# Wave-front sets for $p$-adic Lie algebras

Cheng-Chiang Tsai<EMAIL_ADDRESS>Academia Sinica, Institute of
Mathematics, 6F, Astronomy-Mathematics Building, No. 1, Sec. 4, Roosevelt
Road, Taipei 106319, Taiwan, andNational Sun Yat-Sen University, Department of
Applied Mathematics, No. 70, Lienhai Rd., Kaohsiung 80424, Taiwan

###### Abstract.

We study the wave-front set of an element in a $p$-adic reductive Lie algebra
(for $p\gg\operatorname{rank}$), namely the set of maximal nilpotent orbits
appearing in its Shalika germ expansion. By adapting an algorithm of
Waldspurger that computes orbital integrals, we obtain an inductive algorithm
to compute an invariant that determines the wave-front set. This gives an
algorithm to compute the wave-front sets for regular supercuspidal
representations and reveals examples whose wave-front sets are not contained
in a single geometric orbit, for arbitrarily large $p$ within a fixed rank.

The author is supported by NSTC grant 112-2115-M-001-009.

###### Contents

  1. 1 Introduction
  2. 2 DeBacker’s parametrization
  3. 3 Reduction to twisted Levi
  4. 4 The graph $C_{G}$
  5. 5 Analysis on Moy-Prasad quotients and nilpotent orbits
    1. 5.1 Moy-Prasad quotients à la Reeder-Yu
    2. 5.2 Edges as in (11)
    3. 5.3 Edges as in (10)
  6. 6 Examples
  7. 7 Wave-front sets of representations
  8. A Restrictions on $p$
  9. B Graded Lie algebras and Springer theory
    1. B.1 Existence of rational points

##  1\. Introduction

Let $F$ be a non-archimedean local field with residue field $k$ and
$p:=\operatorname{char}(k)$, $G$ a connected reductive group over $F$ and
$\mathfrak{g}:=\operatorname{Lie}G$. Throughout this paper we enforce some
restrictions on $p:=\mathrm{char}(k)$ for which we refer to Appendix A. In the
study of representations and characters of $G(F)$, it has been long observed
that they demonstrate an inductive structure. For the more recent landmarks,
Jiu-Kang Yu [Yu01] constructed supercuspidal representations inductively from
smaller twisted Levi subgroups. In [KM06], Ju-Lee Kim and Fiona Murnaghan gave
an inductive proof for a formula of the local characters of these and more
representations. After that, Jeff Adler and Loren Spice [AS09] and eventually
Spice [Spi18, Spi21] obtained an inductive description for the full character
of Yu’s supercuspidal representations, which are known to be all supercuspidal
representations [Kim07, Fin21] under restrictions on $p$. Meanwhile, Jean-Loup
Waldspurger [Wal06] proved that the fundamental lemma in mixed and equal
characteristic are equivalent, by developing an inductive algorithm to compute
the orbital integrals needed and showing that the algorithm depends only on
some data defined in terms of the residue field. (This was already done for
$GL_{n}$ in [Wal91].) In [Tsa15] the author mentioned how a pretty similar
algorithm can be used to compute Shalika germs. All aforementioned works
largely employ the same inductive process which we discuss in §3.

Let us recall the theory of Shalika germs as enhanced by the homogeneity
theorems of Waldspurger [Wal95] and DeBacker [DeB02a]. Denote by
$\mathfrak{g}^{nil}(F)/\\!\\!\sim$ the set of nilpotent
$\operatorname{Ad}(G(F))$-orbits in $\mathfrak{g}(F)$. We have

######  Theorem 1.1.

[DeB02a, Thm. 2.1.5] For any $\gamma\in\mathfrak{g}(F)$, there exists
$c_{\mathcal{O}}(\gamma)\in\mathbb{Q}$ indexed by
$\mathcal{O}\in\mathfrak{g}^{nil}(F)/\\!\\!\sim$ and a lattice
$\Lambda_{\gamma}\subset\mathfrak{g}(F)$ such that

$I_{\gamma}(f)=\sum_{\mathcal{O}\in\mathfrak{g}^{nil}(F)/\sim}c_{\mathcal{O}}(\gamma)I_{\mathcal{O}}(f),\;\forall
f\in C_{c}(\mathfrak{g}(F)/\Lambda_{\gamma}).$

Here we denote by $C_{c}(\mathfrak{g}(F)/\Lambda_{\gamma})\subset
C_{c}^{\infty}(\mathfrak{g}(F))$ the space of $\mathbb{C}$-valued compactly
supported functions satisfying $f(X+Y)=f(X)$ for any $Y\in\Lambda_{\gamma}$.
The coefficients $c_{\mathcal{O}}(\gamma)$ are called Shalika germs,
generalizing the theorem of Shalika [Sha72] for $\gamma\in G(F)$ strongly
regular semisimple and sufficiently close to $1$. Consider the partial order
on $\mathfrak{g}^{nil}(F)/\\!\\!\sim$ for which
$\mathcal{O}_{1}\leq\mathcal{O}_{2}$ iff the former is contained in the
closure of the latter in the $p$-adic topology. With Theorem 1.1, we may
define the wave-front set

$\operatorname{WF}(\gamma):=\max\\{\mathcal{O}\in\mathfrak{g}^{nil}(F)/\\!\\!\sim\;|\;c_{\mathcal{O}}(\gamma)\not=0\\}.$

This can be viewed as the linearized version of wave-front sets for
representations of $G(F)$. For example, the wave-front set for a regular
supercuspidal representation [Kal19] is equal to the aforementioned wave-front
set for specific elliptic $\gamma\in\mathfrak{g}(F)$, see Remark 7.1. Given
that wave-front sets of representations of $G(F)$ are important for degenerate
Whittaker models [MW87] and related to the Gan-Gross-Prasad program (see e.g.
[JLZ22, §4]), $\operatorname{WF}(\gamma)$ appears to be a basic object for the
representation theory of $G(F)$, and a natural question is whether we can
compute $\operatorname{WF}(\gamma)$ inductively.

It came as a surprise to us that if we take the question in the literal and
strict sense, then the answer is no. Namely in the inductive process when we
write $\gamma=\gamma_{r}+\gamma_{>r}$ and $H:=Z_{G}(\gamma_{r})$, the set
$\operatorname{WF}^{G}(\gamma)$ cannot be determined by $\gamma_{r}$ and
$\operatorname{WF}^{H}(\gamma_{>r})$; see Example 6.6. This is the subtlety
discussed in [Nev23, Rmk. 3.6]. Nevertheless, we define for any
$s\in\mathbb{R}$ another set (see §2 for the notation):

$c_{G,s}(\gamma):=\\{(\varsigma,c)\;|\;\varsigma\text{ an }s\text{-facet,
}c\in\mathfrak{g}(F)_{\varsigma=s},\;\operatorname{Ad}(G(F))\varsigma\cap\left(c+\mathfrak{g}(F)_{\varsigma>s}\right)\not=\emptyset\\}.$

The set implicitly appeared in [Wal06] and also resembles the residue field
version of $\operatorname{Nil}(\gamma)$ in [Nev23]. The set $c_{G,s}(\gamma)$
determines $\operatorname{WF}(\gamma)$, and we manage to adapt the algorithm
of Waldspurger in [Wal06] to inductively compute $c_{G,r}(\gamma)$. Consider

(1) $C_{G,s}:=\\{(\varsigma,c)\;|\;\varsigma\text{ an }s\text{-facet,
}c\in\mathfrak{g}(F)_{\varsigma=s}\\}$

so that every $c_{G,s}(\gamma)$ is a subset of $C_{G,s}$. There will be a set
$C_{G}$ which is an appropriate union $C_{G}=\bigcup_{s\in\mathbb{R}}C_{G,s}$
and we will define a directed acyclic graph with vertex set $C_{G}$ and some
edges, so that $c_{G,s}(\gamma)$ is equal the subset of those vertices $C_{G}$
who have a path to any vertex in $c_{G,r}(\gamma)$. On the other hand, when
$\gamma=\gamma_{r}+\gamma_{>r}$, $c_{G,r}(\gamma)$ is explicitly determined by
$c_{H,r}(\gamma_{>r})$ where $H=Z_{G}(\gamma_{r})$. Together this gives an
inductive algorithm to determine any $c_{G,s}(\gamma)$.

There are two kinds of complications for this algorithm. For a vertex $v$ in
the graph $C_{G}$, the set of its out-neighbors correspond to the set of
rational orbits in specific graded Lie algebra over $k$ that intersect
specific affine subspace (e.g. Slodowy slices). A good theory about such
orbits - and therefore the edges of the graph - seems very difficult to
achieve. We call this the geometric problem (Problem 5.8). On the other hand,
the graph $C_{G}$ is defined in terms of the Bruhat-Tits building for $G$, or
effectively an apartment of it, which is still sort of complicated.
Furthermore, for a given directed acylic graph it needs not be easy to
determine if there is a path from a specific vertex to another set of vertices
(“does this maze have a solution?”). We call these the combinatorial problem.

We prove some necessary conditions and sufficient conditions for edges in the
graph to exist, mostly to address the geometric problem. Our main existence
result, Corollary 5.21, is based on Theorem B.4 which generalizes the result
established by Dan Ciubotaru and Emile Okada in [CO23, §4]. When the
semisimple part of $\gamma$ is contained in $(\operatorname{Lie}T)(F)$ for
$T\subset G$ an unramified subtorus, it seems possible that necessary and
sufficient conditions are already strong enough to give the correct unramified
wave-front set $\operatorname{WF}^{ur}(\gamma)$; see Conjecture 6.4. In some
cases we analyze our algorithm and confirm that. This includes the case
corresponding to unramified toral supercuspidal representations already worked
out in [CO23], as well as an interesting example (Example 6.2) with $G=U_{6}$
and that $\operatorname{WF}(\gamma)$ contains orbits of type $[4,1,1]$ and
$[3,3]$, for $p$ arbitrarily large.

Let us now explain the organization of the paper. In §2 we explain how to
determine $\operatorname{WF}(\gamma)$ from $c_{G,s}(\gamma)$. In §3 we explain
a result of Kim-Murnaghan that allows us to reduce the computation of
$c_{G,s}(\gamma)$ to twisted Levi subgroups. Then in §4 we present the main
construction (see Theorem 4.7) of the graph $C_{G}$, hence our algorithm for
computing $c_{G,s}(\gamma)$, citing a construction of Waldspurger to be
explained in §5. The longest section §5 analyzes the algorithm, or mostly
analyzes a number of questions arising from [DeB02b] relating Moy-Prasad
quotients and nilpotent orbits. Several results in §5 will refer to Appendix B
regarding (graded) Lie algebras over the residue field. In §6 we present
demonstrations and examples for the algorithm, and lastly in §7 we discuss the
relation between wave-front sets for elements in the Lie algebra and wave-
front sets for representations of $G(F)$.

### Acknowledgment

I wish to express my profound thanks to Monica Nevins who generously explained
to me a subtlety in [Nev23, Thm 3.5] and kindly shared with me an earlier
version of [Nev23]. I benefit a lot from communication with Chi-Heng Lo about
the structure of wave-front sets, for which I deeply appreciate. I am very
grateful to Emile Okada for many illuminating discussions, particularly his
instructive explanation about the theory of $G$-sheets. I also owe many thanks
to Ju-Lee Kim, Jessica Fintzen, Loren Spice, Stephen DeBacker and Masao Oi for
numerous inspiring discussions. I am indebted to the organizers of the 5th
Nisyros Conference on Automorphic Representations & Related Topics where many
of these wonderful discussions happened, and the same for the 2023 Japan-
Taiwan Joint Conference on Number Theory and the University of Bonn.

##  2\. DeBacker’s parametrization

We begin with some notations. Let $\mathcal{B}(G,F)$ be the Bruhat-Tits
building (hereafter always the extended building) for $G/_{F}$. For any
$x\in\mathcal{B}(G,F)$, $r\in\mathbb{R}$, denote by $\mathfrak{g}(F)_{x\geq
r}$ the Moy-Prasad lattice [MP94, §3] of depth $r$, by $\mathfrak{g}(F)_{x>r}$
that of depth $>r$, and by $\mathfrak{g}(F)_{x=r}:=\mathfrak{g}(F)_{x\geq
r}/\mathfrak{g}(F)_{x>r}$ the Moy-Prasad quotient at depth $r$. We also denote
by $G(F)_{x\geq 0}$ the parahoric subgroup and $\mathsf{G}_{x}/_{k}$ the
(algebraic) reductive quotient. In particular $\mathsf{G}_{x}(k)$ is a
quotient of $G(F)_{x\geq 0}$ with pro-$p$ kernel, and
$\operatorname{Lie}\mathsf{G}_{x}=\mathfrak{g}(F)_{x=0}$.

######  Remark 2.1.

Our notations for the Moy-Prasad filtration are different from the usual
notations. The new notations are extensively used in [Spi18] and [Spi21]. They
will also prove convenient for our purpose here.

We also denote by $\mathfrak{g}(F)_{\geq
r}:=\bigcup_{x\in\mathcal{B}(G,F))}\mathfrak{g}(F)_{x\geq r}$ and likewise
$\mathfrak{g}(F)_{>r}:=\bigcup_{x\in\mathcal{B}(G,F))}\mathfrak{g}(F)_{x>r}$.
We define

(2)
$\operatorname{dep}(\gamma):=\max\\{r\in\mathbb{R}\;|\;\gamma\in\mathfrak{g}(F)_{\geq
r}\\}$

Following [DeB02b] we say $x_{1},x_{2}\in\mathcal{B}(G,F)$ are in the same
$r$-facet iff $\mathfrak{g}(F)_{x_{1}\geq r}=\mathfrak{g}(F)_{x_{2}\geq r}$
and $\mathfrak{g}(F)_{x_{1}>r}=\mathfrak{g}(F)_{x_{2}>r}$. Denote by
$\Phi^{G}(r)$ the set of $r$-facets, which form a partition of
$\mathcal{B}(G,F)$. Recall that $\mathfrak{g}^{nil}(F)\subset\mathfrak{g}(F)$
is the nilpotent cone. Let us begin with [DeB02b, Def. 4.1.1]:

######  Definition 2.2.

Let $\varsigma\in\Phi^{G}(r)$ for some $r\in\mathbb{R}$. We say a coset
$c\in\mathfrak{g}(F)_{\varsigma=r}$ is nilpotent iff
$(c+\mathfrak{g}(F)_{\varsigma>r})\cap\mathfrak{g}^{nil}(F)\not=\emptyset$.

In loc. cit. DeBacker (following Moy-Prasad) call such cosets degenerate. We
choose the term “nilpotent” here, since it relates very well with other usage
of nilpotency (but degeneracy can mean several different things). We also
remark that we tend to think of $c$ as if it was an element in
$\mathfrak{g}(F)$. This indeed can be realized if $F$ is a local function
field, and that the coset is nilpotent iff it is nilpotent in
$\mathfrak{g}(F)$. In any case we will usually write
$c+\mathfrak{g}(F)_{\varsigma>r}$ (instead of just $c$) to emphasize the coset
as a subset of $\mathfrak{g}(F)$. Recall the notation
$C_{G,r}=\\{(\varsigma,c)\;|\;\varsigma\in\Phi^{G}(r),c\in\mathfrak{g}(F)_{\varsigma=r}\\}$.
Let us also write

$C_{G,r}^{nil}=\\{(\varsigma,c)\in C_{G,r}\;|\;c\text{ is nilpotent}\\}.$

For any $(\varsigma,c)\in C_{G,r}\in C_{H,r}^{nil}$, [DeB02b, Cor. 5.2.5,
Lemma 5.3.3] asserts the existence of a unique minimal nilpotent orbit
$\mathcal{O}(\varsigma,c)$ such that
$c+\mathfrak{g}(F)_{\varsigma>r}\cap\mathcal{O}(\varsigma,c)\not=\emptyset$.
Whenever $C\subset C_{G,r}^{nil}$ is a subset, we denote by

$\operatorname{DB}_{r}(C)=\\{\mathcal{O}(\varsigma,c)\;|\;(\varsigma,c)\in
C\\}.$

For two subsets $S_{1},S_{2}\in\mathfrak{g}^{nil}(F)/\\!\\!\sim$ we say
$S_{1}\equiv S_{2}$ iff, for every $\mathcal{O}_{i}\in S_{i}$ there exists
$\mathcal{O}_{j}\in S_{j}$ such that $\mathcal{O}_{i}\leq\mathcal{O}_{j}$, for
any $i,j=1,2$. We have

######  Proposition 2.3.

Let $\gamma\in\mathfrak{g}(F)$ with $\operatorname{dep}(\gamma)=r$. For any
$s<r$ we have
$\operatorname{DB}(c_{G,s}(\gamma))\equiv\operatorname{WF}(\gamma)$.

###### Proof.

Suppose $\mathcal{O}_{0}\in\operatorname{DB}_{s}(c_{G,s}(\gamma))$. By
definition this says there exists $\varsigma\in\mathcal{F}(\gamma)$ and
$c\in\mathfrak{g}(F)_{\varsigma=s}$ such that $\gamma\in
c+\mathfrak{g}(F)_{\varsigma>s}$ and
$\mathcal{O}_{0}=\mathcal{O}(\varsigma,c)$. Denote by
$[c+\mathfrak{g}(F)_{\varsigma>s}]$ the characteristic function of the coset.
Since $\operatorname{dep}(\gamma)>s$, by Theorem [DeB02a, Thm. 2.1.5] we have

(3)
$I_{\gamma}([c+\mathfrak{g}(F)_{\varsigma>s}])=\sum_{\mathcal{O}\in\mathfrak{g}^{nil}(F)/\sim}c_{\mathcal{O}}(\gamma)I_{\mathcal{O}}([c+\mathfrak{g}(F)_{\varsigma>s}])$

We have $I_{\gamma}([c+\mathfrak{g}(F)_{\varsigma>s}])>0$. By definition of
$\mathcal{O}(\varsigma,c)$, we also have
$I_{\mathcal{O}}([c+\mathfrak{g}(F)_{\varsigma>s}])>0$ iff
$\mathcal{O}\geq\mathcal{O}(\varsigma,c)$. Hence we have
$c_{\mathcal{O}}(\gamma)\not=0$ for some
$\mathcal{O}\geq\mathcal{O}(\varsigma,c)$.

To conclude that
$\operatorname{DB}_{s}(c_{G,s}(\gamma))\equiv\operatorname{WF}(\gamma)$,
suppose on the contrary we have $\mathcal{O}_{0}\in\operatorname{WF}(\gamma)$
such that $\mathcal{O}_{0}\not\leq\mathcal{O}(\varsigma,c)$ for any
$(\varsigma,c)\in c_{G,s}(\gamma)$. By [DeB02b, Thm. 5.6.1] we can write
$\mathcal{O}_{0}=\mathcal{O}(\varsigma^{\prime},c^{\prime})$ for some
$(\varsigma^{\prime},c^{\prime})\in C_{G,s}$. Applying the last equation to
$[c^{\prime}+\mathfrak{g}(F)_{\varsigma^{\prime}>s}]$, we see that the LHS of
[DeB02a] vanishes while the RHS does not. This gives the desired contradiction
and proves the proposition. ∎

##  3\. Reduction to twisted Levi

In this section we fix

  1. (1)

A semisimple element $\gamma_{r}$ that is good of depth $r$ in the sense of
[AR00, Def. 5.2]. That is, for any torus $T\subset G$ such that
$\gamma_{r}\in\operatorname{Lie}T$, and for every $\alpha\in\Phi(G,T)$, not
necessarily defined over $F$, we have that $d\alpha(\gamma_{r})$ is either $0$
or has normalized valuation $r$, where a uniformizer of $F$ has valuation $1$.
We write $H:=Z_{G}(\gamma_{r})$ and $\mathfrak{h}=\operatorname{Lie}H$.

  2. (2)

An element $\gamma_{>r}\in\mathfrak{h}(F)_{>r}$.

We write $\gamma:=\gamma_{r}+\gamma_{>r}$. Such a decomposition is always
possible for any $\gamma$ and $r=\operatorname{dep}(\gamma)$ by applying
[AR00, Prop. 5.4] to the semisimple part of $\gamma$, and such that
$H\subsetneq G$ is a proper twisted Levi subgroup. There is an embedding
$\mathcal{B}(H,F)\hookrightarrow\mathcal{B}(G,F)$ that is canonical up to the
action of $X_{*}(Z(H)/_{F})\otimes_{\mathbb{Z}}\mathbb{R}$ on
$\mathcal{B}(H,F)$. Here $X_{*}(Z(H)/_{F})$ is the group of cocharacters
$\mathbb{G}_{m}\rightarrow Z(H)$ defined over $F$. We fix any such embedding
$\mathcal{B}(H,F)\hookrightarrow\mathcal{B}(G,F)$.

The key step is to apply [KM03, Lemma 2.3.3] with their $\Gamma$ being our
$\gamma_{r}$, their $X^{\prime}$ being our $\gamma_{>r}$, their $G^{\prime}$
being our $H$, and their $d(\Gamma)$ being our $r$. The reference asserts that
for any $x\in\mathcal{B}(G,F)$:

(4) $\gamma_{r}\in\mathfrak{g}(F)_{x\geq r}\Leftrightarrow
x\in\mathcal{B}(H,F).$ (5) $\gamma_{r}+\gamma_{>r}\in\mathfrak{g}(F)_{x\geq
r}\Rightarrow x\in\mathcal{B}(H,F).$

Note that (4) has the following nice application:

######  Corollary 3.1.

For an $r$-facet $\varsigma\subset\mathcal{B}(G,F)$ for $G$ which intersects
$\mathcal{B}(H,F)$ non-trivially, we have $\varsigma\subset\mathcal{B}(H,F)$.
In other words, $\mathcal{B}(H,F)$ is the union of some $r$-facets in
$\mathcal{B}(G,F)$.

