Formal Degree of Regular Supercuspidals

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 π 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 π 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, Theorem 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 Ψ.The first member of Ψ is an increasing sequence (G i ) 0≤i≤d of twisted Levi subgroups; let R i denote the Date: 3 January 2021.
absolute root system of G i .The second member is a certain point y in the Bruhat-Tits building B(G).The third member is an increasing sequence (r i ) 0≤i≤d of nonnegative real numbers.The fourth member is a certain irreducible representation ρ of the stabilizer G 0 (k) [y] of the image [y] of y in B red (G).We compute the formal degree with respect to a certain Haar measure µ 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 Ψ be a generic cuspidal G-datum with associated supercuspidal representation π.Then The proof of Theorem 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 τ 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 τ /vol(µ, 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 Π(G) of smooth irreducible representations of G(k) is classified by certain homomorphisms ϕ : W ′ k → L G, called L-parameters.Here W k is the Weil group of k, W ′ k := SL 2 (C) × W k is the Weil-Deligne group of k, and L G := G ⋊ W k is the (Weil form of the) L-group of G.The expected classification consists of a partition of Π(G) into finite subsets Π ϕ (G), called L-packets, indexed by (equivalence classes of) Lparameters ϕ.The sets Π ϕ (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 Π(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 Π(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 Π ϕ (G) are parameterized by representations of a certain finite group attached to ϕ. Refining Langlands's proposal, Vogan enhanced [Vog93, Section 9] Lparameters to pairs (ϕ, ρ) consisting of an L-parameter ϕ and an irreducible representation ρ of the finite group π 0 (S ϕ ), where S ϕ is the preimage in G sc of the centralizer in G of ϕ.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 ρ, 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 ρ of S ϕ satisfying the central character condition parameterize the L-packet Π ϕ (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 SL n and 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 µ = µ ψ on G(k) depending on an additive character ψ 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 ϕ and certain additional data -a finite-dimensional representation r of L G and a nontrivial additive character ψ of k -a meromorphic function γ(s, ϕ, r, ψ) of the complex variable s, called a γ-factor of ϕ.The γ-factor is a product of L-and ε-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 g/ z Γ k , where g and z are the Lie algebras of G and of its center and where Γ k is the absolute Galois group of k.Division by z Γ k ensures that the γ-factor is defined at s = 0. We call the factor γ(0, ϕ, Ad, ψ) appearing in the conjecture the the adjoint γ-factor of ϕ.
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 ♮ ϕ denote the centralizer of ϕ in G a , where G a := G/A with A the maximal split central torus of G.
Conjecture 1.Let π be an essentially discrete series representation of G(k) with extended parameter (ϕ, ρ), let ψ be an additive character of k, and let µ ψ be the Gross-Gan measure on G(k) attached to ψ.Then 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 GL n and 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 U 3 , Sp 4 , and 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 γ-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 ε-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 γ-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, θ) consisting of an elliptic maximal torus S of G and a character θ of S(k).On the Galois side, one uses the Langlands-Shelstad theory of χ-data and extensions of Lembeddings [LS87] to construct a L-parameter for G from (S, θ).On the automorphic side, one uses the pair (S, θ) to produce an input to Yu's construction, hence a supercuspidal representation π of G(k).We can thus interpret π as a functorial lift of θ with respect to the embedding S ֒→ G.The L-packet of ϕ consists, roughly, of all π 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 Theorem B, is that they do.Theorem B. Kaletha's regular L-packets satisfy the formal degree conjecture.
To prove Theorem 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 θ and the root system of S. The γ-factor of the character lattice has already been computed in the literature.As for the monomial representations, computing their γ-factors amounts to computing the depths of the inducing characters.The inducing characters are very close to certain characters naturally constructed from θ, 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 χ-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. the concatenation: X → X a → 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∈I is a decreasing, Iindexed filtration of the set X.For i ∈ I, define (2) X i+ := j>i X j .
Let I := I ∪ {i+ : i ∈ I} ∪ {∞} denote Bruhat and Tits's extension of I [BT72, 6.4.1]; their definition is for I = R only, but it is clear how to extend it to arbitrary I. Equation (2) together with the convention defines an extension of the given filtration to a 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 I, We apply this formalism to the Moy-Prasad filtration on a p-adic group G(k) and its Lie algebra 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 R ≥0 , and the filtrations on g are indexed by 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 I is defined by 1.3.Fields.Let k be a nonarchimedean local field of odd residue characteristic p, let O denote the ring of integers of k, and let κ denote the residue field of O. Given a finite algebraic extension ℓ of k, let e ℓ/k denote the ramification degree and f ℓ/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 ord k : k × → Z denote the unique discrete valuation on k with value group Z.We extend ord k to a valuation on the algebraic closure k and denote the extension by ord k as well.Hence the value group for a finite extension ℓ is ord k (ℓ × ) = e −1 ℓ/k Z. Remark 4. Aesthetic reasons might lead one to consider a more general value group for k than Z.Indeed, per our convention, the value group for an extension of k is generally than larger than 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 ε-factor is defined independent of the value group, the relationship between the Artin conductor and the ε-factor, Equation (54), holds only for value group Z.
Given a finite O-module M, let len M denote the length of M. When k has positive characteristic the module M is a κ-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 := |κ|, 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 len M = |M|.
