# Formal Degree of Regular Supercuspidals

David Schwein  530 Church Street, Ann Arbor, MI 48105

(Date: 3 January 2021)

###### Abstract.

Supercuspidal representations are usually infinite-dimensional, so the size of
such a representation cannot be measured by its dimension; the formal degree
is a better alternative. Hiraga, Ichino, and Ikeda conjectured a formula for
the formal degree of a supercuspidal in terms of its $L$-parameter only. Our
first main result is to compute the formal degrees of the supercuspidal
representations constructed by Yu. Our second, using the first, is to verify
that Kaletha’s regular supercuspidal $L$-packets satisfy the conjecture.

Let $G$ be a reductive algebraic group over a nonarchimedean local field $k$.
In the study of the representation theory of the topological group $G(k)$, the
supercuspidal representations are fundamental: there is a precise sense in
which all irreducible (smooth or unitary) representations can be constructed
from supercuspidals. Much recent work has thus focused on the construction and
study of supercuspidal representations.

In 2001, Yu [Yu01], building on earlier work of Howe [How77] and Adler
[Adl98], gave a general construction of supercuspidal representations when $G$
splits over a tamely ramified extension of $k$. Six years later, Kim [Kim07]
proved that Yu’s construction is exhaustive when $k$ has characteristic zero
and the residue characteristic $p$ of $k$ is larger than some ineffective
bound depending on $G$. Recently, Fintzen [Fin18] has improved Kim’s result to
show that Yu’s construction produces all supercuspidals when $p$ does not
divide the order of the absolute Weyl group. Hence Yu’s construction produces
all supercuspidals for many reductive groups, though not all of them.
Moreover, the explicit nature of Yu’s construction makes his supercuspidals
amenable to close study.

The collection of irreducible unitary representations of $G(k)$ carries a
natural topology, the Fell topology, and a natural Borel measure, the
Plancherel measure. The Fell topology is canonical but the Plancherel measure
depends inversely on a choice of Haar measure on $G(k)$. When $G$ is
semisimple, every supercuspidal representation $\pi$ is unitary and thus
appears as a point of the unitary dual. Since supercuspidal representations of
semisimple groups are discrete series, this point is isolated. We may thus ask
for the measure of the point, an interesting numerical invariant of $\pi$
called the _formal degree_.

When $G$ is not semisimple it is no longer necessarily the case that all
supercuspidal representations are unitary. Nonetheless, we can define the
formal degree of an arbitrary supercuspidal representation in a way that
generalizes the formal degree of a unitary supercuspidal. The definition,
given in Section 3.2, makes no reference to the unitary dual, so we may forget
about the unitary representations of $G(k)$ and focus our attention on the
supercuspidal ones.

Our first main result is to compute the formal degrees of Yu’s supercuspidal
representations, A. The formula uses some notation that we must briefly recall
for its statement to be intelligible. Yu’s construction takes as input a
$5$-tuple $\Psi$. The first member of $\Psi$ is an increasing sequence
$(G^{i})_{0\leq i\leq d}$ of twisted Levi subgroups; let $R_{i}$ denote the
absolute root system of $G^{i}$. The second member is a certain point $y$ in
the Bruhat-Tits building $\cal B(G)$. The third member is an increasing
sequence $(r_{i})_{0\leq i\leq d}$ of nonnegative real numbers. The fourth
member is a certain irreducible representation $\rho$ of the stabilizer
$G^{0}(k)_{[y]}$ of the image $[y]$ of $y$ in $\cal B^{\textnormal{red}}(G)$.
We compute the formal degree with respect to a certain Haar measure $\mu$
constructed by Gan and Gross and discussed later in the introduction. Gan and
Gross’s measure depends on a choice of additive character and in the formula
we choose a level-zero character. For simplicity of exposition we assume in
our discussion of the formula that $G$ is semisimple, though this assumption
is relaxed in the paper proper. Finally, let $\exp_{q}(t):=q^{t}$.

######  Theorem A.

Let $G$ be a semisimple $k$-group and let $\Psi$ be a generic cuspidal
$G$-datum with associated supercuspidal representation $\pi$. Then

$\deg(\pi,\mu)=\frac{\dim\rho}{[G^{0}(k)_{[y]}:G^{0}(k)_{y,0+}]}\exp_{q}\biggl{(}\tfrac{1}{2}\dim
G+\tfrac{1}{2}\dim
G^{0}_{y,0:0+}+\tfrac{1}{2}\sum_{i=0}^{d-1}r_{i}(|R_{i+1}|-|R_{i}|)\biggr{)}.$

The proof of A boils down, after several reductions, to computations in
Bruhat-Tits theory. Yu’s supercuspidals are compactly induced from a certain
finite-dimensional irreducible representation $\tau$ of a certain compact-open
subgroup $K$. There is a general formula for the formal degree of a compact
induction which specializes, in this case, to the ratio
$\dim\tau/\\!\operatorname{vol}(\mu,K)$. The main difficulty is to compute the
volume of $K$. We first situate $K$ as a finite-index subgroup of a group of
known measure; this step reduces the problem to computing the index. Using the
theory of the Moy-Prasad filtration, we can translate the computation of this
index into the computation of the length of a certain subquotient of the Lie
algebra. The length computation, Theorem 38, is our key technical result in
the proof of the formal-degree formula.

We can thus compute the formal degree of a broad class of supercuspidal
representations. The other main result of the paper synthesizes this
computation with Langlands’s arithmetic parameterization of supercuspidals.

It is expected that the set $\Pi(G)$ of smooth irreducible representations of
$G(k)$ is classified by certain homomorphisms
$\varphi:W_{k}^{\prime}\to{}^{L}G$, called _$L$ -parameters_. Here $W_{k}$ is
the Weil group of $k$,
$W_{k}^{\prime}:=\operatorname{SL}_{2}(\mathbb{C})\times W_{k}$ is the Weil-
Deligne group of $k$, and ${}^{L}G:=\widehat{G}\rtimes W_{k}$ is the (Weil
form of the) $L$-group of $G$. The expected classification consists of a
partition

$\Pi(G)=\bigsqcup_{\varphi}\Pi_{\varphi}(G)$

of $\Pi(G)$ into finite subsets $\Pi_{\varphi}(G)$, called _$L$ -packets_,
indexed by (equivalence classes of) $L$-parameters $\varphi$. The sets
$\Pi_{\varphi}(G)$ are supposed to satisfy many compatibility conditions, the
simplest of which are summarized in Borel’s Corvallis article [Bor79, Section
10.3]. The resulting partition of $\Pi(G)$ is called a _local Langlands
correspondence_. Although a local Langlands correspondence has been
constructed for many classes of groups and representations, the full
correspondence remains a conjecture. Even after fixing the group $G$, it is
usually quite difficult to establish the correspondence for the entirety of
$\Pi(G)$. Recent work has thus focused on constructing the $L$-packets of
particular $L$-parameters.

Building on the outline of the correspondence, Langlands suggested [Lan,
Chapitre IV] that the elements of the $L$-packet $\Pi_{\varphi}(G)$ are
parameterized by representations of a certain finite group attached to
$\varphi$. Refining Langlands’s proposal, Vogan enhanced [Vog93, Section 9]
$L$-parameters to pairs $(\varphi,\rho)$ consisting of an $L$-parameter
$\varphi$ and an irreducible representation $\rho$ of the finite group
$\pi_{0}(S_{\varphi})$, where $S_{\varphi}$ is the preimage in
$\widehat{G}_{\textnormal{sc}}$ of the centralizer in $\widehat{G}$ of
$\varphi$. Unlike ordinary $L$-parameters, these enhanced parameters keep
track of the inner class of $G$: one imposes an additional condition [HII08b,
Section 1] on the central character of $\rho$, roughly, that it correspond to
the inner class of $G$ via Kottwitz’s classification of inner forms. In this
formulation, it is expected [ABPS17, Section 1.2] that the local Langlands
correspondence becomes a bijection, in other words, that the irreducible
representations $\rho$ of $S_{\varphi}$ satisfying the central character
condition parameterize the $L$-packet $\Pi_{\varphi}(G)$.

Using Vogan’s enhanced $L$-parameters, Hiraga, Ichino, and Ikeda predicted
[HII08b, HII08a] that the formal degree of an essentially discrete series
representation can be computed in terms of its $L$-parameter. They verified
the conjecture in many cases, in particular, for real reductive groups and for
inner forms of $\operatorname{SL}_{n}$ and $\operatorname{GL}_{n}$. We will
state their conjecture in a moment after reviewing two of its inputs.

First, in a paper attaching motives to reductive groups, Gross constructed
[Gro97a, Section 4] a certain Haar measure $\mu=\mu_{\psi}$ on $G(k)$
depending on an additive character $\psi$ of $k$. Two years later, he and Gan
constructed a closely related measure [GG99, Section 5] that conjecturally
agrees with the original one. Hiraga, Ichino, and Ikeda originally predicted
that Gross’s measure was the right one for their conjecture, but realized
later [HII08a] that one should use Gross and Gan’s measure instead.

Second, one can attach to the parameter $\varphi$ and certain additional data
– a finite-dimensional representation $r$ of ${}^{L}G$ and a nontrivial
additive character $\psi$ of $k$ – a meromorphic function
$\gamma(s,\varphi,r,\psi)$ of the complex variable $s$, called a
$\gamma$-factor of $\varphi$. The $\gamma$-factor is a product of $L$\- and
$\varepsilon$-factors; Section 4.1 recalls the precise formula. For the
representation $r$ of ${}^{L}G$ we choose the adjoint representation Ad of
${}^{L}G$ on
$\widehat{\mathfrak{g}}/\,\widehat{\mathfrak{z}}\,^{\Gamma\\!_{k}}$, where
$\widehat{\mathfrak{g}}$ and $\widehat{\mathfrak{z}}$ are the Lie algebras of
$\widehat{G}$ and of its center and where $\Gamma\\!_{k}$ is the absolute
Galois group of $k$. Division by $\widehat{\mathfrak{z}}\,^{\Gamma\\!_{k}}$
ensures that the $\gamma$-factor is defined at $s=0$. We call the factor
$\gamma(0,\varphi,\textnormal{Ad},\psi)$ appearing in the conjecture the the
_adjoint $\gamma$-factor_ of $\varphi$.

We can now state the conjecture [HII08b, Conjecture 1.4] of Hiraga, Ichino,
and Ikeda on the formal degree, referred to in this paper as the “formal
degree conjecture” for the sake of brevity. Let $S_{\varphi}^{\natural}$
denote the centralizer of $\varphi$ in $\widehat{G^{\textnormal{a}}}$, where
$G^{\textnormal{a}}:=G/A$ with $A$ the maximal split central torus of $G$.

######  Conjecture 1.

Let $\pi$ be an essentially discrete series representation of $G(k)$ with
extended parameter $(\varphi,\rho)$, let $\psi$ be an additive character of
$k$, and let $\mu_{\psi}$ be the Gross-Gan measure on $G(k)$ attached to
$\psi$. Then

$\deg(\pi,\mu_{\psi})=\frac{\dim\rho}{|\pi_{0}(S_{\varphi}^{\natural})|}\cdot|\gamma(0,\varphi,\textnormal{Ad},\psi)|.$

Hiraga, Ichino, and Ikeda verified their conjecture in several cases, building
on the work of many others: for an archimedean base field, using Harish-
Chandra’s theory of discrete series [HC75]; for inner forms of
$\operatorname{GL}_{n}$ and $\operatorname{SL}_{n}$, using work of Silberger
and Zink [Zin93, SZ96]; for some Steinberg representations, using work of
Kottwitz [Kot88] and Gross [Gro97a, Gro97b]; for some unipotent discrete
series of adjoint split exceptional groups, using work of Reeder [Ree00]; and
for some depth-zero supercuspidals of pure inner forms of unramified groups,
using work of DeBacker and Reeder [DR09]. In the years following the
announcement of the conjecture, Gan and Ichino [GI14] showed that it holds for
$\textnormal{U}_{3}$, $\textnormal{Sp}_{4}$, and $\textnormal{GSp}_{4}$; Qiu
[Qiu12] showed that it holds for the rank-one metaplectic group; Reeder and Yu
[RY14] and Kaletha [Kal15] showed that it holds for epipelagic supercuspidals;
Gross and Reeder [GR10] showed that it holds for simple supercuspidals;
Ichino, Lapid, and Mao [ILM17] showed that it holds for odd special orthogonal
and metaplectic groups; and Beuzart-Plessis [BP20] showed that it holds for
unitary groups.

The formal degree conjecture is a “meta-conjecture” in the sense that it
depends itself on a conjecture, the local Langlands correspondence. In order
to verify the formal degree conjecture one must first have access to a
candidate local Langlands correspondence, or at least, to candidate
$L$-packets. Strictly speaking, the previous sentence is not entirely true
because some groups admit an analytic construction of the $\gamma$-factor that
bypasses the local Langlands correspondence, though the two are expected to be
compatible. The main example is Godement-Jacquet’s [GJ72] construction of the
$L$\- and $\varepsilon$-factors for representations of the general linear
group, generalizing Tate’s thesis. Their construction explains how Hiraga-
Ichino-Ikeda were able to verify the formal degree conjecture for the general
linear group using work that predated the Henniart [Hen00] and Harris-Taylor
[HT01] constructions of the local Langlands correspondence. Nonetheless, for
the representations we consider in this paper, an analytic theory of the
$\gamma$-factor is not yet available, and so we work with $L$-packets.

Recently, Kaletha has organized into $L$-packets [Kal19b] most of Yu’s
supercuspidal representations, the “regular supercuspidal representations”.
His construction passes through a pair $(S,\theta)$ consisting of an elliptic
maximal torus $S$ of $G$ and a character $\theta$ of $S(k)$. On the Galois
side, one uses the Langlands-Shelstad theory of $\chi$-data and extensions of
$L$-embeddings [LS87] to construct a $L$-parameter for $G$ from $(S,\theta)$.
On the automorphic side, one uses the pair $(S,\theta)$ to produce an input to
Yu’s construction, hence a supercuspidal representation $\pi$ of $G(k)$. We
can thus interpret $\pi$ as a functorial lift of $\theta$ with respect to the
embedding $S\hookrightarrow G$. The $L$-packet of $\varphi$ consists, roughly,
of all $\pi$ produced in this way as we pass through the various conjugacy
classes of embeddings of $S$ in $G$; Section 2 reviews the construction in
more detail.

Since we can compute the formal degree of Yu’s representations, and Kaletha’s
$L$-packets consist of such representations, it is natural to ask whether the
$L$-packets satisfy the formal degree conjecture. Our second main result,
proved in the body of the paper as B, is that they do.

######  Theorem B.

Kaletha’s regular $L$-packets satisfy the formal degree conjecture.

To prove B, we start by computing the adjoint representation attached to a
regular supercuspidal parameter: it is a direct sum of the complexified
character lattice of $S$ and some monomial representations constructed from
$\theta$ and the root system of $S$. The $\gamma$-factor of the character
lattice has already been computed in the literature. As for the monomial
representations, computing their $\gamma$-factors amounts to computing the
depths of the inducing characters. The inducing characters are very close to
certain characters naturally constructed from $\theta$, and whose depth is
usually easy to understand; the difficulty in the proof is to quantify the
difference between the two characters. To quantify it, we prove that
$\chi$-data satisfy a natural base change formula, Theorem 64, and that the
inducing characters are ramified, Lemma 74.

A refinement of the formal degree conjecture due to Gross and Reeder [GR10,
Conjecture 8.3] predicts the root number of the adjoint representation. In
future work, I hope to determine whether Kaletha’s regular $L$-packets also
satisfy this refined conjecture.

The structure of this paper mirrors the formal degree conjecture. After two
preliminary sections that fix notation and review the Langlands correspondence
for regular supercuspidals, we compute the formal degree of a Yu
representation in Section 3, we compute the Galois side of the formal degree
conjecture in Section 4, and we compare the two in the brief Section 5.

###### Contents

  1. 1 Notation
    1. 1.1 Sets
    2. 1.2 Filtrations
    3. 1.3 Fields
    4. 1.4 Groups
    5. 1.5 Root systems
    6. 1.6 Bruhat-Tits theory
    7. 1.7 Base change for groups and vector spaces
    8. 1.8 Base change for characters
  2. 2 Langlands correspondence for regular supercuspidals
    1. 2.1 Tame elliptic regular pairs
    2. 2.2 Regular representations
    3. 2.3 $L$-embeddings
    4. 2.4 Regular parameters
  3. 3 Automorphic side
    1. 3.1 Notation
    2. 3.2 Formal degree
    3. 3.3 Concave-function subgroups: split case
    4. 3.4 Concave-function subgroups: tame case
    5. 3.5 Yu’s groups
    6. 3.6 Length computation
    7. 3.7 Yu’s construction
    8. 3.8 Degree computation
    9. 3.9 Regular supercuspidals
  4. 4 Galois side
    1. 4.1 Review of $L$-, $\varepsilon$\- and $\gamma$-factors
    2. 4.2 Base change for $\chi$-data
    3. 4.3 Adjoint representation
    4. 4.4 Toral summand
    5. 4.5 Root summand
    6. 4.6 Summary
  5. 5 Comparison

##  1\. Notation

###  1.1. Sets

Let $|X|$ denote the cardinality of the set $X$. Given a subset $Y\subseteq
X$, let $\mathbbm{1}_{Y}:X\to\\{0,1\\}$ denote the indicator function of $Y$.

Many operations on sets are expressed by superscripts or subscripts. When we
have several operations denoted this way, say $X\mapsto X_{a}$ and $X\mapsto
X_{b}$, we use a comma to denote the concatenation: $X\mapsto X_{a}\mapsto
X_{a,b}$. This expression is notationally simpler than the longer form
$(X_{a})_{b}$ and should cause no confusion.

