\mathsf{Lus}}_{q^{\frac{1}{2}h^{\vee}}}(\mathbf{G}(\mathsf{k}))$. By
Theorem 2.6.2, we have the bound

$D(\mathbb{O}^{\vee},1)\leq\underline{\mathrm{WF}}(X,\mathbb{C}).$

Thus

(3.0.2) $\underline{\mathrm{WF}}(X,\mathbb{C})\leq
D(\mathbb{O}^{\vee},1)\iff\underline{\mathrm{WF}}(X,\mathbb{C})=D(\mathbb{O}^{\vee},1).$

But by Theorem 2.6.2 we also have

(3.0.3)
$\underline{\mathrm{WF}}(X,\mathbb{C})=D(\mathbb{O}^{\vee}_{\mathsf{AZ}(X)},1).$

Therefore

$\displaystyle
X\in\Pi^{\mathsf{Art}}_{\mathbb{O}^{\vee}}(\mathbf{G}(\mathsf{k}))$
$\displaystyle\iff\mathsf{AZ}(X)\text{ is tempered}$ (Definition 2.7.1)
$\displaystyle\iff\mathbb{O}^{\vee}_{\mathsf{AZ}(X)}=\mathbb{O}^{\vee}$ (Lemma
3.0.2) $\displaystyle\iff
D(\mathbb{O}^{\vee}_{\mathsf{AZ}(X)},1)=D(\mathbb{O}^{\vee},1)$ ([CMO22, Lemma
2.1.2])
$\displaystyle\iff\underline{\mathrm{WF}}(X,\mathbb{C})=D(\mathbb{O}^{\vee},1)$
(Equation 3.0.3) $\displaystyle\iff\underline{\mathrm{WF}}(X,\mathbb{C})\leq
D(\mathbb{O}^{\vee},1)$ $\displaystyle\text{(Equation \ref{eq:bound})}.$

∎

There is a well-known conjecture regarding the (geometric) wavefront sets of
the constituents of an Arthur packet. The following formulation essentially
appears in [JL16, Conjecture 3.2] (in [JL16] it is stated only for classical
groups. The version below is the natural generalization to arbitrary groups).

######  Conjecture 3.0.4 (cf. [JL16, Conjecture 3.2]).

Let $\psi$ be an Arthur parameter and let $\mathbb{O}^{\vee}_{\psi}$ be the
nilpotent $G^{\vee}$-orbit corresponding to the restriction of $\psi$ to
$\mathrm{SL}(2,\mathbb{C})$. Then

  * (i)

For every $X\in\Pi^{\mathsf{Art}}_{\psi}(\mathbf{G}({\mathsf{k}}))$, there is
a bound

$\hphantom{}{}^{{\bar{\mathsf{k}}}}\mathrm{WF}(X,\mathbb{C})\leq
d(\mathbb{O}^{\vee}_{\psi}).$

  * (ii)

The bound in (i) is achieved for some
$X\in\Pi^{\mathsf{Art}}_{\psi}(\mathbf{G}({\mathsf{k}}))$.

We note that a global version of this conjecture appears in [Sha90]. An
immediate consequence of Theorem 3.0.3 is that Conjecture 3.0.4 holds for
basic Arthur packets. In fact, something stronger is true for this class of
packets.

######  Corollary 3.0.5.

Let $\mathbb{O}^{\vee}$ be a nilpotent adjoint $G^{\vee}$-orbit and let
$\psi_{\mathbb{O}^{\vee}}$ be the associated Arthur parameter. Then for every
$X\in\Pi^{\mathsf{Art}}_{\psi_{\mathbb{O}^{\vee}}}(\mathbf{G}({\mathsf{k}}))$,
there is an equality

$\hphantom{}{}^{{\bar{\mathsf{k}}}}\mathrm{WF}(X,\mathbb{C})=d(\mathbb{O}^{\vee}).$

###### Proof.

Let
$X\in\Pi^{\mathsf{Art}}_{\psi_{\mathbb{O}^{\vee}}}(\mathbf{G}({\mathsf{k}}))$.
By Theorem 3.0.3 we have that
$\underline{\mathrm{WF}}(X,\mathbb{C})=D(\mathbb{O}^{\vee}_{\psi},1)$. Recall
from the discussion in Section 2.2 that $\mathrm{pr}_{1}\circ D=d_{S}$ and
that $d_{S}(\mathbb{O}^{\vee},1)=d(\mathbb{O}^{\vee})$. Moreover, by
Proposition 2.6.1, we have that
${}^{{\bar{\mathsf{k}}}}\mathrm{WF}(X,\mathbb{C})=\mathrm{pr}_{1}(\underline{\mathrm{WF}}(X,\mathbb{C}))$.
Therefore