Let W k denote the Weil group of k, let I k ⊆ W k denote the inertia subgroup of W k , let P k ⊆ I k denote the wild inertia subgroup, and for r ≥ 0 let W r k ⊆ W k denote the rth subgroup of W k in its upper numbering filtration, computed with respect to the valuation ord.A representation of the Weil group is a continuous, finite-dimensional, complex representation π of W k .Given a finite extension ℓ of k, let π| ℓ := π| W ℓ .The depth of π is defined as 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 ℓ for a finite extension ℓ of k by using the valuation ord k to define it instead of the valuation ord ℓ .When the extension is tame, as is usually the case, this modification has the effect of scaling the indices of the filtration by e −1 ℓ/k .To make the dependence on k clear, let depth k denote the depth of a representation of W ℓ where its filtration is computed using 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 g and 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 κ-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 κ-groups.So G, S, and T denote κ-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 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 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 G denote the Langlands dual group of G, a complex reductive group, and let L G = G ⋊ W denote the Weil form of the dual group.
Given an extension ℓ of k and an ℓ-group H, let Res ℓ/k H denote the Weil restriction of H from ℓ 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 Gk with respect to Tk, together with its natural Galois action.Let R(G, T ) denote the set of Galois orbits of R(G, T ).I prefer to think of 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 α, β, γ, . . .for elements of R, and their underlines α, β, γ, . . .for elements of R. Given α ∈ R, let α( k) denote the elements of α, a subset of R.
Given α ∈ R, let Γ α denote the stabilizer of α in Γ k and let k α := kΓα denote the fixed field of Γ α .Given α ∈ R, let k α denote a fixed field extension of k that is isomorphic, for some α ∈ α( k), to the extension k α , and let κ α denote the residue field of k α .We can define k α canonically as the inverse limit of the groupoid of extensions k α with α ∈ α( k), in the style of Deligne and Lusztig [DL76, Section 1.1], but since Γ k is nonabelian, it is impossible to canonically identify this limit with any one k α .The notation k α helps us avoid choosing such an α, and is reserved for expressions that depend only on the isomorphism class of the extension, such as the degree [k α : k].
1.6.Bruhat-Tits theory.Given a reductive k-group G, let B(G) and B red (G) be the extended and reduced Bruhat-Tits buildings of G. Let [x] denote the image of x under the canonical reduction map B(G) → B red (G).Given a split maximal torus T of G, let A(G, T ) and A red (G, T ) denote the extended and reduced apartments of T .The apartment A(G, T ) is noncanonically isomorphic to the building 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 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 B(G) or B 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 For each x ∈ B(G), Moy and Prasad defined [MP94, Sections 2 and 3] a canonical decreasing R ≥0 -indexed filtration of G(k) x and a canonical decreasing R-indexed filtration of g, denoted by G(k) x,r and 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+ ≃ 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 × r := {a ∈ k × : ord k (a − 1) ≥ r}.Given a finite separable extension ℓ of k and a reductive ℓ-group H, we compute the Moy-Prasad filtration on H(ℓ) with respect to the norm ord k , not ord ℓ .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(ℓ) agrees with the Moy-Prasad filtration on the identical group (Res ℓ/k G)(k).
The depth of an irreducible admissible representation π of G(k), denoted by depth k π, is the smallest real number r such that for some x ∈ B(G), the representation π 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 to 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 → Y , let X Z denote the base change of X from Y to Z, that is, the pullback X × Y Z.When Z = 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 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 k-group with a k-rational structure.In this common language one can speak, given two k-groups G and H, of morphisms G → H that are not defined over k.For us, a morphism G → 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 ℓ → H ℓ where ℓ is an extension of k, especially ℓ = k.1.8.Base change for characters.Let S be a k-torus, let ℓ be a finite separable extension of k, and let T := Res ℓ/k S ℓ .Since X * (T ) = Ind ℓ/k Res ℓ/k X * (S), there is a canonical map X * (S) → X * (T ) of Galois lattices, the unit of the adjoint pair (Res ℓ/k , Ind ℓ/k ).The dual of this unit is a canonical map N ℓ/k : T → S, called the norm map; we use the same name and notation for the map T (k) = S(ℓ) → S(k) on rational points.Given a character θ : S(k) → C × , define the character θ ℓ/k : S(ℓ) → C × by precomposition with the norm: We call θ ℓ/k the base change of θ from k to ℓ. Contrary to the usual notation for base change of schemes, the notation for base change of characters must include k, not just ℓ, 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 H 1 (W k , S).On the Galois side, we can restrict the L-parameter θ of a character θ to the Weil group W ℓ of a finite separable extension.But by the local Langlands correspondence for S ℓ , this parameter θ| ℓ corresponds to a character of S(ℓ).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, θ| ℓ = θ ℓ/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 ℓ be a finite separable extension of k, let S be a k-torus, and let θ : S(k) → C × be a character.If either (1) ℓ/k is unramified and depth k θ ≥ 0 or (2) ℓ/k tamely ramified and depth k θ > 0 then depth k θ = depth k θ ℓ/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 = G m .The first case is an immediate consequence of local class field.Using the first result, reduce the second to the case where ℓ is obtained from k by adjoining a root of a uniformizer; from here, the result is a straightforward computation.

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, θ) consisting of an elliptic maximal torus S of G and a character θ 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, θ) 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 π (S,θ) of G(k).When (S, θ) 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, θ) as the composition Here the first map corresponds to θ under the local Langlands correspondence for tori and the second map is an extension of a given Galois-stable embedding  : S → G.There is a general procedure, reviewed in Section 2.3, for extending ĵ to L j χ using a certain object χ called a set of χ-data.In our setting one canonically constructs such χ-data from the pair (S, θ), producing a canonical extension L j χ and thus a canonical L-parameter.Using pairs (S, θ), 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 π (jS,θ•j −1 ) as j ranges over G(k)-conjugacy classes of admissible embeddings j : S ֒→ G; in reality, however, we must slightly modify the character θ • j −1 .