Namely every $\varsigma\in\Phi^{G}(r)$ is contained in an $r$-facet in
$\mathcal{B}(H,r)$, which we denote by $\varsigma^{H}\in\Phi^{H}(r)$. Let
$\gamma_{=r}$ be the image of $\gamma_{r}$ in $Z(\mathfrak{h})(F)_{=r}$. We
will derive from (4) and (5) that

######  Proposition 3.2.

The set $c_{G,r}(\gamma)$ is determined by $c_{H,r}(\gamma_{>r})$ is the
following identity

(6)
$c_{G,r}(\gamma)=G(F).\\{(\varsigma,\,c_{H}+\gamma_{=r})\;|\;\varsigma\in\Phi^{G}(r),(\varsigma^{H},c_{H})\in
c_{H,r}(\gamma_{>r})\\}.$

###### Proof.

For the “$\subset$” direction, say we have $(\varsigma,c)\in c_{G,r}(r)$.
Modulo $G(F)$-action we may WLOG assume $\gamma\in
c+\mathfrak{g}(F)_{\varsigma>r}\subset\mathfrak{g}(F)_{\varsigma\geq r}$. By
(5), this gives $\varsigma\subset\mathcal{B}(H,F)$. Since
$\gamma\in\mathfrak{h}(F)$, this shows that $c=c_{H}$ for some
$c_{H}\in\mathfrak{h}(F)_{\varsigma^{H}=r}$, and $\gamma_{>r}\in
c_{H}-\gamma_{=r}+\mathfrak{h}(F)_{\varsigma^{H}=r}$. The latter implies
$(\varsigma^{H},c_{H}-\gamma_{=r})\in\underline{c}_{\mathfrak{h},r}(\gamma_{>r})$,
which by definition says $(\varsigma,c)$ is contained in the RHS of (6).

For the “$\supset$” direction, let $(\varsigma,c)\in C_{G,r}$ be such that
there exists $g\in G(F)$ such that $g^{-1}.\varsigma\subset\mathcal{B}(H,F)$
and there exists $c_{H}$ such that

  1. (1)

$(\varsigma^{H},c_{H})\in c_{H,r}(\gamma_{>r})$, and

  2. (2)

$g^{-1}.c=\gamma_{=r}+c_{H}\in\mathfrak{h}(F)_{\varsigma^{H}=r}$.

That $c_{H}\in c_{H,r}(\gamma_{>r})$ is to say that there exists $h\in H(F)$
such that $\operatorname{Ad}(h)\gamma_{>r}\in
c_{H}+\mathfrak{h}(F)_{\varsigma^{H}>r}$. Since
$\gamma_{r}\in\gamma_{=r}+Z(\mathfrak{h})(F)_{>r}\subset Z(\mathfrak{h})_{\geq
r}$, this implies

$\operatorname{Ad}(g^{-1}h)(\gamma)=\operatorname{Ad}(g^{-1})\operatorname{Ad}(h)\gamma_{>r}+\operatorname{Ad}(g^{-1})\gamma_{r}\in
c_{H}+\gamma_{=r}+\mathfrak{h}(F)_{\varsigma^{H}>r}\subset
c+\mathfrak{g}(F)_{\varsigma>r}$

which gives $(\varsigma,c)\in c_{G,r}(\gamma)$. ∎

##  4\. The graph $C_{G}$

The goal of this section is to define the set $C_{G}$ ($C$ for cosets) as a
glued union $C_{G}=\bigcup_{r\in\mathbb{R}}C_{G,r}$ and a graph with vertex
set $C_{G}$ which we will use to compute $c_{G,r}(\gamma)$. In fact, we will
allow some choices in defining the graph, yet any set of choices produce a
graph that satisfies Theorem 4.7 below. A large part of this section comes
from [Wal06], especially §2.4 and §7 op. cit.. Recall that $\Phi^{G}(r)$ is
the set of $r$-facets and
$C_{G,r}:=\\{(\varsigma,c)\;|\;\varsigma\in\Phi^{G}(r),\;c\in\mathfrak{g}(F)_{\varsigma=r}\\}$.
Note that $C_{G,r}$ is essentially the sets $I_{r}$ in the introduction of
[DeB02b]. We warn the reader that we are not taking the $r$-associativity
relation in [DeB02b, §3.3].

We would like to revisit the above notations by considering all
$r\in\mathbb{R}$ altogether, following the framework in [Wal06, §2.4].
Consider the (underlying) set
$\mathcal{B}^{a}(G,F):=\mathcal{B}(G,F)\times\mathbb{R}$ which we will call
the augmented building for $G(F)$. An augmented apartment is
$\mathcal{A}^{a}:=\mathcal{A}\times\mathbb{R}\subset\mathcal{B}^{a}(G,F)$
where $\mathcal{A}\subset\mathcal{B}(G,F)$ is an apartment in the usual sense.
We decompose $\mathcal{B}^{a}(G,F)$ into augmented facets in the following
way: two elements $(x_{1},r_{1}),(x_{2},r_{2})\in\mathcal{B}^{a}(G,F)$ are in
the same augmented facets iff $\mathfrak{g}(F)_{x_{1}\geq
r_{1}}=\mathfrak{g}(F)_{x_{2}\geq r_{2}}$ and
$\mathfrak{g}(F)_{x_{1}>r_{1}}=\mathfrak{g}(F)_{x_{2}>r_{2}}$. Denote by
$\Phi^{G}$ the set of augmented facets. The group $G(F)$ acts on
$\mathcal{B}^{a}(G,F)$ by $g.(x,r):=(g.x,r)$. This action obviously acts on
$\Phi^{G}$ and acts transitively on the set of augmented apartments. Observe
that the set of $r$-facets can be identified
$\Phi^{G}(r)=\\{\sigma\in\Phi^{G}\;|\;\sigma\cap\left(\mathcal{B}(G,F)\times\\{r\\}\right)\not=\emptyset\\}$
as a subset of $\Phi^{G}$. We will use the two viewpoints on $\Phi^{G}(r)$
interchangeably.

For any augmented apartment $\mathcal{A}^{a}\subset\mathcal{B}^{a}(G,F)$, the
collection

$\Phi^{G}_{\mathcal{A}}:=\\{\sigma\cap(\mathcal{A}^{a})\;|\;\sigma\subset\mathcal{B}^{a}(G,F)\text{
an augmented facet, }\sigma\cap(\mathcal{A}^{a})\not=\emptyset\\}$

is cut out by two types of hyperplanes: the first type is “affine root
hyperplanes” which in the language of [Tit79, §1.6, §1.7] are of the form
$\\{(x,r)\in\mathcal{A}^{a}\;|\;\alpha(x)=r\\}$ for some affine root $\alpha$.
The second type is $\mathcal{A}\times\\{s\\}$ for $s\in\mathbb{Q}$ such that
$\mathfrak{z}_{\geq s}\supsetneq\mathfrak{z}_{>s}$ where $\mathfrak{z}$ is the
Lie algebra of the torus $Z$ associated to $\mathcal{A}$ as in [MP94, §2.2,
§3.2]. Note that the projection map
$\mathcal{A}^{a}\twoheadrightarrow\mathcal{A}$ sends any of the above
hyperplane (of either type) isomorphically onto $\mathcal{A}$.

######  Definition 4.1.

We call both types of the aforementioned hyperplanes on $\mathcal{A}^{a}$
critical, and call the open half-spaces cut out by a critical hyperplane a
critical half-space. For a critical hyperplane $P\subset\mathcal{A}^{a}$, we
say it is above111Since the real number is commonly called depth, our
convention is that $r_{1}$ is “above” $r_{2}$ iff $r_{1}<r_{2}$. (resp. is
below, passes through) a point $(x,r)\in\mathcal{A}^{a}$ if $(x,r^{\prime})\in
P$ for some $r^{\prime}<r$ (resp. $r^{\prime}>r$, $r^{\prime}=r$). We say a
critical hyperplane is above (resp. is below, passes through) an augmented
facet if it is above (resp. is below, passes through) any point in it.

We have that $\Phi^{G}_{\mathcal{A}}$ is the collection of possible
intersections of a finite number of critical hyperplanes and critical half-
spaces. This shows that every augmented facet is convex. For any
$\sigma\in\Phi^{G}$, we denote by
$\mathfrak{g}(F)_{\sigma}:=\mathfrak{g}(F)_{x=r}$ for any
$(x,r)\in\sigma\subset\mathcal{B}^{a}(G,F)\times\mathbb{R}$. Likewise
$\mathfrak{g}(F)_{\geq\sigma}:=\mathfrak{g}(F)_{x\geq r}$ and
$\mathfrak{g}(F)_{>\sigma}:=\mathfrak{g}(F)_{x>r}$. Same for
$\sigma\in\Phi^{G}_{\mathcal{A}}$. From definition of Moy-Prasad filtration,
we have

######  Lemma 4.2.

Let $\sigma_{1},\sigma_{2}\in\Phi^{G}_{\mathcal{A}}$. Then

  1. (1)

$\mathfrak{g}(F)_{\geq\sigma_{1}}\subset\mathfrak{g}(F)_{\geq\sigma_{2}}$ iff
every critical hyperplane on $\mathcal{A}^{a}$ above $\sigma_{2}$ is also
above $\sigma_{1}$.

  2. (2)

$\mathfrak{g}(F)_{>\sigma_{1}}\subset\mathfrak{g}(F)_{>\sigma_{2}}$ iff every
critical hyperplane on $\mathcal{A}^{a}$ below $\sigma_{1}$ is also below
$\sigma_{2}$.

######  Corollary 4.3.

Suppose $\sigma_{1},\sigma_{2}\in\Phi$ are such that $\sigma_{1}$ is contained
in the closure of $\sigma_{2}$. Then

$\mathfrak{g}(F)_{>\sigma_{1}}\subset\mathfrak{g}(F)_{>\sigma_{2}}\subset\mathfrak{g}(F)_{\geq\sigma_{2}}\subset\mathfrak{g}(F)_{\geq\sigma_{1}}.$

######  Remark 4.4.

In [Wal06, §2.4], Waldspurger considered what is essentially our
$\Phi^{G}_{\mathcal{A}}$ and denote it by $\Phi$. There is a natural injection
$\Phi^{G}_{\mathcal{A}}\subset\Phi^{G}$ so that
$G(F).\Phi^{G}_{\mathcal{A}}=\Phi^{G}$. As for most applications, working with
the whole building versus a single apartment (in our notation, $\Phi^{G}$
versus $\Phi^{G}_{\mathcal{A}}$) does not bring any essential difference, and
we choose work with the seeming more general $\Phi^{G}$ merely because the
additional cost is very small anyway. In fact, since the (extended) affine
Weyl group acts transitively on the set of alcoves, we can furthermore
restricts to those augmented facets that intersects a fixed alcove, and will
do so in examples.

For any $\sigma\in\Phi^{G}$, denote by $d(\sigma)$ the image of
$\sigma\subset\mathcal{B}(G,F)\times\mathbb{R}$ under the projection to
$\mathbb{R}$, which is either a point or an open interval. Let us call
$\sigma\in\Phi^{G}$ horizontal of depth $r_{0}$ if $d(\sigma)=\\{r_{0}\\}$,
and non-horizontal of depth $(r_{1},r_{2})$ if $d(\sigma)=(r_{1},r_{2})$. We
have the following contrapositive of [MP94, Prop. 6.4] and its corollary:

######  Lemma 4.5.

Suppose we have $x\in\mathcal{B}(G,F)$, $s\in\mathbb{R}$ and
$\gamma\in\mathfrak{g}(F)_{x\geq s}$ with $\operatorname{dep}(\gamma)>s$. Then
the image of $\gamma$ in the quotient $\mathfrak{g}(F)_{x=s}$ is nilpotent.

######  Corollary 4.6.

Let $\sigma\subset\mathcal{B}^{a}(G,F)$ be a non-horizontal augmented facet.
Then every element in $\mathfrak{g}(F)_{\sigma}$ is nilpotent.

###### Proof.

Say $d_{\sigma}=(r_{1},r_{2})$. For every element
$c\in\mathfrak{g}(F)_{\sigma}$, let $\gamma\in c+\mathfrak{g}(F)_{>\sigma}$ be
an arbitrary lift. By definition of depth we have $0pt(\gamma)\geq r_{2}$.
That $c$ is nilpotent follows from Lemma 4.5 with $s=(r_{1}+r_{2})/2$ and some
$x$ with $(x,s)\in\sigma$. ∎

Write $\operatorname{dep}(\sigma)=\sup d(\sigma)$; this is justified by that
$\operatorname{dep}(\sigma)=\min_{\gamma\in\mathfrak{g}(F)_{\geq\sigma}}\operatorname{dep}(\gamma)$.
Define a partial order on $\Phi^{G}$: for $\sigma_{1},\sigma_{2}\in\Phi^{G}$
we say $\sigma_{1}\prec\sigma_{2}$ iff either
$\operatorname{dep}(\sigma_{1})<\operatorname{dep}(\sigma_{2})$, or if
$\operatorname{dep}(\sigma_{1})=\operatorname{dep}(\sigma_{2})$ and
$\dim(\sigma_{1})<\dim(\sigma_{2})$. Define

(7)
$C_{G}:=\\{(\sigma,c)\;|\;\sigma\in\Phi^{G},\;c\in\mathfrak{g}(F)_{\sigma}\\}.$

as well as

$C_{G}^{nil}:=\\{(\sigma,c)\in C_{G}\;|\;c\text{ is nilpotent}\\}.$

For any $r\in\mathbb{R}$, the natural inclusion $\Phi(r)\subset\Phi$ induces
$C_{G,r}\subset C_{G}$ and $C_{G,r}^{nil}\subset C_{G}^{nil}$. The goal of
this section is

######  Theorem 4.7.

We will define a directed acyclic graph with vertex set $C_{G}$, depending on
some choice made at each vertex, such that for any collection of choices we
have

  1. (1)

There is a edge from $(\sigma_{1},c_{1})$ to $(\sigma_{2},c_{2})$ only if
$c_{1}$ is nilpotent and $\sigma_{1}\prec\sigma_{2}$.

  2. (2)

For any $\gamma\in C_{G}$ and $r=\operatorname{dep}(\gamma)$, the union

$C_{G}(\gamma):=\bigcup_{s\in\mathbb{R}}c_{G,s}(\gamma)=\\{(\sigma,c)\in
C_{G}\;|\;\operatorname{Ad}(G(F))\gamma\cap\left(c+\mathfrak{g}(F)_{>\sigma}\right)\not=\emptyset\\}\subset
C_{G}$

is equal to the set of those vertices in $C_{G}$ which has a path to
$c_{G,r}(\gamma)\subset C_{G,r}\backslash C_{G,r}^{nil}$.

Note that Corollary 4.6 gives $C_{G}(\gamma)\subset C_{G,r}\cup C_{G}^{nil}$.
By the nature of the statement, the theorem can and will be proved inductively
for elements in $C_{G}^{nil}$, i.e. it suffices to show that for any vertex
$(\sigma,c)\in C_{G,s}^{nil}$ of the graph, we have that $(\sigma,c)\in
c_{G}(\gamma)$ iff it has an out-neighbor $(\sigma^{\prime},c^{\prime})$ (i.e.
there is a directed edge from $(\sigma,c)$ to it) that is contained in
$c_{G}(\gamma)$. These out-neighbors will be declared by two different methods
(10) and (11) below. The two method will altogether cover all vertices of the
graph, but they overlap in general. Moreover each method has the freedom given
by some choice. That is, our graph will be defined up to some choices so that
any choice gives a graph for which Theorem 4.7 works, though some choices seem
better when we analyze it in §5.

Our first method of defining the out-neighbors, essentially in [Wal06, §7.3],
applies to a non-horizontal augmented facet $\sigma$ with
$d(\sigma)=(r_{1},r_{2})$. The intersection of
$\mathcal{B}(G,F)\times\\{r_{2}\\}$ with the closure of $\sigma$ is convex and
equals to a union of horizontal augmented facets. They can be described as

######  Definition 4.8.

Suppose $\sigma$ is a non-horizontal facet and $\sigma^{\prime}\in\Phi^{G}$ is
an augmented facet (necessarily horizontal) in the closure of $\sigma$
satisfying $d(\sigma^{\prime})=\\{\operatorname{dep}(\sigma)\\}$. Then we call
$\sigma^{\prime}$ an augmented facet below $\sigma$.

Choose $\sigma^{\prime}\in\Phi^{G}$ below $\sigma$. By Lemma 4.2, we have that

(8)
$\mathfrak{g}(F)_{\geq\sigma^{\prime}}\supset\mathfrak{g}(F)_{\geq\sigma}\supset\mathfrak{g}(F)_{>\sigma}\supset\mathfrak{g}(F)_{>\sigma^{\prime}}.$

We may realize $\mathfrak{g}(F)_{>\sigma}/\mathfrak{g}(F)_{>\sigma^{\prime}}$
as a subspace of $\mathfrak{g}(F)_{\sigma^{\prime}}$, and for any
$c\in\mathfrak{g}(F)_{\sigma}$ we have

(9) $c+\mathfrak{g}(F)_{>\sigma}=\bigsqcup_{c^{\prime}\in
c+\mathfrak{g}(F)_{>\sigma}/\mathfrak{g}(F)_{>\sigma^{\prime}}}c^{\prime}+\mathfrak{g}(F)_{>\sigma^{\prime}}$

We define the set of out-neighbors of $(\sigma,c)$ to be the collection

(10) $\\{(\sigma^{\prime},c^{\prime})\;|\;c^{\prime}\in
c+\mathfrak{g}(F)_{>\sigma}/\mathfrak{g}(F)_{>\sigma^{\prime}}\\}.$

Directly from (9) we see that $(\sigma,c)\in c_{G}(\gamma)$ iff at least one
of $(\sigma^{\prime},c^{\prime})\in c_{G}(\gamma)$. Theorem 4.7 is thus
inductively true for this definition with any choice of $\sigma^{\prime}$ made
before (8).

Our second method is a construction of Waldspurger in [Wal06, Lemma 7.2.2 &
Lemma 7.3.2] that applies to some $(\sigma,c)$ with $c$ nilpotent. We will
review this construction in §5. The point is that whenever $\sigma$ is
horizontal, and sometimes also when $\sigma$ is not, Waldspurger produced from
$(\sigma,c)$ another $(\sigma^{\dagger},c^{\dagger})$ where $\sigma^{\dagger}$
is a non-horizontal coset having $\sigma$ on its boundary and
$\sigma\prec\sigma^{\dagger}$, such that any $\operatorname{Ad}(G(F))$-orbit
in $\mathfrak{g}(F)$ meets $c+\mathfrak{g}(F)_{>\sigma}$ iff it meets
$c^{\dagger}+\mathfrak{g}(F)_{>\sigma^{\dagger}}$; in fact in appropriate
sense $c^{\dagger}=c$, see Lemma 5.6. We then declare, or rather construct the
graph so that

(11) $(\sigma,c)\text{ has a unique out-neighbor
}(\sigma^{\dagger},c^{\dagger}).$

Theorem 4.7 then follows.

######  Remark 4.9.

We can introduce a variant for the graph: that whenever we draw an edge from
$(\sigma,c)$ to $(\sigma^{\prime},c^{\prime})$, we also draw an edge to
$(\sigma,c)$ to $g.(\sigma^{\prime},c^{\prime})$ for any $g\in G(F)$.
Equivalently, we can view our graph as defined on the vertex set $C_{G}/G(F)$.
This obviously does not affect the validity of Theorem 4.7. Also, in (10)
instead of choosing one $\sigma^{\prime}$ below $\sigma$ we can also choose
all augmented facets below $\sigma$. That will not affect Theorem 4.7 and
provides a canonical (though not necessarily desirable) choice.

##  5\. Analysis on Moy-Prasad quotients and nilpotent orbits

From now on, by abuse of notation we will understand $C_{G}$ as a graph with
directed edges constructed as in §4. The goal of this section is to analyze
it, giving criteria for some $(\sigma,c)\in C_{G}$ to have or not have a
directed edge to some other $(\sigma^{\prime},c^{\prime})$. We adapt all the
notations in §4. Let us introduce a convenient notation from [LY17]: suppose
we have a homomorphism
$\lambda:\mathbb{G}_{m}\rightarrow\operatorname{End}(V)$ for a finite-
dimensional vector space $V$ (over some working field), then for any
$i\in\mathbb{Z}$ we write ${}^{\lambda}_{i}V\subset V$ for the eigenspace on
which $\lambda(t).v=t^{i}v$ for $t\in\mathbb{G}_{m}$. We also write
${}^{\lambda}_{\geq j}V:=\bigoplus_{i\geq j}{}^{\lambda}_{i}V$, etc..

###  5.1. Moy-Prasad quotients à la Reeder-Yu

We will need a theory for Moy-Prasad quotients “in the tame case,” for which
we review the results in [RY14, §4] with our language. Denote by
$\mathbb{Z}_{(p)}$ the localization of $\mathbb{Z}$ at $p$, namely the ring of
rational numbers with denominators coprime to $p$. The subset
$\mathcal{B}(G,F)_{(p)}\subset\mathcal{B}(G,F)$ of those points in
$\mathcal{B}(G,F)$ with rational coordinates with denominator coprime to $p$
is well-defined, e.g. they are those points that become hyperspecial vertices
after a tamely ramified base change. When $\mathcal{A}$ corresponds to a
maximal $F$-split torus $S$, we have
$\mathcal{A}_{(p)}:=\mathcal{A}\cap\mathcal{B}(G,F)_{(p)}$ is a torsor of
$X_{*}(S)\otimes_{\mathbb{Z}}\mathbb{Z}_{(p)}$.