###  1.2. Filtrations

Suppose $I$ is a totally ordered index set and $(X_{i})_{i\in I}$ is a
decreasing, $I$-indexed filtration of the set $X$. For $i\in I$, define

(2) $X_{i+}:=\bigcup_{j>i}X_{j}.$

Let $\widetilde{I}:=I\cup\\{i+:i\in I\\}\cup\\{\infty\\}$ denote Bruhat and
Tits’s extension of $I$ [BT72, 6.4.1]; their definition is for $I=\mathbb{R}$
only, but it is clear how to extend it to arbitrary $I$. Equation 2 together
with the convention

$X_{\infty}:=\bigcap_{i}X_{i}$

defines an extension of the given filtration to a $\widetilde{I}$-indexed
filtration. If in addition $X=G$ is a group and each $X_{i}=G_{i}$ is a
subgroup of $G$ then define, for $i<j$ in $\widetilde{I}$,

$G_{i:j}:=G_{i}/G_{j}.$

We apply this formalism to the Moy-Prasad filtration on a $p$-adic group
$G(k)$ and its Lie algebra $\mathfrak{g}$ in Section 1.6, and to the Weil
group $W_{k}$ in Section 1.3. The filtrations on $G(k)$ and $W_{k}$ are
indexed by $\mathbb{R}_{\geq 0}$, and the filtrations on $\mathfrak{g}$ are
indexed by $\mathbb{R}$. In Section 2.1 we consider a filtration (of a root
system) that is increasing, not decreasing. When the filtration on $X$ is
increasing, its extension to $\widetilde{I}$ is defined by

$X_{i+}:=\bigcap_{j>i}X_{j},\qquad\qquad X_{\infty}:=\bigcup_{i}X_{i}.$

###  1.3. Fields

Let $k$ be a nonarchimedean local field of odd residue characteristic $p$, let
$\cal O$ denote the ring of integers of $k$, and let $\kappa$ denote the
residue field of $\cal O$. Given a finite algebraic extension $\ell$ of $k$,
let $e_{\ell/k}$ denote the ramification degree and $f_{\ell/k}$ the residue
degree.

######  Remark 3.

Many of the works this paper is built on, for instance, Kaletha’s construction
of regular supercuspidal $L$-packets [Kal19b], assume that $p$ is odd. For
this reason we, too, assume for the rest of the paper that $p$ is odd.

Let $\operatorname{ord}_{k}:k^{\times}\to\mathbb{Z}$ denote the unique
discrete valuation on $k$ with value group $\mathbb{Z}$. We extend
$\operatorname{ord}_{k}$ to a valuation on the algebraic closure $\bar{k}$ and
denote the extension by $\operatorname{ord}_{k}$ as well. Hence the value
group for a finite extension $\ell$ is
$\operatorname{ord}_{k}(\ell^{\times})=e_{\ell/k}^{-1}\mathbb{Z}$.

######  Remark 4.

Aesthetic reasons might lead one to consider a more general value group for
$k$ than $\mathbb{Z}$. Indeed, per our convention, the value group for an
extension of $k$ is generally than larger than $\mathbb{Z}$. Most of the
computations in this paper can be modified to accommodate a different choice
of value group because their defining object inherit that choice. Many of the
depth computations of Section 3 could be modified to carry through because the
Moy-Prasad filtration inherits its jumps from the value group. Similarly, the
Artin conductor computations of Section 4.1 could be modified to carry through
because the upper numbering filtration inherits its jumps from the value
group. However, this modification would break the connection between the Artin
conductor and the Artin representation. Moreover, since the
$\varepsilon$-factor is defined independent of the value group, the
relationship between the Artin conductor and the $\varepsilon$-factor,
Equation 54, holds only for value group $\mathbb{Z}$.

Given a finite $\cal O$-module $M$, let $\operatorname{len}M$ denote the
length of $M$. When $k$ has positive characteristic the module $M$ is a
$\kappa$-vector space and its length is its dimension, but when $k$ has mixed
characteristic the module $M$ is _not_ a vector space, and we must work
instead with its length.

Let $q:=|\kappa|$, a power of $p$, let $\exp_{q}(t):=t^{q}$, and let
$\log_{q}$ be the functional inverse of $\exp_{q}$. The function $\exp_{q}$ is
related to the length by the equation

$\exp_{q}\operatorname{len}M=|M|.$

Let $W_{k}$ denote the Weil group of $k$, let $I_{k}\subseteq W_{k}$ denote
the inertia subgroup of $W_{k}$, let $P_{k}\subseteq I_{k}$ denote the wild
inertia subgroup, and for $r\geq 0$ let $W_{k}^{r}\subseteq W_{k}$ denote the
$r$th subgroup of $W_{k}$ in its upper numbering filtration, computed with
respect to the valuation $\operatorname{ord}$. A _representation of the Weil
group_ is a continuous, finite-dimensional, complex representation $\pi$ of
$W_{k}$. Given a finite extension $\ell$ of $k$, let
$\pi|_{\ell}:=\pi|_{W_{\ell}}$. The _depth_ of $\pi$ is defined as

$\operatorname{depth}\pi:=\inf\\{r\in\mathbb{R}:\pi(W^{r}_{\\!k})=1\\}.$

In order to make the filtration on the Weil group compatible with the Moy-
Prasad filtration, we need to modify the upper numbering filtration on
$W_{\ell}$ for a finite extension $\ell$ of $k$ by using the valuation
$\operatorname{ord}_{k}$ to define it instead of the valuation
$\operatorname{ord}_{\ell}$. When the extension is tame, as is usually the
case, this modification has the effect of scaling the indices of the
filtration by $e_{\ell/k}^{-1}$. To make the dependence on $k$ clear, let
$\operatorname{depth}_{k}$ denote the depth of a representation of $W_{\ell}$
where its filtration is computed using $\operatorname{ord}_{k}$. The
distinction is crucial for the proof of Lemma 58.

###  1.4. Groups

Let $G$ be a reductive $k$-group, let $Z$ be the center of $G$, and let $A$ be
the maximal split subtorus of $Z$. We reserve the symbols $S$ and $T$ for
tori, often maximal tori of $G$. Let $\mathfrak{g}$ and $\mathfrak{z}$ denote
the Lie algebras of $G$ and $Z$, respectively.

######  Remark 5.

There are three exceptions to the notational convention that $G$ denotes a
reductive group and $S$ and $T$ denotes tori.

First, we sometimes need to work with $\kappa$-groups instead of $k$-groups.
This practice cannot be avoided, but is so rare that we did not see the value
in introducing a separate notational convention for $\kappa$-groups. So $G$,
$S$, and $T$ denote $\kappa$-groups in this setting, which takes place in
small portions of Sections 2.2 and 3.9.

Second, for reasons of notational clarity, in Section 3 the symbol
$\underline{G}$ denotes a $k$-group and the symbol $G$ denotes the topological
group of its rational points, as discussed in Section 3.1.

Third, in Section 3.2, where we discuss the formal degree, $G$ denotes an
arbitrary locally profinite group, the proper setting for that theory.

Given a subgroup $H$ of $G$, let $H^{\textnormal{a}}:=G/A$. The letter a
abbreviates “anisotropic”. The notation hides the dependence on the ambient
group $G$, but the meaning should be clear from context: typically $H$ is a
maximal torus or (twisted) Levi subgroup of $G$.

Let $\widehat{G}$ denote the Langlands dual group of $G$, a complex reductive
group, and let ${}^{L}G=\widehat{G}\rtimes W$ denote the Weil form of the dual
group.

Given an extension $\ell$ of $k$ and an $\ell$-group $H$, let
$\operatorname{Res}_{\ell/k}H$ denote the Weil restriction of $H$ from $\ell$
to $k$.

###  1.5. Root systems

Given a reductive $k$-group $G$ and a maximal torus $T$ of $G$, let $R(G,T)$
be the absolute root system of $G$ with respect to $T$, that is, the root
system of $G_{\bar{k}}$ with respect to $T_{\bar{k}}$, together with its
natural Galois action. Let $\underline{R}(G,T)$ denote the set of Galois
orbits of $R(G,T)$. I prefer to think of $\underline{R}(G,T)$ as the “functor
of roots” in the sense of SGA 3 [DG11, XIX.3], an equivalent but more
elaborate perspective. Reserve the letters $\alpha,\beta,\gamma,\dots$ for
elements of $R$, and their underlines
$\underline{\alpha},\underline{\beta},\underline{\gamma},\dots$ for elements
of $\underline{R}$. Given $\underline{\alpha}\in\underline{R}$, let
$\underline{\alpha}(\bar{k})$ denote the elements of $\underline{\alpha}$, a
subset of $R$.

Given $\alpha\in R$, let $\Gamma\\!_{\alpha}$ denote the stabilizer of
$\alpha$ in $\Gamma\\!_{k}$ and let $k_{\alpha}:=\bar{k}^{\Gamma\\!_{\alpha}}$
denote the fixed field of $\Gamma\\!_{\alpha}$. Given $\underline{\alpha}\in
R$, let $k_{\underline{\alpha}}$ denote a fixed field extension of $k$ that is
isomorphic, for some $\alpha\in\underline{\alpha}(\bar{k})$, to the extension
$k_{\alpha}$, and let $\kappa_{\underline{\alpha}}$ denote the residue field
of $k_{\underline{\alpha}}$. We can define $k_{\underline{\alpha}}$
canonically as the inverse limit of the groupoid of extensions $k_{\alpha}$
with $\alpha\in\alpha(\bar{k})$, in the style of Deligne and Lusztig [DL76,
Section 1.1], but since $\Gamma\\!_{k}$ is nonabelian, it is impossible to
canonically identify this limit with any one $k_{\alpha}$. The notation
$k_{\underline{\alpha}}$ helps us avoid choosing such an $\alpha$, and is
reserved for expressions that depend only on the isomorphism class of the
extension, such as the degree $[k_{\underline{\alpha}}:k]$.

###  1.6. Bruhat-Tits theory

Given a reductive $k$-group $G$, let $\cal B(G)$ and $\cal
B^{\textnormal{red}}(G)$ be the extended and reduced Bruhat-Tits buildings of
$G$. Let $[x]$ denote the image of $x$ under the canonical reduction map $\cal
B(G)\to\cal B^{\textnormal{red}}(G)$. Given a split maximal torus $T$ of $G$,
let $\cal A(G,T)$ and $\cal A^{\textnormal{red}}(G,T)$ denote the extended and
reduced apartments of $T$. The apartment $\cal A(G,T)$ is noncanonically
isomorphic to the building $\cal B(T)$. For us, the words “building” and
“apartment” refer to the extended forms.

The apartment of a split maximal torus is a classical construction, defined in
Bruhat and Tits’s original papers on buildings [BT72, BT84]. More recently, in
a paper establishing tame descent for buildings [PY02], Prasad and Yu showed
for any tame maximal torus $S$ of $G$ how to embed the building of $S$ into
the building of $G$. Although the embedding is not canonical, the image of the
embedding is canonical. We denote this image by $\cal A(G,S)$ and call it the
_apartment_ of $G$ in $S$, though that terminology is typically reserved for
maximal split tori only.

Given a point $x$ of $\cal B(G)$ or $\cal B^{\textnormal{red}}(G)$, let
$G(k)_{x}$ denote the stabilizer of $x$ in $G(k)$. When the center of $G$ is
anisotropic the parahoric group $G(k)_{x,0}$ is of finite index in the
stabilizer $G(k)_{x}$; in general, $A(k)_{0}G(k)_{x,0}$ is of finite index in
$G(k)_{x}$.

For each $x\in\cal B(G)$, Moy and Prasad defined [MP94, Sections 2 and 3] a
canonical decreasing $\mathbb{R}_{\geq 0}$-indexed filtration of $G(k)_{x}$
and a canonical decreasing $\mathbb{R}$-indexed filtration of $\mathfrak{g}$,
denoted by $G(k)_{x,r}$ and $\mathfrak{g}_{x,r}$ and called the _Moy-Prasad
filtrations_. When $G=S$ is a torus the point $x$ is irrelevant and we
suppress it from the notation. Although the filtrations are defined for every
$G$, they are particularly well-behaved when $G$ splits over a tame extension.
In this case, for instance, there is a canonical isomorphism

$G_{x,r:r+}\simeq\mathfrak{g}_{x,r:r+}$

for every $r>0$, called the _Moy-Prasad isomorphism_. Sections 3.3 to 3.5
discuss in much more detail these filtrations and a generalization of them due
to Yu.

The Moy-Prasad filtration is compatible with our chosen discrete valuation on
$k$ in the sense that for $r>0$,

$k^{\times}_{r}:=\\{a\in k^{\times}:\operatorname{ord}_{k}(a-1)\geq r\\}.$

Given a finite separable extension $\ell$ of $k$ and a reductive $\ell$-group
$H$, we compute the Moy-Prasad filtration on $H(\ell)$ with respect to the
norm $\operatorname{ord}_{k}$, not $\operatorname{ord}_{\ell}$. This
convention implies that the Moy-Prasad filtration is a topological invariant
independent of the base field in the following sense: the Moy-Prasad
filtration on the group $G(\ell)$ agrees with the Moy-Prasad filtration on the
identical group $(\operatorname{Res}_{\ell/k}G)(k)$.

The _depth_ of an irreducible admissible representation $\pi$ of $G(k)$,
denoted by $\operatorname{depth}_{k}\pi$, is the smallest real number $r$ such
that for some $x\in\cal B(G)$, the representation $\pi$ has a nonzero vector
fixed by $G(k)_{x,r+}$. The subscript $k$ is a reminder that the depth, via
the Moy-Prasad filtration by which it is defined, depends on the base field
$k$. It is not a priori clear that this minimum is attained, but Moy and
Prasad showed [MP96, Theorem 3.5] that the depth is a nonnegative rational
number. This result makes the depth an indispensable tool in the
representation theory of $p$-adic groups.

######  Remark 6.

Most of Bruhat-Tits theory carries through when the field $k$ is assumed only
to be Henselian. For example, the results of Sections 3.3, 3.4, 3.5, and 3.6
hold at this level of generality. But once representation theory enters the
picture, we must assume $k$ is a local field.

###  1.7. Base change for groups and vector spaces

Given a scheme $X$ over a base scheme $Y$ and a morphism $Z\to Y$, let $X_{Z}$
denote the base change of $X$ from $Y$ to $Z$, that is, the pullback
$X\times_{Y}Z$. When $Z=\operatorname{Spec}A$ is the spectrum of a field $A$,
we write $X_{A}$ for $X_{Z}$. The schemes $X$ that we base change are in
practice always group schemes.

Similar notation can be used for base change of modules. Given a $B$-algebra
$A$ and an $A$-module $M$, let $M_{B}:=M\otimes_{A}B$.

We take the position that an algebraic group carries the information of its
base scheme. This position forces our terminology to differ slightly from
common practice in the literature where a $k$-group is thought of as a
$\bar{k}$-group with a $k$-rational structure. In this common language one can
speak, given two $k$-groups $G$ and $H$, of morphisms $G\to H$ that are not
defined over $k$. For us, a morphism $G\to H$ is automatically defined over
$k$. To speak of a morphism “not defined over $k$” in this common sense, we
would speak of a morphism $G_{\ell}\to H_{\ell}$ where $\ell$ is an extension
of $k$, especially $\ell=\bar{k}$.

###  1.8. Base change for characters

Let $S$ be a $k$-torus, let $\ell$ be a finite separable extension of $k$, and
let $T:=\operatorname{Res}_{\ell/k}S_{\ell}$. Since
$X^{*}(T)=\operatorname{Ind}_{\ell/k}\operatorname{Res}_{\ell/k}X^{*}(S)$,
there is a canonical map $X^{*}(S)\to X^{*}(T)$ of Galois lattices, the unit
of the adjoint pair
$(\operatorname{Res}_{\ell/k},\operatorname{Ind}_{\ell/k})$. The dual of this
unit is a canonical map $N_{\ell/k}:T\to S$, called the _norm map_ ; we use
the same name and notation for the map $T(k)=S(\ell)\to S(k)$ on rational
points. Given a character $\theta:S(k)\to\mathbb{C}^{\times}$, define the
character $\theta_{\ell/k}:S(\ell)\to\mathbb{C}^{\times}$ by precomposition
with the norm:

$\theta_{\ell/k}:=\theta\circ N_{\ell/k},$

We call $\theta_{\ell/k}$ the _base change_ of $\theta$ from $k$ to $\ell$.
Contrary to the usual notation for base change of schemes, the notation for
base change of characters must include $k$, not just $\ell$, because the base
field $k$ cannot be recovered from the topological group $S(k)$.

The base change operation for characters realizes base change in the local
Langlands correspondence. For tori, this correspondence is a bijection between
the complex character group of $S(k)$ and the Galois cohomology group
$\textnormal{H}^{1}(W_{k},\widehat{S})$. On the Galois side, we can restrict
the $L$-parameter $\hat{\theta}$ of a character $\theta$ to the Weil group
$W_{\ell}$ of a finite separable extension. But by the local Langlands
correspondence for $S_{\ell}$, this parameter $\hat{\theta}|_{\ell}$
corresponds to a character of $S(\ell)$. It is well known, and a formal
consequence of the properties of the local Langlands correspondence for tori,
that this character is precisely the base-changed character just defined:
symbolically,

$\hat{\theta}|_{\ell}=\widehat{\theta_{\ell/k}}.$

Yu’s Ottawa article [Yu09] nicely summarizes the local Langlands
correspondence for tori, and proves that for tame tori, the local Langlands
correspondence preserves depth [Yu09, Section 7.10]. He does not discuss base
change, however.

We also need to understand how base change affects depth.

######  Lemma 7.

Let $\ell$ be a finite separable extension of $k$, let $S$ be a $k$-torus, and
let $\theta:S(k)\to\mathbb{C}^{\times}$ be a character. If either

  1. (1)

$\ell/k$ is unramified and $\operatorname{depth}_{k}\theta\geq 0$ or

  2. (2)

$\ell/k$ tamely ramified and $\operatorname{depth}_{k}\theta>0$

then $\operatorname{depth}_{k}\theta=\operatorname{depth}_{k}\theta_{\ell/k}$.

###### Proof.

Using the same trick as in Yu’s proof of the depth-preservation theorem, it
suffices to prove the Lemma in the case where $S=\mathbb{G}_{\textnormal{m}}$.
The first case is an immediate consequence of local class field. Using the
first result, reduce the second to the case where $\ell$ is obtained from $k$
by adjoining a root of a uniformizer; from here, the result is a
straightforward computation. ∎

##  2\. Langlands correspondence for regular supercuspidals

In this section we review the Langlands correspondence for regular
supercuspidal representations, following Kaletha’s article [Kal19b]. Many of
the definitions, for instance, regularity of $L$-parameters, are rather
technical, and instead of restating them, we point to their definitions in the
literature. The description of regular representations and the construction of
their $L$-parameters passes through a pair $(S,\theta)$ consisting of an
elliptic maximal torus $S$ of $G$ and a character $\theta$ of $S(k)$
satisfying certain regularity conditions reviewed in Section 2.1. The primary
goal of this section, then, is to understand, to the extent needed to verify
the formal degree conjecture, how these pairs interface with both sides of the
Langlands correspondence.

On the automorphic side, the pair $(S,\theta)$ produces an input to Yu’s
construction [Yu01] of supercuspidals; we explain how this works in Section
2.2. In this way we produce a supercuspidal representation $\pi_{(S,\theta)}$
of $G(k)$. When $(S,\theta)$ is “tame elliptic regular”, the representations
that arise this way are precisely the regular supercuspidal representations.