${}^{{\bar{\mathsf{k}}}}\mathrm{WF}(X,\mathbb{C})=\mathrm{pr}_{1}(\underline{\mathrm{WF}}(X,\mathbb{C}))=\mathrm{pr}_{1}(D(\mathbb{O}^{\vee},1))=d(\mathbb{O}^{\vee}).$

∎

###  3.1. Weak Arthur packets

Let $\mathbb{O}^{\vee}$ be a nilpotent adjoint $G^{\vee}$-orbit and let
$\psi_{\mathbb{O}^{\vee}}$ be the associated Arthur parameter. Theorem 3.0.3
gives a characterization of the associated Arthur packet
$\Pi^{\mathsf{Art}}_{\mathbb{O}^{\vee}}({\mathbf{G}}(\mathsf{k}))$ in terms of
the _canonical unramified wavefront set_ $\underline{\mathrm{WF}}(X)$. If we
replace $\underline{\mathrm{WF}}(X)$ with the coarser invariant
$\hphantom{}{}^{\bar{{\mathsf{k}}}}\mathrm{WF}(X)$, we get a set which we will
denote by
$\Pi_{\psi_{\mathbb{O}^{\vee}}}^{\mathsf{Weak}}(\mathbf{G}({\mathsf{k}}))$

(3.1.1)
$\Pi_{\psi_{\mathbb{O}^{\vee}}}^{\mathsf{Weak}}(\mathbf{G}({\mathsf{k}})):=\\{X=X(q^{\frac{1}{2}h^{\vee}},n,\rho)\mid\hphantom{}^{\bar{{\mathsf{k}}}}\mathrm{WF}(X)\leq
d(\mathbb{O}^{\vee})\\}.$

Because of the compatibility between $\underline{\mathrm{WF}}(X)$ and
$\hphantom{}{}^{\bar{{\mathsf{k}}}}\mathrm{WF}(X)$, there is a containment

$\Pi_{\psi_{\mathbb{O}^{\vee}}}^{\mathsf{Art}}(\mathbf{G}({\mathsf{k}}))\subseteq\Pi_{\psi_{\mathbb{O}^{\vee}}}^{\mathsf{Weak}}(\mathbf{G}({\mathsf{k}})).$

In contrast to the set
$\Pi^{\mathsf{Art}}_{\psi_{\mathbb{O}^{\vee}}}({\mathbf{G}}(\mathsf{k}))$, the
set $\Pi_{\psi_{\mathbb{O}^{\vee}}}^{\mathsf{Weak}}(\mathbf{G}({\mathsf{k}}))$
is _not_ the $\mathsf{AZ}$-dual of a tempered Arthur packet, nor is it
parameterized (in any simple way) by representations of
$A(\mathbb{O}^{\vee})$. So it is not a reasonable candidate for an Arthur
packet. By analogy with the case of real reductive groups (see [ABV92, Chapter
27]), we call
$\Pi_{\psi_{\mathbb{O}^{\vee}}}^{\mathsf{Weak}}(\mathbf{G}({\mathsf{k}}))$ the
_weak Arthur packet_ attached to $\mathbb{O}^{\vee}$.

Using Theorem 2.6.2 and Proposition 2.6.1, it is easy to describe the
constituents of
$\Pi_{\mathbb{O}^{\vee}}^{\mathsf{Weak}}(\mathbf{G}({\mathsf{k}}))$.

######  Corollary 3.1.1.

The weak Arthur packet
$\Pi_{\mathbb{O}^{\vee}}^{\mathsf{Weak}}(\mathbf{G}({\mathsf{k}}))$ is the set
of irreducible representations
$\mathsf{AZ}(X(q^{\frac{1}{2}h^{\vee}},n,\rho))$, where $n$ belongs to the
special piece (in the sense of [Spa82]) of $\mathbb{O}^{\vee}.$

Guided by the case of real reductive groups, we conjecture that
$\Pi_{\mathbb{O}^{\vee}}^{\mathsf{Weak}}(\mathbf{G}({\mathsf{k}}))$ is a union
of Arthur packets. Recall the notation of Section 2.7.