Denote by $F^{s}/F$ a fixed separable closure and $F^{ur}/F$ the maximal
unramified extension in it, with residue field $\bar{k}$ an algebraic closure
of $k$. Let $n^{\prime}\in\mathbb{Z}_{>0}$ be coprime to $p$,
$r=\frac{a^{\prime}}{m^{\prime}}$ with $a^{\prime}\in\mathbb{Z}$ (we will
replace them by $a$ and $m$ in a moment), and $x\in\mathcal{B}(G,F)_{(p)}$ be
such that it is a hyperspecial vertex in $\mathcal{B}(G,E_{m^{\prime}})$ where
$E_{m^{\prime}}/F^{ur}$ is the unique (up to isomorphism) tame extension with
ramification index $m^{\prime}$. Fix a uniformizer $\varpi_{F}\in F$. For any
$b_{1}\equiv b_{2}(\operatorname{mod}m^{\prime})$, we have
$\mathfrak{g}(F)_{x=\frac{b_{1}}{m^{\prime}}}\cong\mathfrak{g}(F)_{x=\frac{b_{2}}{m^{\prime}}}$
by multiplying an appropriate power of $\varpi_{F}$. Using such
identification, the direct sum

$\mathfrak{g}^{\clubsuit}_{x}:=\bigoplus_{b\in\mathbb{Z}/m^{\prime}}\mathfrak{g}(F)_{x=\frac{b}{m^{\prime}}}$

is a Lie algebra over $k$. Over $\bar{k}$, Reeder-Yu showed [RY14, Thm. 4.1]
that it can be identified with the reductive quotient (Lie algebra version) of
the hyperspecial parahoric of $G/_{E_{m^{\prime}}}$ at $x$. They also consider
the reductive quotient (group version) $\mathsf{G}_{x}^{\clubsuit}$ for which
they give a Frobenius action so that it becomes a connected reductive group
over $k$ with Lie algebra isomorphic to $\mathfrak{g}^{\clubsuit}_{x}$. We
have in particular that $\mathsf{G}_{x}^{\clubsuit}/_{k}$ and $G/_{F}$ has the
same absolute root data. Moreover, there is a $k$-homomorphism
$\rho^{\prime}:\mu_{m^{\prime}}\rightarrow\operatorname{Aut}(\mathsf{G}_{x}^{\clubsuit})$
such that $\mathfrak{g}(F)_{x=\frac{b}{m^{\prime}}}$ is identified with the
eigenspace on which $\rho^{\prime}(z)v=z^{b}v$ for $z\in\mu_{m^{\prime}}$,
$v\in\mathfrak{g}^{\clubsuit}_{x}$, $b\in\mathbb{Z}/m^{\prime}$. We denote
${}^{\rho^{\prime}}_{b}\mathfrak{g}^{\clubsuit}_{x}$ such eigenspace, and
denote by ${}^{\rho^{\prime}}_{0}\mathsf{G}_{x}^{\clubsuit}$ the connected
subgroup invariant by $\rho^{\prime}$ with
$\operatorname{Lie}({}^{\rho^{\prime}}_{0}\mathsf{G}_{x}^{\clubsuit})={}^{\rho^{\prime}}_{0}\mathfrak{g}^{\clubsuit}_{x}$.
Analogous to
${}^{\rho^{\prime}}_{0}\mathfrak{g}_{x}\cong\mathfrak{g}(F)_{x=0}$ we have
${}^{\rho^{\prime}}_{0}\mathsf{G}_{x}^{\clubsuit}\cong\mathsf{G}_{x}$, the
reductive quotient of $G$ over $F$ at $x$ as an algebraic group over $k$. The
following lemma follows from [DeB02b, Lemma 4.1.2]:

######  Lemma 5.1.

For $b\in\mathbb{Z}$, an element in $\mathfrak{g}(F)_{x=\frac{b}{m^{\prime}}}$
is nilpotent in the sense of Definition 2.2 iff it is nilpotent as an element
in the reductive Lie algebra $\mathfrak{g}^{\clubsuit}_{x}$.

Write $m=m^{\prime}/\gcd(a^{\prime},m^{\prime})$,
$a=a^{\prime}/\gcd(a^{\prime},m^{\prime})$. Let

$\mathfrak{g}_{x}^{\heartsuit}:=\bigoplus_{b\in(\gcd(a^{\prime},m^{\prime})\mathbb{Z})/m^{\prime}\mathbb{Z}}\mathfrak{g}(F)_{x=\frac{b}{m^{\prime}}}.$

With

$1\rightarrow\mu_{\gcd(a^{\prime},m^{\prime})}\rightarrow\mu_{a}^{\prime}\rightarrow\mu_{a}\rightarrow
1$

we have
$\mathfrak{g}_{x}^{\heartsuit}=(\mathfrak{g}_{x}^{\clubsuit})^{\rho^{\prime}(\mu_{\gcd(a^{\prime},m^{\prime})})}$.
Let
$\mathsf{G}_{x}^{\heartsuit}:=\left((\mathsf{G}_{x}^{\clubsuit})^{\rho^{\prime}(\mu_{\gcd(a^{\prime},m^{\prime})})}\right)^{o}$
so that we have an induced
$\rho:\mu_{m}\rightarrow\operatorname{Aut}(\mathsf{G}_{x}^{\heartsuit})$. For
any $b\in\mathbb{Z}$ and $b^{\prime}=\gcd(a^{\prime},m^{\prime})\cdot b$, this
gives
$\mathfrak{g}(F)_{x=\frac{b}{m}}=\mathfrak{g}(F)_{x=\frac{b^{\prime}}{m^{\prime}}}={}^{\rho^{\prime}}_{b^{\prime}}\mathfrak{g}_{x}^{\clubsuit}={}^{\rho}_{b}\mathfrak{g}_{x}^{\heartsuit}$.
In particular this holds for $b=a$ and $b=-a$, so that e.g.
$\mathfrak{g}(F)_{x=r}={}^{\rho}_{a}\mathfrak{g}_{x}^{\heartsuit}$. We remark
that $m=m^{\prime}/\gcd(a^{\prime},m^{\prime})$ is the denominator of $r$ in
its reduced expression.

Suppose we are given $c\in\mathfrak{g}(F)_{x=r}$ that is nilpotent. We may
complete $c$ into an $\mathfrak{sl}_{2}$-triple

(12)
$(c,h,d)\in\mathfrak{g}(F)_{x=r}\times\mathfrak{g}(F)_{x=0}\times\mathfrak{g}(F)_{x=-r}\cong{}^{\rho}_{a}\mathfrak{g}^{\heartsuit}_{x}\times{}^{\rho}_{0}\mathfrak{g}^{\heartsuit}_{x}\times{}^{\rho}_{-a}\mathfrak{g}^{\heartsuit}_{x}.$

Viewing the triple in $\mathfrak{g}^{\heartsuit}_{x}$, we can lift it to
$\varphi:SL_{2}/_{k}\rightarrow\mathsf{G}_{x}^{\heartsuit}$ (see e.g. [Car93,
§5.5]). Restricting to the diagonal torus $\mathbb{G}_{m}\subset SL_{2}$ we
get a cocharacter
$\lambda:\mathbb{G}_{m}\rightarrow\mathsf{G}_{x}^{\heartsuit}$. Note that
$\varphi$ and thus $\lambda$ are uniquely determined by the triple, and the
triple is almost invariant under
$\rho:\mu_{m}\rightarrow\operatorname{Aut}(\mathsf{G}_{x}^{\heartsuit})$; more
precisely, every $\rho(z)$ scales $c$ by $z^{a}$ and $d$ by $z^{-a}$, which
has no effect on the diagonal. This implies that $\lambda$ has image in
${}^{\rho}_{0}\mathsf{G}_{x}^{\heartsuit}$. Consequently, we have

$\mathfrak{g}^{\heartsuit}_{x}=\bigoplus_{b\in\mathbb{Z}/m}\bigoplus_{i\in\mathbb{Z}}{}^{\rho}_{b}({}^{\lambda}_{i}\mathfrak{g}^{\heartsuit}_{x}).$

That is, the two gradings given by $\rho$ and $\lambda$ are compatible, and we
may realize

(13)
$(c,h,d)\in{}^{\lambda}_{2}\mathfrak{g}(F)_{x=r}\times{}^{\lambda}_{0}\mathfrak{g}(F)_{x=0}\times{}^{\lambda}_{-2}\mathfrak{g}(F)_{x=-r}={}^{\rho}_{a}({}^{\lambda}_{2}\mathfrak{g}^{\heartsuit}_{x})\times{}^{\rho}_{0}({}^{\lambda}_{0}\mathfrak{g}^{\heartsuit}_{x})\times{}^{\rho}_{-a}({}^{\lambda}_{-2}\mathfrak{g}^{\heartsuit}_{x}).$

Recall that the image of $\lambda$ lies in
${}^{\rho}_{0}\mathsf{G}_{x}^{\heartsuit}$ which we identify as the reductive
quotient $\mathsf{G}_{x}$. Thus there is maximal $k$-split torus
$\mathsf{S}\subset\mathsf{G}_{x}$ that contains the image of $\lambda$. We may
choose a maximal $F$-split torus $S\subset G$ whose corresponding apartment
$\mathcal{A}$ contains $x$ and whose reductive quotient is identified with
$\mathsf{S}$. Under the identification $X_{*}(\mathsf{S})=X_{*}(S)$ we have
that $\lambda$ is identified with
$\tilde{\lambda}:\mathbb{G}_{m}/_{F}\rightarrow S$. By [DeB02b, Cor. 4.3.2] we
can lift $(c,h,d)$ from (12) and (13) to

(14)
$(\tilde{c},\tilde{h},\tilde{d})\in{}^{\tilde{\lambda}}_{2}\mathfrak{g}(F)_{x\geq
r}\times{}^{\tilde{\lambda}}_{0}\mathfrak{g}(F)_{x\geq
0}\times{}^{\tilde{\lambda}}_{-2}\mathfrak{g}(F)_{x\geq-r}$

As before let us denote by $\mathcal{O}(x,r,c)$ the minimal nilpotent
$\operatorname{Ad}(G(F))$-orbit that intersects $c+\mathfrak{g}(F)_{x>r}$. By
[DeB02b, Cor. 5.2.4], we have that
$\mathcal{O}(x,r,c)=\operatorname{Ad}(G(F))\tilde{c}$. Since the weighted
Dynkin diagram associated to $c$ and $\tilde{c}$ are determined by $\lambda$
and $\tilde{\lambda}$ respectively. Hence the geometric orbits
$\operatorname{Ad}(\mathsf{G}_{x}^{\clubsuit}(\bar{k}))c$ and
$\operatorname{Ad}(G(F^{s}))\tilde{c}$, although defined over different
fields, are “the same” as they are classified by the same weighted Dynkin
diagram; we refer to [CM93] for the notion of weighted Dynkin diagrams. For
example in classical groups they correspond to the same partition. Note that
we have to use $\mathsf{G}_{x}^{\clubsuit}$ instead of
$\mathsf{G}_{x}^{\heartsuit}$ cause only the former has the same absolute root
data as $G$. We highlight this result:

######  Proposition 5.2.

Let $\tilde{c}=\mathcal{O}(x,r,c)$. The geometric orbits
$\operatorname{Ad}(\mathfrak{g}_{x}^{\clubsuit}(\bar{k}))c$ and
$\operatorname{Ad}(G(F^{s}))\tilde{c}$ are “the same” in the sense that they
have the same weighted Dynkin diagram.

For any finite extension $E/F$, denote by $\mathcal{O}^{E}(x,r,c)$ the minimal
$\operatorname{Ad}(G(E))$-orbit that intersects $c+\mathfrak{g}(E)_{x>r}$.

######  Corollary 5.3.

$\operatorname{Ad}(G(E))\mathcal{O}(x,r,c)=\mathcal{O}^{E}(x,r,c)$ for any
finite $E/F$.

###### Proof.

By definition the
$\operatorname{Ad}(G(E))\mathcal{O}(x,r,c)\geq\mathcal{O}^{E}(x,r,c)$. However
Proposition 5.2 says they live in the same geometric orbits. Hence they are
equal. ∎

With the corollary we may also denote by $\mathcal{O}^{ur}(x,r,c)$ (resp.
$\mathcal{O}^{s}(x,r,c)$) the limit of $\mathcal{O}^{E}(x,r,c)$ for $E/F$
unramified (resp. any $E/F$) and big enough, which necessarily stabilizes and
is the minimal $\operatorname{Ad}(G(F^{ur}))$-orbit (resp.
$\operatorname{Ad}(G(F^{s}))$-orbit) which meets
$c+\mathfrak{g}(F^{ur})_{x>r}$ (resp. $c+\mathfrak{g}(F^{s})_{x>r}$).

Let $S\subset G$ be a maximal $F$-split torus with corresponding apartment
$\mathcal{A}$ such that $x\in\mathcal{A}$. Let
$\tilde{\lambda}:\mathbb{G}_{m}/_{F}\rightarrow S$ be any cocharacter. As
before we identify the reductive quotient $\mathsf{S}$ of $S$ as a maximal
$k$-split torus of $\mathsf{G}_{x}$, and we denote by
$\lambda:\mathbb{G}_{m}/_{k}\rightarrow\mathsf{S}\subset\mathsf{G}_{x}$ the
corresponding cocharacter. For any $v\in
X_{*}(S)\otimes_{\mathbb{Z}}\mathbb{Z}_{(p)}$ we denote by
$x+v\in\mathcal{A}_{(p)}$ the action of $v$ on $x$. Unfolding the definition
of Moy-Prasad filtration, we have for any‘$\ell\in\mathbb{Z}$ and
$s\in\mathbb{R}$ that

(15)
${}^{\lambda}_{\ell}\mathfrak{g}(F)_{x=r}={}^{\lambda}_{\ell}\mathfrak{g}(F)_{(x+s\tilde{\lambda})=r+\ell
s},\;\;{}^{\tilde{\lambda}}_{\ell}\mathfrak{g}(F)_{x\geq
r}={}^{\tilde{\lambda}}_{\ell}\mathfrak{g}(F)_{(x+s\tilde{\lambda})\geq r+\ell
s},$

######  Lemma 5.4.

For any $t\in\mathbb{R}$ we have
$\mathcal{O}(x,r,c)=\mathcal{O}(x+t\tilde{\lambda},r+\ell t,c)$.

###### Proof.

Observe that $\mathcal{O}(x+s\tilde{\lambda},r+\ell s,c)$ depends only on the
augmented facet that $(x+s\tilde{\lambda},r+\ell s)$ lives in. By standard
continuity argument it suffices to prove that there exists $\delta>0$ such
that for all $s\in(-\delta,\delta)\cap\mathbb{Z}_{(p)}$, we have
$\mathcal{O}(x,r,c)=\mathcal{O}(x+s\tilde{\lambda},r+\ell s,c)$. Take $\delta$
small enough so that for any $s\in(-\delta,\delta)$, the augmented facet
containing $(x+s\tilde{\lambda},r+\ell s)$ has its closure contains $(x,r)$.
This implies that $c+\mathfrak{g}(F)_{x>r}\subset
c+\mathfrak{g}(F)_{x+s\tilde{\lambda}>r+\ell s}$ and therefore
$\mathcal{O}(x,r,c)\leq\mathcal{O}(x+s\tilde{\lambda},r+\ell s,c)$.

Proposition 5.2 asserts that $\mathcal{O}(x,r,c)$ and
$\mathcal{O}(x+s\tilde{\lambda},r+\ell s,c)$ have the same weighted Dynkin
diagram as that of $c$ in the corresponding $\mathfrak{g}_{x}^{\clubsuit}$ and
$\mathfrak{g}_{y}^{\clubsuit}$ where $y=x+s\tilde{\lambda}$. Suppose
$s=\frac{b}{n}$ with $b,n\in\mathbb{Z}$ and $n$ coprime to $p$. Then
$\tilde{\lambda}(\varpi^{b/n})$ sends $\mathfrak{g}_{x}^{\clubsuit}$ to
$\mathfrak{g}_{y}^{\clubsuit}$ and maps $c$ to $c$. This shows that the
weighted Dynkin diagram of $c\in\mathfrak{g}_{x}^{\clubsuit}$ and
$c\in\mathfrak{g}_{y}^{\clubsuit}$ can be identified by an element in
$G(F^{s})$, and consequently $\mathcal{O}(x,r,c)$ and
$\mathcal{O}(x+s\tilde{\lambda},r+\ell s,c)$ have the same weighted Dynkin
diagram, i.e. they are in the same $\operatorname{Ad}(G(F^{s}))$-orbit. This
together with $\mathcal{O}(x,r,c)\leq\mathcal{O}(x+s\tilde{\lambda},r+\ell
s,c)$ shows that they are the same $\operatorname{Ad}(G(F))$-orbit. ∎

Suppose we have another nilpotent element $c^{\prime}\in
c+{}^{\rho}_{a}({}^{\lambda}_{<\ell}\mathfrak{g}_{x}^{\heartsuit})$. We end
this subsection with another lemma:

######  Lemma 5.5.

We have that $\mathcal{O}(x,r,c)\leq\mathcal{O}(x,r,c^{\prime})$. If there is
an equality, then
$c^{\prime}\in\operatorname{Ad}({}^{\rho}_{0}\mathsf{G}_{x}^{\heartsuit}(k))c$.

###### Proof.

By definition of Moy-Prasad filtration, for $s\in\mathbb{R}_{>0}$ small enough
we have

$\mathfrak{g}(F)_{x+s\tilde{\lambda}>r+\ell
s}=\mathfrak{g}(F)_{x=r}\times_{{}^{\rho}_{a}\mathfrak{g}_{x}^{\heartsuit}}{}^{\rho}_{a}({}^{\lambda}_{<\ell}\mathfrak{g}_{x}^{\heartsuit})$

and thus

$c^{\prime}+\mathfrak{g}(F)_{x>r}\subset
c+\mathfrak{g}(F)_{x+s\tilde{\lambda}>r+\ell s}.$

By definition of $\mathcal{O}(x,r,c^{\prime})$, this gives
$\mathcal{O}(x,r,c^{\prime})\geq\mathcal{O}(x+s\tilde{\lambda},r+\ell
s,c)=\mathcal{O}(x,r,c)$ where the last equality follows from Lemma 5.4. Now
suppose $\mathcal{O}(x,r,c^{\prime})=\mathcal{O}(x,r,c)$. By Proposition 5.2,
this shows that $c,c^{\prime}\in\mathfrak{g}_{x}^{\clubsuit}$ have the same
weighted Dynkin diagram, i.e.
$\operatorname{Ad}(\mathsf{G}_{x}^{\clubsuit}(\bar{k}))c=\operatorname{Ad}(\mathsf{G}_{x}^{\clubsuit}(\bar{k}))c^{\prime}$.
By Lemma B.2, this implies
$c^{\prime}\in\operatorname{Ad}({}^{\rho}_{0}\mathsf{G}_{x}^{\heartsuit}(k))c$
as desired. ∎

###  5.2. Edges as in (11)

We continue the setup in the previous subsection and study $C_{G}$. Recall
that one of the two constructions of the edges, namely (11), refers to the
proof of [Wal06, Lemma 7.3.2]. We review and study it below. Suppose we are
given $(\sigma,c)\in C_{G}^{nil}$. [Wal06, Lemma 2.4.1] asserts that
$\sigma\cap(\mathcal{B}(G,F)_{(p)}\times\mathbb{Z}_{(p)})$ is dense in
$\sigma$. Hence we may pick
$(x,r)\in\mathcal{B}(G,F)_{(p)}\times\mathbb{Z}_{(p)}$ with $r=\frac{m}{n}$ is
the reduced expression as in the previous subsection. We had (after some
choices) tori $\mathsf{S}\subset\mathsf{G}_{x}$, $S\subset G$ and its
corresponding apartment $\mathcal{A}$, and $\tilde{\lambda}\in X_{*}(S)$.

Let $\mathcal{A}^{a}=\mathcal{A}\times\mathbb{R}$ be the augmented apartment,
which is a torsor of
$(X_{*}(S)\otimes_{\mathbb{Z}}\mathbb{R})\times\mathbb{R}$. The idea is to
“walk” towards the direction of
$(\tilde{\lambda},2)\in(X_{*}(S)\otimes_{\mathbb{Z}}\mathbb{R})\times\mathbb{R}$
from $(x,r)\in\mathcal{A}^{a}$. There are two possible scenarios:

  1. (A)

There exists $\delta>0$ such that for any $0\leq s<\delta$,
$(x,r)+s(\tilde{\lambda},2)=(x+s\tilde{\lambda},r+2s)\in\sigma$, or

  2. (B)

There exists a non-horizontal augmented facet $\sigma^{\dagger}$ for which

    1. (i)

$\sigma$ is contained in the boundary of $\sigma^{\dagger}$,

    2. (ii)

$\sigma\prec\sigma^{\dagger}$ by the partial order defined before (7), and

    3. (iii)

There exists $\delta>0$ such that for any $0<s<\delta$,
$(x,r)+s(\tilde{\lambda},2)=(x+s\tilde{\lambda},r+2s)\in\sigma^{\dagger}$.

Apparently, if $\sigma$ is horizontal we are in scenario (B). The construction
of the edges in (11) can be applied whenever we are in scenario (B), which we
assume so in the rest of this subsection. Using the definition of Moy-Prasad
quotient, just like in (15) we have

(16)
$\mathfrak{g}(F)_{\geq\sigma^{\dagger}}=\mathfrak{g}(F)_{\geq\sigma}\times_{\mathfrak{g}(F)_{\sigma}}{}^{\lambda}_{\geq
2}\mathfrak{g}(F)_{\sigma},\;\;\mathfrak{g}(F)_{>\sigma^{\dagger}}=\mathfrak{g}(F)_{\geq\sigma}\times_{\mathfrak{g}(F)_{\sigma}}{}^{\lambda}_{>2}\mathfrak{g}(F)_{\sigma},\;\;\mathfrak{g}(F)_{\sigma^{\dagger}}={}^{\lambda}_{2}\mathfrak{g}(F)_{\sigma}$

We have

######  Lemma 5.6.

Every element in
$c+{}^{\lambda}_{>2}\mathfrak{g}(F)_{\sigma}+\mathfrak{g}(F)_{>\sigma}=c+\mathfrak{g}(F)_{>\sigma^{\dagger}}$
is conjugate under $G(F)$ to an element in $c+\mathfrak{g}(F)_{>\sigma}$.

###### Proof.

We sketch essentially the same proof in [Wal06, Lemma 7.2.2 & 7.3.2]. Note
that $c\in\mathfrak{g}(F)_{\sigma}={}^{\rho}_{a}\mathfrak{g}_{x}^{\heartsuit}$
and
${}^{\lambda}_{>2}\mathfrak{g}(F)_{\sigma}={}^{\rho}_{a}({}^{\lambda}_{>2}\mathfrak{g}_{x}^{\heartsuit})$.
By the property of $\mathfrak{sl}_{2}$-triple as in (13), we have
${}^{\rho}_{a}({}^{\lambda}_{>2}\mathfrak{g}_{x}^{\heartsuit})=[c,{}^{\rho}_{0}({}^{\lambda}_{>0}\mathfrak{g}_{x}^{\heartsuit})]$.
From this one deduces that every element in
$c+{}^{\lambda}_{>2}\mathfrak{g}(F)_{\sigma}$ is conjugate under
$\exp({}^{\lambda}_{>0}\mathfrak{g}(F)_{x=0})\subset\mathsf{G}_{x}(k)$ to $c$.
Consequently, every element in
$c+{}^{\lambda}_{>2}\mathfrak{g}(F)_{\sigma}+\mathfrak{g}(F)_{>\sigma}$ is
conjugate under the parahoric $G(F)_{x\geq 0}$ to an element in
$c+\mathfrak{g}(F)_{>\sigma}$. ∎

The lemma is exactly what we claimed and used for Theorem 4.7. That is, in
(11) we put $c^{\dagger}\in{}^{\lambda}_{2}\mathfrak{g}(F)_{\sigma^{\dagger}}$
to be the element identified with $c$. We highlight that Lemma 5.4 gives

(17) $\mathcal{O}(\sigma,c)=\mathcal{O}(\sigma^{\dagger},c^{\dagger}).$

######  Remark 5.7.

One can show that once $S$ and $\mathcal{A}$ are fixed, $\sigma^{\dagger}$
does not depend on the choice of $(x,r)\in\sigma\cap\mathcal{A}^{a}_{(p)}$.
Based on this, one can furthermore show that up to
$(\sigma^{\dagger},c^{\dagger})$ depends only on $(\sigma,c)$ up to the action
of the stabilizer of $(\sigma,c)$ in $G(F)$. As in Remark 4.9, one can declare
edges from $(\sigma,c)$ to all such $(\sigma^{\dagger},c^{\dagger})$, so that
we don’t need to make any choice at $(\sigma,c)$.