2.1.Tame elliptic regular pairs.Pairs (S, θ), consisting of a k-torus S and a character θ : S(k) → C × 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, θ) 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, θ) 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, θ) and G. Let R = R(G, S) and let ℓ be the splitting field of S. Given a real number r > 0, consider the set of roots 0 be the breaks of this filtration, and let r −1 = 0 and 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) the action of the inertia group on R 0 preserves a set of positive roots, and (2) the stabilizer of the action of the group (2 ′ ) the stabilizer of the action of the group Ω(G 0 , S)(k) on θ| S(k) 0 is trivial.Here N(G 0 , S) is the normalizer of G 0 in S and Ω(G 0 , S) is the Weyl group.
When we compute in Section 4 the formal degree of the regular parameter attached to (S, θ), half of the problem (the "root summand" of Section 4.5) boils down to knowing for each coroot α ∨ the depth of the character . Therefore, our main goal in this subsection is to compute the depth of this character.To carry out the computation, we systematically decompose θ 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 φ : H(k) → C × 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 ⊆ H be a tame maximal torus, and let φ : H(k) → C × be a character of positive depth r whose restriction to H sc (k) is trivial.Then φ is G-generic if and only if for every root α ∈ R(G, S) \ R(H, S) and every finite tame extension ℓ of k α , the character Proof.Kaletha proved this lemma in the case where ℓ is a fixed splitting field of S [Kal19b, Lemma 3.6.8].We can deduce our result in the case where ℓ = k α 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, 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 θ 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.Lemma 12. Let (S, θ) be a tame pair and let α ∈ R i , where Since α ∨ factors through G i , condition (2) of a Howe factorization implies that the factors of this product are trivial for j ≥ i.By Lemma 9, the jth remaining factor has depth r j , and since the sequence j → r j is strictly increasing, the product has depth r i−1 .
Lemma 12 conspicuously omits the case where α ∈ R 0 .We have more to say about this in Section 4.5, especially in Lemma 74 and Remark 75.
2.2.Regular representations.Yu's seminal paper [Yu01] constructs a broad class of supercuspidal representations starting from a certain triple ( G, π −1 , φ), 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 κgroup.Representations of G(κ) are well understood through the work of Deligne and Lusztig [DL76].They attached to an elliptic pair (S, θ) over κ (that is, S is an elliptic maximal torus of G) a virtual representation R (S,θ) , the Deligne-Lusztig induction.If the character θ of the κ-torus S is regular [Kal19b, Definition 3.4.16]then ±R (S,θ) is an irreducible representation.We say that a representation ρ of G(κ) 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 π of G(k), there is a vertex x ∈ B red (G) such that π| G(k) x,0 contains the inflation to G(k) x,0 of an irreducible cuspidal representation ρ of G(k) x,0:0+ .Furthermore, we can recover π by compact induction: there is some representation ρ of G(k) x such that ρ| G(k) x,0 contains the inflation of ρ and such that π = c-Ind First, there is a canonical bijection between regular depth-zero supercuspidals and conjugacy classes of regular tame elliptic pairs (S, θ) 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 π.
Second, it turns out [Kal19b, Sections 3.4.4and 3.4.5]that π (S,θ) can be compactly induced from a representation η (S,θ) 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 π 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 ℓ of k splitting G such that H ℓ is a Levi subgroup of G ℓ , and H is tame if ℓ can be taken to be a tame extension of k.A twisted Levi sequence in G is an increasing sequence of twisted Levi subgroups of G; it is tame if each of its members is tame.The first component G of a Yu datum is a tame twisted Levi sequence; the second component π −1 of a Yu datum is a depth-zero supercuspidal representation of G 0 (k); and the third component φ of a Yu datum is a sequence of characters 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 ( G, π −1 , φ) is regular if π −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 ( G, π −1 , θ), we can find a maximally unramified torus S of G 0 and a regular depth-zero character then produces a tame elliptic regular pair (S, θ).Conversely, given such a pair (S, θ), with Howe factorization (φ i : 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 π (S,θ) from a tame elliptic regular pair (S, θ).

L-embeddings.
In this subsection we explain and study a formalism of Langlands and Shelstad for extending an embedding  : S → G with Galois-stable G-conjugacy class to an L-embedding L j : L S → L G. Fix a Γ k -pinning of G with maximal torus T .
The first difficulty in extending  to L j is to reconcile the Galois actions on S and G. Specifically, let τ G denote the action (homomorphism) of W k on T through its action on G; let τ S denote the action of W k on T through its action on S, transferred using ; let N be the normalizer of T in G; and let Ω = Ω( G, T ) be the Weyl group.Given a Weil element w ∈ W k , thought of as an element of L S, its image under an extension L j has the form nw where n ∈ N lifts the Weyl element ω S,G (w) := τ S (w)τ G (w) −1 ∈ Ω, so that nw acts on T by the S-action.The lift exists precisely because the G-conjugacy class of  is Galois-stable.To define the extension L j, then, we need only choose a specific lift n of ω S,G (w) to N.
For many reductive groups, in particular, the special linear group, the projection map N → Ω 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 : Ω → 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 To get around this problem, Langlands and Shelstad studied the failure of this formula to define a homomorphism as measured by the function t : Using an object χ called a set of χ-data, whose definition we review momentarily, they constructed a function r χ : W k → S that negates this failure in the sense that where ∂ is the coboundary operator.Hence the modified formula does define a homomorphism W k → L G, and in total, the extension L j χ of  is given by the formula 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 χ-data and the function r χ .