On the Galois side, one can define a certain class of “regular supercuspidal
parameters” and show that each arises from a pair $(S,\theta)$ as the
composition

$W_{k}\xrightarrow{{}^{L}\theta}{}^{L}S\xrightarrow{{}^{L}j_{\chi}}{}^{L}G.$

Here the first map corresponds to $\theta$ under the local Langlands
correspondence for tori and the second map is an extension of a given Galois-
stable embedding $\hat{\jmath}:\widehat{S}\to\widehat{G}$. There is a general
procedure, reviewed in Section 2.3, for extending $\hat{j}$ to
${}^{L}j_{\chi}$ using a certain object $\chi$ called a set of $\chi$-data. In
our setting one canonically constructs such $\chi$-data from the pair
$(S,\theta)$, producing a canonical extension ${}^{L}j_{\chi}$ and thus a
canonical $L$-parameter. Using pairs $(S,\theta)$, we organize regular
supercuspidal parameters into $L$-packets in Section 2.4. To first
approximation a regular supercuspidal $L$-packet consists of the regular
supercuspidal representations $\pi_{(jS,\theta\circ j^{-1})}$ as $j$ ranges
over $G(k)$-conjugacy classes of admissible embeddings $j:S\hookrightarrow G$;
in reality, however, we must slightly modify the character $\theta\circ
j^{-1}$.

###  2.1. Tame elliptic regular pairs

Pairs $(S,\theta)$, consisting of a $k$-torus $S$ and a character
$\theta:S(k)\to\mathbb{C}^{\times}$ subject to certain conditions, mediate the
local Langlands correspondence for regular supercuspidals. This subsection
reviews these conditions, following the discussion in Kaletha’s article
[Kal19b, Sections 3.6 and 3.7], and uses them to compute the depths of certain
auxiliary characters that arise in Section 4.5.

The simplest condition is tameness: the pair $(S,\theta)$ is _tame_ if $S$ is
tame, that is, if $S$ splits over a tamely ramified extension of $k$. In this
paper $S$ is always assumed tame, though we sometimes repeat the hypothesis
for emphasis.

All other conditions on our pair require that $S$ be embedded as a maximal
torus of a reductive group $G$. This requirement is extremely natural on the
automorphic side, but on the Galois side, we must reinterpret it carefully
since the embedding is allowed to vary.

So assume in the rest of this subsection that $S$ is a maximal torus of $G$.
The pair $(S,\theta)$ is _elliptic_ if $S$ is elliptic, that is, if the torus
$S/Z$ (where $Z$ is the center of $G$) is anisotropic.

For the final condition, regularity, we need to probe more deeply the
relationship between $(S,\theta)$ and $G$. Let $R=R(G,S)$ and let $\ell$ be
the splitting field of $S$. Given a real number $r>0$, consider the set of
roots

$R^{r}:=\\{\alpha\in
R\mid(\theta_{\ell/k}\circ\alpha^{\vee})(\ell_{r}^{\times})=1\\}.$

The assignment $r\mapsto R^{r}$ is an increasing, Galois-stable,
$\mathbb{R}$-indexed filtration of $R$. Let $R_{r+}:=\bigcup_{s>r}R_{s}$, let
$r_{d-1}>\cdots>r_{0}>0$ be the breaks of this filtration, and let $r_{-1}=0$
and $r_{d}=\operatorname{depth}_{k}\theta$. For each $0\leq i\leq d$, let
$R_{i}:=R^{(r_{i-1})+}$. It turns out [Kal19b, Lemma 3.6.1] that $R_{i}$ is a
Levi subsystem of $R$. Let $G^{i}$ be the connected reductive subgroup of $G$
containing $S$ whose root system with respect to $S$ is $R_{i}$; the group
$G^{i}$ can be constructed by Galois descent, for example. Let $G^{-1}:=S$.

######  Definition 8 ([Kal19b, Definition 3.7.5]).

A tame elliptic pair is _regular_ if

  1. (1)

the action of the inertia group on $R_{0}$ preserves a set of positive roots,
and

  2. (2)

the stabilizer of the action of the group $N(G^{0},S)(k)/S(k)$ on
$\theta|_{S(k)_{0}}$ is trivial.

It is _extra regular_ if, in addition,

  1. ($2^{\prime}$)

the stabilizer of the action of the group $\Omega(G^{0},S)(k)$ on
$\theta|_{S(k)_{0}}$ is trivial.

Here $N(G^{0},S)$ is the normalizer of $G^{0}$ in $S$ and $\Omega(G^{0},S)$ is
the Weyl group.

When we compute in Section 4 the formal degree of the regular parameter
attached to $(S,\theta)$, half of the problem (the “root summand” of Section
4.5) boils down to knowing for each coroot $\alpha^{\vee}$ the depth of the
character

$\theta_{k_{\alpha}/k}\circ\alpha^{\vee};$

here $\alpha^{\vee}$ is interpreted as a homomorphism $k_{\alpha}^{\times}\to
S(k_{\alpha})$. Therefore, our main goal in this subsection is to compute the
depth of this character. To carry out the computation, we systematically
decompose $\theta$ as a product of characters of known depth using Kaletha’s
notion of a Howe factorization, after reviewing an important component of that
definition, due to Yu.

The definition of a Howe factorization relies on a definition of Yu [Yu01,
Section 9] for a character $\phi:H(k)\to\mathbb{C}^{\times}$ of a twisted Levi
subgroup $H$ of $G$ to be _$G$ -generic of depth $r$_. We need not concern
ourselves with the precise definition of $G$-genericity, but we do need one of
its consequences, which approximates the full definition.

######  Lemma 9.

Let $G$ be a reductive $k$-group, let $H$ be a tame twisted Levi subgroup of
$G$, let $S\subseteq H$ be a tame maximal torus, and let
$\phi:H(k)\to\mathbb{C}^{\times}$ be a character of positive depth $r$ whose
restriction to $H_{\textnormal{sc}}(k)$ is trivial. Then $\phi$ is $G$-generic
if and only if for every root $\alpha\in R(G,S)\setminus R(H,S)$ and every
finite tame extension $\ell$ of $k_{\alpha}$, the character
$\phi_{\ell/k}\circ\alpha_{\ell}^{\vee}$ of $\ell^{\times}$ has depth $r$.

###### Proof.

Kaletha proved this Lemma in the case where $\ell$ is a fixed splitting field
of $S$ [Kal19b, Lemma 3.6.8]. We can deduce our result in the case where
$\ell=k_{\alpha}$ from his result using the naturality of the norm map, and
from there, we can deduce the result in general using naturality and Lemma 7.
∎

######  Corollary 10.

In the setting of Lemma 9,
$\operatorname{depth}_{k}\phi\in\operatorname{ord}(k_{\alpha}^{\times})$ for
each $\alpha\in R(G,S)\setminus R(H,S)$.

Regularity is much less restrictive than genericity, but we need to know
something about genericity in order to understand the depths of various
auxiliary characters constructed from $\theta$ in Section 4.5. Roughly
speaking, any character, regular or not, can be decomposed as a product of
generic characters related to the filtration of the root system. This
decomposition is called a Howe factorization.

######  Definition 11.

A _Howe factorization_ of $(S,\theta)$ is a sequence
$(\phi_{i}:G^{i}(k)\to\mathbb{C}^{\times})_{-1\leq i\leq d}$ of characters
satisfying the following properties.

  1. (1)

$\displaystyle\theta=\prod_{i=-1}^{d}\phi_{i}|_{S(k)}$.

  2. (2)

The character $\phi_{i}$ is trivial on $G^{i}_{\textnormal{sc}}(k)$ for $0\leq
i\leq d$.

  3. (3)

The character $\phi_{i}$ is $G^{i+1}$-generic of depth $r_{i}$ for $0\leq
i\leq d-1$; the character $\phi_{d}$ is trivial if $r_{d}=r_{d-1}$ and has
depth $r_{d}$ otherwise; and the character $\phi_{-1}$ is trivial if $G^{0}=S$
and otherwise satisfies $\phi_{-1}|_{S(k)_{0+}}=1$.

It turns out [Kal19b, Proposition 3.6.7] that every tame pair admits a Howe
factorization. When $\alpha\notin R_{0}$, we can use this factorization to
compute the depth of $\theta_{k_{\alpha}/k}\circ\alpha^{\vee}$.

######  Lemma 12.

Let $(S,\theta)$ be a tame pair and let $\alpha\in R_{i}$, where $1\leq i\leq
d$. Then the character
$\theta_{k_{\alpha}/k}\circ\alpha^{\vee}:k_{\alpha}^{\times}\to\mathbb{C}^{\times}$
has depth $r_{i-1}$.

###### Proof.

Let $(\phi_{j}:G^{j}(k)\to\mathbb{C}^{\times})_{-1\leq j\leq d}$ be a Howe
factorization of $(S,\theta)$. Then

$\theta_{k_{\alpha}/k}\circ\alpha^{\vee}=\prod_{j=-1}^{d}\phi_{j,k_{\alpha}/k}\circ\alpha^{\vee}.$

Since $\alpha^{\vee}$ factors through $G^{i}$, condition (2) of a Howe
factorization implies that the factors of this product are trivial for $j\geq
i$. By Lemma 9, the $j$th remaining factor has depth $r_{j}$, and since the
sequence $j\mapsto r_{j}$ is strictly increasing, the product has depth
$r_{i-1}$. ∎

Lemma 12 conspicuously omits the case where $\alpha\in R_{0}$. We have more to
say about this in Section 4.5, especially in Lemmas 74 and 75.

###  2.2. Regular representations

Yu’s seminal paper [Yu01] constructs a broad class of supercuspidal
representations starting from a certain triple
$(\vec{G},\pi_{-1},\vec{\phi})$, which we call, for reference, a _Yu datum_.
This subsection reviews these triples and explains how a tame elliptic regular
pair gives rise to a Yu datum. Later, Section 3.7 explains in more detail how
to construct supercuspidal representations from Yu data.

There are three stages of representations used in in Yu’s construction, each
informing the previous one: representations of finite groups of Lie type;
depth-zero supercuspidal representations; and supercuspidal representations of
arbitrary depth. Since the definition of regular supercuspidal passes through
each of these stages, we start by reviewing each stage in turn.

The first stage is finite groups of Lie type. In this paragraph only, let $G$
be a reductive $\kappa$-group. Representations of $G(\kappa)$ are well
understood through the work of Deligne and Lusztig [DL76]. They attached to an
elliptic pair $(S,\theta)$ over $\kappa$ (that is, $S$ is an elliptic maximal
torus of $G$) a virtual representation $\textnormal{R}_{(S,\theta)}$, the
Deligne-Lusztig induction. If the character $\theta$ of the $\kappa$-torus $S$
is _regular_ [Kal19b, Definition 3.4.16] then $\pm\textnormal{R}_{(S,\theta)}$
is an irreducible representation. We say that a representation $\rho$ of
$G(\kappa)$ is _regular_ if it is isomorphic to such a Deligne-Lusztig
representation.

Passing to the second stage, depth-zero supercuspidals, the result that
initiated general constructions of supercuspidals was the following
classification theorem [MP96, Proposition 6.8] of Moy and Prasad: for every
depth-zero supercuspidal representation $\pi$ of $G(k)$, there is a vertex
$x\in\cal B^{\textnormal{red}}(G)$ such that $\pi|_{G(k)_{x,0}}$ contains the
inflation to $G(k)_{x,0}$ of an irreducible cuspidal representation
$\widetilde{\rho}$ of $G(k)_{x,0:0+}$. Furthermore, we can recover $\pi$ by
compact induction: there is some representation $\rho$ of $G(k)_{x}$ such that
$\rho|_{G(k)_{x,0}}$ contains the inflation of $\widetilde{\rho}$ and such
that $\pi=\operatorname{c-Ind}_{G(k)_{x}}^{G(k)}\rho$. The depth-zero
supercuspidal representation $\pi$ is _regular_ if the representation $\rho$
of $G(k)_{x,0:0+}$ is regular. Regular depth-zero supercuspidals $\pi$ enjoy
two pleasant properties.

First, there is a canonical bijection between regular depth-zero
supercuspidals and conjugacy classes of regular tame elliptic pairs
$(S,\theta)$ of depth zero in which the torus $S$ is _maximally unramified_ in
$G$ [Kal19b, Definition 3.4.1]. In particular, we can recover $S$ from $\pi$.

Second, it turns out [Kal19b, Sections 3.4.4 and 3.4.5] that
$\pi_{(S,\theta)}$ can be compactly induced from a representation
$\eta_{(S,\theta)}$ of the group $S(k)G(k)_{x,0}$. This is an improvement over
Moy and Prasad’s theorem, which uses the larger stabilizer group $G(k)_{x}$
instead; generally $S(k)G(k)_{x,0}$ is easier to understand than $G(k)_{x}$.
The fact that $\pi$ is compactly induced from this smaller group plays a
crucial role in our final computation, in Section 3.9, of the formal degree of
a regular supercuspidal.

We can now discuss the third stage, Yu’s general construction of
supercuspidals.

To start, we recall the definition of a Yu datum. A subgroup $H$ of $G$ is a
_twisted Levi_ subgroup if there is a finite separable extension $\ell$ of $k$
splitting $G$ such that $H_{\ell}$ is a Levi subgroup of $G_{\ell}$, and $H$
is _tame_ if $\ell$ can be taken to be a tame extension of $k$. A _twisted
Levi sequence_ in $G$ is an increasing sequence

$\vec{G}=(G^{0}\subsetneq G^{1}\subsetneq\cdots\subsetneq G^{d})$

of twisted Levi subgroups of $G$; it is _tame_ if each of its members is tame.
The first component $\vec{G}$ of a Yu datum is a tame twisted Levi sequence;
the second component $\pi_{-1}$ of a Yu datum is a depth-zero supercuspidal
representation of $G^{0}(k)$; and the third component $\vec{\phi}$ of a Yu
datum is a sequence of characters

$\vec{\phi}=(\phi_{i}:G^{i}(k)\to\mathbb{C}^{\times})_{0\leq i\leq d}.$

These three objects are required to satisfy certain conditions that Section
3.7 spells out in detail. In fact, in that section we work with a certain
five-tuple instead of a Yu datum, but the two objects are closely related
[HM08, Section 3.1].

To simplify the following definition, we assume in the rest of this subsection
that $p$ does not divide the order of the fundamental group of $G$. Kaletha
defined regularity in general using $z$-extensions [Kal19b, Section 3.7.4],
but we have no need to understand how this works.

######  Definition 13.

A Yu datum $(\vec{G},\pi_{-1},\vec{\phi})$ is _regular_ if $\pi_{-1}$ is a
regular depth-zero supercuspidal representation. A supercuspidal
representation is _regular_ if it is isomorphic to a supercuspidal
representation constructed from a regular Yu datum.

We have thus defined the supercuspidal representations of interest to us; the
next matter is to connect them to torus-character pairs.

Given a Yu datum $(\vec{G},\pi_{-1},\vec{\theta})$, we can find a maximally
unramified torus $S$ of $G^{0}$ and a regular depth-zero character $\phi_{-1}$
of $S(k)$ such that $\pi_{-1}=\pi_{(S,\phi_{-1})}$. Setting

$\theta=\prod_{i=-1}^{d}\phi_{i}|_{S(k)}$

then produces a tame elliptic regular pair $(S,\theta)$. Conversely, given
such a pair $(S,\theta)$, with Howe factorization
$(\phi_{i}:G^{i}(k)\to\mathbb{C}^{\times})_{-1\leq i\leq d}$, the triple

$(\vec{G}=(G^{i})_{0\leq i\leq
d},\pi_{-1}=\pi_{(S,\phi_{-1})},\vec{\phi}=(\phi_{i})_{0\leq i\leq d})$

is a Yu datum. It turns out [Kal19b, Proposition 3.7.8] that these assignments
are bijections modulo the appropriate equivalences. In this way, we can form a
regular supercuspidal representation $\pi_{(S,\theta)}$ from a tame elliptic
regular pair $(S,\theta)$.

###  2.3. $L$-embeddings

In this subsection we explain and study a formalism of Langlands and Shelstad
for extending an embedding $\hat{\jmath}:\widehat{S}\to\widehat{G}$ with
Galois-stable $\widehat{G}$-conjugacy class to an $L$-embedding
${}^{L}j:{}^{L}S\to{}^{L}G$. Fix a $\Gamma\\!_{k}$-pinning of $\widehat{G}$
with maximal torus $\widehat{T}$.

The first difficulty in extending $\hat{\jmath}$ to ${}^{L}j$ is to reconcile
the Galois actions on $\widehat{S}$ and $\widehat{G}$. Specifically, let
$\tau_{G}$ denote the action (homomorphism) of $W_{k}$ on $\widehat{T}$
through its action on $\widehat{G}$; let $\tau_{S}$ denote the action of
$W_{k}$ on $\widehat{T}$ through its action on $\widehat{S}$, transferred
using $\hat{\jmath}$; let $N$ be the normalizer of $\widehat{T}$ in
$\widehat{G}$; and let $\Omega=\Omega(\widehat{G},\widehat{T})$ be the Weyl
group. Given a Weil element $w\in W_{k}$, thought of as an element of
${}^{L}S$, its image under an extension ${}^{L}j$ has the form $nw$ where
$n\in N$ lifts the Weyl element

$\omega_{S,G}(w):=\tau_{S}(w)\tau_{G}(w)^{-1}\in\Omega,$

so that $nw$ acts on $\widehat{T}$ by the $\widehat{S}$-action. The lift
exists precisely because the $\widehat{G}$-conjugacy class of $\hat{\jmath}$
is Galois-stable. To define the extension ${}^{L}j$, then, we need only choose
a specific lift $n$ of $\omega_{S,G}(w)$ to $N$.

For many reductive groups, in particular, the special linear group, the
projection map $N\to\Omega$ does not admit a homomorphic section. Nonetheless,
by finding a good way to lift fundamental reflections, Tits [Tit66] defined a
canonical set-theoretic section $n:\Omega\to N$, which we call the _Tits
lift_. The precise definition of the lift [LS87, Section 2.1] is not so
important for us.

Since the Tits lift is not a homomorphism, the candidate formula

$w\mapsto n(\omega_{S,G}(w))w$

for ${}^{L}j|_{W_{k}}$ does not define a homomorphism $W_{k}\to{}^{L}G$. To
get around this problem, Langlands and Shelstad studied the failure of this
formula to define a homomorphism as measured by the function $t:W_{k}\times
W_{k}\to\widehat{T}$ given by

$t(w_{1},w_{2}):=(n(w_{1})w_{1})(n(w_{2})w_{2})(n(w_{1}w_{2})w_{1}w_{2})^{-1}.$

Using an object $\chi$ called a set of $\chi$-data, whose definition we review
momentarily, they constructed a function $r_{\chi}:W_{k}\to\widehat{S}$ that
negates this failure in the sense that $\partial(\hat{\jmath}\circ
r_{\chi})=t^{-1}$, where $\partial$ is the coboundary operator. Hence the
modified formula

$w\mapsto\hat{\jmath}(r_{\chi}(w))n(\omega_{S,G}(w))w$

does define a homomorphism $W_{k}\to{}^{L}G$, and in total, the extension
${}^{L}j_{\chi}$ of $\hat{\jmath}$ is given by the formula

${}^{L}j_{\chi}(sw)=\hat{\jmath}(s\,r_{\chi}(w))n(\omega_{S,G}(w))w.$

This concludes our outline of the Langlands-Shelstad procedure for extending
an embedding of a torus to an $L$-embedding. Later, in Section 4.2, we recall
the definitions of $\chi$-data and the function $r_{\chi}$.