###  5.3. Edges as in (10)

We turn to the edges defined in (10) which are actually the crucial ones
despite the seemingly innocent construction. We are given a non-horizontal
augmented facet $\sigma$ and another horizontal augmented facet
$\sigma^{\prime}$ below it, following the choice made after Definition 4.8.
From now on, let $(x,r)\in\sigma^{\prime}\cap\mathcal{A}^{a}_{(p)}$ be any
choice. Here $\mathcal{A}^{a}_{(p)}:=\mathcal{A}_{(p)}\times\mathbb{Z}_{(p)}$
is any augmented apartment intersecting $\sigma$ and thus $\sigma^{\prime}$.
Say $S\subset G$ is the corresponding maximal $F$-split torus. There exists
$\tilde{\lambda}\in X_{*}(S)$ and $\ell\in\mathbb{Z}_{>0}$ such that there
exists $\delta>0$ such that $0<s<\delta\implies(x,r)+(s\tilde{\lambda},-\ell
s)\in\sigma$. We have $\mathsf{S}\subset\mathsf{G}_{x}$ the reductive quotient
of $S$, with $X_{*}(S)=X_{*}(\mathsf{S})$ so that we denote by $\lambda\in
X_{*}(\mathsf{S})$ the image of $\tilde{\lambda}$. Unfolding the definition of
Moy-Prasad filtration we have

(18)
$\mathfrak{g}(F)_{\geq\sigma}=\mathfrak{g}(F)_{\geq\sigma^{\prime}}\times_{\mathfrak{g}(F)_{\sigma^{\prime}}}{}^{\lambda}_{\leq\ell}\mathfrak{g}(F)_{\sigma^{\prime}},\;\;\mathfrak{g}(F)_{>\sigma}=\mathfrak{g}(F)_{\geq\sigma^{\prime}}\times_{\mathfrak{g}(F)_{\sigma^{\prime}}}{}^{\lambda}_{<\ell}\mathfrak{g}(F)_{\sigma^{\prime}},\;\;\mathfrak{g}(F)_{\sigma}={}^{\lambda}_{\ell}\mathfrak{g}(F)_{\sigma^{\prime}}.$

By (10), we draw an edge from $(\sigma,c)$ to $(\sigma^{\prime},c^{\prime})$
for any $c^{\prime}\in
c+\mathfrak{g}(F)_{>\sigma}/\mathfrak{g}(F)_{>\sigma^{\prime}}$, i.e. for any
$c^{\prime}\in c+{}^{\lambda}_{<\ell}\mathfrak{g}(F)_{\sigma^{\prime}}$. Let
$m\in\mathbb{Z}_{>0}$ be the denominator of $r=\frac{a}{m}$ as in §5.1, we
have that
$c\in{}^{\lambda}_{\ell}\mathfrak{g}(F)_{\sigma^{\prime}}={}^{\rho}_{a}({}^{\lambda}_{\ell}\mathfrak{g}_{x}^{\heartsuit})$
and
${}^{\lambda}_{<\ell}\mathfrak{g}(F)_{\sigma^{\prime}}={}^{\rho}_{a}({}^{\lambda}_{<\ell}\mathfrak{g}_{x}^{\heartsuit})$.
Recall also that ${}^{\rho}_{0}\mathsf{G}_{x}^{\heartsuit}=\mathsf{G}_{x}$. As
in Remark 5.7 we are happy to conjugate our elements by $\mathsf{G}_{x}$
whenever applicable. This brings us to the geometric problem, which we
highlight below:

######  Problem 5.8.

Given

  1. (1)

A connected reductive group $\mathsf{G}_{x}^{\heartsuit}$ over our finite
field $k$ with
$\mathfrak{g}_{x}^{\heartsuit}:=\operatorname{Lie}\mathsf{G}_{x}^{\heartsuit}$,

  2. (2)

An integer $m>0$ prime to $p$ and an algebraic group homomorphism
$\rho:\mu_{m}\rightarrow\operatorname{Aut}(\mathsf{G}_{x}^{\heartsuit})$,

  3. (3)

A cocharcter
$\lambda:\mathbb{G}_{m}\rightarrow{}^{\rho}_{0}\mathsf{G}_{x}^{\heartsuit}$, a
residue class $a\in\mathbb{Z}/m$, an integer $\ell>0$, and an element
$c\in{}^{\rho}_{a}({}^{\lambda}_{\ell}\mathfrak{g}^{\heartsuit}_{x})$.

What are the
$\operatorname{Ad}({}^{\rho}_{0}\mathsf{G}_{x}^{\heartsuit}(k))$-orbits in
${}^{\rho}_{a}\mathfrak{g}_{x}^{\heartsuit}$ that intersects
$c+{}^{\rho}_{a}({}^{\lambda}_{<\ell}\mathfrak{g}^{\heartsuit}_{x})$?

Consider any $c^{\prime}\in
c+{}^{\rho}_{a}({}^{\lambda}_{<\ell}\mathfrak{g}^{\heartsuit}_{x})$. We begin
by the simpler situation when $c^{\prime}$ is nilpotent. Recall that
$\mathcal{O}(\sigma^{\prime},c^{\prime})$ is the minimal
$\operatorname{Ad}(G(F))$-orbit that intersects
$c^{\prime}+\mathfrak{g}(F)_{>\sigma^{\prime}}=c^{\prime}+\mathfrak{g}(F)_{x>r}$.
We have

######  Corollary 5.9.

Suppose $c^{\prime}\in
c+{}^{\rho}_{a}({}^{\lambda}_{<\ell}\mathfrak{g}^{\heartsuit}_{x})$ is
nilpotent. Then
$\mathcal{O}(\sigma^{\prime},c^{\prime})\geq\mathcal{O}(\sigma^{\prime},c)=\mathcal{O}(\sigma,c)$.
When the equality holds, we have
$c^{\prime}\in\operatorname{Ad}({}^{\rho}_{0}\mathsf{G}_{x}^{\heartsuit}(k))c=\operatorname{Ad}(\mathsf{G}_{x}(k))c$.

###### Proof.

The equality $\mathcal{O}(\sigma^{\prime},c)=\mathcal{O}(\sigma,c)$ comes from
Lemma 5.4 while the rest of the corollary come from Lemma 5.5. ∎

######  Corollary 5.10.

Under the identification
$\mathfrak{g}(F)_{\sigma}={}^{\lambda}_{\ell}\mathfrak{g}(F)_{\sigma^{\prime}}$
there is an edge from $(\sigma,c)$ to $(\sigma^{\prime},c)$, with
$\mathcal{O}(\sigma,c)=\mathcal{O}(\sigma^{\prime},c)$. Any edge from
$(\sigma,c)$ to $(\sigma^{\prime},c^{\prime})$ such that $c^{\prime}$ is
nilpotent and $\mathcal{O}(\sigma,c)=\mathcal{O}(\sigma^{\prime},c^{\prime})$
is $G(F)$-conjugate to the edge in the first sentence.

###### Proof.

That there is an edge from $(\sigma,c)$ to $(\sigma^{\prime},c)$ follows
immediately from definition. The rest of the statements follows from Corollary
5.9. ∎

Now we turn the case of non-nilpotent $c^{\prime}\in
c+{}^{\rho}_{a}({}^{\lambda}_{<\ell}\mathfrak{g}^{\heartsuit}_{x})$, for which
we can get weaker results. Firstly we have a minor generalization of [CO23,
Prop. 3.6] (in loc. cit. it is assumed that $\lambda$ is associated to an
$\mathfrak{sl}_{2}$-triple completing $c$), which is just a rephrasement of
Lemma B.2:

######  Lemma 5.11.

The element $c$ is contained in the Zariski closure of
$\bar{k}\cdot\operatorname{Ad}(({}^{\rho}_{0}\mathsf{G}_{x}^{\heartsuit})(\bar{k}))c^{\prime}$.

Because the dimension of centralizer for an element in a Lie algebra is lower
semi-continuous (it is the rank of the adjoint action), the lemma implies

######  Corollary 5.12.

We have
$\dim\operatorname{Ad}(\mathsf{G}_{x}^{\heartsuit}(\bar{k}))c^{\prime}\geq\dim\operatorname{Ad}(\mathsf{G}_{x}^{\heartsuit}(\bar{k}))c$.

Let $c^{\prime}=c_{s}^{\prime}+c_{n}^{\prime}$ be the Jordan decomposition.
Consider the nilpotent geometric
$\operatorname{Ad}(\mathsf{G}_{x}^{\heartsuit})(\bar{k})$-orbit
$N(c^{\prime})\subset\mathfrak{g}_{x}^{\heartsuit}$ given by the Lusztig-
Spaltenstein induction of the orbit
$\operatorname{Ad}(\mathsf{G}_{x}^{\heartsuit}(\bar{k}))c_{n}^{\prime}$ from
$Z_{\mathsf{G}_{x}^{\heartsuit}}(c_{s}^{\prime})$ to
$\mathsf{G}_{x}^{\heartsuit}$; see Definition B.3. Ciubotaru and Okada
observed [CO23, Lemma 4.2] that the closure of $N(c^{\prime})$ is the
intersection of
$\bar{k}\cdot\operatorname{Ad}(({}^{\rho}_{0}\mathsf{G}_{x}^{\heartsuit})(\bar{k}))c^{\prime}$
with the nilpotent cone of $\mathfrak{g}_{x}^{\heartsuit}$. With Lemma 5.11 we
have

######  Corollary 5.13.

([CO23, Lemma 4.2]) The element $c$ is contained in the Zariski closure of
$N(c^{\prime})$.

Note that $N(c^{\prime})$ is only a
$\mathsf{G}_{x}^{\heartsuit}(\bar{k})$-orbit, not a
$({}^{\rho}_{0}\mathsf{G}_{x}^{\heartsuit})(\bar{k})$-orbit. In general
$N(c^{\prime})\cap{}^{\rho}_{a}(\mathfrak{g}_{x}^{\heartsuit})(\bar{k})$ is
the closure of the union of a number of
$\mathsf{G}_{x}^{\heartsuit}(\bar{k})$-orbits, say
$\\{N^{\rho}_{1}(c^{\prime}),...,N^{\rho}_{n}(c^{\prime})\\}$. Recall that
$(x,r)\in\sigma^{\prime}$. Similar to before we denote by
$\mathcal{O}^{ur}(\sigma^{\prime},N^{\rho}_{i}(c^{\prime}))$ the minimal
nilpotent $\operatorname{Ad}(G(F^{ur}))$-orbit that meets
$N^{\rho}_{i}(c^{\prime})+\mathfrak{g}(F^{ur})_{>\sigma^{\prime}}$. We have
the following corollary to Corollary 5.13, which is implicit in [CO23, §5]:

######  Corollary 5.14.

We have $\mathcal{O}(\sigma,c)=\mathcal{O}(\sigma^{\prime},c)$ is contained in
the $p$-adic closure of one of
$\mathcal{O}^{ur}(\sigma^{\prime},N^{\rho}_{i}(c^{\prime}))$ ($1\leq i\leq n$)
above.

###### Proof.

By definition $c$ is contained in the Zariski closure of one of
$N^{\rho}_{i}(c^{\prime})$. Thanks to Corollary 5.3, we have
$\operatorname{Ad}(G(F^{ur}))\mathcal{O}(\sigma,c)=\mathcal{O}^{ur}(\sigma,c)$.
The latter is contained in the $p$-adic closure of
$\mathcal{O}^{ur}(\sigma^{\prime},N^{\rho}_{i}(c^{\prime}))$ thanks to Lemma
5.5 applied with $F$ replaced by $F^{ur}$. ∎

Corollary 5.9 to Corollary 5.14 all give necessary conditions for the
existence of an edge in $C_{G}$ from $(\sigma,c)$ to
$(\sigma^{\prime},c^{\prime})$, with the exception of Corollary 5.10 which
also contains a rather naive existence result. When $r\in\mathbb{Z}$, we have
$m=1$ so that $\rho$ is trivial. In this case we will prove a stronger
existence result. To accurately state what we can achieve we need more
notions. Define an equivalence relation $\sim_{semi}$ on
$\mathfrak{g}^{nil}(F)/\\!\\!\sim$ as follows:

######  Definition 5.15.

We denote by $\sim_{semi}$ the equivalence relation on
$\mathfrak{g}^{nil}(F)/\\!\\!\sim$ generated by the following relation: For
$\mathcal{O}_{1},\mathcal{O}_{2}\in\mathfrak{g}^{nil}(F)/\\!\\!\sim$ we
declare $\mathcal{O}_{1}\sim_{semi}\mathcal{O}_{2}$ if there exist
$r\in\mathbb{Z}$, $x\in\mathcal{B}(G,F)$ and
$c_{1},c_{2}\in\mathfrak{g}(F)_{x=r}$ such that
$\mathcal{O}_{i}=\mathcal{O}(x,r,c_{i})$ and
$\operatorname{Ad}(\mathsf{G}_{x}(\bar{k}))c_{1}=\operatorname{Ad}(\mathsf{G}_{x}(\bar{k}))c_{2}$.

The following lemma will not be used in this article; it is just there so that
Definition 5.15 looks slightly simpler.

######  Lemma 5.16.

In Definition 5.15, it suffices to take $r=0$, i.e. those relations given by
$r=0$ generate the same equivalence relation.

###### Proof.

Suppose $\mathcal{O}_{1},\mathcal{O}_{2}\in\mathfrak{g}^{nil}(F)/\\!\\!\sim$,
$r\in\mathbb{Z}$, $x\in\mathcal{B}(G,F)$ and
$c_{1},c_{2}\in\mathfrak{g}(F)_{x=r}$ are such that
$\mathcal{O}_{i}=\mathcal{O}(x,r,c_{i})$ and
$\operatorname{Ad}(\mathsf{G}_{x}(\bar{k}))c_{1}=\operatorname{Ad}(\mathsf{G}_{x}(\bar{k}))c_{2}$.
Then $c_{1},c_{2}$ has the same weighted Dynkin diagram in
$\mathfrak{g}(F)_{x=r}\cong\mathfrak{g}(F)_{x=0}\cong\operatorname{Lie}\mathsf{G}_{x}$.
Hence their $\mathfrak{sl}_{2}$-triples have the same associated cocharacter.
In particular there exists
$\lambda:\mathbb{G}_{m}/_{k}\rightarrow\mathsf{S}\subset\mathsf{G}_{x}$ such
that $c_{1},c_{2}\in{}^{\lambda}_{2}\mathfrak{g}(F)_{x=r}$, and that
$\operatorname{Ad}({}^{\lambda}_{0}\mathsf{G}_{x}(\bar{k}))c_{1}=\operatorname{Ad}({}^{\lambda}_{0}\mathsf{G}_{x}(\bar{k}))c_{2}$.
Identify $\lambda$ as $\tilde{\lambda}\in X_{*}(S)$ where $S\subset G$ is a
maximal $F$-split torus with reductive quotient $\mathsf{S}$ and such that the
corresponding apartment contains $x$. Then Lemma 5.4 gives
$\mathcal{O}(x,r,c_{i})=\mathcal{O}(x-\frac{r}{2}\tilde{\lambda},0,c_{i})$.
Write $y=x-\frac{r}{2}\tilde{\lambda}$. We have
${}^{\lambda}_{0}\mathsf{G}_{y}={}^{\lambda}_{0}\mathsf{G}_{x}$. Thus
$\operatorname{Ad}(\mathsf{G}_{y}(\bar{k}))c_{1}=\operatorname{Ad}(\mathsf{G}_{y}(\bar{k}))c_{2}$
and by replacing $x$ with $y$ we have reduced to the case $r=0$. ∎

######  Definition 5.17.

We denote by $\mathfrak{g}^{nil}(F)/\\!\\!\sim_{semi}$ the quotient set of
$\mathfrak{g}^{nil}(F)/\\!\\!\sim$ by $\sim_{semi}$. We give
$\mathfrak{g}^{nil}(F)/\\!\\!\sim_{semi}$ the partial order generated by that
in $\mathfrak{g}^{nil}$, i.e. for any $\mathcal{O}_{1}\leq\mathcal{O}_{2}$ we
declare $\mathcal{O}_{1}/\\!\sim_{semi}\,\leq\mathcal{O}_{2}/\\!\sim_{semi}$
and consider the partial order relation this generates.

For $(\sigma,c)\in C_{G}$ or $(x,r)\in\mathcal{B}^{a}(G,F)$,
$c\in\mathfrak{g}(F)_{x=r}$ we denote by $\mathcal{O}^{semi}(\sigma,c)$ (resp.
$\mathcal{O}^{semi}(x,r,c)$) the image of $\mathcal{O}(\sigma,c)$ (resp.
$\mathcal{O}(x,r,c)$) under the quotient map. For $\gamma\in\mathfrak{g}(F)$
we also denote by
$\operatorname{WF}^{semi}(\gamma)\subset\mathfrak{g}^{nil}(F)/\\!\\!\sim_{semi}$
the set of maximal elements in $\mathfrak{g}^{nil}(F)/\\!\\!\sim_{semi}$ among
the image of $\operatorname{WF}(\gamma)$ under the quotient map. We call
$\operatorname{WF}^{semi}(\gamma)$ the semi-unramified wave-front set.

Denote by $\mathfrak{g}^{nil}(F^{ur})/\\!\\!\sim$ the set of
$\operatorname{Ad}(G(F^{ur}))$-orbits in $\mathfrak{g}^{nil}(F^{ur})$, with
the partial order given by $p$-adic closure. We denote by
$\operatorname{WF}^{ur}(\gamma)$ the set of maximal elements in the image of
$\operatorname{WF}(\gamma)$ under
$\mathfrak{g}^{nil}(F)/\\!\\!\sim\;\rightarrow\mathfrak{g}^{nil}(F^{ur})/\\!\\!\sim$.
Note that the example in [Tsa23] shows that an element in
$\operatorname{WF}(\gamma)$ needs not match with an element in
$\operatorname{WF}^{ur}(\gamma)$.

######  Remark 5.18.

In the example when $G=U_{2}$ is an unramified unitary group,
$\mathfrak{g}^{nil}(F)/\\!\\!\sim_{semi}$ has two regular orbits while
$\mathfrak{g}^{nil}(F^{ur})/\\!\\!\sim$ has only one. In general, thanks to
Corollary 5.3 there is an obvious map from
$\mathfrak{g}^{nil}(F)/\\!\\!\sim_{semi}$ to
$\mathfrak{g}^{nil}(F^{ur})/\\!\\!\sim$. Unfolding definition we see that
$\operatorname{WF}^{ur}(\gamma)$ is again the set of maximal elements among
the image of $\operatorname{WF}^{semi}(\gamma)$.

We have the following variant of Corollary 5.14 whose proof is identical.

######  Corollary 5.19.

Suppose $r\in\mathbb{Z}$. Then we have
$\mathcal{O}(\sigma,c)=\mathcal{O}(\sigma^{\prime},c)$ is contained in the
closure of one of
$\mathcal{O}^{semi}(\sigma^{\prime},N^{\rho}_{i}(c^{\prime}))$ ($1\leq
i\leq\ell$).

######  Theorem 5.20.

(Corollary B.16) Suppose $r\in\mathbb{Z}$ so that $m=1$. Fix
$\sigma^{\prime}\in\Phi^{G}(r)$ horizontal and
$(x,r)\in\sigma^{\prime}\cap\mathcal{A}^{a}_{(p)}$. Let
$c^{\prime}=c_{s}^{\prime}+c_{n}^{\prime}\in{}^{\rho}_{a}\mathfrak{g}_{x}^{\heartsuit}=\mathfrak{g}_{x}^{\heartsuit}$
and $N(c^{\prime})$ be as before. Then there exists $c\in N(c^{\prime})(k)$
such that for any
$\lambda:\mathbb{G}_{m}\rightarrow\mathsf{G}_{x}^{\heartsuit}$ and
$\ell\in\mathbb{Z}_{>0}$ with
$c\in{}^{\lambda}_{\ell}(\mathfrak{g}_{x}^{\heartsuit})$, we have
$c^{\prime}\in c+{}^{\lambda}_{<\ell}(\mathfrak{g}_{x}^{\heartsuit})$.

With the notations in Theorem 5.20, up to conjugating by an element in
$\mathsf{G}_{x}^{\heartsuit}(k)=\mathsf{G}_{x}(k)$ we may assume that
$\lambda$ has image in $\mathsf{S}$ and thus corresponds to
$\tilde{\lambda}\in X_{*}(S)$. We remark that such $\lambda$ always exists,
e.g. the one associated to an $\mathfrak{sl}_{2}$-triple completing $c$ with
$\ell=2$. For any such $\lambda$ and $\tilde{\lambda}$, there exists
$\sigma\in\Phi^{G}$ and $\delta>0$ such that for $0<s<\delta$ we have
$(x,r)-(s\tilde{\lambda},\ell s)\in\sigma$. This leads to

######  Corollary 5.21.

Let $c$ be as in Theorem 5.20. If in the construction of the graph $C_{G}$ in
(10) we make the choice $\sigma^{\prime}$ for $\sigma$ right after Definition
4.8. Then there is an edge from $\mathcal{O}(\sigma,c)$ to
$\mathcal{O}(\sigma^{\prime},c^{\prime})$. We have
$\mathcal{O}^{semi}(\sigma,c)=\mathcal{O}^{semi}(\sigma^{\prime},N(c^{\prime}))$.