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 ϕ 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 ϕ and S ♮ ϕ 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 ρ 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, , χ, θ) consisting of • a tame torus S of dimension the absolute rank of G, • an embedding  : S → G of complex reductive groups whose G-conjugacy class is Galois stable, • a minimally ramified set of χ-data for R(G, S), and For the meaning of minimally ramified χ-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 G-conjugacy class of  gives rise to a Galois-stable G( k)-conjugacy class of embeddings Sk ֒→ Gk whose elements are called admissible (with respect to ) [Kal19b, Section 5.1].Since S is elliptic, this G( 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, , θ, χ), its parameter is the composition where L j χ 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 γ-factor.Let (S, θ) 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 θ to jS, we need to modify it slightly: define where j : S ֒→ G ranges over the G(k)-conjugacy classes of admissible embeddings.

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 compactinduction of a finite-dimensional representation of a compact-open (or really, compact-modcenter) 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 to 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 G, H, . . .and their groups of rational points by G, H, . . .: In particular, we write G for a twisted Levi sequence and G for the sequence of topological groups obtained from it by taking k-points.The convention extends to O-and κ-schemes as well.For instance, we write G x,r for the smooth O-group, constructed by Yu [Yu02, Section 8], whose group of O-points is the Moy-Prasad group G x,r .Similarly, G x,0:0+ denotes the maximal reductive quotient of the special fiber of G x , a κ-group.

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 (π, V ) be a smooth irreducible representation of G. Let A be a closed subgroup of Z such that Z/A is compact, and let µ 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 π with respect to v ∈ V and v ∨ ∈ V ∨ (where for all z ∈ Z. Assume that the central character is unitary.In this case, the function x → |π v,v ∨ (x)| 2 is constant on cosets of A, and hence defines a function on G/A.We write for the integral of this function, and say that (π, V ) is discrete series (with 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 V and V ∨ defines a matrix coefficient π v,w for v, w ∈ V .Set π v := π v,v .
For a discrete series representation (π, 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 π 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 (π, V ) be an essentially discrete series representation and let µ be a Haar measure on G/A.If π is in addition discrete series then there exists a positive real constant deg(π, µ), called the formal degree of π, such that for all v ∈ V , .
In general, we define the formal degree of π as the formal degree of any discrete series representation of G obtained from π by twisting by a character of G.
Evidently the formal degree scales inversely with the Haar measure used to define it: 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 µ on G.An irreducible unitary representation (π, V ) is discrete series if and only if it is an isolated point of positive measure, and that measure is the formal degree of π with respect to µ.In particular, if G is compact and we choose the Haar measure on G giving it volume one then the formal degree of (π, 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 (ρ, W ) be a smooth irreducible unitary representation of K, and let (π, V ) be the representation of G compactly induced from (ρ, W ).That is, V is the space of smooth functions f : G → W whose support is compact-mod-K and that satisfy f (hx) = ρ(h)f (x) for all h ∈ K and x ∈ G.The representation (π, V ) is unitary; in fact, an invariant scalar product is given by the formula where µ K\G is any positive G-invariant Radon measure on K\G, for instance, the counting measure.
Lemma 18.Let (ρ, W ) be a finite-dimensional unitary representation of K, let (π, V ) be the compact induction of W to G, and let µ be a Haar measure on G/A.If π is irreducible then Proof.We start by defining an isometric embedding W ֒→ V .Given a vector w ∈ W , define ẇ ∈ V by The space K\G is discrete because K is open, so we can take the measure µ K\G in Eq. ( 17) to be the counting measure.With this choice, the map W → V defined by w → ẇ is an isometric embedding.It follows that the matrix coefficient of ẇ 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 ∩ A) to be the restriction of µ, denoted also by µ.Then for any nonzero Finally, since the formal degree scales inversely to the Haar measure used to define it, deg(ρ, µ) = vol(K/(K ∩ A), µ 0 ) −1 deg(ρ, µ 0 ) where µ 0 is the measure on K/(K ∩ A) assigning it total volume one.The degree of ρ with respect to this measure is just the dimension of W .

3.3.
Concave-function subgroups: split case.Suppose G is split with split maximal torus T and root system R = R(G, T ), so that R = R.In this setting, Bruhat and Tits showed [BT72, Section 6.4] how to construct from a function f : R ∪ {0} → R and point x ∈ A(G, T ) a subgroup G x,f of G and a subgroup g x,f in 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 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 α x : U α (k) → R) α∈R , where U α is the root group of α.Since U α is canonically isomorphic to the root line g α , we may also think of v α x as a valuation g α → R. Now let U α x,r := {u ∈ U α : v α x (u) ≥ r}, g α x,r := {X ∈ g α : v α x (X) ≥ r}.As for the point α = 0, we can think of T as the root group U 0 and its Lie algebra t as the root space g 0 .These objects carry their own filtrations: let T r := {t ∈ T : ∀χ ∈ X * (T ), ord(χ(t) − 1) ≥ r} and let t r := {X ∈ T : ∀χ ∈ X * (T ), ord(dχ(X)) ≥ r}.The objects T r and t r do not depend on x, but we reserve the right to denote them by T x,r and 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
for any α ∈ R ∪ {0}.The group G x,f is then defined as the subgroup of G generated by the subgroups U α x,f with α ∈ R ∪ {0}, and the lattice g x,f is defined as the subgroup of g spanned by the subgroups g α x,f with α ∈ R ∪ {0}.Remark 20.When r = ∞ the group U α x,∞ is trivial, and when in addition α = 0 we can recover the filtrations on the root groups and root lines as g α x,r = g x,f and U α x,r = G x,f where 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 (α i ) i∈I of elements of R ∪ {0}, whenever i∈I α i ∈ R∪{0}.We can define g x,f for any f whatsoever, but when f is concave, g x,f is a sub Lie algebra of 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 G is split, only that it split over a tamely ramified extension.Hence we must distinguish between R := R(G, T ) and R := R(G, T ).In this setting, Yu showed how to generalize the constructions of the previous subsection, he constructed for each function f : R ∪ {0} → R and point x ∈ A(G, 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 ℓ ⊇ k be some fixed tame Galois extension of k splitting the maximal torus T and let Γ ℓ/k be the Galois group of ℓ over k.A function f : R ∪ {0} → R can be interpreted as a Galois-invariant function R ∪ {0} → R, and we say that f is concave if the associated Galois-invariant function is concave.