######  Conjecture 3.1.2.

The weak Arthur packet
$\Pi_{\psi_{\mathbb{O}^{\vee}}}^{\mathsf{Weak}}(\mathbf{G}({\mathsf{k}}))$ is
a (possibly non-disjoint) union of several Arthur packets

(3.1.2)
$\Pi_{\psi_{\mathbb{O}^{\vee}}}^{\mathsf{Weak}}(\mathbf{G}({\mathsf{k}}))=\bigcup_{i=1}^{n}\Pi_{\widetilde{\psi}_{i}}^{\mathsf{Art}}(\mathbf{G}({\mathsf{k}})),$

corresponding to a collection of simplified Arthur parameters

(3.1.3)
$\\{\widetilde{\psi}_{i}:W_{\mathsf{k}}\times\mathrm{SL}(2,\mathbb{C})_{\mathsf{Lan}}\times\mathrm{SL}(2,\mathbb{C})_{\mathsf{Art}}\stackrel{{\scriptstyle}}{{\to}}G^{\vee}\mid\
1\leq i\leq n\\},$

including the anti-tempered parameter $\widetilde{\psi}_{\mathbb{O}^{\vee}}$,
but also several others. Each $\widetilde{\psi}_{i}$ is trivial on the Weil
group $W_{k}$ and thus corresponds to an algebraic homomorphism

$\widetilde{\psi}_{i}:\mathrm{SL}(2,\mathbb{C})_{\mathrm{Lan}}\times\mathrm{SL}(2,\mathbb{C})_{\mathsf{Art}}\stackrel{{\scriptstyle}}{{\to}}G^{\vee}$

Via the Jacobson-Morozov theorem, each such $\widetilde{\psi}_{i}$ can be
identified with a pair of nilpotent adjoint $G^{\vee}$-orbits

(3.1.4)
$(\mathbb{O}^{\vee}_{i,\mathsf{Art}},\mathbb{O}^{\vee}_{i,\mathsf{Lan}}),$

and a pair of semisimple elements in a single fixed Cartan subalgebra of
$\mathfrak{g}^{\vee}$

(3.1.5) $(h^{\vee}_{i,\mathsf{Art}},h^{\vee}_{i,\mathsf{Lan}}).$

These elements should satisfy

(3.1.6) $\frac{1}{2}h^{\vee}\in
G^{\vee}(\frac{1}{2}h^{\vee}_{i,\mathsf{Art}}+\frac{1}{2}h^{\vee}_{i,\mathsf{Lan}}).$

One might be tempted to guess that the set $\\{\widetilde{\psi}_{i}\\}$ of
parameters appearing in Conjecture 3.1.2 consists of _all_ possible parameters
satisfying the infinitesimal character condition (3.1.6). But nothing quite so
simple is true. The tempered Arthur packet corresponding to the pair
$(\mathbb{O}^{\vee}_{\mathsf{Art}},\mathbb{O}^{\vee}_{\mathsf{Lan}})=(0,\mathbb{O}^{\vee})$
is almost never contained in
$\Pi_{\mathbb{O}^{\vee}}^{\mathsf{Weak}}(\mathbf{G}({\mathsf{k}}))$ (see
Section 4 for a detailed example). Thus, the problem of describing the Arthur
packets in $\Pi_{\mathbb{O}^{\vee}}^{\mathsf{Weak}}(\mathbf{G}({\mathsf{k}}))$
seems interesting and nontrivial. We note that Conjecture 3.1.2 together with
Corollary 3.1.1 implies the following conjecture (noting that $\mathsf{AZ}$
preserves unitarity):

######  Conjecture 3.1.3.

Let $(q^{\frac{1}{2}h^{\vee}},n,\rho)$ be a Deligne-Langlands-Lusztig
parameter such that $n$ belongs to the special piece of $\mathbb{O}^{\vee}$.
Then the irreducible representation $X(q^{\frac{1}{2}h^{\vee}},n,\rho)$ is
unitary.