###### Proof.

The setting in this subsection applies so that we have an edge from
$(\sigma,c)$ to $(\sigma^{\prime},c^{\prime})$. By construction $c^{\prime}$
and $N(c)$ are in the same
$\operatorname{Ad}(\mathsf{G}_{x}^{\heartsuit})$-orbit and therefore
$\mathcal{O}^{semi}(\sigma^{\prime},c^{\prime})=\mathcal{O}^{semi}(\sigma^{\prime},N(c))$.
∎

######  Remark 5.22.

Although the statement of Corollary 5.21 seems to rely on some choice, it
implies that whenever $(\sigma^{\prime},c^{\prime})\in c_{G}(\gamma)$, then
$\operatorname{WF}(\gamma)$ must contain an orbit in
$\mathcal{O}^{semi}(\sigma,c)$ or some larger orbit, because Theorem 4.7 holds
for any choice. Frequently in examples, Theorem 4.7 will reversely force the
choice in Corollary 5.21 to be the unique one because the wave-front set in
the end does not depend on the choice.

######  Remark 5.23.

Corollary B.14 and B.15 will allow us to reduce the geometric problem -
Problem 5.8 \- to the case when $\lambda$ is associated to an
$\mathfrak{sl}_{2}$-triple completing $c$. Even with the reduction, the
problem seems rather formidable in general. For example, the construction of
[Tsa24] is based on some rather specific computer experiment of this problem
(in [Tsa24, Lemma 2.2, 2.3, 2.4]) when $m=2$,
$\mathsf{G}_{x}^{\heartsuit}=GL_{n}$ and the action $\rho$ is given by
$g\mapsto(g^{t})^{-1}$. In this case Problem 5.8 involves at least some
geometry of Jacobians of hyperelliptic curves studied in [Tsa20] based on
[Wan18].

The author also gave some discussion [Tsa24, §4] that this problem might be
related to graded Springer theory (which is still largely being developed).
Meanwhile, Corollary B.16 is concerned with the “ungraded” case $m=1$, namely
the case when graded Springer theory becomes the well-understood generalized
Springer theory. Indeed, Theorem B.4 behinds Corollary B.16 can be proved with
Springer theory, though we take an elementary approach with a key step
Proposition B.6 inspired by Springer theory.

##  6\. Examples

We now collect all our results so far to produce precise results about semi-
unramified wave-front set (Definition 5.17, an invariant stronger than
$\operatorname{WF}^{ur}(\gamma)$) in some “unramified” situation. Let
$\gamma=\gamma_{r}+\gamma_{>r}$ be as in §3. We remark that this implies that
$\gamma_{r}\in\operatorname{Lie}T$ for some torus $T\subset G$ that splits
over an unramified extension. Again write $H=Z_{G}(\gamma_{r})$. Our goal is
to compute $\operatorname{WF}^{semi}(\gamma)$, using $c_{H,r}(\gamma_{>r})$ as
a black box. The set $c_{G,r}(\gamma)$ is determined by $c_{H,r}(\gamma_{>r})$
via (6). To apply Proposition 2.3 we need to determine $c_{G,s}(\gamma)$ for
any slightly smaller $s$. Indeed, take $s=r-\epsilon$ with
$\epsilon\in\mathbb{Z}_{(p)}$, $\epsilon>0$ and small enough so that there is
no horizontal facet in $\Phi^{G}$ of depth within $[s,r)$, i.e. all
$\sigma\in\Phi^{G}(s)$ are non-horizontal with $0pt(\sigma)\geq r$. Hence by
Theorem 4.7, $c_{G,s}(\gamma)$ are precisely those $(\sigma,c)$ with an edge
to $(\sigma^{\prime},c^{\prime})\in c_{G,r}(\gamma)$ in the manner of (10). In
this case $\sigma^{\prime}$ is the horizontal augmented facet below $\sigma$
and the datum $(\sigma,c,\sigma^{\prime},c^{\prime})$ satisfies the setup in
(18), Problem 5.8 and the paragraph in between. We can apply Corollary 5.14,
which shows that $\mathcal{O}(\sigma,c)$ is contained in the $p$-adic closure
of $\mathcal{O}^{ur}(\sigma^{\prime},N(c^{\prime}))$. This gives

(19) $\begin{array}[]{rcl}\operatorname{WF}^{ur}(\gamma)&=&\text{the set of
maximal elements in }\\{\mathcal{O}^{ur}(\sigma,c)\;|\;(\sigma,c)\in
c_{G,s}(\gamma)\\}\\\ &\leq&\text{the set of maximal elements in
}\\{\mathcal{O}^{ur}(\sigma^{\prime},N(c^{\prime}))\;|\;(\sigma^{\prime},c^{\prime})\in
c_{G,r}(\gamma)\\}.\end{array}$

where $\leq$ means that every element in $\operatorname{WF}^{ur}(\gamma)$ is
$\leq\mathcal{O}$ for some $\mathcal{O}$ in the RHS. When we impose the
additional assumption that $r\in\mathbb{Z}$, we can replace $\mathcal{O}^{ur}$
by $\mathcal{O}^{semi}$ using Corollary 5.19. Corollary 5.21 says for every
$(\sigma^{\prime},c^{\prime})\in c_{G,r}(\gamma)$, we can choose to have
$(\sigma,c)$ as above with an edge to $(\sigma^{\prime},c^{\prime})$ such that
$\mathcal{O}^{semi}(\sigma,c)=\mathcal{O}^{semi}(\sigma^{\prime},c)$ is
precisely $\mathcal{O}^{semi}(\sigma^{\prime},N(c^{\prime}))$. Therefore when
$r\in\mathbb{Z}$ we have

(20) $\operatorname{WF}^{semi}(\gamma)=\text{the set of maximal elements in
}\\{\mathcal{O}^{semi}(\sigma^{\prime},N(c^{\prime}))\;|\;(\sigma^{\prime},c^{\prime})\in
c_{G,r}(\gamma)\\}.$

We apply (20) to a number of cases.

######  Example 6.1.

Let $\gamma\in\mathfrak{g}(F)$ be a good element of depth $r$. We don’t assume
$r\in\mathbb{Z}$ yet. We would like to compute $\operatorname{WF}(\gamma)$.
Again write $H=Z_{G}(\gamma)$. We have $\gamma=\gamma+0$ is the decomposition
in §3 and

(21) $c_{H,r}(0)=\\{(\varsigma_{H},0)\;|\;\sigma_{H}\in\Phi^{H}(r)\\}$

Denote by $\gamma_{r}$ the image of $\gamma$ to the sub-quotient
$Z(\mathfrak{h})(F)_{=r}$. Note that $Z(\mathfrak{h})(F)_{=r}$ is a subspace
of $\mathfrak{g}(F)_{x=r}$ for any $x\in\mathcal{B}(H,F)$, so we can realize
$\gamma_{r}$ in $\mathfrak{g}(F)_{x=r}$ for any $x\in\mathcal{B}(H,F)$. By (6)
and (21) we have

(22)
$c_{G,r}(\gamma)=G(F).\\{(\varsigma,\gamma_{=r})\;|\;\varsigma\in\Phi^{G}(r),\;\varsigma\cap\mathcal{B}(H,F)\not=\emptyset\\}.$

We plug (22) to the RHS of (20). We remark that since (22) is apparently
$G(F)$-invariant, we can discard the $G(F)$-action part in (22) to make it
simpler. For example, when $H/Z(G)$ is anisotropic, we have that
$\mathcal{B}(H,F)$ is a unique point modulo the central direction of
$\mathcal{B}(G,F)$. By Corollary 3.1, this makes $\mathcal{B}(H,F)$ a single
$r$-facet $\sigma^{\prime}$ in $\mathcal{B}(G,F)$. When $r\in\mathbb{Z}$, (20)
gives

(23)
$\operatorname{WF}^{semi}(\gamma)=\mathcal{O}^{semi}(\sigma^{\prime},N(\gamma_{=r}))$

which recovers [CO23, Cor. 6.1]. In general when $r\not\in\mathbb{Z}$,
$\operatorname{WF}^{ur}(\gamma)$ is the set of maximal elements among
$\mathcal{O}^{ur}(\sigma,c)$ for those $(\sigma,c)$ satisfying Problem 5.8
where we have a non-trivial $\rho$.

######  Example 6.2.

Denote by $\varpi_{F}\in F$ be a fixed uniformizer. Let $G=U_{6}$ be an
unramified unitary group over $F$ that splits over the quadratic unramified
extension $E/F$, given by the following chosen hermitian form:

(24) $\langle
x,y\rangle=x_{1}\bar{y}_{1}+x_{2}\bar{y}_{2}+x_{3}\bar{y}_{3}+x_{4}\bar{y}_{4}+x_{5}\bar{y}_{5}+\varpi_{F}x_{6}\bar{y}_{6}.$

Note that there is an additional coefficient $\varpi_{F}$ for the 6th
coordinate, making $G$ non-quasi-split. Fix
$\lambda_{1},\lambda_{2},...,\lambda_{6}\in\mathcal{O}_{E}^{\times}$ with
distinct non-zero residues such that $\operatorname{Tr}_{E/F}\lambda_{i}=0$.
Consider the following diagonal elements

$\gamma_{-1}=\varpi_{F}^{-1}\cdot\left[\begin{matrix}0&&&&&\\\ &0&&&&\\\
&&\lambda_{3}&&&\\\ &&&\lambda_{4}&&\\\ &&&&\lambda_{5}&\\\
&&&&&\lambda_{6}\end{matrix}\right],$
$\gamma_{0}=\left[\begin{matrix}\lambda_{1}&&&&&\\\ &\lambda_{2}&&&&\\\
&&0&&&\\\ &&&0&&\\\ &&&&0&\\\ &&&&&0\end{matrix}\right]$

######  Proposition 6.3.

In Example 6.2, $\operatorname{WF}^{semi}(\gamma_{-1}+\gamma_{0})$ contains of
a nilpotent orbit of Jordan type $[4,1,1]$ and another of Jordan type $[3,3]$.

This corresponds, for example, to a tame supercuspidal
$\pi\in\operatorname{Irr}(G(F))$ whose Yu datum [Yu01, §3] has $d=1$,
$\vec{G}=(H,G)$ where $H=Z_{G}(\gamma_{-1})$, and $r_{0}=r_{1}=1$. That is,
$\operatorname{WF}(\pi)$ consists of a nilpotent orbit of Jordan type
$[4,1,1]$ and another of Jordan type $[3,3]$.

###### Proof of Proposition 6.3.

Note that $H=Z_{G}(\gamma_{-1})$ is a quasi-split $U_{2}$ times four copies of
$U_{1}$. By the algorithm in §3, we have to begin with
$c_{H,{-1}}(\gamma_{0})$. The set $\Phi^{H}(-1)$ of $(-1)$-facets is just the
set of usual facets in the building $\mathcal{B}(H,F)$, which is the same as
in the building for $SU_{2}\cong SL_{2}$. The building has three $H(F)$-orbits
of facets: the two orbits of hyperspecial vertices and the orbit of alcoves.
Let $\\{e_{1},...,e_{6}\\}$ be the basis of our hermitian space on which we
equip the hermitian form (24). Suppose $\\{e_{1}^{\prime},e_{2}^{\prime}\\}$
is a different basis of $\operatorname{span}_{E}(e_{1},e_{2})$ such that with
respect to $\\{e_{1}^{\prime},e_{2}^{\prime},e_{3},...,e_{6}\\}$ the form (24)
becomes

(25) $\langle
x,y\rangle=\varpi_{F}x_{1}\bar{y}_{1}+\varpi_{F}x_{2}\bar{y}_{2}+x_{3}\bar{y}_{3}+x_{4}\bar{y}_{4}+x_{5}\bar{y}_{5}+\varpi_{F}x_{6}\bar{y}_{6}.$

Let $H(F)_{y\geq 0}$ be the hyperspecial subgroup in $H(F)$ that stabilizes
$\operatorname{span}_{\mathcal{O}_{E}}(e_{1},e_{2})$ and $H(F)_{z\geq 0}$ be
the hyperspecial subgroup that stabilizes
$\operatorname{span}_{\mathcal{O}_{E}}(e_{1}^{\prime},e_{2}^{\prime})$. These
are exactly the two $H(F)$-orbits of hyperspecial subgroups. Also let
$\varsigma\subset\mathcal{B}(H,F)$ be any alcove. One checks that

$c_{H,-1}(\gamma_{0})=H(F).\\{((y,-1),0),((\varsigma,-1),0),((z,-1),c_{n})\\}$

where $c_{n}\in\mathfrak{h}(F)_{z=-1}$ is the unique $H(F)_{z\geq 0}$-orbit of
non-trivial (regular) nilpotent elements. As before by $\gamma_{=-1}$ the
image of $\gamma_{-1}$ in
$Z(\mathfrak{h})(F)_{=-1}\subset\mathfrak{g}(h)_{x=-1}\subset\mathfrak{g}(F)_{x=-1}$
for any $x\in\mathcal{B}(H,F)$. Thanks to (20), we have

(26) $\begin{array}[]{l}\operatorname{WF}^{semi}(\gamma)=\text{ the set of
maximal elements in }\\\
\\{\mathcal{O}^{semi}((y,-1),N(\gamma_{-1})),\;\;\mathcal{O}^{semi}((\varsigma,-1),N(\gamma_{-1})),\;\;\mathcal{O}^{semi}((z,-1),N(\gamma_{-1}+c_{n}))\\}.\end{array}$

We have to compute the groups $\mathsf{G}_{x}^{\heartsuit}=\mathsf{G}_{x}$ for
$x=y,z$ and $x\in\varsigma$. This is an exercise about computing the embedding
of buildings $\mathcal{B}(H,F)\rightarrow\mathcal{B}(G,F)$. The result is
quite intuitive:

  1. (i)

Thanks to (24), $G(F)_{y\geq 0}$ is the parahoric subgroup of unitary
operators stabilizing $\operatorname{span}_{\mathcal{O}_{E}}(e_{1},...,e_{6})$
and
$\operatorname{span}_{\mathcal{O}_{E}}(e_{1},e_{2},...,e_{5},\varpi_{F}^{-1}e_{6})$.
The reductive quotient $\mathsf{G}_{y}\cong U_{5}\times U_{1}/_{k}$.

  2. (ii)

Thanks to (25), $G(F)_{z\geq 0}$ is the parahoric subgroup of unitary
operators stabilizing $\operatorname{span}_{\mathcal{O}_{E}}(e_{1},...,e_{6})$
and
$\operatorname{span}_{\mathcal{O}_{E}}(\varpi_{F}^{-1}e_{1},\varpi_{F}^{-1}e_{2},e_{3},e_{4},e_{5},\varpi_{F}^{-1}e_{6})$.
The reductive quotient $\mathsf{G}_{z}\cong U_{3}\times U_{3}/_{k}$.

  3. (iii)

$G(F)_{\varsigma\geq 0}$ is a parahoric subgroup whose reductive quotient is
$U_{3}\times U_{1}^{3}$.

When $x=y$, the element $\gamma_{=-1}$ is a semsimple element in
$\mathfrak{g}_{y}^{\heartsuit}(k)=\mathfrak{g}(F)_{y=-1}\cong\operatorname{Lie}(U_{5}\times
U_{1})(k)$ whose $\operatorname{Lie}U_{5}$-component has four distinct
eigenvalues and one of them has multiplicity two. We denote nilpotent
geometric orbits in $\mathfrak{g}_{y}^{\heartsuit}$ by a pair of partitions of
$5$ and $1$ respectively. For example the regular nilpotent orbit is
$([5],[1])$ while the zero orbit is $([1,1,1,1,1],[1])$. Then $N(c^{\prime})$
corresponds to $([4,1],[1])$. Under $\mathsf{G}_{y}^{\heartsuit}=U_{5}\times
U_{1}\hookrightarrow U_{6}=\mathsf{G}_{y}^{\clubsuit}$ it goes to the orbit
corresponding to $[4,1,1]$. By Proposition 5.2,
$\mathcal{O}^{semi}((y,-1),N(\gamma_{=-1}))$ has Jordan type $[4,1,1]$ as
well.

On the other hand, when $x=z$ we have that
$\gamma_{=-1}+c_{n}\in\mathfrak{g}_{z}^{\heartsuit}(k)\cong\mathfrak{u}_{3}\times\mathfrak{u}_{3}$
is regular and thus by Corollary 5.12
$N(c^{\prime})\subset\mathfrak{g}_{z}^{\heartsuit}(\bar{k})$ is again the
regular nilpotent orbit, i.e. it corresponds to the partition $([3],[3])$.
Under $\mathsf{G}_{z}^{\heartsuit}\hookrightarrow\mathsf{G}_{z}^{\clubsuit}$
which is $U_{3}\times U_{3}\hookrightarrow U_{6}$, it goes to the orbit
corresponding to $[3,3]$. By Proposition 5.2,
$\mathcal{O}^{semi}((z,-1),N(\gamma_{=-1}+c_{n}))$ has Jordan type $[3,3]$.

Lastly, as $\mathfrak{g}_{\varsigma=-1}$ has reductive quotient
$\operatorname{Lie}(U_{3}\times U_{1}^{3})$,
$\mathcal{O}^{semi}((\varsigma,-1),N(\gamma_{=-1}))$ is of type as most (in
fact equal to) $[3,1,1,1]$. By (26) we have that
$\operatorname{WF}^{semi}(\gamma)$ has an element of type $[4,1,1]$ and
another of type $[3,3]$. ∎

More generally than in Example 6.2, consider
$\gamma=\gamma_{d-1}+\gamma_{d-2}+...+\gamma_{0}+\gamma_{-1}$ where

  1. (1)

$[r_{i},r_{j}]=0$ for $-1\leq i,j\leq d-1$

  2. (2)

For $0\leq i\leq d-1$, $\gamma_{i}$ is good of depth $r_{i}\in\mathbb{Z}$ for
a sequence of integers $r_{d-1}<r_{d-2}<...<r_{0}$.

  3. (3)

$\gamma_{-1}$ is nilpotent. (It might be less distracting to start with the
case $\gamma_{-1}=0$.)

An element $\gamma$ can be written into this form iff its semisimple part lies
in the Lie algebra of an unramified subtorus. Define $G^{d}:=G$ and
inductively $G^{i}:=Z_{G^{i+1}}(\gamma_{i})$ for $i=d-1,d-2,...,0$. The
notations are chosen so that the sequence $G^{0}\subsetneq
G^{1}\subsetneq...\subsetneq G^{d}=G$ plays the same role as in [Yu01]. We
would like to compute $\operatorname{WF}^{semi}(\gamma)$. Write
$\gamma_{i}^{\dagger}:=\gamma_{i}+\gamma_{i-1}+...+\gamma_{-1}$ for
$i=-1,0,...,d-1$. Following our algorithm, for $i=0,1,...,d-1$ we should
inductively

  1. (i)

From $c_{G^{i},r_{i}}(\gamma_{i-1}^{\dagger})$ we deduce
$c_{G^{i+1},r_{i}}(\gamma_{i}^{\dagger})$ via §3.

  2. (ii)

From $c_{G^{i+1},r_{i}}(\gamma_{i}^{\dagger})$ we deduce
$c_{G^{i+1},r_{i}-\epsilon}(\gamma_{i}^{\dagger})$ as in §5.3.

  3. (iii)

Use the algorithm in §4 to further compute
$c_{G^{i+1},r_{i+1}}(\gamma_{i}^{\dagger})$. This means we backtrack possible
paths in the graph $C_{G^{i+1}}$. (This step is not needed for $i=d-1$ though
it does not harm too.)

For step (ii), Corollary 5.14 offers an upper bound for
$c_{G^{i+1},r_{i}-\epsilon}(\gamma_{i}^{\dagger})$ for which Corollary 5.21
offers some similar (but different) existence result. On the other hand, in
step (iii) we have nilpotent $c^{\prime}$ which can be simpler. However, it
involves tracing paths in $C_{G^{i+1}}$ from depth $r_{i}-\epsilon$ to depth
$r_{i+1}$, hence the “combinatorial problem.”

We propose an estimate to the result of step (ii) and (iii) altogether. In
step (ii), we already obtain the semi-unramified wave-front set
$\operatorname{WF}^{semi,G^{i+1}}(\gamma_{i}^{\dagger})$ thanks to Theorem
5.20 and Appendix B. Then, Proposition 2.3 can be used backward to bound
$c_{G^{i+1},r_{i+1}}(\gamma_{i}^{\dagger})$. That is

(27)
$c_{G^{i+1},r_{i+1}}(\gamma_{i}^{\dagger})\subset\\{(\sigma,c)\;|\;\sigma\in\Phi^{G^{i+1}}(r_{i+1}),\;\mathcal{O}^{semi}(\sigma)(\sigma,c)\leq\mathcal{O}\text{
for some
}\mathcal{O}\in\operatorname{WF}^{semi,G^{i+1}}(\gamma_{i}^{\dagger})\\}$

We should have plugged $c_{G^{i+1},r_{i+1}}(\gamma_{i}^{\dagger})$ to step (i)
for $i=i+1$ and continue. Had we instead plug the RHS of (27), we would get a
potentially larger $c_{G^{i+2},r_{i+1}}(\gamma_{i+1}^{\dagger})$. We can
continue the algorithm, using (27) to replace step (ii) and (iii) for each
$i=0,...,d-2$. By doing so we eventually get an upper bound for
$\operatorname{WF}^{semi,G^{d}}(\gamma_{d-1}^{\dagger})=\operatorname{WF}^{semi,G}(\gamma)$.
We propose that

######  Conjecture 6.4.

This upper bound is always equal to $\operatorname{WF}^{semi,G}(\gamma)$.

The rather few reasons we have to believe Conjecture 6.4 is that it is true in
Example 6.2 and a few similar examples, and some vague intuition from the
geometry of affine Springer fibers (which a large part of our algorithm from
Waldspurger can be used to study too; the idea was discussed in [Tsa15], and
many improvements and results in our §4 and §5 applies to the method there).
Conjecture 6.4 is very close to saying that

$\operatorname{WF}^{semi,G}(\gamma)=\operatorname{WF}^{semi,G}(\gamma_{d-1}+\gamma_{d-2}^{\dagger})\text{
is determined by }\gamma_{d-1}\text{ and
}\operatorname{WF}^{semi,G^{d-1}}(\gamma_{d-2}^{\dagger}).$

However, as discussed in the introduction, this is not true at least without
the assumption that the semisimple part $\gamma$ lies in an unramified torus;
see Example 6.6.