Since f is Galois-invariant, the subgroups (g ℓ ) x,f and G(ℓ) x,f are Galois-invariant and we define g x,f := (g ℓ ) x,f .These groups do not depend on the choice of ℓ.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 α in R ∪ {0} a root space When α = 0, the root space g 0 is just the Lie algebra of T .Combining this construction with Remark 20, we see that each root space g α admits a natural filtration: define It follows immediately from the definitions that Warning 22.In comparison to the function of Remark 20, the function f : R ∪ {0} → R defined, for a fixed α ∈ R, by is generally not concave.Failure of concavity relates to the fact that when T is not split, there is generally no "root group" (or even "root variety") U α whose Lie algebra is g α .
Remark 23.If we assume that f is finite, or without loss of generality, that f takes values in R, then the group G x,f can be interpreted as the integral points of an O-group: that is, there is a canonical O-group G x,f , an integral model of G, such that as subgroups of G.The construction is due to Yu [Yu02, Section 8].
Having defined the group G x,f and Lie algebra 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 G be a reductive k-group, let T be a tame maximal torus of G, let f : R(G, T ) → R be a positive concave function, let A ⊆ T be a split central torus of G, and let y ∈ A(T , G).Then the groups G y,f /(A ∩ 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 : R ∪ {0} → R be positive concave functions.Then 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 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 α x,r ∩ U α x,s = U α x,max(r,s) .Third and lastly, we compare subgroup indices between G and 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 : R ∪ {0} → R be positive concave functions such that f ≤ g.Assume in addition that the following condition is satisfied: for all non-empty finite sequences (a i ) i∈I and (b j ) j∈J of elements of R ∪ {0} such that a := , so that the group G x,f :g is abelian, and (2) there is a canonical isomorphism g x,f :g ≃ G x,f :g of abelian groups.
Remark 27.In Lemma 26, it is tempting to instead impose the simpler condition that g(a) ≤ i∈I f (a i ) for all non-empty finite sequences (a i ) i∈I such that i∈I a i = a.However, this stronger condition would significantly weaken the conclusion: the condition implies, by taking the constant sequence, that g ≤ f , so that g = f .
Corollary 28.Let f, g : R ∪ {0} → R be positive concave functions such that f ≤ g, and suppose there is a chain of concave functions 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.
The admissible sequence is weakly increasing if, in addition, r i ≤ r i+1 for all i with 0 ≤ i ≤ d − 1.
Recall the definition of a (tame) twisted Levi sequence G from Section 2.2.We assume that the tame maximal torus T of G is contained in G 0 , so that x ∈ B(G i ) for each i.For each i with 0 It is sometimes necessary to work with truncated twisted Levi sequences.Given integers a and b such that 0 given an integer i with 0 Given an admissible sequence r and a twisted Levi sequence G of the same length d, define the function f r : R → R by Since r is admissible, the function f r is concave [Yu01, Lemma 1.2].Hence we may define Remark 29.The group G x, r depends only on G, x, and r; in particular, it is independent of the choice of torus T , provided that x ∈ 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 B(G).
Lemma 30.Let G be a tame Levi sequence of length d and let r and s be admissible sequences of length d such that for all i, (31) 0 < r i ≤ s i ≤ min(r i , . . ., r d ) + min( r).
Then G x, r: s is an abelian group canonically isomorphic to g x, r: s .
Proof.Condition 31 implies that the pair (f r , f s ) satisfies the conditions of Lemma 26.
Corollary 32.Let G be a tame Levi sequence of length d and let r and s be weakly increasing admissible sequences of length d such that 0 < r i ≤ s i < ∞ for all i.
Proof.Since r is weakly increasing, Condition 31 simplifies to the condition that (33) It is now an elementary but tedious exercise to construct a chain ( s (j) ) 0≤j≤N of weakly increasing admissible sequences s (j) = (s , where N ≫ 0, such that s (0) = r and s (d) = s, such that s for all i and j with 0 ≤ j ≤ N − 1, and such that each pair ( s (j−1) , s (j) ) satisfies Condition 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 G be a tame Levi sequence of length d and let r be a weakly increasing admissible sequence of length d.Then 3.6.Length computation.Retaining the notation of Section 3.4, let f : R → R be a positive function and let R ′ ⊆ R be a subset.Our goal in this subsection is to compute the length of the O-module g R ′ x,0+:f , culminating in Theorem 38.We start by studying the jumps in the filtration on g α x .For α ∈ R, consider the set ord x α := {t ∈ R : g α x,t:t+ = 0} of jumps in the Moy-Prasad filtration of g α , defined and studied by DeBacker and Spice [DS18, Definition 3.6].A full description of ord x α requires an understanding of the point x, and thus the way in which B(T ) embeds in B(G).This is quite difficult in general.But for us it is enough to know several weak properties of these sets.