###  2.4. Regular parameters

This subsection is largely an expository account of Sections 5.2 and 6.1 of
Kaletha’s article [Kal19b]. Our goal is to describe the regular supercuspidal
$L$-parameters and their construction from tame elliptic extra regular pairs.

The definition of regularity for $L$-parameters [Kal19b, Definition 5.2.3] is
not important for us, so we omit the precise statement: roughly speaking, a
parameter $\varphi$ is regular if it takes the wild subgroup to a torus and
the centralizer of the inertia subgroup is abelian (in fact, this is the
definition of a “strongly regular” parameter). Consequently, the groups
$S_{\varphi}$ and $S_{\varphi}^{\natural}$ appearing in the statement of the
formal degree conjecture are abelian, so that their irreducible
representations are one-dimensional. This means that we can ignore the factor
$\dim\rho$ appearing in the Galois side of the formal degree conjecture, since
it equals $1$.

However, we should explain the relationship between regularity and torus-
character pairs. To classify regular parameters, Kaletha introduced an
auxiliary category of _regular supercuspidal $L$-packet data_ whose objects
are quadruples $(S,\hat{\jmath},\chi,\theta)$ consisting of

  * •

a tame torus $S$ of dimension the absolute rank of $G$,

  * •

an embedding $\hat{\jmath}:\widehat{S}\to\widehat{G}$ of complex reductive
groups whose $\widehat{G}$-conjugacy class is Galois stable,

  * •

a minimally ramified set of $\chi$-data for $R(G,S)$, and

  * •

a character $\theta:S(k)\to\mathbb{C}^{\times}$.

For the meaning of _minimally ramified $\chi$-data_, see Definitions 59 and 68
in Section 4.2. These objects are required to satisfy additional conditions
[Kal19b, Definition 5.2.4] that do not concern us here. One can also define a
morphism of such data, organizing them into a category in which all morphisms
are isomorphisms.

######  Warning 14.

In the definition of regular supercuspidal $L$-packet data, we do not assume
that $S$ is a maximal torus of $G$. In general, the $\widehat{G}$-conjugacy
class of $\hat{\jmath}$ gives rise to a Galois-stable $G(\bar{k})$-conjugacy
class of embeddings $S_{\bar{k}}\hookrightarrow G_{\bar{k}}$ whose elements
are called _admissible_ (with respect to $\hat{\jmath}$) [Kal19b, Section
5.1]. Since $S$ is elliptic, this $G(\bar{k})$-conjugacy class contains
embeddings defined over $k$. However, there is no canonical such embedding, or
even $G(k)$-conjugacy class of embeddings. This failure is related to the need
to organize supercuspidal representations into $L$-packets.

The key property of the category of regular supercuspidal $L$-packet data is
that the isomorphism classes of its objects are in natural bijection with
equivalence classes of regular supercuspidal parameters. Given a regular
supercuspidal $L$-packet datum $(S,\hat{\jmath},\theta,\chi)$, its parameter
is the composition

${}^{L}j_{\chi}\circ{}^{L}\theta,$

where ${}^{L}j_{\chi}$ is the $L$-embedding of Section 2.3; this is the
direction of the correspondence that we need to understand when we compute, in
Section 4, the absolute value of the adjoint $\gamma$-factor.

Let $(S,\theta)$ be a tame elliptic extra regular pair. Assume that there is
at least one admissible embedding $j$ of $S$ as a maximal torus of $G$.
Instead of just pulling back the character $\theta$ to $jS$, we need to modify
it slightly: define

$j\theta^{\prime}:=\theta\circ j^{-1}\cdot\varepsilon$

where
$\varepsilon=\varepsilon_{f,\textnormal{ram}}\cdot\varepsilon^{\textnormal{ram}}$
is a certain tamely-ramified Weyl-invariant character of $jS$ [Kal19b, Section
5.3]. Kaletha used the character formula of Adler, DeBacker, and Spice [AS08,
AS09, DS18] to construct from $(S,\theta,j)$ a certain minimally ramified set
$\chi$ of $\chi$-data, which appears, for one, in the definition of
$\varepsilon$. Then the $L$-packet corresponding to
$(S,\hat{\jmath},\chi,\theta)$ consists of the set of regular supercuspidal
representations

$\pi_{(jS,j\theta^{\prime})}$

where $j:S\hookrightarrow G$ ranges over the $G(k)$-conjugacy classes of
admissible embeddings.

##  3\. Automorphic side

In Section 2.2, we outlined Yu’s construction of supercuspidal
representations. In this section we calculate the formal degree of such a
representation. This result is of independent interest, and could be used to
verify the formal degree conjecture for broader classes of supercuspidal
representations than those considered in this paper.

The basic idea of the computation is quite simple, but various technical
complications arise in the process. As Section 3.7 explains, Yu’s
representations are obtained by compact-induction of a finite-dimensional
representation of a compact-open (or really, compact-mod-center) subgroup of
$G(k)$. There is a general formula for the formal degree of such a
representation in terms of the dimension of the starting representation and
the volume of the subgroup. Section 3.2 explains this formula and reviews the
notion of formal degree. To compute the formal degree of a Yu representation,
then, one need only compute two numbers, a dimension and a volume.

The dimension comes from Deligne-Lusztig theory and is straightforward to
compute in our case. We work it out in Section 3.9, where we specialize the
formal degree computation to the case of a regular supercuspidal
representation.

The volume comes from Bruhat-Tits theory, and is much more difficult to
compute. Still, the basic idea is clear. Computing the volume of a compact-
open subgroup amounts to computing its index in a larger group of known
volume, so the volume computation boils down to an index computation. Using
the Moy-Prasad isomorphism, that index computation, in turn, boils down to a
computation of the subquotients in the Moy-Prasad filtration on the Lie
algebra.

The groups used in Yu’s construction generalize the subgroups of the Moy-
Prasad filtration. In Sections 3.3, 3.4, and 3.5 we review their construction
and explain various ways in which Yu’s theory is an elaboration of the theory
of Moy and Prasad [MP94, MP96], or going back even farther, the theory of
Bruhat and Tits [BT72, BT84], Our goal in that lengthy section is to
generalize the Moy-Prasad isomorphism to Yu’s groups, and to understand the
extent to which the Lie algebra of one of Yu’s groups decomposes as a direct
sum of root lines. After these preliminaries, we compute the dimension of such
a Lie algebra in Section 3.6. The Moy-Prasad isomorphism then translates this
dimension into the subgroup index that we need to compute. At this point the
main steps are in place, and in Section 3.8 we walk up the staircase and
finish the computation.

To make statements like Lemma 48 easier to read, in Section 3.1 we introduce
notation particular to Section 3 for reductive groups and their topological
groups of rational points.

###  3.1. Notation

It is important for many reasons to distinguish between a linear algebraic
$k$-group and the group of its rational points. The former is a $k$-scheme and
the latter is just an abstract group, a topological group if $k$ has a
topology. Many constructions are easier on the level of schemes, but for the
kind of representation theory we consider in this paper, one can work only
with the group of rational points.

It is conventional in algebraic geometry to denote a $k$-group by $G$ and the
group of its rational points by $G(k)$. Following the convention in this
section, however, would create a confusing proliferation of “$(k)$” suffixes.
In Section 3 only, therefore, we underline, denoting $k$-groups by
$\underline{G},\underline{H},\dots$ and their groups of rational points by
$G,H,\dots$:

$G:=\underline{G}(k).$

In particular, we write $\vec{\underline{G}}$ for a twisted Levi sequence and
$\vec{G}$ for the sequence of topological groups obtained from it by taking
$k$-points. The convention extends to $\cal O$\- and $\kappa$-schemes as well.
For instance, we write $\underline{G}_{x,r}$ for the smooth $\cal O$-group,
constructed by Yu [Yu02, Section 8], whose group of $\cal O$-points is the
Moy-Prasad group $G_{x,r}$. Similarly, $\underline{G}_{x,0:0+}$ denotes the
maximal reductive quotient of the special fiber of $\underline{G}_{x}$, a
$\kappa$-group.

###  3.2. Formal degree

Here we review the formal degree, following the relevant section of Renard’s
monograph on representations of $p$-adic groups [Ren10, IV.3], then calculate
the formal degree of a compactly induced representation. Contrary to the
conventions of the rest of the paper, in this subsection only, let $G$ be a
unimodular locally profinite group, let $Z$ be the center of $G$, and let
$(\pi,V)$ be a smooth irreducible representation of $G$. Let $A$ be a closed
subgroup of $Z$ such that $Z/A$ is compact, and let $\mu$ be a Haar measure on
$G/A$. Several of our results use $A$ in the statement but are independent of
the choice of $A$.

The _matrix coefficient_ of $\pi$ with respect to $v\in V$ and $v^{\vee}\in
V^{\vee}$ (where $V^{\vee}$ is the smooth dual of $V$) is the function
$\pi_{v,v^{\vee}}:G\to\mathbb{C}$ defined by
$\pi_{v,v^{\vee}}(x)=\langle\pi(x)v,v^{\vee}\rangle$. Since $\pi$ is
irreducible, there is a character $\chi$ of $Z$, called the _central
character_ of $\pi$, such that $\pi(z)=\chi(z)$ for all $z\in Z$. Assume that
the central character is unitary. In this case, the function
$x\mapsto|\pi_{v,v^{\vee}}(x)|^{2}$ is constant on cosets of $A$, and hence
defines a function on $G/A$. We write
$|\pi_{v,v^{\vee}}|_{L^{2}(G/A,\mu)}^{2}$ for the integral of this function,
and say that $(\pi,V)$ is _discrete series_ (with respect to $A$) if
$|\pi_{v,v^{\vee}}|_{L^{2}(G/A,\mu)}<\infty$ for all $v\in V$ and $v^{\vee}\in
V^{\vee}$. This condition is independent of $A$, and also of the choice of
Haar measure on $G/A$.

In practice, it is useful to slightly weaken the definition of a discrete
series representation, and to define a representation to be _essentially
discrete series_ if it becomes discrete series after twisting by some
character of the group.

It can be shown that every discrete series representation is _unitary_ , not
in the sense of Hilbert space representations, but in the sense that it admits
a positive-definite $G$-invariant Hermitian product. The resulting isomorphism
between $\overline{V}$ and $V^{\vee}$ defines a matrix coefficient $\pi_{v,w}$
for $v,w\in V$. Set $\pi_{v}:=\pi_{v,v}$.

For a discrete series representation $(\pi,V)$, one would hope for a
relationship between the norm of a vector and the $L^{2}$-norm of its matrix
coefficient. Although these norms are not equal in general, it turns out that
they differ by a multiplicative constant depending only on $\pi$ and the Haar
measure on $G/A$, not on the vector. This constant is called the _formal
degree_.

######  Definition 15 ([Ren10, IV.3.3]).

Let $(\pi,V)$ be an essentially discrete series representation and let $\mu$
be a Haar measure on $G/A$. If $\pi$ is in addition discrete series then there
exists a positive real constant $\deg(\pi,\mu)$, called the _formal degree_ of
$\pi$, such that for all $v\in V$,

$|\pi_{v}|^{2}_{L^{2}(G/A,\mu)}=\frac{|v|^{2}}{\deg(\pi,\mu)}.$

In general, we define the formal degree of $\pi$ as the formal degree of any
discrete series representation of $G$ obtained from $\pi$ by twisting by a
character of $G$.

Evidently the formal degree scales inversely with the Haar measure used to
define it:

$\deg(\pi,c\mu)=c^{-1}\deg(\pi,\mu).$

######  Remark 16.

Assume $Z$ is compact in this remark for simplicity. In harmonic analysis, one
studies the set of irreducible unitary representations of $G$ by endowing it
with a certain topology and measure, called the _Plancherel measure_. The
measure, though not the topology, depends on a choice of Haar measure $\mu$ on
$G$. An irreducible unitary representation $(\pi,V)$ is discrete series if and
only if it is an isolated point of positive measure, and that measure is the
formal degree of $\pi$ with respect to $\mu$. In particular, if $G$ is compact
and we choose the Haar measure on $G$ giving it volume one then the formal
degree of $(\pi,V)$ is just the dimension of $V$.

Discrete series representations are closely related to supercuspidal
representations, but neither notion implies the other. Recall that a smooth
irreducible representation is _supercuspidal_ if its matrix coefficients have
compact support modulo the center, or equivalently, modulo $A$. If a
supercuspidal representation has unitary central character then it is
certainly discrete series, since compactly supported functions are square-
integrable; but since not every compactly supported function is square-
integrable, in general, there are discrete series representations that are not
supercuspidal. At the same time, since supercuspidal representations need not
have unitary central character, not all supercuspidal representations are
discrete series. However, when $G$ is a $p$-adic reductive group, every
supercuspidal representation of $G$ is essentially discrete series, and we may
thus speak of the representation’s formal degree.

Since Yu’s supercuspidal representations are compactly induced, we compute
their formal degree using a general formula for the formal degree of a
compactly induced representation. As a preliminary step, we define a natural
Hermitian product on a compactly induced representation.

Let $K$ be an open, compact-mod-$A$ subgroup of $G$ (this condition is
independent of $A$), let $(\rho,W)$ be a smooth irreducible unitary
representation of $K$, and let $(\pi,V)$ be the representation of $G$
compactly induced from $(\rho,W)$. That is, $V$ is the space of smooth
functions $f:G\to W$ whose support is compact-mod-$K$ and that satisfy
$f(hx)=\rho(h)f(x)$ for all $h\in K$ and $x\in G$. The representation
$(\pi,V)$ is unitary; in fact, an invariant scalar product is given by the
formula

(17) $\langle f_{1},f_{2}\rangle=\int_{K\backslash G}\langle
f_{1}(x),f_{2}(x)\rangle\mathop{}\\!\mathrm{d}\mu_{K\backslash G}(x),$

where $\mu_{K\backslash G}$ is any positive $G$-invariant Radon measure on
$K\backslash G$, for instance, the counting measure.

######  Lemma 18.

Let $(\rho,W)$ be a finite-dimensional unitary representation of $K$, let
$(\pi,V)$ be the compact induction of $W$ to $G$, and let $\mu$ be a Haar
measure on $G/A$. If $\pi$ is irreducible then

$\deg(\pi,\mu)=\frac{\dim\rho}{\operatorname{vol}(K/(K\cap A),\mu)}.$

###### Proof.

We start by defining an isometric embedding $W\hookrightarrow V$. Given a
vector $w\in W$, define $\dot{w}\in V$ by

$\dot{w}(x)=\mathbbm{1}_{H}(x)\rho(x)w.$

The space $K\backslash G$ is discrete because $K$ is open, so we can take the
measure $\mu_{K\backslash G}$ in Eq. 17 to be the counting measure. With this
choice, the map $W\to V$ defined by $w\mapsto\dot{w}$ is an isometric
embedding. It follows that the matrix coefficient of $\dot{w}$ is the
extension by zero of the matrix coefficient of $w$, and that their
$L^{2}$-norms coincide provided that we take the Haar measure on $K/(K\cap A)$
to be the restriction of $\mu$, denoted also by $\mu$. Then for any nonzero
$w\in W$,

$\deg(\pi,\mu)=|\dot{w}|^{2}\cdot|\pi_{\dot{w}}|_{L^{2}(G/A,\mu)}^{-2}=|w|^{2}\cdot|\rho_{w}|_{L^{2}(K/(K\cap
A),\mu)}^{-2}=\deg(\rho,\mu).$

Finally, since the formal degree scales inversely to the Haar measure used to
define it,

$\deg(\rho,\mu)=\operatorname{vol}(K/(K\cap
A),\mu_{0})^{-1}\deg(\rho,\mu_{0})$

where $\mu_{0}$ is the measure on $K/(K\cap A)$ assigning it total volume one.
The degree of $\rho$ with respect to this measure is just the dimension of
$W$. ∎

###  3.3. Concave-function subgroups: split case

Suppose $\underline{G}$ is split with split maximal torus $\underline{T}$ and
root system $R=R(\underline{G},\underline{T})$, so that $R=\underline{R}$. In
this setting, Bruhat and Tits showed [BT72, Section 6.4] how to construct from
a function $f:R\cup\\{0\\}\to\widetilde{\mathbb{R}}$ and point $x\in\cal
A(\underline{G},\underline{T})$ a subgroup $G_{x,f}$ of $G$ and a subgroup
$\mathfrak{g}_{x,f}$ in $\mathfrak{g}$. In this subsection we review Bruhat
and Tits’s construction. The eventual goal, in later subsections, is to
explain how their construction generalizes to the construction of subgroups
that appear in Yu’s construction of supercuspidals, and to then study Yu’s
subgroups.

The definitions of $G_{x,f}$ and $\mathfrak{g}_{x,f}$ are quite natural. The
point $x$ defines, or “is”, depending on one’s point of view, a family of
additive valuations
$(v^{\alpha}_{x}:\underline{U}^{\alpha}(k)\to\mathbb{R})_{\alpha\in R}$, where
$\underline{U}^{\alpha}$ is the root group of $\alpha$. Since
$\underline{U}^{\alpha}$ is canonically isomorphic to the root line
$\mathfrak{g}^{\alpha}$, we may also think of $v^{\alpha}_{x}$ as a valuation
$\mathfrak{g}^{\alpha}\to\mathbb{R}$. Now let

$U^{\alpha}_{x,r}:=\\{u\in U^{\alpha}:v_{x}^{\alpha}(u)\geq
r\\},\qquad\mathfrak{g}^{\alpha}_{x,r}:=\\{X\in\mathfrak{g}^{\alpha}:v_{x}^{\alpha}(X)\geq
r\\}.$

As for the point $\alpha=0$, we can think of $\underline{T}$ as the root group
$\underline{U}^{0}$ and its Lie algebra $\mathfrak{t}$ as the root space
$\mathfrak{g}^{0}$. These objects carry their own filtrations: let

$T_{r}:=\\{t\in T:\forall\chi\in X^{*}(T),\;\operatorname{ord}(\chi(t)-1)\geq
r\\}$

and let

$\mathfrak{t}_{r}:=\\{X\in T:\forall\chi\in
X^{*}(T),\;\operatorname{ord}(\mathop{}\\!\mathrm{d}\chi(X))\geq r\\}.$

The objects $T_{r}$ and $\mathfrak{t}_{r}$ do not depend on $x$, but we
reserve the right to denote them by $T_{x,r}$ and $\mathfrak{t}_{x,r}$ for
uniformity of notation.