######  Remark 6.5.

The example in [Tsa24] and Example 6.2 are sort of two extreme examples for
which $\operatorname{WF}(\pi)$ contains two elements whose geometric orbits
are incomparable. In the case of [Tsa24], our element $\gamma$ is what is
denoted by $\tilde{A}$ in [§4.1, op. cit.]. It is a good element of depth
$-\frac{1}{2}$ with $H=Z_{G}(\gamma)$ an anisotropic torus so that the
discussion in Example 6.1 applies and the only problem is the geometric
problem. On the other hand, in Example 6.2 the geometric problem is much
simpler in the sense that the grading $\rho$ is trivial, and one might say
that the complication of Example 6.2 is combinatorial. At the time of writing
[Tsa24] the author had expected, apparently wrongly, that the geometric
problem is the only source for the geometric wave-front set to be not a
singleton. Maybe a correct philosophy is that “a random complicated enough
(supercuspidal) representation has a non-singleton geometric wave-front set.”

######  Example 6.6.

Let $E/F$ be a ramified quadratic extension with common residue field
$k=\mathbb{F}_{23}$ and $G=U_{7}/_{F}$ splitting over $E$ be given by the
hermitian form

(28) $\langle
x,y\rangle=x_{1}\bar{y}_{7}+...+x_{7}\bar{y}_{1},\;x_{i},y_{j}\in E$

where $\bar{y}$ denotes the $E/F$-conjugate of $y$. Fix $\varpi_{E}\in E$ an
uniformizer with $\bar{\varpi}_{E}=-\varpi_{E}$. Consider
$\gamma=\gamma_{0}+\gamma_{1/2}\in\mathfrak{g}(F)$ with

$\gamma_{0}=\left[\begin{matrix}0&&\tilde{5}\varpi_{E}\\\ &\mathbf{0}&\\\
\varpi_{E}^{-1}&&0\end{matrix}\right]$

where $\mathbf{0}$ in the middle indicates a $5\times 5$ zero matrix, and
$\tilde{5}\in F^{\times}$ is any element whose image in $k=\mathbb{F}_{23}$ is
$5$, or really any non-square. Also

(29) $\gamma_{1/2}=\varpi_{E}\cdot\left[\begin{matrix}0&&&&&&0\\\ &0&3&&1&&\\\
&1&0&&&1&\\\ &&1&0&&&\\\ &&&1&0&3&\\\ &&&&1&0&\\\
0&&&&&&0\end{matrix}\right],\;\;\gamma_{1/2}^{\prime}=\varpi_{E}\cdot\left[\begin{matrix}0&&&&&&0\\\
&0&1&&1&&\\\ &1&0&&&1&\\\ &&1&0&&&\\\ &&&1&0&1&\\\ &&&&1&0&\\\
0&&&&&&0\end{matrix}\right].$

The centralizer $H:=Z_{G}(\gamma_{0})$ is the product of a $U_{5}$ (again
splits over $E$) and an anisotropic torus $T$. The torus $T$ has rank $2$ and
its reductive quotient $\mathsf{T}/_{k}$ is non-split and of rank $1$. There
is a special vertex $y\in\mathcal{B}(H,F)$ for which

  1. (1)

The stabilizer of $y$ in $U_{5}(F)$ is the stabilizer of the lattice
$\mathcal{O}_{E}^{5}$,

  2. (2)

$y$ is the unique point in
$\mathcal{B}(Z_{H}(\gamma_{1/2}),F)\hookrightarrow\mathcal{B}(H,F)$,

  3. (3)

$\mathsf{H}_{y}\cong SO_{5}\times\mathsf{T}$.

From this we see that there exists $(y,c_{y})\in c_{H,0}(\gamma_{1/2})\cap
c_{H,0}(\gamma_{1/2}^{\prime})$ such that
$\mathcal{O}(y,c_{y})\in\operatorname{Lie}U_{5}$ has Jordan type $[5]$;
indeed, this comes from that for the $U_{5}$-part we have

$\operatorname{Ad}(\left[\begin{matrix}\varpi_{E}^{2}&&&&\\\ &\varpi_{E}&&&\\\
&&1&&\\\ &&&-\varpi_{E}^{-1}&\\\
&&&&\varpi_{E}^{-2}\end{matrix}\right])\gamma_{1/2}=\left[\begin{matrix}0&&&&\\\
1&0&&&\\\ &1&0&&\\\ &&-1&0&\\\ &&&-1&0\end{matrix}\right]+R$

where $R\in\operatorname{Lie}U_{5}(F)$ has all its entries in
$\mathfrak{m}_{E}=\varpi_{E}\mathcal{O}_{E}$. Consequently
$\operatorname{WF}^{H}(\gamma_{1/2})$ and
$\operatorname{WF}^{H}(\gamma_{1/2})$ are both the singleton set of the
regular nilpotent orbit in $U_{5}$. On the other hand, $\mathcal{B}(H,F)$ has
another special vertex $z$ for which $\mathsf{H}_{z}\cong
Sp_{4}\times\mathsf{T}$. For any regular nilpotent
$c_{z}\in\operatorname{Lie}\mathsf{H}_{z}(k)=\mathfrak{h}(F)_{z=0}$, we have
that $\mathcal{O}(z,c_{z})\in\operatorname{Lie}U_{5}$ has Jordan type $[4,1]$.
The highlight is that

######  Proposition 6.7.

For any regular nilpotent
$c_{z}\in\operatorname{Lie}\mathsf{H}_{z}(k)=\mathfrak{h}(F)_{z=0}$, we have
$(z,c_{z})\not\in c_{H,0}(\gamma_{1/2})$ while $(z,c_{z})\in
c_{H,0}(\gamma_{1/2}^{\prime})$

With the proposition we can compute $\operatorname{WF}^{semi}(\gamma)$ using
(20). We outline the result:

  1. (1)

$y$ is a vertex in $\mathcal{B}(G,F)$ with $\mathsf{G}_{y}\cong SO_{5}\times
Sp_{2}$ so that $\mathsf{H}_{y}\hookrightarrow\mathsf{G}_{y}$ is
$SO_{5}\times\mathsf{T}\hookrightarrow SO_{5}\times Sp_{2}$ given by
$\mathsf{T}\hookrightarrow Sp_{2}$.

  2. (2)

$z$ is a vertex in $\mathcal{B}(G,F)$ with $\mathsf{G}_{z}\cong SO_{1}\times
Sp_{6}$ so that $\mathsf{H}_{z}\hookrightarrow\mathsf{G}_{z}$ is
$Sp_{4}\times\mathsf{T}\hookrightarrow Sp_{6}$ (as a twisted Levi subgroup).

Thus if $(z,c_{z})\in c_{H,0}(\gamma_{1/2})$,
$\mathcal{O}^{semi}((z,0),N(\gamma_{=0}+c_{z}))$ will be an orbit of type
$[6,1]$. Meanwhile $\mathcal{O}(y,c_{y})$ contributes an orbit in
$\mathfrak{g}(F)$ of Jordan type $[5,2]$. All other facets also contribute
orbits of type $<[5,2]$. Thus despite that
$\operatorname{WF}^{H}(\gamma_{1/2})=\operatorname{WF}^{H}(\gamma_{1/2}^{\prime})$
are both regular, $\gamma_{1/2}$ somewhat misses a subregular contribution,
resulting in that $\operatorname{WF}^{ur,G}(\gamma_{0}+\gamma_{1/2})$ is
smaller than $\operatorname{WF}^{ur,G}(\gamma_{0}+\gamma_{1/2}^{\prime})$. It
remains to prove Proposition 6.7.

###### Proof of Proposition 6.7.

Suppose on the contrary that $(z,c_{z})\in c_{H,0}(\gamma_{1/2})$. By our
algorithm, namely Theorem 4.7, there exists a path

(30)
$(\sigma_{0},c_{0})\rightarrow(\sigma_{1},c_{1})\rightarrow...(\sigma_{\ell-1},c_{\ell-1})\rightarrow(\sigma_{\ell},c_{\ell})$

in $C_{H}$ where $(\sigma_{0},c_{0})=(z,c_{z})$, $(\sigma_{\ell},c_{\ell})\in
c_{H,\frac{1}{2}}(\gamma_{1/2})$, and the arrows indicate directed edges in
$C_{H}$. Note that $c_{0},...,c_{\ell-1}$ has to be nilpotent while $c_{\ell}$
will be the image of $\gamma_{1/2}$ and is necessarily non-nilpotent. We
compute these $\sigma_{i}$ explicitly. Let $S\subset U_{5}\subset H$ be the
diagonal torus. To be precise, $U_{5}(F)$ is the group of unitary operators on
$E^{5}$ equipped with the hermitian form

(31) $\langle
x,y\rangle=x_{1}\bar{y}_{5}+...+x_{5}\bar{y}_{1},\;x_{i},y_{j}\in E.$

Let $S\subset U_{5}$ be the diagonal maximal $F$-split torus, i.e.

$S(F)=\\{\left[\begin{matrix}t_{1}&&&&\\\ &t_{2}&&&\\\ &&1&&\\\
&&&t_{2}^{-1}&\\\ &&&&t_{1}^{-1}\end{matrix}\right]\;|\;t_{1},t_{2}\in
F^{\times}\\}.$

Identify $X_{*}(S)\cong\mathbb{Z}^{2}$ with standard basis $e_{1},e_{2}\in
X_{*}(S)$ such that

$e_{1}(t)=\left[\begin{matrix}t&&&&\\\ &1&&&\\\ &&1&&\\\ &&&1&\\\
&&&&t^{-1}\end{matrix}\right],\;\;e_{2}(t)=\left[\begin{matrix}1&&&&\\\
&t&&&\\\ &&1&&\\\ &&&t^{-1}&\\\ &&&&1\end{matrix}\right].$

From now on we pretend that $H=U_{5}$ instead of $U_{5}\times T$, as the
anisotropic factor $T$ will play no role. We identify the apartment
corresponding to $S$ as $\mathcal{A}\cong\mathbb{R}^{2}$, with the origin
$(0,0)=y$, the point whose stabilizer in $U_{5}(F)$ is the stabilizer of
$\mathcal{O}_{E}^{5}\subset E^{5}$. At the same time, one can verify that that
$H(F)$-orbit of $z$ is the $H(F)$-orbit of
$(\frac{3}{4},\frac{1}{4})\in\mathcal{A}$. Without loss of generality we
suppose $z=(\frac{3}{4},\frac{1}{4})$. We have

$\mathfrak{h}(F)_{z=0}=\\{\left[\begin{matrix}a_{5}&a_{3}&0&a_{2}&a_{1}\\\
a_{7}&a_{6}&0&a_{4}&-a_{2}\\\ 0&0&0&0&0\\\ a_{9}&a_{8}&0&-a_{6}&a_{3}\\\
a_{10}&-a_{9}&0&a_{7}&-a_{5}\end{matrix}\right]\;|\;a_{1}\in\mathfrak{m}_{E}^{-3}/\mathfrak{m}_{E}^{-2},\;a_{2}\in\mathfrak{m}_{E}^{-2}/\mathfrak{m}_{E}^{-1},\;a_{3},a_{4}\in\mathfrak{m}_{E}^{-1}/\mathcal{O}_{E},$
$a_{5},a_{6}\in\mathcal{O}_{E}/\mathfrak{m}_{E},\;a_{7},a_{8}\in\mathfrak{m}_{E}/\mathfrak{m}_{E}^{2},\;a_{9}\in\mathfrak{m}_{E}^{2}/\mathfrak{m}_{E}^{3},\;a_{10}\in\mathfrak{m}_{E}^{3}/\mathfrak{m}_{E}^{4}\\}$

Any regular nilpotent $c_{z}\in\mathfrak{h}(F)_{z=0}$ is conjugate under
$\mathsf{H}_{z}(k)$ to an element of the form

(32) $c_{a}:=\left[\begin{matrix}0&&&&\\\ \varpi_{E}&0&&&\\\ &&0&&\\\
&a&&0&\\\
&&&\varpi_{E}&0\end{matrix}\right],\;a\in\mathfrak{m}_{E}/\mathfrak{m}_{E}^{2}.$

Let $\tilde{\lambda}\in X_{*}(S)$ be $\tilde{\lambda}=(-3,-1)$, and
$\lambda\in X_{*}(\mathsf{S})$ the corresponding cocharacter to the reductive
quotient. Then
$c_{a}\in{}^{\lambda}_{2}\mathfrak{g}(F)_{(\frac{3}{4}-3s,\frac{1}{4}-s)=2s}$
for any $s\in\mathbb{R}$.

######  Lemma 6.8.

In (30), we must have $\ell=10$. For each $0\leq i\leq\ell$, up to
$H(F)$-conjugate we have $\sigma_{i}$ is given by the table below and
$c_{i}=c_{a}$ for some
$a\in\varpi_{E}\mathcal{O}_{E}/\varpi_{E}^{2}\mathcal{O}_{E}$ if $i<\ell$.

$0\leq i\leq 10$ | $\sigma_{i}$
---|---
$0$ |  $\sigma_{0}=\\{z\\}$, $z:=((\frac{3}{4},\frac{1}{4}),0)$
$1$ | $\\{((\frac{3}{4}-3s,\frac{1}{4}-s),2s)\;|\;0<s<\frac{1}{20}\\}$
$2$ |  $\sigma_{2}=\\{x_{2}\\}$, $x_{2}:=((\frac{3}{5},\frac{1}{5}),\frac{1}{10})$
$3$ | $\\{((\frac{3}{4}-3s,\frac{1}{4}-s),2s)\;|\;\frac{1}{20}<s<\frac{1}{12}\\}$
$4$ |  $\sigma_{4}=\\{x_{4}\\}$, $x_{4}:=((\frac{1}{2},\frac{1}{6}),\frac{1}{6})$
$5$ | $\\{((\frac{3}{4}-3s,\frac{1}{4}-s),2s)\;|\;\frac{1}{12}<s<\frac{1}{18}\\}$
$6$ |  $\sigma_{6}=\\{x_{6}\\}$, $x_{6}:=((\frac{3}{8},\frac{1}{8}),\frac{1}{4})$
$7$ | $\\{((\frac{3}{4}-3s,\frac{1}{4}-s),2s)\;|\;\frac{1}{8}<s<\frac{3}{20}\\}$
$8$ |  $\sigma_{8}=\\{x_{8}\\}$, $x_{8}:=((\frac{3}{10},\frac{1}{10}),\frac{3}{10})$
$9$ | $\\{((\frac{3}{4}-3s,\frac{1}{4}-s),2s)\;|\;\frac{3}{20}<s<\frac{1}{4}\\}$
$10$ |  $\sigma_{10}=\\{y\\}$, $y=((0,0),\frac{1}{2})$

###### Proof.

We prove the lemma step by step, i.e. by induction on $i$. The case $i=0$ is
given. For each odd $i$, we are in the situation of (11) analyzed in §5.2,
where $\sigma=\sigma_{i-1}$, one computes that $\sigma^{\dagger}=\sigma_{i}$
is the same as in the above table. Also the construction gives
$c_{i}=c_{i-1}$.

For even $i$, we are in the situation of (10), for which
$\sigma^{\prime}=\sigma_{i}=\\{x_{i}\\}$ in the above table is the unique
augmented facet below $\sigma=\sigma_{i-1}$ and thus the correct one. The
challenge is that a priori we have many $c^{\prime}$ in (10). For $i=2,4,6,8$,
we have to show that every $c^{\prime}\in
c_{a}+\mathfrak{h}(F)_{>\sigma_{i-1}}/\mathfrak{h}(F)_{>\sigma_{i}}$ is either
in $\operatorname{Ad}(\mathsf{G}_{x_{i}}(k))c_{a}$ or non-nilpotent (which we
can discard). A useful input is Corollary 5.9, which asserts that
$\mathcal{O}(\sigma_{i},c_{i})$ has to be increasing, and when
$\mathcal{O}(\sigma_{i-1},c_{i-1})=\mathcal{O}(\sigma_{i},c_{i})$ we must have
$c_{i}=c_{i-1}$ up to conjugate. Since by induction
$\mathcal{O}(\sigma_{i-1},c_{i})=\mathcal{O}(\sigma_{i-1},c_{a})$ has Jordan
type $[4,1]$, we only have to rule out the case when
$\mathcal{O}(\sigma_{i},c_{i})$ might have Jordan type $[5]$, which by
Proposition 5.2 must also be the Jordan type of $c_{i}$ itself.

The rest is direct computation. For $i=4,6$, the Moy-Prasad quotient
$\mathfrak{h}(F)_{\sigma_{i}}$ simply has no nilpotent element of Jordan type
$[5]$, so we have nothing left to rule out. For $i=2,8$ there is. But in both
cases the Moy-Prasad quotient looks like (using $i=2$ as example)

$\mathfrak{h}(F)_{\sigma_{2}}=\\{\left[\begin{matrix}&&b&&\\\ a_{7}&&&&\\\
&&&&b\\\ &a_{8}&&&\\\
&&&a_{7}&\end{matrix}\right]\;|\;a_{7},a_{8}\in\mathfrak{m}_{E}/\mathfrak{m}_{E}^{2},\;b\in\mathfrak{m}_{E}^{-1}/\mathcal{O}_{E}\\}$

and $\mathfrak{h}(F)_{>\sigma_{1}}/\mathfrak{h}(F)_{>\sigma_{2}}$ is the
“$b$-part,” i.e. those in $\mathfrak{h}(F)_{\sigma_{2}}$ with $a_{7}=a_{8}=0$.
With $c_{a}$ as in (32), we see that $c^{\prime}\in
c_{a}+\mathfrak{h}(F)_{>\sigma_{1}}/\mathfrak{h}(F)_{>\sigma_{2}}$ is either
$c_{a}$ itself (which has type $[4,1]$) or non-nilpotent, in particular never
nilpotent of type $[5]$. This finishes the proof. ∎

We continue the proof of Proposition 6.7. Thanks to the Lemma, it remains to
look at the edge $(\sigma_{9},c_{a})\rightarrow(\sigma_{10},c_{10})$. In
$\sigma_{10}=(y,\frac{1}{4})$. The question is whether
$c_{a}+\mathfrak{h}(F)_{>\sigma_{9}}/\mathfrak{h}(F)_{>\sigma_{10}}$ contains
an element in $\operatorname{Ad}(\mathsf{H}_{y}(k))\gamma_{=1/2}$ or
$\operatorname{Ad}(\mathsf{H}_{y}(k))\gamma_{=1/2}$, where $\gamma_{=1/2}$
(resp. $\gamma_{=1/2}$) is the reduction of $\gamma_{1/2}$ (resp.
$\gamma_{1/2}^{\prime}$) in $\mathfrak{h}(F)_{y=\frac{1}{2}}$. One verifies
that
$\mathfrak{h}(F)_{>\sigma_{9}}/\mathfrak{h}(F)_{>\sigma_{10}}\subset\mathfrak{h}(F)_{y=1/2}$
is of the form

$\\{\left[\begin{matrix}b_{2}&b_{5}&b_{7}&b_{9}&b_{10}\\\
&b_{3}&b_{6}&b_{8}&b_{9}\\\ &b_{1}&b_{4}&b_{6}&b_{7}\\\ &&b_{1}&b_{3}&b_{5}\\\
&&&&b_{2}\end{matrix}\right]\;|\;b_{i}\in\varpi_{E}\mathcal{O}_{E}/\varpi^{2}\mathcal{O}_{E}\\}$

With (32), to say $\operatorname{Ad}(\mathsf{H}_{y}(k))\gamma_{=1/2}$ does not
intersect $c_{a}+\mathfrak{h}(F)_{>\sigma_{9}}/\mathfrak{h}(F)_{>\sigma_{10}}$
is to say that the affine variety

(33) $\\{\bar{g}\in
SO_{5}\;|\;\operatorname{Ad}(\bar{g})\left[\begin{matrix}0&3&&1&\\\ 1&0&&&1\\\
&1&0&&\\\ &&1&0&3\\\
&&&1&0\end{matrix}\right]\in\left[\begin{matrix}*&*&*&*&*\\\ \not=0&*&*&*&*\\\
0&*&*&*&*\\\ 0&\not=0&*&*&*\\\ 0&0&0&\not=0&*\end{matrix}\right]\\}$

has no $k$-point (resp. has a $k$-point with $3$ replaced by $1$ in (33)).
Here $*$ means that the entry can be anything and $\not=0$ means that the
entry has to be non-zero. This can really be viewed as a subvariety $SO_{5}/B$
as in [Tsa24, App. A]; in fact it’s an affine curve in $SO_{5}/B$. That it
doesn’t have $k$-point (or it has) can be checked by a computer program as in
loc. cit.. This finishes the analysis of Example 6.6. ∎

##  7\. Wave-front sets of representations

######  Remark 7.1.

Let us sketch the relation between $\operatorname{WF}(\gamma)$ and
$\operatorname{WF}(\pi)$ for regular supercuspidal representations $\pi$
[Kal19]. Essentially the result is in [KM06], but maybe the best reference is
[FKS21, Thm. 4.3.5]. Consider any $\gamma\in\mathfrak{g}^{rs}(F)$ that is
regular semisimple and elliptic (i.e. $Z_{G}(\gamma)/Z(G)$ is anisotropic).
Say $S=Z_{G}(\gamma)$. We assume (beyond Appendix A) that the exponential map
converges on $\mathfrak{t}(F)_{>r}$ giving $\mathfrak{t}(F)_{>r}\cong
S(F)_{>r}$ for some $r>0$, and for any $\alpha\in\Phi(G,S)$ (over $F^{s}$) we
have $\operatorname{val}(d\alpha(\gamma))<-r$ so that [FKS21, Thm. 4.3.5]
applies222To the best of my knowledge, there is a typo in the statement of
[FKS21, Thm. 4.3.5] that the condition $C^{\leq 0}_{G}(X)=S$ in the language
of [Spi21, §4] is missing. This should not affect their result nor our result;
with large enough $p$ that condition for us is
$\operatorname{val}(d\alpha(\gamma))\leq 0$. with their $X$ being our
$\gamma$. We have a character

$\Theta_{s,\gamma}:S(F)_{>s}\rightarrow\mathbb{C}^{\times},\;\;t\mapsto\psi(B(\log(t),\gamma))$

where $B(\cdot,\cdot)$ is a $G$-invariant non-degenerate symmetric bilinear
form normalized as in [DK06, §2.5] and $\psi:F\rightarrow\mathbb{C}^{\times}$
an additive character trivial on $\mathfrak{m}_{F}$ but non-trivial on
$\mathcal{O}_{F}$. Suppose $\theta:S(F)\rightarrow\mathbb{C}^{\times}$ is any
character extending $\theta_{s,\gamma}$. Then any such $\theta$ is regular in
the sense that [Kal19] gives a class of regular supercuspidal representations
$\pi$. For any of them, near the identity the character $\Theta_{\pi}$ is
proportional to the Fourier transform $\hat{I}_{\gamma}$; see [FKS21, Thm.
4.3.5] where their $\gamma_{0}=\mathrm{id}$, their $J$ is our $G$, their $X$
is our $\gamma$ and their $\hat{O}^{J}_{X^{g}}=\hat{O}^{G}_{X}$ is our
$\hat{I}_{\gamma}$. This gives
$\operatorname{WF}(\pi)=\operatorname{WF}(\gamma)$.