##  4\. Example (split $F_{4}$)

Let $\mathbf{G}({\mathsf{k}})$ be the split form of the (unique) simple
exceptional group of type $F_{4}$, and let $\mathbb{O}^{\vee}=F_{4}(a_{3})$,
the minimal distinguished orbit. There are 20 representations
$X_{1},...,X_{20}$ in
$\Pi^{\mathsf{Lan}}_{q^{\frac{1}{2}h^{\vee}}}(\mathbf{G}({\mathsf{k}}))$,
enumerated in [Ciu20, Section 5]. All but one of these representations
(namely, the representation labeled $X_{5}$) are Iwahori-spherical. For each
$X_{k}=X(q^{\frac{1}{2}h^{\vee}},n,\rho)$ we record in Table LABEL:table:f4a3
below the nilpotent orbit $G^{\vee}n\subset\mathcal{N}^{\vee}$ and the
representation $\rho$ of $A(s,n)$ (in all cases, $A(s,n)$ is a symmetric
group, so $\rho$ corresponds to a partition). We also record
$\mathsf{AZ}(X_{k})$ and indicate whether $X_{k}$ is unitary. For these
calculations, we refer the reader to [Ciu20, Proposition 5.7]. Using (i) of
Theorem 2.4.1, it is easy to determine which representations are tempered
(there are 5 such representations, labeled $X_{1}$,…,$X_{5}$). Finally, we
compute the canonical unramified wavefront set
$\underline{\mathrm{WF}}(X_{k})$. We have by Theorem 2.6.2 that

$\underline{\mathrm{WF}}(\mathsf{AZ}(X_{k}))=D(G^{\vee}n,1).$

The right hand side can be computed using the tables in [Ach03, Section 6].

Table 1. Irreducible representations of split $F_{4}$ with infinitesimal character $q^{h^{\vee}/2}$ for $\mathbb{O}^{\vee}=F_{4}(a_{3})$. | $G^{\vee}n$ | $\rho$ | I-spherical? | Tempered? | Unitary? | $\mathsf{AZ}$ | $\underline{\mathrm{WF}}$
---|---|---|---|---|---|---|---
$X_{1}$ | $F_{4}(a_{3})$ | $(4)$ | yes | yes | yes | $X_{20}$ | $(F_{4},1)$
$X_{2}$ | $F_{4}(a_{3})$ | $(31)$ | yes | yes | yes | $X_{19}$ | $(F_{4}(a_{1}),(12))$
$X_{3}$ | $F_{4}(a_{3})$ | $(2^{2})$ | yes | yes | yes | $X_{17}$ | $(F_{4}(a_{1}),1)$
$X_{4}$ | $F_{4}(a_{3})$ | $(21^{2})$ | yes | yes | yes | $X_{13}$ | $(C_{3},1)$
$X_{5}$ | $F_{4}(a_{3})$ | $(1^{4})$ | no | yes | yes | $X_{5}$ | $(F_{4}(a_{3}),1)$
$X_{6}$ | $C_{3}(a_{1})$ | $(2)$ | yes | no | yes | $X_{15}$ | $(F_{4}(a_{2}),1)$
$X_{7}$ | $C_{3}(a_{1})$ | $(1^{2})$ | yes | no | yes | $X_{9}$ | $(F_{4}(a_{3}),(1234)$
$X_{8}$ | $A_{1}+\widetilde{A}_{2}$ | $(1)$ | yes | no | yes | $X_{8}$ | $(F_{4}(a_{3}),(123))$
$X_{9}$ | $\widetilde{A}_{1}+A_{2}$ | $(1)$ | yes | no | yes | $X_{7}$ | $(F_{4}(a_{3}),(12))$
$X_{10}$ | $B_{2}$ | $(2)$ | yes | no | yes | $X_{18}$ | $(F_{4}(a_{1}),1)$
$X_{11}$ | $B_{2}$ | $(1^{2})$ | yes | no | yes | $X_{11}$ | $(F_{4}(a_{3}),(12)(34))$
$X_{12}$ | $A_{2}$ | $(2)$ | yes | no | yes | $X_{14}$ | $(B_{3},1)$
$X_{13}$ | $A_{2}$ | $(1^{2})$ | yes | no | yes | $X_{4}$ | $(F_{4}(a_{3}),1)$
$X_{14}$ | $\widetilde{A}_{2}$ | $(1)$ | yes | no | yes | $X_{12}$ | $(C_{3},1)$
$X_{15}$ | $A_{1}+\widetilde{A}_{1}$ | $(1)$ | yes | no | yes | $X_{6}$ | $(F_{4}(a_{3}),(12))$
$X_{16}$ | $A_{1}+\widetilde{A}_{1}$ | $(1)$ | yes | no | no | $X_{16}$ | $(F_{4}(a_{2}),1)$
$X_{17}$ | $\widetilde{A}_{1}$ | $(2)$ | yes | no | yes | $X_{3}$ | $(F_{4}(a_{3}),1)$
$X_{18}$ | $\widetilde{A}_{1}$ | $(1^{2})$ | yes | no | yes | $X_{10}$ | $(F_{4}(a_{3}),(12)(34))$
$X_{19}$ | $A_{1}$ | $(1)$ | yes | no | yes | $X_{2}$ | $(F_{4}(a_{3}),1)$
$X_{20}$ | $0$ | $(1)$ | yes | no | yes | $X_{1}$ | $(F_{4}(a_{3}),1)$