######  Warning 19.

The group $G^{0}$ written here is unrelated to the zeroth group in a twisted
Levi sequence, even though the notation for the two is the same. The two
notations never appear in the same subsection, however, so there is little
risk of confusion.

Given a function $f:R\cup\\{0\\}\to\widetilde{\mathbb{R}}$, let

$U^{\alpha}_{x,f}:=U^{\alpha}_{x,f(\alpha)},\qquad\mathfrak{g}^{\alpha}_{x,f}:=\mathfrak{g}^{\alpha}_{x,f(\alpha)}$

for any $\alpha\in R\cup\\{0\\}$. The group $G_{x,f}$ is then defined as the
subgroup of $G$ generated by the subgroups $U^{\alpha}_{x,f}$ with $\alpha\in
R\cup\\{0\\}$, and the lattice $\mathfrak{g}_{x,f}$ is defined as the subgroup
of $\mathfrak{g}$ spanned by the subgroups $\mathfrak{g}^{\alpha}_{x,f}$ with
$\alpha\in R\cup\\{0\\}$.

######  Remark 20.

When $r=\infty$ the group $U^{\alpha}_{x,\infty}$ is trivial, and when in
addition $\alpha\neq 0$ we can recover the filtrations on the root groups and
root lines as $\mathfrak{g}^{\alpha}_{x,r}=\mathfrak{g}_{x,f}$ and
$U^{\alpha}_{x,r}=G_{x,f}$ where

$f(\beta)=\begin{cases}r&\textnormal{if }\beta=\alpha\\\ \infty&\textnormal{if
not.}\end{cases}$

In order for the construction of $G_{x,f}$ to behave nicely we must assume
that $f$ is nonnegative and _concave_ , that is, that for all finite families
$(\alpha_{i})_{i\in I}$ of elements of $R\cup\\{0\\}$,

$f\Bigl{(}\sum_{i\in I}\alpha_{i}\Bigr{)}\leq\sum_{i\in I}f(\alpha_{i})$

whenever $\sum_{i\in I}\alpha_{i}\in R\cup\\{0\\}$. We can define
$\mathfrak{g}_{x,f}$ for any $f$ whatsoever, but when $f$ is concave,
$\mathfrak{g}_{x,f}$ is a sub Lie algebra of $\mathfrak{g}$.

This completes our discussion of the split case. We next generalize the split
case to the tame case, a simple exercise in Galois descent.

######  Remark 21.

For simplicity of narrative we attributed the construction of $G_{x,f}$ to
Bruhat and Tits, but Yu is partially responsible for the construction, even in
the split case. Bruhat and Tits worked only with functions $f$ such that
$f(0)=0$, but Yu extended their theory to all $f$.

###  3.4. Concave-function subgroups: tame case

. We no longer assume that $\underline{G}$ is split, only that it split over a
tamely ramified extension. Hence we must distinguish between
$\underline{R}:=\underline{R}(\underline{G},\underline{T})$ and
$R:=R(\underline{G},\underline{T})$. In this setting, Yu showed how to
generalize the constructions of the previous subsection, he constructed for
each function $f:R\cup\\{0\\}\to\widetilde{\mathbb{R}}$ and point $x\in\cal
A(\underline{G},\underline{T})$, a subgroup $G_{x,f}$. In fact, Yu defined his
construction only for a special class of functions, which we review in the
next subsection. However, for aesthetic reasons, we found it helpful to review
the theory in this moderately greater generality.

Let $\ell\supseteq k$ be some fixed tame Galois extension of $k$ splitting the
maximal torus $\underline{T}$ and let $\Gamma\\!_{\ell/k}$ be the Galois group
of $\ell$ over $k$. A function
$f:\underline{R}\cup\\{0\\}\to\widetilde{\mathbb{R}}$ can be interpreted as a
Galois-invariant function $R\cup\\{0\\}\to\widetilde{\mathbb{R}}$, and we say
that $f$ is _concave_ if the associated Galois-invariant function is concave.

Since $f$ is Galois-invariant, the subgroups $(\mathfrak{g}_{\ell})_{x,f}$ and
$\underline{G}(\ell)_{x,f}$ are Galois-invariant and we define

$\mathfrak{g}_{x,f}:=(\mathfrak{g}_{\ell})_{x,f}^{\Gamma\\!_{\ell/k}},\qquad
G_{x,f}:=\underline{G}(\ell)_{x,f}^{\Gamma\\!_{\ell/k}}.$

These groups do not depend on the choice of $\ell$. We need not assume that
$f$ is concave to define these objects, although they are best-behaved in that
case. We can also construct for each $\underline{\alpha}$ in
$\underline{R}\cup\\{0\\}$ a root space

$\mathfrak{g}^{\underline{\alpha}}:=\Bigl{(}\bigoplus_{\alpha\in\alpha(\bar{k})}\mathfrak{g}_{\ell}^{\alpha}\Bigr{)}^{\Gamma\\!_{\ell/k}}.$

When $\alpha=0$, the root space $\mathfrak{g}^{0}$ is just the Lie algebra of
$\underline{T}$. Combining this construction with Remark 20, we see that each
root space $\mathfrak{g}^{\underline{\alpha}}$ admits a natural filtration:
define

$\mathfrak{g}^{\underline{\alpha}}_{x,f}=\Bigl{(}\bigoplus_{\alpha\in\underline{\alpha}(\bar{k})}(\mathfrak{g}_{\ell}^{\alpha})_{x,f}\Bigr{)}^{\Gamma\\!_{\ell/k}}.$

It follows immediately from the definitions that

$\mathfrak{g}_{x,f}=\bigoplus_{\underline{\alpha}\in\underline{R}\cup\\{0\\}}\mathfrak{g}^{\underline{\alpha}}_{x,f}.$

More generally, given a subset
$\underline{R}^{\prime}\subseteq\underline{R}\cup\\{0\\}$, let

$\mathfrak{g}^{\underline{R}^{\prime}}:=\bigoplus_{\underline{\alpha}\in\underline{R}^{\prime}}\mathfrak{g}^{\underline{\alpha}},\qquad\mathfrak{g}^{\underline{R}^{\prime}}_{x,f}:=\bigoplus_{\underline{\alpha}\in\underline{R}^{\prime}}\mathfrak{g}^{\underline{\alpha}}_{x,f}.$

######  Warning 22.

In comparison to the function of Remark 20, the function
$f:\underline{R}\cup\\{0\\}\to\widetilde{\mathbb{R}}$ defined, for a fixed
$\underline{\alpha}\in\underline{R}$, by

$f(\underline{\beta})=\begin{cases}r&\textnormal{if
}\underline{\beta}=\underline{\alpha}\\\ \infty&\textnormal{if
not}\end{cases}$

is generally not concave. Failure of concavity relates to the fact that when
$\underline{T}$ is not split, there is generally no “root group” (or even
“root variety”) $\underline{U}^{\underline{\alpha}}$ whose Lie algebra is
$\mathfrak{g}^{\underline{\alpha}}$.

######  Remark 23.

If we assume that $f$ is finite, or without loss of generality, that $f$ takes
values in $\mathbb{R}$, then the group $G_{x,f}$ can be interpreted as the
integral points of an $\cal O$-group: that is, there is a canonical $\cal
O$-group $\underline{G}_{x,f}$, an integral model of $\underline{G}$, such
that

$\underline{G}_{x,f}(\cal O)=G_{x,f}$

as subgroups of $G$. The construction is due to Yu [Yu02, Section 8].

Having defined the group $G_{x,f}$ and Lie algebra $\mathfrak{g}_{x,f}$, we
now study them. There are three areas of interest for our later applications.

First, in our calculation of the formal degree, it greatly simplifies notation
to reduce to the case where $G$ has anisotropic center. We record here the
lemma effecting this reduction.

######  Lemma 24.

Let $\underline{G}$ be a reductive $k$-group, let $\underline{T}$ be a tame
maximal torus of $\underline{G}$, let
$f:\underline{R}(\underline{G},\underline{T})\to\widetilde{\mathbb{R}}$ be a
positive concave function, let $\underline{A}\subseteq\underline{T}$ be a
split central torus of $\underline{G}$, and let $y\in\cal
A(\underline{T},\underline{G})$. Then the groups $G_{y,f}/(A\cap G_{y,f})$ and
$(G/A)_{y,f}$ are identical subgroups of $G/A$.

###### Proof.

This follows in the split case using Hilbert’s Theorem 90, and the general
case follows immediately from the split case by taking Galois invariants. ∎

Second, when we compute certain subgroup indices in Section 3.8, we need to
understand how to intersect groups of the form $G_{x,f}$, for fixed $x$.

######  Lemma 25.

Let $f,g:\underline{R}\cup\\{0\\}\to\widetilde{\mathbb{R}}$ be positive
concave functions. Then

$G_{x,f}\cap G_{x,g}=G_{x,\max(f,g)}.$

###### Proof.

In the case where $G$ is split, a classical result of Bruhat and Tits [BT72,
(6.4.48)] can be used to show [Yu02, 8.3.1] that the natural multiplication
map

$\prod_{\alpha\in R\cup\\{0\\}}U^{\alpha}_{x,f(\alpha)}\to G_{x,f}$

is a bijection for a certain ordering of the factors. Using tame descent, this
observation reduces the proof to the obvious fact (still in the split case)
that $U^{\alpha}_{x,r}\cap U^{\alpha}_{x,s}=U^{\alpha}_{x,\max(r,s)}$. ∎

Third and lastly, we compare subgroup indices between $G$ and $\mathfrak{g}$,
generalizing the Moy-Prasad isomorphism. In Section 3.8, this comparison
reduces a volume computation to a length computation, which we carry out in
Section 3.6.

######  Lemma 26.

Let $f,g:\underline{R}\cup\\{0\\}\to\widetilde{\mathbb{R}}$ be positive
concave functions such that $f\leq g$. Assume in addition that the following
condition is satisfied:

$g(a)\leq\sum_{i\in I}f(a_{i})+\sum_{j\in J}f(b_{j})$

for all non-empty finite sequences $(a_{i})_{i\in I}$ and $(b_{j})_{j\in J}$
of elements of $\underline{R}\cup\\{0\\}$ such that $a:=\sum_{i\in
I}a_{i}+\sum_{j\in J}b_{j}\in\underline{R}\cup\\{0\\}$. Then

  1. (1)

$[G_{x,f},G_{x,f}]\subseteq G_{x,g}$, so that the group $G_{x,f:g}$ is
abelian, and

  2. (2)

there is a canonical isomorphism $\mathfrak{g}_{x,f:g}\simeq G_{x,f:g}$ of
abelian groups.

###### Proof.

This follows from [BT72, 6.4.44] and [BT72, 6.4.48] in the split case and
[Yu02, Section 2] in general. ∎

######  Remark 27.

In Lemma 26, it is tempting to instead impose the simpler condition that
$g(a)\leq\sum_{i\in I}f(a_{i})$ for all non-empty finite sequences
$(a_{i})_{i\in I}$ such that $\sum_{i\in I}a_{i}=a$. However, this stronger
condition would significantly weaken the conclusion: the condition implies, by
taking the constant sequence, that $g\leq f$, so that $g=f$.

######  Corollary 28.

Let $f,g:\underline{R}\cup\\{0\\}\to\mathbb{R}$ be positive concave functions
such that $f\leq g$, and suppose there is a chain of concave functions

$f=f_{0}\leq f_{1}\leq\cdots\leq f_{n}=g$

such that for each $i$ with $1\leq i\leq n$, the pair $(f_{i-1},f_{i})$
satisfies the conditions of Lemma 26. Then

$|G_{x,f:g}|=|\mathfrak{g}_{x,f:g}|.$

###  3.5. Yu’s groups

, In our application, it is enough to work with the subgroups constructed from
a certain restricted class of concave functions, those constructed from
admissible sequences and tame twisted Levi sequences. The construction
specializes that of Section 3.4, so we work in the same setting. After
reviewing Yu’s construction of these groups, we specialize the theory of
Section 3.4 to show that they admit a Moy-Prasad isomorphism.

A sequence $\vec{r}=(r_{i})_{i=0}^{d}$ in $\widetilde{\mathbb{R}}$ is
_admissible_ if there is some $j$ with $0\leq j\leq d$ such that $0\leq
r_{0}=\cdots=r_{j}$ and $\tfrac{1}{2}r_{j}\leq r_{j+1}\leq\cdots r_{d}$. The
admissible sequence is _weakly increasing_ if, in addition, $r_{i}\leq
r_{i+1}$ for all $i$ with $0\leq i\leq d-1$.

Recall the definition of a (tame) twisted Levi sequence $\vec{\underline{G}}$
from Section 2.2. We assume that the tame maximal torus $\underline{T}$ of
$\underline{G}$ is contained in $\underline{G}^{0}$, so that $x\in\cal
B(\underline{G}^{i})$ for each $i$. For each $i$ with $0\leq i\leq d$, let
$\underline{R}_{i}:=\underline{R}(\underline{G}^{i},\underline{T})$.

It is sometimes necessary to work with truncated twisted Levi sequences. Given
integers $a$ and $b$ such that $0\leq a\leq b\leq d$, let

$\vec{\underline{G}}^{[a,b]}:=(\underline{G}^{a}\subsetneq\cdots\subsetneq\underline{G}^{b});$

given an integer $i$ with $0\leq i\leq d$, define
$\vec{\underline{G}}^{(i)}:=\vec{\underline{G}}^{[0,i]}$.

Given an admissible sequence $\vec{r}$ and a twisted Levi sequence
$\vec{\underline{G}}$ of the same length $d$, define the function
$f_{\vec{r}}:\underline{R}\to\widetilde{\mathbb{R}}$ by

$f_{\vec{r}}(\underline{\alpha}):=\begin{cases}r_{0}&\textnormal{if
}\underline{\alpha}\in\underline{R}_{0}\cup\\{0\\}\\\ r_{i}&\textnormal{if
}\underline{\alpha}\in\underline{R}_{i}\setminus\underline{R}_{i-1},\,1\leq
i\leq d.\end{cases}$

Since $\vec{r}$ is admissible, the function $f_{\vec{r}}$ is concave [Yu01,
Lemma 1.2]. Hence we may define

$\vec{G}_{x,\vec{r}}:=G_{x,f_{\vec{r}}},\qquad\vec{\mathfrak{g}}_{x,\vec{r}}:=\mathfrak{g}_{x,f_{\vec{r}}}.$

Given a second admissible sequence $\vec{s}$ of length $d$ with $r_{i}\leq
s_{i}$ for all $i$, define

$\vec{G}_{x,\vec{r}:\vec{s}}:=G_{x,f_{\vec{r}}:f_{\vec{s}}},\qquad\vec{\mathfrak{g}}_{x,\vec{r}:\vec{s}}:=\mathfrak{g}_{x,f_{\vec{r}}:f_{\vec{s}}}.$

######  Remark 29.

The group $\vec{G}_{x,\vec{r}}$ depends only on $\vec{\underline{G}}$, $x$,
and $\vec{r}$; in particular, it is independent of the choice of torus $T$,
provided that $x\in\cal B(T)$ [Yu01, Section 1]. The same cannot necessarily
be said for a general group of the form $G_{x,f}$, however, because the domain
of definition of $f$ knows something about the torus $T$, namely, the Galois
action on the root system. This Galois action can vary among maximal tori
whose buildings are identical subsets of $\cal B(G)$.

In this setting, Yu generalized [Yu01, Lemma 1.3 and Corollary 2.4] the Moy-
Prasad isomorphism.

######  Lemma 30.

Let $\vec{\underline{G}}$ be a tame Levi sequence of length $d$ and let
$\vec{r}$ and $\vec{s}$ be admissible sequences of length $d$ such that for
all $i$,

(31) $0<r_{i}\leq s_{i}\leq\min(r_{i},\dots,r_{d})+\min(\vec{r}).$

Then $\vec{G}_{x,\vec{r}:\vec{s}}$ is an abelian group canonically isomorphic
to $\vec{\mathfrak{g}}_{x,\vec{r}:\vec{s}}$.

###### Proof.

Eq. 31 implies that the pair $(f_{\vec{r}},f_{\vec{s}})$ satisfies the
conditions of Lemma 26. ∎

######  Corollary 32.

Let $\vec{\underline{G}}$ be a tame Levi sequence of length $d$ and let
$\vec{r}$ and $\vec{s}$ be weakly increasing admissible sequences of length
$d$ such that $0<r_{i}\leq s_{i}<\infty$ for all $i$. Then

$|\vec{G}_{x,\vec{r}:\vec{s}}|=|\vec{\mathfrak{g}}_{x,\vec{r}:\vec{s}}|.$

###### Proof.

Since $\vec{r}$ is weakly increasing, Eq. 31 simplifies to the condition that

(33) $s_{i}\in[r_{i},r_{i}+r_{0}]\quad\textnormal{ for all~{}$i$.}$

It is now an elementary but tedious exercise to construct a chain
$(\vec{s}\,^{(j)})_{0\leq j\leq N}$ of weakly increasing admissible sequences
$\vec{s}\,^{(j)}=(s^{(j)}_{0}\leq\cdots\leq s^{(j)}_{d})$, where $N\gg 0$,
such that $s^{(0)}=\vec{r}$ and $s^{(d)}=\vec{s}$, such that $s^{(j)}_{i}\leq
s^{(j+1)}_{i}$ for all $i$ and $j$ with $0\leq j\leq N-1$, and such that each
pair $(\vec{s}\,^{(j-1)},\vec{s}\,^{(j)})$ satisfies Eq. 31 of Lemma 30. After
completing this exercise, we invoke Corollary 28. ∎

We need the Corollary in the following special case only.

######  Corollary 34.

Let $\vec{\underline{G}}$ be a tame Levi sequence of length $d$ and let
$\vec{r}$ be a weakly increasing admissible sequence of length $d$. Then

$|\vec{G}_{x,0+:\vec{r}}|=|\vec{\mathfrak{g}}_{x,0+:\vec{r}}|.$

###  3.6. Length computation

Retaining the notation of Section 3.4, let $f:\underline{R}\to\mathbb{R}$ be a
positive function and let $\underline{R}^{\prime}\subseteq\underline{R}$ be a
subset. Our goal in this subsection is to compute the length of the $\cal
O$-module $\mathfrak{g}^{\underline{R}^{\prime}}_{x,0+:f}$, culminating in
Theorem 38.