Corollary 36.Let T be a tame maximal torus of G, let x ∈ A(G, T ), let f : R ∪ {0} → R be a positive function, and let R ′ ⊆ R. Then To simplify the sum in Corollary 36 we prove a more general result about summation of discretely supported functions h : R → N. Given a lattice Λ ⊂ R, that is, a nontrivial cyclic subgroup, say that h is Λ-periodic if h(t + λ) = h(t) for all t ∈ R and λ ∈ Λ.Given a bounded interval I ⊂ R with endpoints a < b, define where [•] is the Iverson bracket.Because of the normalization at the endpoints, this summation operator enjoys the property that whenever I, J, and I ∪ J are compact intervals.
Lemma 37. Let Λ ⊂ R be a free abelian group of rank one, let λ 0 := min(Λ ∩ R >0 ), let h : R → N be a discretely supported function, and let H(s) := ′ 0≤t≤s h(t) for s > 0. Suppose h is even and Λ-periodic.Then for all s ∈ 1 2 Λ ∩ R >0 , Proof.Since H(s + λ) = H(s) + H(λ), induction reduces the proof to the case where s = λ 0 or s = 1 2 λ 0 .The first case is obvious; for the second, use that h(t) = h(λ 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 ∈ A(G, T ), let f : R ∪ {0} → R be a positive even function, and let R ′ ⊆ R be a subset closed under negation.Suppose that Proof.By Corollary 36, the lefthand side of the theorem is len(g Since f is even and κ α = κ −α , the righthand side above is By Lemma 35, the function ½ ord x α + ½ ordx(−α) is even and ord(k × α )-periodic.Hence we may apply Lemma 37 to conclude that A cuspidal G-datum is a 5-tuple Ψ = ( G, y, r, ρ, φ) consisting of: • a tame twisted Levi sequence G such that Z 0 /Z is anisotropic, where Z is the center of G and Z 0 is the center of G 0 ; • a point y in the apartment of a tame maximal torus of G 0 ; • an increasing sequence r = (0 then we only require that 0 ≤ r 0 ); • an irreducible representation ρ of G 0 [y] whose restriction to G 0 y,0+ is 1-isotypic and for whom the compact induction c-Ind G 0 K 0 ρ is irreducible (hence supercuspidal); • a sequence φ = (φ 0 , . . ., φ d ) of characters, with φ i a character of G i , such that φ d = 1 if r d = r d−1 and otherwise φ i has depth r i for all i.The datum is generic if for each i = d the character φ i of G i is G i+1 -generic in the sense of Section 2.1, and the datum is regular if, in addition, the depth-zero supercuspidal representation c-Ind G 0 K 0 ρ is regular.Many of the objects used in Yu's construction and built from a cuspidal G-datum do not depend on the representations ρ and φ.To make this independence explicit, we define a cuspidal G-datum without representations to be a 3-tuple ( G, y, r) consisting of the first three components of a cuspidal G-datum.
Let Ψ = ( G, r, y) be a cuspidal G-datum without representations.From Ψ we can construct the following subgroups: y,(0+,s 0 ,...,s i ) y,(0+,s 0 +,...,s i +) Here 0 ≤ i ≤ d−1 and s i := r i /2.Generally the dependence of these objects on Ψ is implicit, but if we wish to make the dependence explicit we indicate it with a subscript, for instance, K = K Ψ .When Ψ 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.
We can now outline Yu's construction.Let Ψ = ( G, y, r, ρ, φ) be a generic cuspidal Gdatum.For each 0 ≤ i ≤ d−1 there is a certain finite-dimensional irreducible representation1 ρ i of K i+1 constructed from φ i .To specify ρ i precisely one uses the theory of the Weil-Heisenberg representation, but for our purposes, it is enough to know that The representation ρ i is then inflated to a representation2 τ 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 τ −1 to be the inflation (by the same procedure) of ρ to K, and in the edge case i = d take τ d to be the restriction of φ d to K; we could also handle the case i = −1 in the same way as the case 0 ≤ i ≤ d − 1 by defining ρ −1 := ρ.Finally, define the supercuspidal representation π attached to Ψ as the compact induction In summary, then, by Lemma 18 the formal degree of the supercuspidal representation π attached to a cuspidal generic Yu-datum Ψ has formal degree where A is the maximal split central subtorus of G and µ 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 G is anisotropic, that is, Lemma 42.Let Ψ be a generic cuspidal G-datum without representations and let Ψ be the reduction of Ψ modulo A. Then (1) the groups K Ψ and K Ψ /A are identical as subgroups of G/A, and For the first part, since , and this follows immediately from the surjectivity of the map G → (G/A)(k) and the natural identification of the reduced buildings of G and G/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 G be a reductive k-group let Ψ = ( G, y, r, ρ, φ) be a cuspidal G-datum, let A be the maximal split central subtorus of G, and let µ 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, Lemma 44.Let G be a group, let H be a subgroup of G, and let N ⊆ M be subgroups of G normalized by H.3 Then provided that all three indices in the expression are finite.
Proof.The inclusion M ∩H ֒→ M induces an injective map M ∩H/N ∩H ֒→ 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 ∩ H/N ∩ 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 ∩ H.So M ∩ H/N ∩ H is the stabilizer of the identity element, and the orbit-stabilizer theorem concludes the proof.