Conversely, every regular supercuspidal $\pi$ is associated to some elliptic
$\gamma\in\mathfrak{g}(F)$ (non-unique) so that
$\operatorname{WF}(\pi)=\operatorname{WF}(\gamma)$. Hence the study of
$\operatorname{WF}(\gamma)$ for elliptic $\gamma$ is more-or-less equivalent
to the study of wave-front sets of regular supercuspidal representations. By
LLC for tori, the character $\theta$ as above gives rises to
$\varphi_{\theta}:W_{F}\rightarrow{}^{L}S$. With any $L$-embedding
${}^{L}S\hookrightarrow{}^{L}G$ constructed in [Kal19, §5] we get a Langlands
parameter for which the L-packet loc. cit. consists of regular supercuspidal
representations whose wave-front sets are given by
$\operatorname{WF}(\gamma^{\prime})$ for some $\gamma^{\prime}$ stably
conjugate to our $\gamma$ above, with each $\gamma^{\prime}$ appearing at
least once.

######  Remark 7.2.

In [CJLZ23, Thm. 1.4], Cheng Chen, Dihua Jiang, Dongwen Liu and Lei Zhang
achieved a significant step towards the conjecture [JLZ22, Conj. 1.3] that
describes the wave-front sets for representations in generic $L$-packets for
classical groups via an algorithm regarding the Langlands parameter on the
Galois side. We would love to compare the algorithm in [JLZ22] and our
algorithm in the special case for regular supercuspidal representations, and
learn from [JLZ22] how to analyze and/or simplify our algorithm.

######  Remark 7.3.

Beyond regular supercuspidal representations, [Spi21, Thm. 10.2.2] actually
reduces the character of a general supercuspidal representation $\pi$ of
$G(F)$ to the character of a depth-zero supercuspidal representation $\pi_{0}$
of $G^{0}(F)$, a twisted Levi subgroup. One can define similar invariants
$c_{G,r}(\pi)$ and $c_{G^{0},s}(\pi_{0})$ but only for
$r<-\operatorname{dep}(\pi)$ and $s<0$, namely

$c_{G,r}(\pi)=\\{(\sigma,c)\in
C_{G,r}\;|\;\Theta_{\pi}(FT[c+\mathfrak{g}(F)_{>\sigma}])>0\\}.$

Here $[c+\mathfrak{g}(F)_{>\sigma}]$ is the characteristic function of the
coset and $FT$ denotes Fourier transform. Note that
$\Theta_{\pi}(FT[c+\mathfrak{g}(F)_{>\sigma}])$ counts (up to normalization)
the multiplicity of $c$-isotypic factors in the fixed subspace $\pi^{K}$ where
$K={G(F)_{x\geq-r}}$ for $(x,r)\in\sigma$. Hence
$\Theta_{\pi}(FT[c+\mathfrak{g}(F)_{>\sigma}])$ is always non-negative. With
[Spi21, Thm. 10.2.2] and above definition, our algorithm again computes
$c_{G,r}(\pi)$ in terms of $c_{G^{0},s}(\pi_{0})$. When $s=-\epsilon$ for very
small $\epsilon>0$, $c_{G^{0},s}(\pi_{0})$ is determined by whether the
cuspidal representation of the finite reductive group behind $\pi_{0}$ appears
in the generalised Gelfand–Graev representations [Kaw85, §1.3] associated to
all nilpotent orbits, not necessarily the maximal orbits in the wave-front
set.

In general, one can hope that a similar reduction exists from general
irreducible admissible representations of $G(F)$ to depth-zero representations
of some $G^{0}(F)$ for which some generalization of [Spi21, Thm. 10.2.2]
holds. If this is indeed the case, then our algorithm could again compute
$c_{G,r}(\pi)$ in terms of $c_{G^{0},s}(\pi_{0})$.

##  Appendix A Restrictions on $p$

Throughout this article we enforce the hypotheses in [Wal06, §1.1]. In [Wal06,
App. 3] the hypotheses were shown to hold when $p>6N-1$ where $N$ is the
absolute rank and furthermore $p\geq 271$ if there is an exceptional factor.
We remark that $p>6N-1$ does not hold in Example 6.6 but in that case it’s not
hard to verify the original hypotheses in [Wal06, §1.1].

We highlight that the hypotheses ensure a number of properties:

  1. (1)

The exponential map is defined on any nilpotent element.

  2. (2)

The theory of $\mathfrak{sl}_{2}$-triples always work well; we always get a
semisimple representation of the $\mathfrak{sl}_{2}$ and a homomorphism from
the group $SL_{2}$.

  3. (3)

Any reductive group over the local field ever appears splits over a tame
extension.

  4. (4)

All important coordinates on the Bruhat-Tits building have denominator coprime
to $p$.

##  Appendix B Graded Lie algebras and Springer theory

In this appendix we use a different set of notations from the rest of this
article. Let $k$ be a finite field with a fixed algebraic closure $\bar{k}$,
again with assumption on $p$ as in Appendix A. Fix a connected reductive group
$G$ over $k$ and $\mathfrak{g}:=\operatorname{Lie}G$. Fix
$m\in\mathbb{Z}_{>0}$ invertible in $k$ and
$\rho:\mu_{m}/_{k}\rightarrow\operatorname{Aut}(G)$ be a homomorphism of
algebraic groups. For $a\in\mathbb{Z}/m$, we denote by $\mathfrak{g}_{a}$ the
$k$-subspace of $\mathfrak{g}=\operatorname{Lie}G$ on which $\rho$ acts with
weight $a$; they were denoted ${}^{\rho}_{a}\mathfrak{g}$ in §5, but
thankfully we will have less confusing subscripts in this appendix. Let also
$G_{0}=(G^{\rho(\mu_{m})})^{o}$. Let $\lambda:\mathbb{G}_{m}\rightarrow G_{0}$
be any cocharacter. Fix $\ell\in\mathbb{Z}$, $a\in\mathbb{Z}/m$, and a
nilpotent element $c\in{}^{\lambda}_{\ell}\mathfrak{g}_{a}(k)$. For any
$i\in\mathbb{Z}$, $i<\ell$, choose
$V_{i}\subset{}^{\lambda}_{i}\mathfrak{g}_{a}$ a $k$-subspace complementary to
$[c,{}^{\lambda}_{i-\ell}\mathfrak{g}_{0}]$. Let $V=\bigoplus_{i<\ell}V_{i}$.

######  Lemma B.1.

For every $c^{\prime}\in c+V$ we have that $c$ is in the Zariski closure of
$\bar{k}\cdot\operatorname{Ad}(G_{0}(\bar{k}))c^{\prime}$. If moreover $c$ is
in $\operatorname{Ad}(G(\bar{k}))c^{\prime}$, then $c^{\prime}=c$.

###### Proof.

Consider the $\mathbb{G}_{m}$-action on $c+V$ given by
$t.u=t^{-\ell}\operatorname{Ad}(\lambda(t))u$ for $t\in\bar{k}^{\times},u\in
c+V$. This $\mathbb{G}_{m}$-action retracts $c+V$ to $\\{c\\}$ and preserves
$\bar{k}\cdot\operatorname{Ad}(G_{0}(\bar{k}))c^{\prime}$. Hence if
$c^{\prime}\in c+V$ then $c$ is in the Zariski closure of
$\bar{k}\cdot\operatorname{Ad}(G_{0}(\bar{k}))c^{\prime}$. It remains to
consider the case $c\in\operatorname{Ad}(G(\bar{k}))c^{\prime}$, i.e.
$c^{\prime}\in\operatorname{Ad}(G(\bar{k}))c\cap(c+V)$. Then we also have
$t^{-\ell}\operatorname{Ad}(\lambda(t))c^{\prime}\in\operatorname{Ad}(G(\bar{k}))c\cap(c+V)$
for any $t\in\bar{k}^{\times}$. Suppose on the contrary $c^{\prime}\not=c$.
Then the Zariski closure of the infinite set

$\\{t^{-\ell}\operatorname{Ad}(\lambda(t))c^{\prime}\;|\;t\in\bar{k}^{\times}\\}$

is a positive dimensional subvariety of the Zariski closure of
$\operatorname{Ad}(G(\bar{k}))c\cap(c+V)$, which is
$\operatorname{Ad}(G(\bar{k}))c\cap(c+V)$ itself because any nilpotent element
in $c+V$ lives in an equal or larger orbit. The existence of this positive
dimensional subvariety of $\operatorname{Ad}(G(\bar{k}))c\cap(c+V)$ shows that
the tangent space $T_{c}(\operatorname{Ad}(G(\bar{k}))c)$ and $T_{c}(c+V)=V$
has non-trivial intersection. But
$T_{c}(\operatorname{Ad}(G(\bar{k}))c)=[c,\mathfrak{g}]$ while

$[c,\mathfrak{g}]\cap V=[c,\mathfrak{g}_{0}]\cap
V=[c,{}^{\lambda}_{<0}\mathfrak{g}_{0}]\cap V=0,$

a contradiction. ∎

Denote by ${}^{\lambda}_{<0}G_{0}\subset G_{0}$ the connected unipotent
subgroup whose Lie algebra is ${}^{\lambda}_{<0}\mathfrak{g}_{0}$.

######  Lemma B.2.

Suppose $c^{\prime}\in c+{}^{\lambda}_{<\ell}\mathfrak{g}_{a}(k)$. Then $c$ is
in the Zariski closure of
$\bar{k}\cdot\operatorname{Ad}(G_{0}(\bar{k}))c^{\prime}$. If furthermore
$c^{\prime}\in\operatorname{Ad}(G(\bar{k}))c$, then
$c^{\prime}\in\operatorname{Ad}({}^{\lambda}_{<0}G_{0}(k))c$.

###### Proof.

We claim that any element in $c+{}^{\lambda}_{<\ell}\mathfrak{g}_{a}$ is in
the same $\operatorname{Ad}({}^{\lambda}_{<0}G_{0}(k))$-orbit as some element
in $c+V$. The claim together with Lemma B.1 finishes the proof. We use
induction for $j\in\mathbb{Z}_{\geq 1}$ on the statement that any element in
$c+{}^{\lambda}_{<\ell}\mathfrak{g}_{a}$ is in the same
$\operatorname{Ad}({}^{\lambda}_{<0}G_{0}(k))$-orbit as some element in
$c+{}^{\lambda}_{\leq\ell-j}\mathfrak{g}_{a}+V$. For $j$ large enough this
proves the lemma. The statement is trivially true for $j=1$. For a general
element in $c+{}^{\lambda}_{<\ell}\mathfrak{g}_{a}$ for which we already
conjugate it by ${}^{\lambda}_{<0}G_{0}(k)$ to $c_{j}\in
c+{}^{\lambda}_{\leq\ell-j}\mathfrak{g}_{a}+V$, we would like to conjugate it
again so that it lives in $c+{}^{\lambda}_{\leq\ell-j-1}\mathfrak{g}_{a}+V$.
Denote by ${}^{\lambda}_{\ell-j}(c_{j})$ the projection of $c_{j}$ from
$\mathfrak{g}_{a}=\bigoplus{}^{\lambda}_{i}\mathfrak{g}_{a}$ to
${}^{\lambda}_{\ell-j}\mathfrak{g}_{a}$. By the construction of $V_{\ell-j}$,
there exists an $y\in{}^{\lambda}_{-j}\mathfrak{g}_{0}(k)$ such that
${}^{\lambda}_{\ell-j}(c_{j})+[y,c]\in
V_{\ell-j}\subset{}^{\lambda}_{\ell-j}\mathfrak{g}_{a}$. This implies that
$\exp(y)c_{j}\in c+{}^{\lambda}_{\leq\ell-j-1}\mathfrak{g}_{a}+V$, as desired.
∎

###  B.1. Existence of rational points

From now on until Proposition B.13, we will not make use of the action
$\rho:\mu_{m}\rightarrow\operatorname{Aut}(G)$. Consider $x\in\mathfrak{g}(k)$
with the Jordan decomposition $x=x_{s}+x_{n}$. We say $x$ splits over a field
extension $k^{\prime}/k$ if $L:=Z_{G}(x_{s})$ is a Levi subgroup over
$k^{\prime}$, i.e. $L$ is contained in a parabolic subgroup $P\subset G$
defined over $k^{\prime}$, and just say $x$ splits if it does so over $k$.
Apparently $x$ splits over $\bar{k}$. Let $P$ be as above (defined over
$\bar{k}$) and $\mathfrak{u}_{P}$ be the Lie algebra of its unipotent radical.

######  Definition B.3.

[LS79] We denote by $N(x)$ unique nilpotent
$\operatorname{Ad}(G(\bar{k}))$-orbit whose intersection with
$x_{n}+\mathfrak{u}_{P}(\bar{k})$ is a dense open subset of the latter.

The orbit $N(x)$ is the Lusztig-Spaltenstein induction of $x_{n}$ from $L$ to
$G$. Following [CO23, §4], consider the variety $\Theta(x)$ of
$\mathfrak{sl}_{2}$-triples $(c,h,d)\in\mathfrak{g}(\bar{k})^{3}$ such that

  1. (1)

$c,d\in N(x)$, and

  2. (2)

$x\in c+Z_{\mathfrak{g}}(d)$ is in the so-called Slodowy slice.

The first goal of this section is to show the following generalization of
[CO23, Lemma 4.5].

######  Theorem B.4.

The connected centralizer $Z_{G}(x)^{o}$ acts transitively on $\Theta(x)$.
Hence $\Theta(x)(k)$ is non-empty.

######  Remark B.5.

The second sentence of the theorem follows from the first thanks to Lang’s
theorem. The result is essentially that every element is good in the sense of
[CO23, §4]. It also gives a different proof of [Kat82, Thm. 0.3] (also used in
[CO23, §4]), which is equivalent to that $\Theta(x)$ is a $Z_{G}(x)$-orbit.

Let $C(\mathfrak{g}(k))$ be the space of $\mathbb{C}$-valued functions on
$\mathfrak{g}(k)$. We fix a $G(k)$-invariant non-degenerate symmetric bilinear
form $B(\cdot,\cdot):\mathfrak{g}\times\mathfrak{g}\rightarrow\mathbb{G}_{a}$
and a non-trivial additive character $\psi:k\rightarrow\mathbb{C}^{\times}$.
For $f,f^{\prime}\in C(\mathfrak{g}(k))$, write

$\langle
f,f^{\prime}\rangle:=\sum_{x\in\mathfrak{g}(k)}f(x)f^{\prime}(x),\;\;\hat{f}(y):=\sum_{x\in\mathfrak{g}(k)}\psi(B(x,y))f(x).$

Let $J(\mathfrak{g}(k))\subset C(\mathfrak{g}(k))$ be the subspace of
$\operatorname{Ad}(G(k))$-invariant functions, and
$J(\mathfrak{g}(k))^{nil}\subset J(\mathfrak{g}(k))$ be the subspace of those
supported on $\mathfrak{g}^{nil}(k)$, the nilpotent cone. The Fourier
transform $f\mapsto\hat{f}$ maps $J(\mathfrak{g}(k))$ isomorphically to
itself, and let $J(\mathfrak{g}(k))^{conil}$ be the image of
$J(\mathfrak{g}(k))^{nil}$ under the Fourier transform. For any
sub-$k$-variety $X\subset\mathfrak{g}$, let $I_{X}$ be the function that takes
the value $1$ on $X(k)$ and zero elsewhere. The idea of the following
proposition comes from Springer theory, particularly [Spr76, Thm. 4.4]. See
also the MathOverflow question [Tsa].

######  Proposition B.6.

Let $x$ be split with $L=Z_{G}(x_{s})$ and $P$ a parabolic subgroup containing
$L$ and defined over $k$. We have

(34) $|U_{P}(k)|\cdot\langle I_{x},\xi\rangle=\langle
I_{x_{n}+\mathfrak{u}_{P}},\xi\rangle,\;\forall\xi\in
J(\mathfrak{g}(k))^{conil}.$

###### Proof.

We have $I_{x+\mathfrak{u}_{P}}(y)=\sum_{u\in
U_{P}(k)}I_{x}(\operatorname{Ad}(u)y)$ and thus

(35) $|\mathfrak{u}_{P}(k)|\cdot\langle I_{x},\xi\rangle=\langle
I_{x+\mathfrak{u}_{P}},\xi\rangle,\;\forall\xi\in J(\mathfrak{g}(k)).$

We compare $I_{x+\mathfrak{u}_{P}}$ and $I_{x_{n}+\mathfrak{u}_{P}}$ by
looking at Fourier transforms. The two functions are apparently invariant
under translation by $\mathfrak{u}_{P}(k)$. Thus their Fourier transforms are
supported on $\mathfrak{p}=\mathfrak{l}\oplus\mathfrak{u}_{P}$ where
$\mathfrak{p}=\operatorname{Lie}P$ and $\mathfrak{l}=\operatorname{Lie}L$.
Note that $x-x_{n}=x_{s}\in Z(\mathfrak{l})$. Since $Z(\mathfrak{l})$ is
necessarily perpendicular to $\mathfrak{l}^{der}+\mathfrak{u}_{P}$ under our
bilinear form $B(\cdot,\cdot)$, we have that $\hat{I}_{x+\mathfrak{u}_{P}}$
and $\hat{I}_{x_{n}+\mathfrak{u}_{P}}$ have the same restriction on
$\mathfrak{l}^{der}(k)+\mathfrak{u}_{P}(k)$. Since all nilpotent elements in
$\mathfrak{p}$ lies in $\mathfrak{l}^{der}(k)+\mathfrak{u}_{P}(k)$, this shows
that $\hat{I}_{x+\mathfrak{u}_{P}}$ and $\hat{I}_{x_{n}+\mathfrak{u}_{P}}$
have the same restriction to the nilpotent cone $\mathfrak{g}^{nil}(k)$. This
implies that

$\langle\hat{I}_{x+\mathfrak{u}_{P}},\xi\rangle=\langle\hat{I}_{x_{n}+\mathfrak{u}_{P}},\xi\rangle,\;\forall\xi\in
J(\mathfrak{g}(k))^{nil}.$

which by Fourier inversion is equivalent to

$\langle I_{x+\mathfrak{u}_{P}},\xi\rangle=\langle
I_{x_{n}+\mathfrak{u}_{P}},\xi\rangle,\;\forall\xi\in
J(\mathfrak{g}(k))^{conil}$

which together with (35) implies (34). ∎

By our assumption on $\operatorname{char}(k)$, for any
$\mathfrak{sl}_{2}$-triples $(c,h,d)$ there is a cocharacter
$\lambda:\mathbb{G}_{m}\rightarrow G$ associated to $(c,h,d)$ such that
$c\in{}^{\lambda}_{2}\mathfrak{g}$ (the weight $2$ subspace),
$h\in{}^{\lambda}_{0}\mathfrak{g}$ and $d\in{}^{\lambda}_{-2}\mathfrak{g}$,
and

  1. (1)

$\operatorname{ad}(c)$ maps ${}^{\lambda}_{\leq-1}\mathfrak{g}$ injectively
into ${}^{\lambda}_{\leq 1}\mathfrak{g}$.

  2. (2)

${}^{\lambda}_{\leq
1}\mathfrak{g}=Z_{\mathfrak{g}}(d)\oplus[c,{}^{\lambda}_{\leq-1}\mathfrak{g}]$.

Let ${}^{\lambda}_{\leq-1}G$ be the unipotent subgroup of $G$ whose Lie
algebra is ${}^{\lambda}_{\leq-1}\mathfrak{g}$; we can also characterize
${}^{\lambda}_{\leq-1}G$ as the unipotent subgroup of the parabolic subgroup
determined by $\lambda$. The following lemma is somewhat standard and can be
proved analogous to Lemma B.2 (with extra care to prove the uniqueness part).

######  Lemma B.7.

For every element $y\in c+{}^{\lambda}_{\leq 1}\mathfrak{g}(k)$, there is a
unique element $u\in({}^{\lambda}_{\leq-1}G)(k)$ such that
$\operatorname{Ad}(u)y\in c+Z_{\mathfrak{g}}(d)$.

For any $\mathfrak{sl}_{2}$-triples $(c,h,d)$ denote by $f_{c,h,d}\in
C(\mathfrak{g}(k))$ the function with $f_{c,h,d}(x)=\sum_{g\in
G(k)}I_{c+Z_{\mathfrak{g}}(d)}(\operatorname{Ad}(g)x)$.

######  Lemma B.8.

We have $f_{c,h,d}\in J(\mathfrak{g}(k))^{conil}$.