We start by studying the jumps in the filtration on
$\mathfrak{g}^{\underline{\alpha}}_{x}$. For
$\underline{\alpha}\in\underline{R}$, consider the set

$\operatorname{ord}_{x}\underline{\alpha}:=\\{t\in\mathbb{R}:\mathfrak{g}^{\underline{\alpha}}_{x,t:t+}\neq
0\\}$

of jumps in the Moy-Prasad filtration of $\mathfrak{g}^{\underline{\alpha}}$,
defined and studied by DeBacker and Spice [DS18, Definition 3.6]. A full
description of $\operatorname{ord}_{x}\underline{\alpha}$ requires an
understanding of the point $x$, and thus the way in which $\cal
B(\underline{T})$ embeds in $\cal B(\underline{G})$. This is quite difficult
in general. But for us it is enough to know several weak properties of these
sets.

######  Lemma 35.

Let $\underline{T}$ be a tame maximal torus of $\underline{G}$, let $x\in\cal
A(\underline{G},\underline{T})$, and let $\underline{\alpha}\in\underline{R}$.
Then

  1. (1)

$\operatorname{ord}_{x}(-\underline{\alpha})=-\operatorname{ord}_{x}\underline{\alpha}$,

  2. (2)

$\operatorname{ord}_{x}\underline{\alpha}$ is an
$\operatorname{ord}_{k}(k_{\underline{\alpha}}^{\times})$-torsor, and

  3. (3)

$\operatorname{len}\mathfrak{g}^{\underline{\alpha}}_{x,t:t+}=[\kappa_{\underline{\alpha}}:\kappa]\mathbbm{1}_{\operatorname{ord}_{x}\underline{\alpha}}(t)$.

###### Proof.

Property (1) follows from a $\operatorname{PGL}_{2}$ calculation [DS18,
Corollary 3.11]. As for the other properties, earlier we mentioned that
$\mathfrak{g}^{\underline{\alpha}}$ was isomorphic to
$k_{\underline{\alpha}}$, though not canonically. Choose one such isomorphism
$\phi:k_{\underline{\alpha}}\to\mathfrak{g}_{\underline{\alpha}}$. This
isomorphism is compatible with the Moy-Prasad filtration in the following
sense: there is a real number $r_{0}$ such that for all $r\in\mathbb{R}$, the
isomorphism $\phi$ restricts to an isomorphism
$k_{\underline{\alpha},r+r_{0}}\simeq\mathfrak{g}_{\underline{\alpha},r}$ of
$\cal O$-modules. Properties (2) and (3) now follow immediately. ∎

######  Corollary 36.

Let $\underline{T}$ be a tame maximal torus of $\underline{G}$, let $x\in\cal
A(\underline{G},\underline{T})$, let $f:\underline{R}\cup\\{0\\}\to\mathbb{R}$
be a positive function, and let
$\underline{R}^{\prime}\subseteq\underline{R}$. Then

$\operatorname{len}\mathfrak{g}^{\underline{R}^{\prime}}_{x,0+:f}=\sum_{\underline{\alpha}\in\underline{R}^{\prime}}\sum_{0<t<f(\underline{\alpha})}[\kappa_{\underline{\alpha}}:\kappa]\mathbbm{1}_{\operatorname{ord}_{x}\underline{\alpha}}(t)$

###### Proof.

It suffices to prove the Corollary in the case where
$\underline{R}^{\prime}=\\{\underline{\alpha}\\}$. Then both sides equal

$\sum_{\begin{subarray}{c}t\in\operatorname{ord}_{\underline{\alpha}}x\\\
0<t<f(\underline{\alpha})\end{subarray}}\operatorname{len}\mathfrak{g}^{\underline{\alpha}}_{x,t:t+}$

by the third part of Lemma 35. ∎

To simplify the sum in Corollary 36 we prove a more general result about
summation of discretely supported functions $h:\mathbb{R}\to\mathbb{N}$. Given
a lattice $\Lambda\subset\mathbb{R}$, that is, a nontrivial cyclic subgroup,
say that $h$ is _$\Lambda$ -periodic_ if $h(t+\lambda)=h(t)$ for all
$t\in\mathbb{R}$ and $\lambda\in\Lambda$. Given a bounded interval
$I\subset\mathbb{R}$ with endpoints $a<b$, define

$\sideset{}{{}^{\prime}}{\sum}_{I}h=\sideset{}{{}^{\prime}}{\sum}_{t\in
I}h(t)=\sum_{a<t<b}h(t)+\tfrac{1}{2}[a\in I]h(a)+\tfrac{1}{2}[b\in I]h(b),$

where $[\cdot]$ is the Iverson bracket. Because of the normalization at the
endpoints, this summation operator enjoys the property that

$\sideset{}{{}^{\prime}}{\sum}_{I}h+\sideset{}{{}^{\prime}}{\sum}_{J}h=\sideset{}{{}^{\prime}}{\sum}_{I\cup
J}h$

whenever $I$, $J$, and $I\cup J$ are compact intervals.

######  Lemma 37.

Let $\Lambda\subset\mathbb{R}$ be a free abelian group of rank one, let
$\lambda_{0}:=\min(\Lambda\cap\mathbb{R}_{>0})$, let
$h:\mathbb{R}\to\mathbb{N}$ be a discretely supported function, and let
$H(s):=\sum_{0\leq t\leq s}^{\prime}h(t)$ for $s>0$. Suppose $h$ is even and
$\Lambda$-periodic. Then for all $s\in\tfrac{1}{2}\Lambda\cap\mathbb{R}_{>0}$,

$H(s)=\frac{s}{\lambda_{0}}H(\lambda_{0}).$

###### Proof.

Since $H(s+\lambda)=H(s)+H(\lambda)$, induction reduces the proof to the case
where $s=\lambda_{0}$ or $s=\tfrac{1}{2}\lambda_{0}$. The first case is
obvious; for the second, use that $h(t)=h(\lambda_{0}-t)$. ∎

We can now compute a certain sum that appears in the formal degree.

######  Theorem 38.

Let $T$ be a tame maximal torus of $G$, let $x\in\cal
A(\underline{G},\underline{T})$, let $f:\underline{R}\cup\\{0\\}\to\mathbb{R}$
be a positive even function, and let
$\underline{R^{\prime}}\subseteq\underline{R}$ be a subset closed under
negation. Suppose that
$f(\underline{\alpha})\in\tfrac{1}{2}\operatorname{ord}(k_{\underline{\alpha}}^{\times})$
for all $\underline{\alpha}\in\underline{R}$. Then

$\operatorname{len}(\mathfrak{g}_{x,0+:f}^{\underline{R^{\prime}}})+\tfrac{1}{2}\operatorname{len}(\mathfrak{g}_{x,0:0+}^{\underline{R^{\prime}}})+\tfrac{1}{2}\operatorname{len}(\mathfrak{g}_{x,f:f+}^{\underline{R^{\prime}}})=\sum_{\underline{\alpha}\in\underline{R^{\prime}}}[k_{\underline{\alpha}}:k]f(\underline{\alpha}).$

###### Proof.

By Corollary 36, the lefthand side of the Theorem is

$\operatorname{len}(\mathfrak{g}_{x,0+:f}^{\underline{R^{\prime}}})+\tfrac{1}{2}\operatorname{len}(\mathfrak{g}_{x,0:0+}^{\underline{R^{\prime}}})+\tfrac{1}{2}\operatorname{len}(\mathfrak{g}_{x,f:f+}^{\underline{R^{\prime}}})=\sum_{\underline{\alpha}\in\underline{R^{\prime}}}\sideset{}{{}^{\prime}}{\sum}_{0\leq
t\leq
f(\underline{\alpha})}[\kappa_{\underline{\alpha}}:\kappa]\mathbbm{1}_{\operatorname{ord}_{x}\underline{\alpha}}(t).$

Since $f$ is even and
$\kappa_{\underline{\alpha}}=\kappa_{-\underline{\alpha}}$, the righthand side
above is

$\sum_{\underline{\alpha}\in\underline{R^{\prime}}}\sideset{}{{}^{\prime}}{\sum}_{0\leq
t\leq
f(\underline{\alpha})}\tfrac{1}{2}[\kappa_{\underline{\alpha}}:\kappa]\bigl{(}\mathbbm{1}_{\operatorname{ord}_{x}\underline{\alpha}}(t)+\mathbbm{1}_{\operatorname{ord}_{x}(-\underline{\alpha})}(t)\bigr{)}.$

By Lemma 35, the function
$\mathbbm{1}_{\operatorname{ord}_{x}\underline{\alpha}}+\mathbbm{1}_{\operatorname{ord}_{x}(-\underline{\alpha})}$
is even and $\operatorname{ord}(k_{\underline{\alpha}}^{\times})$-periodic.
Hence we may apply Lemma 37 to conclude that

$\sideset{}{{}^{\prime}}{\sum}_{0\leq t\leq
f(\underline{\alpha})}\tfrac{1}{2}[\kappa_{\underline{\alpha}}:\kappa]\bigl{(}\mathbbm{1}_{\operatorname{ord}_{x}\underline{\alpha}}(t)+\mathbbm{1}_{\operatorname{ord}_{x}(-\underline{\alpha})}(t)\bigr{)}=[k_{\underline{\alpha}}:k]f(\underline{\alpha}),$

using the fact that

$\sideset{}{{}^{\prime}}{\sum}_{0\leq t\leq
1}[\kappa_{\underline{\alpha}}:\kappa]\mathbbm{1}_{\operatorname{ord}_{x}\underline{\alpha}}(t)=\sum_{0\leq
t<1}[\kappa_{\underline{\alpha}}:\kappa]\mathbbm{1}_{\operatorname{ord}_{x}\underline{\alpha}}(t)=[\kappa_{\underline{\alpha}}:\kappa]\operatorname{ord}_{k}(k_{\alpha}^{\times})=[k_{\underline{\alpha}}:k].\qed$

###  3.7. Yu’s construction

In this subsection we describe Yu’s supercuspidal representations, following
Hakim and Murnaghan’s expanded exposition [HM08, Section 3] of Yu’s original
paper [Yu01]. Yu’s full construction is quite elaborate, but fortunately, it
is enough for us to understand only the parts of the construction needed to
calculate the formal degree.

A _cuspidal $\underline{G}$-datum_ is a 5-tuple
$\Psi=(\vec{\underline{G}},y,\vec{r},\rho,\vec{\phi})$ consisting of:

  * •

a tame twisted Levi sequence $\vec{\underline{G}}$ such that
$\underline{Z}^{0}/\underline{Z}$ is anisotropic, where $\underline{Z}$ is the
center of $\underline{G}$ and $\underline{Z}^{0}$ is the center of
$\underline{G}^{0}$;

  * •

a point $y$ in the apartment of a tame maximal torus of $\underline{G}^{0}$;

  * •

an increasing sequence $\vec{r}=(0<r_{0}<r_{1}<\cdots<r_{d-1}\leq r_{d})$ of
real numbers (if $d=0$ then we only require that $0\leq r_{0}$);

  * •

an irreducible representation $\rho$ of $G^{0}_{[y]}$ whose restriction to
$G^{0}_{y,0+}$ is $1$-isotypic and for whom the compact induction
$\operatorname{c-Ind}_{K^{0}}^{G^{0}}\rho$ is irreducible (hence
supercuspidal);

  * •

a sequence $\vec{\phi}=(\phi_{0},\dots,\phi_{d})$ of characters, with
$\phi_{i}$ a character of $G^{i}$, such that $\phi_{d}=1$ if $r_{d}=r_{d-1}$
and otherwise $\phi_{i}$ has depth $r_{i}$ for all $i$.

The datum is _generic_ if for each $i\neq d$ the character $\phi_{i}$ of
$G^{i}$ is $\underline{G}^{i+1}$-generic in the sense of Section 2.1, and the
datum is _regular_ if, in addition, the depth-zero supercuspidal
representation $\operatorname{c-Ind}_{K^{0}}^{G^{0}}\rho$ is regular.

Many of the objects used in Yu’s construction and built from a cuspidal
$\underline{G}$-datum do not depend on the representations $\rho$ and
$\vec{\phi}$. To make this independence explicit, we define a _cuspidal
$\underline{G}$-datum without representations_ to be a 3-tuple
$(\vec{\underline{G}},y,\vec{r})$ consisting of the first three components of
a cuspidal $\underline{G}$-datum.

Let $\Psi=(\vec{\underline{G}},\vec{r},y)$ be a cuspidal $\underline{G}$-datum
without representations. From $\Psi$ we can construct the following subgroups:

$\displaystyle K^{0}$ $\displaystyle=G^{0}_{[y]}$ $\displaystyle K^{0}_{+}$
$\displaystyle=G^{0}_{y,0+}$ $\displaystyle K^{i+1}$
$\displaystyle=K^{0}\vec{G}^{(i+1)}_{y,(0+,s_{0},\dots,s_{i})}$ $\displaystyle
K^{i+1}_{+}$ $\displaystyle=\vec{G}^{(i+1)}_{y,(0+,s_{0}+,\dots,s_{i}+)}$
$\displaystyle J^{i+1}$ $\displaystyle=(G^{i},G^{i+1})_{y,(r_{i},s_{i})}$
$\displaystyle J^{i+1}_{+}$ $\displaystyle=(G^{i},G^{i+1})_{y,(r_{i},s_{i}+)}$
$\displaystyle K$ $\displaystyle:=K^{d}:=K^{d+1}$ $\displaystyle K_{+}$
$\displaystyle:=K^{d}_{+}:=K^{d+1}_{+}.$

Here $0\leq i\leq d-1$ and $s_{i}:=r_{i}/2$. Generally the dependence of these
objects on $\Psi$ is implicit, but if we wish to make the dependence explicit
we indicate it with a subscript, for instance, $K=K_{\Psi}$. When $\Psi$ is
regular, an additional group can be constructed: the maximal torus $S$ of
Section 2.2, maximally unramified in $G^{0}$. The groups $K^{i+1}$ and
$K^{i+1}_{+}$ are particularly important; later on, we need to express them in
the following alternative ways.

(39) $\displaystyle K^{i+1}$
$\displaystyle=K^{0}\vec{G}^{[1,i+1]}_{y,s_{0},\dots,s_{i}}$ $\displaystyle
K^{i+1}_{+}$
$\displaystyle=K^{0}_{+}\vec{G}^{[1,i+1]}_{y,s_{0}+,\dots,s_{i}+}$ (40)
$\displaystyle K^{i+1}$ $\displaystyle=K^{i}J^{i+1}$ $\displaystyle
K^{i+1}_{+}$ $\displaystyle=K^{i}_{+}J^{i+1}.$

We can now outline Yu’s construction. Let
$\Psi=(\vec{\underline{G}},y,\vec{r},\rho,\vec{\phi})$ be a generic cuspidal
$\underline{G}$-datum. For each $0\leq i\leq d-1$ there is a certain finite-
dimensional irreducible representation111Our $\rho_{i}$ is Hakim and
Murnaghan’s $\phi_{i}^{\prime}$. $\rho_{i}$ of $K^{i+1}$ constructed from
$\phi_{i}$. To specify $\rho_{i}$ precisely one uses the theory of the Weil-
Heisenberg representation, but for our purposes, it is enough to know that

$\dim\rho_{i}=[J^{i+1}:J^{i+1}_{+}]^{1/2}.$

The representation $\rho_{i}$ is then inflated to a representation222 Our
$\tau_{i}$ is Hakim and Murnaghan’s $\kappa_{i}$. $\tau_{i}$ of $K$. Section
3.9 explains in more detail how the inflation procedure works, but at the
moment, it is enough to know that the inflation procedure preserves dimension.
In the edge case $i=-1$ take $\tau_{-1}$ to be the inflation (by the same
procedure) of $\rho$ to $K$, and in the edge case $i=d$ take $\tau_{d}$ to be
the restriction of $\phi_{d}$ to $K$; we could also handle the case $i=-1$ in
the same way as the case $0\leq i\leq d-1$ by defining $\rho_{-1}:=\rho$.
Finally, define the supercuspidal representation $\pi$ attached to $\Psi$ as
the compact induction

$\pi=\operatorname{c-Ind}_{K}^{G}\tau,\qquad\tau:=\tau_{-1}\otimes\tau_{0}\otimes\cdots\otimes\tau_{d}.$

In summary, then, by Lemma 18 the formal degree of the supercuspidal
representation $\pi$ attached to a cuspidal generic Yu-datum $\Psi$ has formal
degree

(41)
$\deg(\pi_{\Psi},\mu)=\frac{\dim\rho\cdot\prod_{i=0}^{d-1}[J^{i+1}:J^{i+1}_{+}]^{1/2}}{\operatorname{vol}(K/A,\mu)}$

where $\underline{A}$ is the maximal split central subtorus of $\underline{G}$
and $\mu$ is a Haar measure on $G/A$.

It greatly simplifies the notation in our computation of the formal degree to
reduce to the case where the center of $\underline{G}$ is anisotropic, that
is, $\underline{A}=1$. Given a generic cuspidal $\underline{G}$-datum without
representations $\Psi=(\vec{\underline{G}},y,\vec{r})$, let
$\overline{\Psi}=(\overrightarrow{\underline{G}^{\textnormal{a}}},y,\vec{r}\,)$
denote the reduction of $\Psi$ modulo $\underline{A}$: that is,
$\underline{G}^{\textnormal{a},i}:=\underline{G}^{i,\textnormal{a}}=\underline{G}^{i}\\!/\\!\underline{A}$.

######  Lemma 42.

Let $\Psi$ be a generic cuspidal $\underline{G}$-datum without representations
and let $\overline{\Psi}$ be the reduction of $\Psi$ modulo $\underline{A}$.
Then

  1. (1)

the groups $K_{\overline{\Psi}}$ and $K_{\Psi}/A$ are identical as subgroups
of $G/A$, and

  2. (2)

$[J_{\Psi}^{i+1}:J^{i+1}_{\Psi,+}]=[J_{\overline{\Psi}}^{i+1}:J^{i+1}_{\overline{\Psi},+}]$
for all $0\leq i\leq d-1$.

###### Proof.

For the first part, since

$\frac{K_{\Psi}}{A}=\frac{G^{0}_{[y]}}{A}\cdot\frac{A\vec{G}^{(d)}_{y,(0+,\vec{s})}}{\vec{G}^{(d)}_{y,(0+,\vec{s})}}=\frac{G^{0}_{[y]}}{A}\cdot\frac{\vec{G}^{(d)}_{y,(0+,\vec{s})}}{\vec{G}^{(d)}_{y,(0+,\vec{s})}\cap
A},$

it suffices by Lemma 24 to show that $G^{0}_{[y]}/\\!A=(G^{0}\\!/\\!A)_{[y]}$,
and this follows immediately from the surjectivity of the map
$\underline{G}\to(\underline{G}/\\!\underline{A})(k)$ and the natural
identification of the reduced buildings of $\underline{G}$ and
$\underline{G}/\\!\underline{A}$.