The volume of G y,0+ is known from the literature.
Proof.DeBacker and Reeder [DR09, Section 5.1] defined a Haar measure ν on G(k) such that for any x ∈ B(G), in particular, ν does not depend on the choice of x.Kaletha showed [Kal15, Lemma 5.15] that ν = q (dim G)/2 µ.Combining these results proves the lemma.
At this point, we can say that We can now simplify Eq. (47) using our earlier results on concave functions.
We now make the reduction promised in Lemma 42.Recall the notation for Ψ defined immediately before Lemma 42.
Theorem A. Let G be a reductive k-group and let Ψ be a generic cuspidal G-datum with associated supercuspidal representation π.Then Proof.When A = 1, the formula follows from Eq. (47), Lemma 48, and the fact that g 0 y,0:0+ is the Lie algebra of G a,0 y,0:0+ .In general, according to Eq. (41), with the exception of the factor dim ρ, the formal degree depends only on the underlying G-datum without representations Ψ = ( G, y, r).It now suffices to observe that by Lemma 42, the expression is the same for both Ψ and the reduced datum Ψ := ( − → G a , y, 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 π is compactly induced from a finite-dimensional representation ρ of the group G [y] , but when the supercuspidal π is regular, we can recover a maximal torus S from π, and π is induced from a finite-dimensional representation η of the smaller group SG y,0 .In fact, the former representation is induced from the latter: SG y,0 η.In the depth zero case, replacing ρ by η in the formula for the formal degree thus multiplies the rest of the formula by the index [G [y] : SG y,0 ].And since η 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 ρ of K i satisfying the following condition. (50) The restriction of ρ to K i ∩ J i+1 is 1-isotypic.
Ultimately we will apply the following analysis to the representation ρ i , which satisfies Condition 50.Hence ρ may be interpreted as (the inflation of) a representation of the quotient group K i /(K i ∩J i+1 ).Since K i normalizes J i+1 , the decomposition of Equation (40) becomes a semidirect product after dividing by K i ∩ J i+1 .We can now inflate this representation of the quotient first to the semidirect product K i+1 /(K i ∩ J i+1 ), then to the full group K i+1 .Let Inf K i+1 K i ρ denote the resulting representation.Since this representation is 1-isotypic on J i+1 , and since K i+1 ∩ J i+2 ⊆ J i+1 , the representation satisfies Condition 50 with i replaced by i + 1.By induction, we can therefore define for any j ≥ i + 1, and in particular for j = d (if i < d), the inflated representation Lemma 51.Let Ψ be a cuspidal G-datum; recall the notations of Section 3.7 for the various objects attached to Ψ. Let H i := SG 0 y,0 K i + and let H := H d .Suppose that there is a (necessarily irreducible) representation η of H 0 such that ρ := Ind K 0 H 0 η.Then σ −1 := Inf H H 0 η is defined and there is a canonical identification Proof.The notation inf H j H i mimics the notation inf K j K i : the same construction works if the symbol K is replaced everywhere by H. Since ρ is 1-isotypic on J 1 , so is η; hence σ −1 := inf H H 0 η is defined.There is a canonical identification The result now follows from the well-known [Ren10, Section III.2.11] formula ( , where π 1 is a representation of H and π 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 ρ has the property that ρ = Ind K 0 SG 0 y,0 η where η 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 η using Deligne-Lusztig theory.
Let's briefly recall the dimension formula for a Deligne-Lusztig induction.In this paragraph only, let G be a reductive κ-group, let S be a maximal elliptic torus of G, and let θ : S → C × be a character.Deligne We can now assemble these results to make our final formula.Recall the notation of Theorem A.
Corollary 52.Let Ψ be a regular generic cuspidal G-datum with resulting supercuspidal representation π and let S be the maximally unramified maximal torus of G 0 resulting from Ψ, as explained in Section 2.2.Then Proof.By Lemma 51, Lemma 18, and Kaletha's description of ρ, the formula of Theorem A remains true if dim ρ is replaced by The dimension formula for η discussed in the paragraph above shows that = [G a,0 y,0 : S a 0 G a,0 y,0+ ] exp q 1 2 (dim G a,0 y,0:0+ − rank G a,0 y,0:0+ ) −1 .
The formula now follows.

Galois side
Let (S, θ) be a tame elliptic pair.We saw in Section 2.4 that when θ is extra regular, such a pair can be extended to a regular supercuspidal L-packet datum (S, , χ, θ), and that the resulting set of χ-data can then be used to form the regular supercuspidal parameter ϕ (S,θ) := L j χ • L θ.
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 ϕ = ϕ (S,θ) .As we mentioned in Section 2.4, the group S ♮ ϕ is abelian and thus has only one-dimensional irreducible representations, so that the Galois side of the conjecture simplifies to Moreover, the factor |π 0 (S ♮ ϕ )| has been computed in the literature.So our task is to compute the absolute value |γ(0, ϕ, Ad, ψ)| of the adjoint γ-factor.
We start by reviewing in Section 4.1 the general definition of the γ-factor.As a second preliminary step, we work out in Section 4.2 how to base change the function r χ used to solve the extension problem of Section 2.3.
To compute the adjoint γ-factor, we give an explicit description of the adjoint representation attached to the L-parameter ϕ.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 γ-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 γ-factor computation proper, we must fix  and χ in the L-packet datum (S, , χ, θ) extending (S, θ).where ψ is a nontrivial additive character of k and µ is an additive Haar measure on k.
Hence the γ-factor is built from two quantities, the L-factor and the ε-factor.