Note that $D(\mathbb{O}^{\vee},1)=(F_{4}(a_{3}),1)$, see [Ach03, Section 6].
So the Arthur packet attached to $\mathbb{O}^{\vee}$ is the set

(4.0.1)
$\Pi_{\mathbb{O}^{\vee}}^{\mathsf{Art}}(\mathbf{G}(\mathsf{k}))=\\{X_{5},X_{13},X_{17},X_{19},X_{20}\\}$

Note that this is precisely the set of anti-tempered representations in
$\Pi^{\mathrm{Lus}}_{q^{\frac{1}{2}h^{\vee}}}(\mathbf{G}({\mathsf{k}}))$, as
predicted by Theorem 3.0.3.

On the other hand, the _weak Arthur packet_ attached to $\mathbb{O}^{\vee}$,
see (3.1.1), is the much larger set

(4.0.2)
$\Pi^{\mathsf{Weak}}_{\mathbb{O}^{\vee}}(\mathbf{G}({\mathsf{k}}))=\\{X_{5},,X_{7},X_{8},X_{9},X_{11},X_{13},X_{15},X_{17},X_{18},X_{19},X_{20}\\}$

In this case, we can attempt to verify Conjecture 3.1.2 by hand.

There are 10 simplified Arthur parameters
$\widetilde{\psi}:\mathrm{SL}(2,\mathbb{C})_{\mathsf{Lan}}\times\mathrm{SL}(2,\mathbb{C})_{\mathsf{Art}}\stackrel{{\scriptstyle}}{{\to}}G^{\vee}$
satisfying the infinitesimal character condition (3.1.6). The corresponding
pairs $(\mathbb{O}_{\mathsf{Lan}}^{\vee},\mathbb{O}_{\mathsf{Art}}^{\vee})$ of
nilpotent orbits are

$(0,F_{4}(a_{3})),\ (A_{1},C_{3}(a_{1})),\ (\widetilde{A}_{1},B_{2}),\
(A_{1}+\widetilde{A}_{1},A_{1}+\widetilde{A}_{2}),\
(\widetilde{A}_{1}+A_{2},\widetilde{A}_{1}+A_{2}).$

together with the ‘flips’ of these pairs. We believe that the Arthur packets
contained in the weak Arthur packet (4.0.2) are as in Table 2. This is our
naive guess for the decomposition of the weak Arthur packet; it is compatible
with (2.7.1), Conjecture 1.0.3(i),(ii), and the expectation for the occurrence
of the supercuspidal representation in the Arthur packets.

$\mathbb{O}_{\mathsf{Lan}}^{\vee}$ | $\mathbb{O}_{\mathsf{Art}}^{\vee}$ | Arthur packet
---|---|---
$0$ | $F_{4}(a_{3})$ | {$X_{5},X_{13},X_{17},X_{19},X_{20}\\}$
$\widetilde{A}_{1}$ | $B_{2}$ | $\\{X_{5},X_{17},X_{18},X_{11}\\}$
$A_{1}+\widetilde{A}_{1}$ | $A_{1}+\widetilde{A}_{2}$ | $\\{X_{5},X_{15},X_{8}\\}$
$\widetilde{A}_{1}+A_{2}$ | $\widetilde{A}_{1}+A_{2}$ | $\\{X_{5},X_{9},X_{7}\\}$
Table 2. Arthur packets in weak packet attached to $F_{4}(a_{3})$.

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