The second part follows by an argument similar to the proof of Lemma 24. ∎

###  3.8. Degree computation

In this subsection we compute the formal degree of Yu’s supercuspidal
representation. Let $\underline{G}$ be a reductive $k$-group let
$\Psi=(\vec{\underline{G}},y,\vec{r},\rho,\vec{\phi})$ be a cuspidal
$\underline{G}$-datum, let $\underline{A}$ be the maximal split central
subtorus of $\underline{G}$, and let $\mu$ be the Haar measure on $G\\!/\\!A$
attached by Gan and Gross [GG99, HII08a] to a level-zero additive character of
$k$.

Starting from Eq. 41, we will reduce the problem of computing the formal
degree to the problem of computing certain subgroup indices. We will then be
in a position to apply Theorem 38, finishing the calculation. By Lemma 42, we
may assume for now that the center of $G$ is anisotropic, though of course
this restriction will have to be relaxed in the final formula.

We will start by simplifying the volume of $K$. To begin with,

(43)
$\operatorname{vol}(K,\mu)^{-1}=\operatorname{vol}(G_{y,0+},\mu)^{-1}\frac{[KG_{y,0+}:K]}{[KG_{y,0+}:G_{y,0+}]}.$

######  Lemma 44.

Let $G$ be a group, let $H$ be a subgroup of $G$, and let $N\subseteq M$ be
subgroups of $G$ normalized by $H$.333This condition is needed for $MH$ and
$NH$ to be groups; it might be possible to weaken it. Then

$[MH:NH]=\frac{[M:N]}{[M\cap H:N\cap H]},$

provided that all three indices in the expression are finite.

###### Proof.

The inclusion $M\cap H\hookrightarrow M$ induces an injective map $M\cap
H/N\cap H\hookrightarrow M/N$ which we can use to interpret the former as a
subgroup of the latter. The group $M/N$ acts transitively by left
multiplication on the coset space $MH/NH$. Consider the stabilizer of the
identity coset under this action. Clearly $M\cap H/N\cap H$ lies in the
stabilizer, and conversely, it is easy to see that any representative of an
element of the stabilizer can be translated by an element of $N$ to lie in
$M\cap H$. So $M\cap H/N\cap H$ is the stabilizer of the identity element, and
the orbit-stabilizer theorem concludes the proof. ∎

By Equations 39, 44, and 25,

(45)
$[KG_{y,0+}:K]=[K^{0}G_{y,0+}:K^{0}\vec{G}^{[1,d]}_{y,(s_{0},\dots,s_{d-1})}]=\frac{[G_{y,0+}:\vec{G}^{[1,d]}_{y,(s_{0},\dots,s_{d-1})}]}{[G^{0}_{y,0+}:G^{0}_{y,s_{0}}]}.$

It follows from Lemma 25 again that $[KG_{y,0+}:G_{y,0+}]=[K^{0}:K^{0}_{+}]$.
Combining this calculation with Eqs. 39, 43, and 45 yields

$\operatorname{vol}(K,\mu)^{-1}=\operatorname{vol}(G_{y,0+},\mu)^{-1}\frac{[KG_{y,0+}:K]}{[KG_{y,0+}:G_{y,0+}]}=\operatorname{vol}(G_{y,0+},\mu)^{-1}\frac{[G_{y,0+}:\vec{G}^{[1,d]}_{y,(s_{0},\dots,s_{d-1})}]}{[G^{0}_{[y]}:G^{0}_{y,s_{0}}]}.$

The volume of $G_{y,0+}$ is known from the literature.

######  Lemma 46.

Let $x\in\cal B(G)$. Then $\operatorname{vol}(G(k)_{x,0+},\mu)^{-1}=q^{(\dim
G)/2}|\mathfrak{g}_{x,0:0+}|^{1/2}$.

###### Proof.

DeBacker and Reeder [DR09, Section 5.1] defined a Haar measure $\nu$ on $G(k)$
such that for any $x\in\cal B(G)$,

$\operatorname{vol}(G(k)_{x,0},\nu)=\frac{|G_{x,0:0+}|}{|\mathfrak{g}_{x,0:0+}|^{1/2}};$

in particular, $\nu$ does not depend on the choice of $x$. Kaletha showed
[Kal15, Lemma 5.15] that $\nu=q^{(\dim G)/2}\mu$. Combining these results
proves the lemma. ∎

At this point, we can say that

(47) $\deg(\pi,\mu)=\frac{\dim\rho}{[G^{0}_{[y]}:G^{0}_{y,0+}]}q^{(\dim
G)/2}|\mathfrak{g}_{y,0:0+}|^{1/2}\frac{[G_{y,0+}:\vec{G}^{[1,d]}_{y,(s_{0},\dots,s_{d-1})}]}{[G^{0}_{[y]}:G^{0}_{y,s_{0}}]}\prod_{i=0}^{d-1}[J^{i+1}:J^{i+1}_{+}]^{1/2}.$

We can now simplify Eq. 47 using our earlier results on concave functions.

######  Lemma 48.

$|\mathfrak{g}_{y,0:0+}|^{1/2}\frac{[G_{y,0+}:\vec{G}^{[1,d]}_{y,(s_{0},\dots,s_{d-1})}]}{[G^{0}_{y,0+}:G^{0}_{y,s_{0}}]}\prod_{i=0}^{d-1}[J^{i+1}:J^{i+1}_{+}]^{1/2}=\exp_{q}\biggl{(}\tfrac{1}{2}\operatorname{len}\mathfrak{g}^{0}_{y,0:0+}+\tfrac{1}{2}\sum_{i=0}^{d-1}r_{i}(|R_{i+1}|-|R_{i}|)\biggr{)}$

###### Proof.

Let $f=f_{(s_{0},\dots,s_{d-1})}$ for the twisted Levi sequence
$\vec{\underline{G}}^{[1,d]}$. By Lemma 30,

$[J^{i+1}:J^{i+1}_{+}]=|(\mathfrak{g}^{i},\mathfrak{g}^{i+1})_{y,(r_{i},s_{i}):(r_{i},s_{i}+)}|=\bigl{|}(\mathfrak{g}^{i+1})^{\underline{R}_{i+1}\setminus\,\underline{R}_{i}}_{y,s_{i}:s_{i}+}\bigr{|}=\exp_{q}\Bigl{(}\sum_{\underline{\alpha}\in\underline{R}_{i+1}\\!\setminus\,\underline{R}_{i}}\operatorname{len}\mathfrak{g}^{\underline{\alpha}}_{y,f:f+}\Bigr{)}.$

By Corollary 34,

$[G_{y,0+}:\vec{G}^{[1,d]}_{y,(s_{0},\dots,s_{d-1})}]=|\vec{G}^{[1,d]}_{y,(0+,\dots,0+):(s_{0},\dots,s_{d-1})}|=\exp_{q}\Bigl{(}\operatorname{len}\mathfrak{g}^{0}_{y,0:s_{0}}+\sum_{i=0}^{d-1}\sum_{\underline{R}_{i+1}\\!\setminus\,\underline{R}_{i}}\operatorname{len}\mathfrak{g}^{\underline{\alpha}}_{y,0:f}\Bigr{)}$

and
$[G^{0}_{y,0+}:G^{0}_{y,s_{0}}]=\exp_{q}\operatorname{len}\mathfrak{g}^{0}_{y,0:s_{0}}$,
so that

$\frac{[G_{y,0+}:\vec{G}^{[1,d]}_{y,(s_{0},\dots,s_{d-1})}]}{[G^{0}_{y,0+}:G^{0}_{y,s_{0}}]}=\exp_{q}\Bigl{(}\sum_{i=0}^{d-1}\sum_{\underline{R}_{i+1}\\!\setminus\,\underline{R}_{i}}\operatorname{len}\mathfrak{g}^{\underline{\alpha}}_{y,0:f}\Bigr{)}=\exp_{q}\Bigl{(}\sum_{\underline{\alpha}\in\underline{R}\setminus\,\underline{R}_{0}}\operatorname{len}\mathfrak{g}^{\underline{\alpha}}_{y,0:f}\Bigr{)}.$

We now recognize that

$\frac{|\mathfrak{g}_{y,0:0+}|^{1/2}}{|\mathfrak{g}^{0}_{y,0:0+}|^{1/2}}\cdot\frac{[G_{y,0+}:\vec{G}^{[1,d]}_{y,(s_{0},\dots,s_{d-1})}]}{[G^{0}_{y,0+}:G^{0}_{y,s_{0}}]}\prod_{i=0}^{d-1}[J^{i+1}:J^{i+1}_{+}]^{1/2}$

equals $\exp_{q}$ of

$\sum_{\underline{\alpha}\in\underline{R}\setminus\,\underline{R}_{0}}\sideset{}{{}^{\prime}}{\sum}_{0\leq
t\leq
f(\underline{\alpha})}\operatorname{len}\mathfrak{g}^{\underline{\alpha}}_{y,t:t+}.$

By Corollary 10 the hypotheses of Theorem 38 are satisfied, and the expression
above becomes

$\tfrac{1}{2}\sum_{i=0}^{d-1}r_{i}(|R_{i+1}|-|R_{i}|)$

The proof is finished by recalling that
$|\mathfrak{g}^{0}_{y,0:0+}|^{1/2}=\exp_{q}\bigl{(}\tfrac{1}{2}\operatorname{len}(\mathfrak{g}^{0}_{y,0:0+})\bigr{)}$.
∎

We now make the reduction promised in Lemma 42. Recall the notation for
$\overline{\Psi}$ defined immediately before Lemma 42.

######  Theorem A.

Let $\underline{G}$ be a reductive $k$-group and let $\Psi$ be a generic
cuspidal $\underline{G}$-datum with associated supercuspidal representation
$\pi$. Then

$\deg(\pi,\mu)=\frac{\dim\rho}{[G^{\textnormal{a},0}_{[y]}:G^{\textnormal{a},0}_{y,0+}]}\exp_{q}\biggl{(}\tfrac{1}{2}\dim\underline{G}^{\textnormal{a}}+\tfrac{1}{2}\dim\underline{G}^{\textnormal{a},0}_{y,0:0+}+\tfrac{1}{2}\sum_{i=0}^{d-1}r_{i}(|R_{i+1}|-|R_{i}|)\biggr{)}.$

###### Proof.

When $\underline{A}=1$, the formula follows from Eq. 47, Lemma 48, and the
fact that $\mathfrak{g}^{0}_{y,0:0+}$ is the Lie algebra of
$\underline{G}^{\textnormal{a},0}_{y,0:0+}$. In general, according to Eq. 41,
with the exception of the factor $\dim\rho$, the formal degree depends only on
the underlying $G$-datum without representations
$\Psi=(\vec{\underline{G}},y,\vec{r})$. It now suffices to observe that by
Lemma 42, the expression

$\frac{\prod_{i=0}^{d-1}[J^{i+1}:J^{i+1}_{+}]^{1/2}}{\operatorname{vol}(K/A,\mu)}$

is the same for both $\Psi$ and the reduced datum
$\overline{\Psi}:=(\overrightarrow{\underline{G}^{\textnormal{a}}},y,\vec{r}\,)$.
∎

###  3.9. Regular supercuspidals

In the special case where the supercuspidal representation is regular, we can
further simplify the expression for its formal degree. We have already seen,
in Section 2.2, the source of this simplification: an arbitrary depth-zero
supercuspidal $\pi$ is compactly induced from a finite-dimensional
representation $\rho$ of the group $G_{[y]}$, but when the supercuspidal $\pi$
is regular, we can recover a maximal torus $S$ from $\pi$, and $\pi$ is
induced from a finite-dimensional representation $\eta$ of the smaller group
$SG_{y,0}$. In fact, the former representation is induced from the latter:

(49) $\rho=\operatorname{Ind}^{G_{[y]}}_{SG_{y,0}}\eta.$

In the depth zero case, replacing $\rho$ by $\eta$ in the formula for the
formal degree thus multiplies the rest of the formula by the index
$[G_{[y]}:SG_{y,0}]$. And since $\eta$ is an extension of a Deligne-Lusztig
representation, the literature provides a formula for its dimension. These
observations simplify the formal degree for depth-zero regular supercuspidals.

When the regular supercuspidal has positive depth, however, there is a slight
complication: Equation 49 must be propagated from $G_{[y]}=K^{0}$ to $K$. And
to propagate the formula, we need to understand the inflation procedure
mentioned in passing in Section 3.7. Nonetheless, inflation is compatible with
induction in the most straightforward way, and in the end, the effect on the
formula for the formal degree is the same.

Yu’s inflation procedure is quite simple; we explain it following Hakim and
Murnaghan [HM08, Section 3.4]. Recall Equation 40, that
$K^{i+1}=K^{i}J^{i+1}$. Suppose we are given a representation $\rho$ of
$K^{i}$ satisfying the following condition.

(50) The restriction of $\rho$ to $K^{i}\cap J^{i+1}$ is $1$-isotypic.

Ultimately we will apply the following analysis to the representation
$\rho_{i}$, which satisfies Equation 50. Hence $\rho$ may be interpreted as
(the inflation of) a representation of the quotient group $K^{i}/(K^{i}\cap
J^{i+1})$. Since $K^{i}$ normalizes $J^{i+1}$, the decomposition of Equation
40 becomes a semidirect product after dividing by $K^{i}\cap J^{i+1}$. We can
now inflate this representation of the quotient first to the semidirect
product $K^{i+1}/(K^{i}\cap J^{i+1})$, then to the full group $K^{i+1}$. Let

$\operatorname{Inf}_{K^{i}}^{K^{i+1}}\rho$

denote the resulting representation. Since this representation is $1$-isotypic
on $J^{i+1}$, and since $K^{i+1}\cap J^{i+2}\subseteq J^{i+1}$, the
representation satisfies Equation 50 with $i$ replaced by $i+1$. By induction,
we can therefore define for any $j\geq i+1$, and in particular for $j=d$ (if
$i<d$), the inflated representation

$\operatorname{Inf}^{K^{j}}_{K_{i}}\rho:=\operatorname{Inf}^{K^{j}}_{K^{j-1}}\cdots\operatorname{Inf}^{K^{i+1}}_{K^{i}}\rho.$

######  Lemma 51.

Let $\Psi$ be a cuspidal $\underline{G}$-datum; recall the notations of
Section 3.7 for the various objects attached to $\Psi$. Let
$H^{i}:=SG_{y,0}^{0}K^{i}_{+}$ and let $H:=H^{d}$. Suppose that there is a
(necessarily irreducible) representation $\eta$ of $H^{0}$ such that
$\rho:=\operatorname{Ind}_{H^{0}}^{K^{0}}\eta$. Then
$\sigma_{-1}:=\operatorname{Inf}_{H^{0}}^{H}\eta$ is defined and there is a
canonical identification

$\tau=\operatorname{Ind}_{H}^{K}(\sigma_{-1}\otimes\tau_{0}|_{H}\otimes\cdots\otimes\tau_{d}|_{H})$

of representations of $K$.

###### Proof.

The notation $\inf_{H^{i}}^{H^{j}}$ mimics the notation
$\inf_{K^{i}}^{K^{j}}$: the same construction works if the symbol $K$ is
replaced everywhere by $H$. Since $\rho$ is $1$-isotypic on $J^{1}$, so is
$\eta$; hence $\sigma_{-1}:=\inf_{H^{0}}^{H}\eta$ is defined. There is a
canonical identification

$\operatorname{Ind}^{K}_{H}\operatorname{Inf}^{H}_{H^{0}}\eta=\operatorname{Inf}^{K}_{K^{0}}\operatorname{Ind}^{K^{0}}_{H^{0}}\eta,$

so that
$\tau=(\operatorname{Ind}^{K}_{H}\sigma_{-1})\otimes\tau_{0}\otimes\cdots\otimes\tau_{d}$.
The result now follows from the well-known [Ren10, Section III.2.11] formula
$(\operatorname{Ind}^{K}_{H}\pi_{1})\otimes\pi_{2}=\operatorname{Ind}^{K}_{H}(\pi_{1}\otimes(\pi_{2}|_{H}))$,
where $\pi_{1}$ is a representation of $H$ and $\pi_{2}$ is a representation
of $K$. ∎

Kaletha showed [Kal19b, Section 3.4.4, proof of Lemma 3.4.20] that for a
regular Yu datum, the finite-dimensional representation $\rho$ has the
property that $\rho=\operatorname{Ind}_{SG^{0}_{y,0}}^{K^{0}}\eta$ where
$\eta$ extends the inflation to $G^{0}_{y,0}$ of a Deligne-Lusztig induced
representation of the group $G^{0}_{y,0:0+}$. We can thus compute $\dim\eta$
using Deligne-Lusztig theory.

Let’s briefly recall the dimension formula for a Deligne-Lusztig induction. In
this paragraph only, let $\underline{G}$ be a reductive $\kappa$-group, let
$\underline{S}$ be a maximal elliptic torus of $\underline{G}$, and let
$\theta:S\to\mathbb{C}^{\times}$ be a character. Deligne and Lusztig computed
the dimension of the virtual representation $\textnormal{R}_{(S,\theta)}$ in
their original paper [DL76, Corollary 7.2]; it is

$\dim\textnormal{R}_{(S,\theta)}=\frac{[G:S]}{\dim\textnormal{St}_{G}}$

where $\textnormal{St}_{G}$ is the Steinberg representation of $G$. In his
classic book on representations of finite groups of Lie type, Carter computed
[Car85, Corollary 6.4.3] the dimension of the Steinberg representation; it is

$\log_{q}\dim\textnormal{St}_{G}=\tfrac{1}{2}(\dim\underline{G}-\dim\underline{S}).$

We can now assemble these results to make our final formula. Recall the
notation of A.