###### Proof.

As before let $I_{c+{}^{\lambda}_{\leq 1}\mathfrak{g}}\in C(\mathfrak{g}(k))$
be the function that takes the value $1$ on $c+{}^{\lambda}_{\leq
1}\mathfrak{g}(k)$ and $0$ elsewhere. Since $I_{c+{}^{\lambda}_{\leq
1}\mathfrak{g}}$ is invariant under translation by ${}^{\lambda}_{\leq
1}\mathfrak{g}(k)$, its Fourier transform $\hat{I}_{c+{}^{\lambda}_{\leq
1}\mathfrak{g}}$ is supported on ${}^{\lambda}_{\leq-2}\mathfrak{g}(k)$, in
particular in $\mathfrak{g}^{nil}(k)$. Lemma B.7 implies that

$f_{c,h,d}(x)=\sum_{g\in
G(k)}I_{c+Z_{\mathfrak{g}}(d)}(\operatorname{Ad}(g)x)=\frac{1}{|({}^{\lambda}_{\leq-1}G)(k)|}\sum_{g\in
G(k)}I_{c+{}^{\lambda}_{\leq 1}\mathfrak{g}}(\operatorname{Ad}(g)x).$

The Fourier transform $\hat{\xi}$ is thus a sum of conjugates of
$\hat{I}_{c+{}^{\lambda}_{\leq 1}\mathfrak{g}}$, all supported in
$\mathfrak{g}^{nil}(k)$, i.e. $\hat{f}_{c,h,d}\in J(\mathfrak{g}(k))^{nil}$
and thus $f_{c,h,d}\in J(\mathfrak{g}(k))^{conil}$. ∎

We also need more lemmas regarding $\mathfrak{sl}_{2}$-triples:

######  Lemma B.9.

The centralizer $Z_{G}(c)$ acts transitively on the set of
$\mathfrak{sl}_{2}$-triples $(c,h^{\prime},d^{\prime})$ completing $c$, and
for any such we have
$\pi_{0}(Z_{G}(c,h^{\prime},d^{\prime}))\xrightarrow{\sim}\pi_{0}(Z_{G}(c))$.

Lastly, let us recall the following which relies on a little bit of Weil
Conjectures.

######  Lemma B.10.

A variety $X$ over $k$ has dimension $\delta$ and only one geometric component
of dimension $\delta$ if and only if there exists a constant $C$ and a finite
extension $k^{\dagger}/k$ such that
$|X(k^{\prime})-|k^{\prime}|^{\delta}|<C\cdot|k^{\prime}|^{\delta-1}$ for any
finite extension $k^{\prime}/k^{\dagger}$.

For our convenience, let us denote the second condition in the lemma by
$|X(k^{\prime})|\sim|k^{\prime}|^{\delta}$. To prove Theorem B.4, we first
prove a weaker version of it:

######  Proposition B.11.

Let $c\in N(x)$. The variety $\Theta(x)$ has dimension $\delta_{c}:=\dim
Z_{G}(c)/Z_{G}(c,h,d)$ and has only one geometric component of this dimension.

###### Proof.

Consider the case $x$ is nilpotent, i.e. $x=c$. Then
$N(c)=\operatorname{Ad}(G(\bar{k}))c$. By Lemma B.1 applied to our case
without $\rho$ (i.e. $\rho$ is trivial), $\Theta(c)$ is equal to the variety
of $\mathfrak{sl}_{2}$-triples $(c,h^{\prime},d^{\prime})$ completing $c$, on
which $Z_{G}(c)$ acts transitively by Lemma B.9. Moreover, we have
$\pi_{0}(Z_{G}(c,h^{\prime},d^{\prime}))\xrightarrow{\sim}\pi_{0}(Z_{G}(c))$
and that $Z_{G}(c)$ acts transitively on $\Theta(c)$ with stabilizer at
$(c,h,d)\in\Theta(c)$ equal to $Z_{G}(c,h,d)$, i.e. we have
$Z_{G}(c)/Z_{G}(c,h,d)\cong Z_{G}(c)^{o}/Z_{G}(c,h,d)^{o}\cong\Theta(c)$.
Thanks to Lemma B.10 this proves the proposition for $x=c$, i.e.
$|\Theta(c)(k^{\prime})|\sim|k^{\prime}|^{\delta_{c}}$.

For $y\in\mathfrak{g}(k)$, denote by $\Xi_{x}(y)$ the set of
$\mathfrak{sl}_{2}$-triples $(c,h,d)$ with $c\in N(x)$ and such that $y\in
c+Z_{\mathfrak{g}}(d)$. Hence $\Theta(x)=\Xi_{x}(x)$. The $k$-points in $N(x)$
is a finite number of $\operatorname{Ad}(G(k))$-orbits, say
$c_{1},c_{2},..,c_{m}$. We can complete each of them to
$\mathfrak{sl}_{2}$-triples $(c_{i},h_{i},d_{i})$. By Lemma B.9,
$\Xi_{x}(y)(k)$ is the disjoint union of the set of
$\operatorname{Ad}(G(k))$-conjugates $(c,h,d)$ of $(c_{i},h_{i},d_{i})$ such
that $x\in c+Z_{\mathfrak{g}}(d)$. Hence for any $y\in\mathfrak{g}(k)$ we have

(36)
$|\Xi_{x}(y)(k)|=\sum_{i=1}^{m}\frac{1}{|Z_{G}(c_{i},h_{i},d_{i})(k)|}\cdot\langle
I_{y},f_{c_{i},h_{i},d_{i}}\rangle.$

For general $x$, the statement to prove is geometric and there is no harm to
perform a base change on the ground field $k$. In particular, we may and shall
assume that $x$ is split. Plug $y=x$ into (36) and apply Proposition B.6 and
Lemma B.8 to the RHS, we have

(37)
$|\mathfrak{u}_{P}(k)|\cdot\sum_{i=1}^{m}\frac{1}{|Z_{G}(c_{i},h_{i},d_{i})(k)|}\cdot\langle
I_{x},f_{c_{i},h_{i},d_{i}}\rangle=\sum_{u\in\mathfrak{u}_{P}(k)}\sum_{i=1}^{m}\frac{1}{|Z_{G}(c_{i},h_{i},d_{i})(k)|}\cdot\langle
I_{x_{n}+u},f_{c_{i},h_{i},d_{i}}\rangle.$

Using (36) for both $y=x$ and $y=x_{n}+u$, we have

$|\mathfrak{u}_{P}(k)|\cdot|\Xi_{x}(x)(k)|=\sum_{u\in\mathfrak{u}_{P}(k)}|\Xi_{x}(x_{n}+u)(k)|$

By definition of the Lusztig-Spaltenstein induction $N(x)$, the element
$x_{n}+u$ is either in $N(x)$ (for $u$ in a dense subset of
$\mathfrak{u}_{P}$) or is contained in an orbit of a smaller dimension. In the
latter case, the orbit of $x_{n}+u$ must not meet
$c_{i}+Z_{\mathfrak{g}}(d_{i})$ for any nilpotent orbit meeting
$c_{i}+Z_{\mathfrak{g}}(d_{i})$ contains $c_{i}$ in its closure, i.e.
$\Xi_{x}(x_{n}+u)=\emptyset$ when $x_{n}+u$ is in an orbit of a smaller
dimension. For the other “generic” $u$ we have
$\Xi_{x}(x_{n}+u)=\Theta(x_{n}+u)$ with
$|\Theta(x_{n}+u)(k^{\prime})|\sim|k^{\prime}|^{\delta_{c}}$. Hence we have

$|\mathfrak{u}_{P}(k^{\prime})|\cdot|\Theta(x)(k^{\prime})|\sim|k^{\prime}|^{\delta_{c}+\dim\mathfrak{u}_{P}}\implies|\Theta(x)(k^{\prime})|\sim|k^{\prime}|^{\delta_{c}},$

which proves the proposition. ∎

###### Proof of Theorem B.4.

The centralizer $Z_{G}(x)$ acts on $\Theta(x)$. Since $\dim Z_{G}(x)=\dim
Z_{G}(c)$ for $c\in N(x)$ [LS79, Thm. 1.3(a)], any $Z_{G}(x)$-orbit on
$\Theta(x)$ is a subvariety of dimension at least $\dim Z_{G}(x)-\dim
Z_{G}(c,h,d)=\delta_{c}$ (for $(c,h,d)\in\Theta(x)$). By Proposition B.11
there can be at most one such orbit, and the orbit has to be geometrically
connected, i.e. it has to be a $Z_{G}(x)^{o}$-orbit. Hence we are done. ∎

We view Theorem B.4 as saying that any $x\in\mathfrak{g}(k)$ is “supported” by
a nilpotent element in the following sense:

######  Definition B.12.

For $x\in\mathfrak{g}(k)$, $c\in\mathfrak{g}^{nil}(k)$, let us say $c$
supports $x$ if we can complete it to an $\mathfrak{sl}_{2}$-triple
$(c,h,d)\in\Theta(x)(k)$.

Let us consider the homomorphism
$\rho:\mu_{m}\rightarrow\operatorname{Aut}(G)$ again and fix
$a\in\mathbb{Z}/m$. The $G$-invariant form
$B(\cdot,\cdot):\mathfrak{g}\times\mathfrak{g}\rightarrow\mathbb{G}_{a}$
necessarily restricts to a perfect pairing
$\mathfrak{g}_{a}\times\mathfrak{g}_{-a}\rightarrow\mathbb{G}_{a}$. (Here
$\mathbb{G}_{a}$ is the standard additive group scheme and has a priori
nothing to do with $\mathfrak{g}_{a}$; we apologize for this funny notational
clash.) Let $C(\mathfrak{g}_{a}(k))$ be the set of $\mathbb{C}$-valued
functions on $\mathfrak{g}_{a}(k)$. For $f\in C(\mathfrak{g}_{a}(k))$ we put

$\hat{f}(y)=\sum_{x\in\mathfrak{g}_{a}(k)}\psi(B(x,y))f(x),\;\forall
y\in\mathfrak{g}_{-a}(k)$

so that $f\mapsto\hat{f}$ defines a $G_{0}(k)$-equivariant isomorphism
$C(\mathfrak{g}_{a}(k))\xrightarrow{\sim}C(\mathfrak{g}_{-a}(k))$. Let
$J(\mathfrak{g}_{a}(k))\subset C(\mathfrak{g}_{a}(k))$ be the subspace of
$G_{0}(k)$-invariant functions so that Fourier transform again gives
$J(\mathfrak{g}_{a}(k))\xrightarrow{\sim}J(\mathfrak{g}_{-a}(k))$. Denote
again by $J(\mathfrak{g}_{a}(k))^{nil}\subset J(\mathfrak{g}_{a}(k))$ the
subspace of those supported on nilpotent elements and
$J(\mathfrak{g}_{a}(k))^{conil}$ the image of $J(\mathfrak{g}_{-a}(k))^{nil}$
under Fourier transform (note the subscript $-a$).

######  Proposition B.13.

The map $f\mapsto f|_{\mathfrak{g}_{a}^{nil}(k)}$ induces an isomorphism
$J(\mathfrak{g}_{a}(k))^{conil}\xrightarrow{\sim}J(\mathfrak{g}_{a}(k))^{nil}$.

###### Proof.

Let $c_{1},c_{2},...$ be a set (apparently finite) of representatives of
$G_{0}(k)$-orbits in $\mathfrak{g}_{a}^{nil}(k)$, ordered so that
$\dim\operatorname{Ad}(G_{0})c_{i}\leq\dim\operatorname{Ad}(G_{0})c_{j}$ for
$i\leq j$. We complete each $c_{i}$ into an $\mathfrak{sl}_{2}$-triple
$(c_{i},h_{i},d_{i})$ with $h_{i}\in\mathfrak{g}_{0}$,
$d_{i}\in\mathfrak{g}_{-a}$. Consider $I_{c_{i}+Z_{\mathfrak{g}_{a}}(d_{i})}$
the function with value $1$ on $c_{i}+Z_{\mathfrak{g}_{a}(k)}d_{i}$ and $0$
elsewhere, and $f_{c_{i},h_{i},d_{i}}\in J(\mathfrak{g}_{a}(k))$ with
$f_{c_{i},h_{i},d_{i}}(x)=\sum_{g\in
G_{0}(k)}I_{c_{i}+Z_{\mathfrak{g}_{a}}}(\operatorname{Ad}(g)x)$. By the same
proof as in Lemma B.8 we have $f_{c_{i},h_{i},d_{i}}\in
J(\mathfrak{g}_{a}(k))^{conil}$.

By Lemma B.1,
$\operatorname{supp}(f_{c_{i},h_{i},d_{i}})\cap\mathfrak{g}_{a}^{nil}$ is the
union of $\operatorname{Ad}(G_{0}(k))c_{i}$ and some
$\operatorname{Ad}(G_{0}(k))c_{j}$ lying in larger
$\operatorname{Ad}(G_{0})$-orbits. In particular
$\operatorname{supp}(f_{c_{i},h_{i},d_{i}})$ must be disjoint from
$\operatorname{Ad}(G_{0}(k))c_{i}$ for $j<i$. This proves that
$f_{c_{i},h_{i},d_{i}}|_{\mathfrak{g}_{a}^{nil}(k)}$ form a basis of
$J(\mathfrak{g}_{a}(k))^{nil}$. In particular, the map
$J(\mathfrak{g}_{a}(k))^{conil}\rightarrow J(\mathfrak{g}_{a}(k))^{nil}$ given
by restriction is surjective. Hence it is an isomorphism as both sides have
the same dimension over the coefficient field $\mathbb{C}$. ∎

######  Corollary B.14.

Suppose a function $f\in J(\mathfrak{g}_{a}(k))^{conil}$ takes values in
$\mathbb{R}_{\geq 0}$. Suppose we have some
$c\in\operatorname{supp}(f)\cap\mathfrak{g}_{a}^{nil}$, and that
$c^{\prime}\in\operatorname{supp}(f)\cap\mathfrak{g}_{a}^{nil}$ only if
$c\in\operatorname{Ad}(G_{0}(k))c^{\prime}$ or $c$ is contained in the Zariski
boundary of $\operatorname{Ad}(G_{0}(\bar{k}))c^{\prime}$. Then there exist

  1. (1)

Nilpotent elements $c_{1},...,c_{r}$ such that $c$ is contained in the Zariski
boundary of $\operatorname{Ad}(G_{0}(\bar{k}))c_{i}$ for each $1\leq i\leq r$.
Each is completed into an $\mathfrak{sl}_{2}$-triple $(c_{i},h_{i},d_{i})$. We
also complete $c$ to $(c,h,d)$. All these $\mathfrak{sl}_{2}$-triples are
defined over $k$.

  2. (2)

Constants $\alpha\in\mathbb{R}_{>0}$ and
$\alpha_{1},...,\alpha_{r}\in\mathbb{R}$

such that

$f=\alpha\cdot f_{c,h,d}+\sum_{i=1}^{r}\alpha_{i}\cdot f_{c_{i},h_{i},d_{i}}.$

###### Proof.

This follows from the property established in the proof of Proposition B.13
that $f_{c_{i},h_{i},d_{i}}|_{\mathfrak{g}_{a}^{nil}(k)}$ is supported at
$\operatorname{Ad}(G_{0}(k))c_{i}$ and a number of
$\operatorname{Ad}(G_{0}(k))$-orbits with strictly larger geometric orbits. ∎

The point of the previous and the next corollary is that they give our
template for test functions:

######  Corollary B.15.

Let $\ell\in\mathbb{Z}_{>0}$ and $\lambda:\mathbb{G}_{m}/_{k}\rightarrow
G_{0}$ be any cocharacter. Suppose $c\in{}^{\lambda}_{\ell}\mathfrak{g}_{a}$.
Let $f_{\lambda,c}\in J(\mathfrak{g}_{a}(k))$ be given by
$f_{\lambda,c}(x)=\sum_{g\in
G_{0}(k)}I_{c+{}^{\lambda}_{<\ell}\mathfrak{g}_{a}}(\operatorname{Ad}(g)x)$.
Then $f_{\lambda,c}$ satisfies the hypotheses of Corollary B.14 and therefore
the conclusions for the same $c$.

###### Proof.

The Fourier transform of $I_{c+{}^{\lambda}_{<\ell}\mathfrak{g}_{a}}$ is
supported on ${}^{\lambda}_{\leq-\ell}\mathfrak{g}_{-a}$ and thus
$f_{\lambda,c}\in J(\mathfrak{g}_{a}(k))^{conil}$. The rest follows from Lemma
B.2. ∎

In the next corollary we go back to the case when $m=1$ and $\rho$ is trivial.

######  Corollary B.16.

Let $\ell\in\mathbb{Z}_{>0}$, $\lambda:\mathbb{G}_{m}/_{k}\rightarrow G_{0}$
be any cocharacter, and $c\in{}^{\lambda}_{\ell}\mathfrak{g}(k)$. Suppose $c$
supports $x$ (Definition B.12). Then
$\operatorname{Ad}(G(k))x\cap(c+{}^{\lambda}_{<\ell}\mathfrak{g})\not=\emptyset$.

###### Proof.

Apply Corollary B.14 to $\langle I_{x},f_{\lambda,c}\rangle$ where
$f_{\lambda,c}$ is as in Corollary B.15 and get

$\langle I_{x},f_{\lambda,c}\rangle=\alpha\cdot\langle
I_{x},f_{c,h,d}\rangle+\sum_{i=1}^{r}\alpha_{i}\cdot\langle
I_{x},f_{c_{i},h_{i},d_{i}}\rangle.$

Since the orbit of $c_{i}$ is larger than that of $c$, we have
$\dim\operatorname{Ad}(G)c_{i}>\dim\operatorname{Ad}(G)c=\dim\operatorname{Ad}(G)x$.
Hence either $x$ is non-nilpotent and
$\dim(\mathbb{G}_{a}\cdot\operatorname{Ad}(G)x)=\dim(\operatorname{Ad}(G)x)+1\leq\dim\operatorname{Ad}(G)c_{i}$,
or $x$ is nilpotent and
$\dim(\mathbb{G}_{a}\cdot\operatorname{Ad}(G)x)=\dim\operatorname{Ad}(G)x\leq\dim\operatorname{Ad}(G)c_{i}$.
By Lemma B.2 this gives $\langle I_{x},f_{c_{i},h_{i},d_{i}}\rangle=0$. At the
same time to say $c$ supports $x$ is to say $\langle
I_{x},f_{c,h,d}\rangle>0$. Hence $\langle I_{x},f_{\lambda,c}\rangle>0$ which
proves the corollary. ∎

## References

  * [AR00] Jeffrey D. Adler and Alan Roche, _An intertwining result for $p$-adic groups_, Canad. J. Math. 52 (2000), no. 3, 449–467. MR 1758228 (2001m:22032)
  * [AS09] Jeffrey D. Adler and Loren Spice, _Supercuspidal characters of reductive $p$-adic groups_, Amer. J. Math. 131 (2009), no. 4, 1137–1210. MR 2543925 (2011a:22018)
  * [Car93] Roger W. Carter, _Finite groups of Lie type_ , Wiley Classics Library, John Wiley & Sons, Ltd., Chichester, 1993, Conjugacy classes and complex characters, Reprint of the 1985 original, A Wiley-Interscience Publication. MR 1266626
  * [CJLZ23] Cheng Chen, Dihua Jiang, Dongwen Liu, and Lei Zhang, _Arithmetic branching law and generic $L$-packets_, 2023, https://arxiv.org/abs/2309.12430v1.
  * [CM93] David H. Collingwood and William M. McGovern, _Nilpotent orbits in semisimple Lie algebras_ , Van Nostrand Reinhold Mathematics Series, Van Nostrand Reinhold Co., New York, 1993. MR 1251060
  * [CO23] Dan Ciubotaru and Emile Okada, _Local character expansions via positive depth Barbasch-Moy theory_ , 2023, https://arxiv.org/abs/2307.06780v1.
  * [DeB02a] Stephen DeBacker, _Homogeneity results for invariant distributions of a reductive $p$-adic group_, Ann. Sci. École Norm. Sup. (4) 35 (2002), no. 3, 391–422. MR 1914003 (2003i:22019)
  * [DeB02b] by same author, _Parametrizing nilpotent orbits via Bruhat-Tits theory_ , Ann. of Math. (2) 156 (2002), no. 1, 295–332. MR 1935848 (2003i:20086)
  * [DK06] Stephen DeBacker and David Kazhdan, _Stable distributions supported on the nilpotent cone for the group $G_{2}$_, The unity of mathematics, Progr. Math., vol. 244, Birkhäuser Boston, Boston, MA, 2006, pp. 205–262. MR 2181807 (2006h:22012)
  * [Fin21] Jessica Fintzen, _Types for tame $p$-adic groups_, Ann. of Math. (2) 193 (2021), no. 1, 303–346. MR 4199732
  * [FKS21] Jessica Fintzen, Tasho Kaletha, and Loren Spice, _A twisted Yu construction, Harish-Chandra characters, and endoscopy_ , 2021, https://arxiv.org/abs/2106.09120v2.
  * [JLZ22] Dihua Jiang, Dongwen Liu, and Lei Zhang, _Arithmetic wavefront sets and generic $L$-packets_, 2022, https://arxiv.org/abs/2207.04700v2.
  * [Kal19] Tasho Kaletha, _Regular supercuspidal representations_ , J. Amer. Math. Soc. 32 (2019), no. 4, 1071–1170. MR 4013740
  * [Kat82] P. I. Katsylo, _Sections of sheets in a reductive algebraic Lie algebra_ , Izv. Akad. Nauk SSSR Ser. Mat. 46 (1982), no. 3, 477–486, 670\. MR 661143
  * [Kaw85] N. Kawanaka, _Generalized Gelfand-Graev representations and Ennola duality_ , Algebraic groups and related topics (Kyoto/Nagoya, 1983), Adv. Stud. Pure Math., vol. 6, North-Holland, Amsterdam, 1985, pp. 175–206. MR 803335
  * [Kim07] Ju-Lee Kim, _Supercuspidal representations: an exhaustion theorem_ , J. Amer. Math. Soc. 20 (2007), no. 2, 273–320. MR 2276772
  * [KM03] Ju-Lee Kim and Fiona Murnaghan, _Character expansions and unrefined minimal $K$-types_, Amer. J. Math. 125 (2003), no. 6, 1199–1234. MR 2018660 (2004k:22024)
  * [KM06] by same author, _K-types and $\Gamma$-asymptotic expansions_, J. Reine Angew. Math. 592 (2006), 189–236. MR 2222734 (2007m:22016)
  * [LS79] G. Lusztig and N. Spaltenstein, _Induced unipotent classes_ , J. London Math. Soc. (2) 19 (1979), no. 1, 41–52. MR 527733 (82g:20070)
  * [LY17] George Lusztig and Zhiwei Yun, _$\mathbf{Z}/m$ -graded Lie algebras and perverse sheaves, I_, Represent. Theory 21 (2017), 277–321. MR 3697026
  * [MP94] Allen Moy and Gopal Prasad, _Unrefined minimal $K$-types for $p$-adic groups_, Invent. Math. 116 (1994), no. 1-3, 393–408. MR 1253198 (95f:22023)