######  Corollary 52.

Let $\Psi$ be a regular generic cuspidal $\underline{G}$-datum with resulting
supercuspidal representation $\pi$ and let $\underline{S}$ be the maximally
unramified maximal torus of $\underline{G}^{0}$ resulting from $\Psi$, as
explained in Section 2.2. Then

(53)
$\deg(\pi,\mu)=|S^{\textnormal{a}}_{0:0+}|^{-1}\exp_{q}\Bigl{(}\tfrac{1}{2}\dim\underline{G}^{\textnormal{a}}+\tfrac{1}{2}\operatorname{rank}\underline{G}^{\textnormal{a},0}_{y,0:0+}+\sum_{i=0}^{d-1}s_{i}(|R_{i+1}|-|R_{i}|)\Bigr{)}.$

###### Proof.

By Lemma 51, Lemma 18, and Kaletha’s description of $\rho$, the formula of A
remains true if $\dim\rho$ is replaced by

$[K^{0}:SG^{0}_{y,0}]\dim\eta=[G^{\textnormal{a}}_{[y]}:S^{\textnormal{a}}G^{\textnormal{a},0}_{y,0}]\dim\eta.$

The dimension formula for $\eta$ discussed in the paragraph above shows that

$\displaystyle\dim\eta$
$\displaystyle=[G^{0}_{y,0}:S_{0}G^{0}_{y,0+}]\exp_{q}\bigl{(}\tfrac{1}{2}(\dim\underline{G}^{0}_{y,0:0+}-\dim\underline{S}_{0:0+})\bigr{)}^{-1}$
$\displaystyle=[G^{\textnormal{a},0}_{y,0}:S^{\textnormal{a}}_{0}G^{\textnormal{a},0}_{y,0+}]\exp_{q}\bigl{(}\tfrac{1}{2}(\dim\underline{G}^{\textnormal{a},0}_{y,0:0+}-\operatorname{rank}\underline{G}^{\textnormal{a},0}_{y,0:0+})\bigr{)}^{-1}.$

The formula now follows. ∎

##  4\. Galois side

Let $(S,\theta)$ be a tame elliptic pair. We saw in Section 2.4 that when
$\theta$ is extra regular, such a pair can be extended to a regular
supercuspidal $L$-packet datum $(S,\hat{\jmath},\chi,\theta)$, and that the
resulting set of $\chi$-data can then be used to form the regular
supercuspidal parameter

$\varphi_{(S,\theta)}:={}^{L}j_{\chi}\circ{}^{L}\theta.$

Moreover, every regular supercuspidal parameter arises in this way. Our goal
in this section is to compute the Galois side of the formal degree conjecture
for the parameter $\varphi=\varphi_{(S,\theta)}$. As we mentioned in Section
2.4, the group $S_{\varphi}^{\natural}$ is abelian and thus has only one-
dimensional irreducible representations, so that the Galois side of the
conjecture simplifies to

$\frac{|\gamma(0,\varphi,\textnormal{Ad},\psi)|}{|\pi_{0}(S_{\varphi}^{\natural})|}.$

Moreover, the factor $|\pi_{0}(S_{\varphi}^{\natural})|$ has been computed in
the literature. So our task is to compute the absolute value
$|\gamma(0,\varphi,\textnormal{Ad},\psi)|$ of the adjoint $\gamma$-factor.

We start by reviewing in Section 4.1 the general definition of the
$\gamma$-factor. As a second preliminary step, we work out in Section 4.2 how
to base change the function $r_{\chi}$ used to solve the extension problem of
Section 2.3.

To compute the adjoint $\gamma$-factor, we give an explicit description of the
adjoint representation attached to the $L$-parameter $\varphi$. It turns out
that this representation decomposes as a direct sum of two representations,
one coming from the maximal torus of the dual group and the other from its
root system. We can thus compute the adjoint $\gamma$-factors separately, in
Sections 4.4 and 4.5, and multiply them together for the final answer, in
Section 4.6. Beginning in Section 4.3, the start of the $\gamma$-factor
computation proper, we must fix $\hat{\jmath}$ and $\chi$ in the $L$-packet
datum $(S,\hat{\jmath},\chi,\theta)$ extending $(S,\theta)$.

###  4.1. Review of $L$-, $\varepsilon$\- and $\gamma$-factors

The $\gamma$-factor of a representation $(\pi,V)$ of the Weil group $W_{k}$ is
defined by the formula

$\gamma(s,\pi,\psi,\mu):=\varepsilon(s,\pi,\psi,\mu)\frac{L(1-s,\pi^{\vee})}{L(s,\pi)}$

where $\psi$ is a nontrivial additive character of $k$ and $\mu$ is an
additive Haar measure on $k$. Hence the $\gamma$-factor is built from two
quantities, the $L$-factor and the $\varepsilon$-factor.

In this subsection we recall the definitions of the $L$-factor and the
$\varepsilon$-factor, following Tate’s Corvallis notes [Tat79]. Roughly
speaking, the $L$-factor carries information about the unramified part of the
representation and the $\varepsilon$-factor carries information about the
ramified part of the representation. Since computing the absolute value of an
$\varepsilon$-factor amounts to computing an Artin conductor, we also explain
how to compute this quantity in our application, following Chapter VI of
Serre’s _Local Fields_ [Ser79].

The _$L$ -factor_ of $\pi$ is the holomorphic function

$L(s,\pi):=\det\bigl{(}1-q^{-s}\pi(\textnormal{Frob})\mid
V^{I_{k}}\bigr{)}^{-1}$

where $I_{k}\subset W_{k}$ is the inertia group and $\textnormal{Frob}\in
W_{k}$ is a Frobenius element. Later, we use the fact that the $L$-factor is
_inductive_ : if $\ell\supseteq k$ is a field extension of $k$ and $(\pi,V)$
is a finite-dimensional complex representation of $W_{k}$ then

$L(s,\operatorname{Ind}_{\ell/k}(\pi))=L(s,\pi),\qquad\operatorname{Ind}_{\ell/k}:=\operatorname{Ind}^{W_{k}}_{W_{\ell}}.$

The $\varepsilon$-factor is more subtle to define than the $L$-factor, and
most of the subtlety resides in its complex argument. Fortunately, since we
are interested in only the absolute value of the $\gamma$-factor, not its
complex argument, we content ourselves with a description of the absolute
value of the $\varepsilon$-factor instead.

Changing $s$, $\psi$, or $\mu$ scales the $\varepsilon$-factor by a known
quantity. We may thus define with no loss of information the simplified
$\varepsilon$-factor

$\varepsilon(\pi):=\varepsilon(0,\pi,\psi,\mu)$

where $\psi$ has level zero, that is, $\max\\{n\mid\psi(\pi^{-n}\cal
O)=1\\}=0$, and where the Haar measure $\mu$ is self-dual with respect to
$\psi$. With these conventions, the absolute value of the $\varepsilon$-factor
is

(54) $|\varepsilon(\pi)|^{2}=q^{\operatorname{cond}\pi}$

where $\operatorname{cond}\pi$ is the Artin conductor of $\pi$. So computing
the absolute value of the $\varepsilon$-factor amounts to computing the Artin
conductor.

The Artin conductor is defined by the following procedure. Given a Galois
representation $(\pi,V)$, choose a Galois extension $\ell$ of $k$ such that
$\pi|_{W_{\ell}}$ is trivial, and let $\Gamma\\!_{\ell/k}$ be the Galois group
of $\ell$ over $k$. Then the Artin conductor satisfies the formula

$\operatorname{cond}\pi=\sum_{i\geq
0}\frac{\operatorname{codim}(V^{\Gamma\\!_{\ell/k,i}})}{[\Gamma\\!_{\ell/k,0}:\Gamma\\!_{\ell/k,i}]},$

where $i\mapsto G_{\ell/k,i}$ is the lower numbering filtration. This formula
is independent of the choice of $\ell$. We can now extend the definition to
all complex representations of the Weil group, not necessarily those of Galois
type, by stipulating that the Artin conductor be unchanged by unramified
twists. In particular, $\operatorname{cond}\pi=0$ if and only if $\pi$ is
unramified, and

(55) $\operatorname{cond}\pi=\operatorname{codim}V^{I_{k}}$

if $\pi$ is tamely ramified. Heuristically, the numerical invariant
$\operatorname{cond}\pi$ is an enhancement of Equation 55 that takes wild
ramification into account and measures the extent to which $\pi$ ramifies.

When $\pi$ is irreducible, this heuristic is made precise by the formula

(56) $\operatorname{cond}\pi=(\dim\pi)(1+\operatorname{depth}_{k}\pi).$

In particular, Equation 56 holds if $\pi$ is a character. For our application,
we need only understand how to compute the Artin conductor of a tamely induced
representation. Unlike the $L$-factor, the Artin conductor is not invariant
under induction. The best we can say in general is that given a finite
extension $\ell$ of $k$ and a representation $\pi$ of $W_{\ell}$, the induced
representation has conductor

(57)
$\operatorname{cond}\operatorname{Ind}_{\ell/k}\pi=\operatorname{ord}_{k}(\textnormal{disc}_{\ell/k})\dim\pi+f_{\ell/k}\operatorname{cond}\pi,$

where $\textnormal{disc}_{\ell/k}$ is the discriminant of $\ell$ over $k$. But
when $\ell$ is tamely ramified over $k$ and $\pi=\chi$ is a character the
formula simplifies considerably, even if, unlike in Equation 56, the induced
representation is reducible.

######  Lemma 58.

Let $\ell\supseteq k$ be a finite tame extension and let
$\chi:W_{\ell}\to\mathbb{C}^{\times}$ be a character. Then

$\operatorname{cond}\operatorname{Ind}_{\ell/k}\chi=[\ell:k](1+\operatorname{depth}_{k}\chi).$

###### Proof.

Check, using the tameness of the extension, that
$\operatorname{ord}_{k}\textnormal{disc}_{\ell/k}=[\ell:k]-f_{\ell/k}$. This
computation together with Equations 56 and 57 yields

$\operatorname{cond}\operatorname{Ind}_{\ell/k}\chi=[\ell:k]+f_{\ell/k}\operatorname{depth}_{\ell}\chi.$

Now use that
$\operatorname{depth}_{\ell}\chi=e_{\ell/k}\operatorname{depth}_{k}\chi$. ∎

The $L$-factor, $\varepsilon$-factor, and Artin conductor are additive in the
sense that

$L(s,\pi)=L(s,\pi_{1})L(s,\pi_{2}),\qquad\varepsilon(\pi)=\varepsilon(\pi_{1})\varepsilon(\pi_{2}),\qquad\operatorname{cond}\pi=\operatorname{cond}\pi_{1}+\operatorname{cond}\pi_{2}$

where $\pi=\pi_{1}\oplus\pi_{2}$. Hence the $\gamma$-factor is additive as
well. This simple but crucial fact allows us to restrict our attention to
summands of the adjoint representation.

###  4.2. Base change for $\chi$-data

The main goal of this subsection is to determine how $\chi$-data behave under
base change, that is, restriction to the Weil group of a finite separable
extension of $k$. Once we understand the effect of base change for arbitrary
$\chi$-data, we study its effect on minimally ramified $\chi$-data.

Most of the definitions of this subsection are due to Langlands and Shelstad
[LS87, Section 2.5], but our treatment is also influenced by Kaletha’s recent
reinterpretation of Langlands and Shelstad’s formalism [Kal19a].

Let $\ell$ be a separable quadratic extension of $k$. Local class field theory
shows that the quotient $k^{\times}/N_{\ell/k}(\ell^{\times})$ is cyclic of
order two. The _quadratic sign character_ of the extension $\ell\supseteq k$
is the character $k^{\times}\to\\{\pm 1\\}$ given by projection onto this
quotient.

A root $\alpha\in R(G,S)$ is _symmetric_ if it is Galois-conjugate to
$-\alpha$, and is _asymmetric_ otherwise. A symmetric root $\alpha$ is
_unramified_ if the quadratic extension $k_{\alpha}\supset k_{\pm\alpha}$ is
unramified and is _ramified_ otherwise. Letting $k_{\pm\alpha}$ denote the
fixed field of the stabilizer in $\Gamma\\!_{k}$ of $\\{\pm\alpha\\}$, the
extension $k_{\pm\alpha}\subseteq k_{\alpha}$ has degree two if $\alpha$ is
symmetric and degree one if $\alpha$ is asymmetric.

######  Definition 59.

Let $R=R(G,S)$. A _set of $\chi$-data_ for $(S,G)$ (or just $S$ if $G$ is
understood) is a collection
$\chi=(\chi_{\alpha}:k_{\alpha}^{\times}\to\mathbb{C}^{\times})_{\alpha\in R}$
of characters satisfying the following properties.

  1. (1)

$\chi_{-\alpha}=\chi_{\alpha}^{-1}$.

  2. (2)

$\chi_{\sigma\alpha}=\chi_{\alpha}\circ\sigma^{-1}$ for all
$\sigma\in\Gamma\\!_{k}$.

  3. (3)

If $\alpha$ is symmetric then $\chi_{\alpha}$ extends the quadratic sign
character of $k_{\alpha}\supset k_{\pm\alpha}$.

Kaletha has interpreted a set of $\chi$-data as giving rise to a character of
a certain double cover of a torus, and the function $r_{\chi}$ as the
$L$-parameter of this character [Kal19a, Section 3]. In light of that
interpretation, we would expect that restricting $r_{\chi}$ to an extension of
$k$ corresponds to composing the $\chi$-data with the norm map, in analogy
with the discussion from Section 1.8. This turns out to be the case, as
Theorem 64 shows.

######  Definition 60.

Let $\chi$ be a set of $\chi$-data and let $\ell$ be a finite separable
extension of $k$. The _base change_ of $\chi$ to $\ell$ is the $\chi$-datum
$\chi_{\ell}$ defined by
$\chi_{\ell,\alpha}:=\chi_{\alpha,\ell_{\alpha}/k_{\alpha}}$.

The definition of base change makes sense only if the formula for
$\chi_{\ell}$ defines a $\chi$-datum. We should immediately check this.

######  Lemma 61.

The function $\chi_{\ell}$ of Definition 60 is a set of $\chi$-data.

###### Proof.

Negation equivariance is clear. Compatibility with the Galois group follows
from the easily verified formula

$\sigma^{-1}\circ
N_{\ell_{\sigma\alpha}/k_{\sigma\alpha}}=N_{\ell_{\alpha}/k_{\alpha}}\circ\sigma^{-1}.$

As for the third property, if $\alpha$ is symmetric over $\ell$ then it is
also symmetric over $k$ and the canonical map
$\Gamma_{\ell_{\alpha}/\ell_{\pm\alpha}}\to\Gamma_{k_{\alpha}/k_{\pm\alpha}}$
is an isomorphism. Now recall [Tat79, (1.2.2)] that the local class field
theory homomorphism $W_{k}\to k^{\times}$ (whose abelianization is the Artin
reciprocity isomorphism) intertwines the inclusion $W_{\ell}\hookrightarrow
W_{k}$ with the norm map $\ell^{\times}\to k^{\times}$. It follows that the
canonical map

$N_{\ell_{\alpha}/k_{\alpha}}(\ell_{\alpha}^{\times})/N_{\ell_{\alpha}/k_{\alpha}}(\ell_{\pm\alpha}^{\times})\to
k_{\alpha}^{\times}/k_{\pm\alpha}^{\times}$

is an isomorphism, and hence that $\chi_{\ell,\alpha}$ extends the quadratic
sign character of $\ell_{\alpha}\supset\ell_{\pm\alpha}$. ∎

It is time to start defining the function $r_{\chi}$. The definition requires
a brief preliminary discussion of abstract group theory. Let $G$ be a group
and $K$ a subgroup. A section $u:K\backslash G\to G$ of the projection $G\to
K\backslash G$ – in other words, a choice of coset representatives – gives
rise to a $K\backslash G$-indexed family of set maps $u_{x}:G\to K$, for $x\in
K\backslash G$. To define the maps $u_{x}$, we write down the element $u(x)g$
and decompose it as a product of an element of $K$ followed by its coset
representative in $K\backslash G$; that is, $u_{x}$ is defined by the
following equation:

$u_{x}(g)u(xg)=\mathop{u}(x)g.$

Before defining $r_{\chi}$, a word is on order on the exact nature of the
object we are defining, since that object depends on many arbitrary choices. A
_gauge_ is a function $p:R\to\\{\pm 1\\}$ such that $p(-\alpha)=-p(\alpha)$
for all $\alpha\in R$. Our construction produces for each gauge $p$ a
cohomology class $r_{\chi,p}$ of $1$-cochains $W_{k}\to\widehat{S}$. As the
gauge $p$ varies the cohomology classes $r_{\chi,p}$ are not identical.
However, there is a canonical means of relating one class to the other. For
any two gauges $p$ and $q$, Langlands and Shelstad constructed a canonical
$1$-cochain $s_{q/p}$, depending only on $\underline{R}$ and not on the choice
of $\chi$-data. By definition, the cohomology classes $r_{\chi,p}$, are
related by the equation

(62) $r_{\chi,q}=s_{q/p}r_{\chi,p},$

and the $1$-cochains $s_{q/p}$ satisfy the right compatibility conditions to
make these equations consistent [LS87, Corollary 2.4.B]. In the construction
that follows we therefore define $r_{\chi,p}$ for a particular choice of
gauge, and Equation 62 then defines $r_{\chi,q}$ for every other $q$.

We can now write down the formula defining $r_{\chi}$, making several
arbitrary choices along the way. First, choose

  1. (1)

a section $[\underline{\alpha}]\mapsto\alpha$ of the orbit map
$R\to\underline{R}/\\{\pm 1\\}$.

Each $[\underline{\alpha}]\in\underline{R}/\\{\pm 1\\}$ thus gives rise to two
subgroups $W_{\alpha}$ and $W_{\pm\alpha}$, the stabilizers of $\alpha$ and
$\\{\pm\alpha\\}$ in $W_{k}$. For each $[\underline{\alpha}]$, choose in
addition

  1. (2)

a section $u^{\alpha}:W_{\pm\alpha}\backslash W_{k}\to W_{k}$, and

  2. (3)

a section $v^{\alpha}:W_{\alpha}\backslash W_{\pm\alpha}\to W_{\pm\alpha}$.

What we call “choosing a section” is more commonly called “choosing coset
representatives”. Using these choices, define the element $r_{\chi}(w)$ of
$\widehat{S}=X^{*}(S)_{\mathbb{C}}$ by

(63)
$r_{\chi}(w)=\prod_{[\underline{\alpha}],x}\chi_{\alpha}(v^{\alpha}_{0}(u^{\alpha}_{x}(w)))^{u^{\alpha}(x)^{-1}\alpha},\qquad[\underline{\alpha}]\in\underline{R}/\\{\pm
1\\},\,x\in W_{\pm\alpha}\backslash W_{k}.$

We still have to explain the dependence on the gauge. Use choices (1) and (2)
above to define the gauge $p:R\to\\{\pm 1\\}$ by setting $p(\beta)=1$ if and
only if $\beta=u^{\alpha}(x)^{-1}\alpha$ for some $x\in
W_{\pm\alpha}\backslash W_{k}$. Then Equation 63 defines
$r_{\chi,p}:=r_{\chi}$. Now use Equation 62 to extend the definition to all
gauges.

######  Theorem 64.

Let $p:R\to\\{\pm 1\\}$ be a gauge, let $\chi$ be a set of $\chi$-data for
$S$, and let $\ell$ be a finite separable extension of $k$. Then

$r_{\chi,p}|_{\ell}=r_{\chi_{\ell},p}$

for some set of auxiliary choices ((1), (2), and (3) above) in the definition