In this subsection we recall the definitions of the L-factor and the ε-factor, following Tate's Corvallis notes [Tat79].Roughly speaking, the L-factor carries information about the unramified part of the representation and the ε-factor carries information about the ramified part of the representation.Since computing the absolute value of an ε-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 π is the holomorphic function where I k ⊂ W k is the inertia group and Frob ∈ W k is a Frobenius element.Later, we use the fact that the L-factor is inductive: if ℓ ⊇ k is a field extension of k and (π, V ) is a finite-dimensional complex representation of W k then The ε-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 γ-factor, not its complex argument, we content ourselves with a description of the absolute value of the ε-factor instead.
Changing s, ψ, or µ scales the ε-factor by a known quantity.We may thus define with no loss of information the simplified ε-factor ε(π) := ε(0, π, ψ, µ) where ψ has level zero, that is, max{n | ψ(π −n O) = 1} = 0, and where the Haar measure µ is self-dual with respect to ψ.With these conventions, the absolute value of the ε-factor is where cond π is the Artin conductor of π.So computing the absolute value of the ε-factor amounts to computing the Artin conductor.The Artin conductor is defined by the following procedure.Given a Galois representation (π, V ), choose a Galois extension ℓ of k such that π| W ℓ is trivial, and let Γ ℓ/k be the Galois group of ℓ over k.Then the Artin conductor satisfies the formula , where i → G ℓ/k,i is the lower numbering filtration.This formula is independent of the choice of ℓ.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, cond π = 0 if and only if π is unramified, and (55) cond π = codim V I k if π is tamely ramified.Heuristically, the numerical invariant cond π is an enhancement of Equation (55) that takes wild ramification into account and measures the extent to which π ramifies.When π is irreducible, this heuristic is made precise by the formula (56) cond π = (dim π)(1 + depth k π).
In particular, Equation (56) holds if π 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 ℓ of k and a representation π of W ℓ , the induced representation has conductor where disc ℓ/k is the discriminant of ℓ over k.But when ℓ is tamely ramified over k and π = χ is a character the formula simplifies considerably, even if, unlike in Equation (56), the induced representation is reducible.
Now use that depth ℓ χ = e ℓ/k depth k χ.
The L-factor, ε-factor, and Artin conductor are additive in the sense that where π = π 1 ⊕ π 2 .Hence the γ-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 χ-data.The main goal of this subsection is to determine how χdata 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 χ-data, we study its effect on minimally ramified χ-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 ℓ be a separable quadratic extension of k.Local class field theory shows that the quotient k × /N ℓ/k (ℓ × ) is cyclic of order two.The quadratic sign character of the extension ℓ ⊇ k is the character k × → {±1} given by projection onto this quotient.
A root α ∈ R(G, S) is symmetric if it is Galois-conjugate to −α, and is asymmetric otherwise.A symmetric root α is unramified if the quadratic extension k α ⊃ k ±α is unramified and is ramified otherwise.Letting k ±α denote the fixed field of the stabilizer in Γ k of {±α}, the extension k ±α ⊆ k α has degree two if α is symmetric and degree one if α is asymmetric.
Kaletha has interpreted a set of χ-data as giving rise to a character of a certain double cover of a torus, and the function r χ as the L-parameter of this character [Kal19a, Section 3].In light of that interpretation, we would expect that restricting r χ to an extension of k corresponds to composing the χ-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 χ be a set of χ-data and let ℓ be a finite separable extension of k.The base change of χ to ℓ is the χ-datum χ ℓ defined by χ ℓ,α := χ α,ℓα/kα .The definition of base change makes sense only if the formula for χ ℓ defines a χ-datum.We should immediately check this.
Lemma 61.The function χ ℓ of Definition 60 is a set of χ-data.
Proof.Negation equivariance is clear.Compatibility with the Galois group follows from the easily verified formula Use the sections u and v to compute the L-parameter of χ: Assume now that w = w ′ ∈ W ′ .Use Claim 65 to simplify the expression to To complete the argument, use the property cited above that the local class field theory homomorphism intertwines inclusion with the norm map.
Remark 66.Unlike most of the other parts of this paper, Theorem 64 and the definitions preceding it do not require the torus S to split over a tamely ramified extension; they hold for any torus whatsoever.
We can use the base change formula to bound the ramification of r χ .
Corollary 67.Let χ be a set of χ-data for a tamely ramified torus S. If each χ α is tamely ramified of finite order then r χ is tamely ramified of finite order.
Proof.By hypothesis, there is a finite tamely ramified extension ℓ of k such that for each α the character χ α,ℓ is trivial, so that χ ℓ is trivial.Then r χ restricts trivially to ℓ by Theorem 64 and is therefore tamely ramified of finite order.
To conclude this subsection we define minimally ramified χ-data following Kaletha [Kal19b, Definition 4.6.1].The definition is relevant to this subsection because a minimally ramified set of χ-data satisfies the hypotheses, and thus the conclusion, of Corollary 67; we use this observation in the proof of Lemma 70.Definition 68.A χ-datum χ for S is minimally ramified if S is tame and in addition χ α is trivial for asymmetric α, unramified for unramified symmetric α, and tamely ramified for ramified symmetric α. 4.3.Adjoint representation.Our goal in this subsection is to describe the adjoint representation of a regular parameter, specifically, its decomposition into a "toral summand" coming from the torus S and a "root summand" coming from the root system R(G, S).In subsequent subsections, we compute the ε-factor of both summands.
Recall from Section 2.3 that the regular parameter ϕ (S,θ) is given by the formula where  : S → G is an admissible embedding with image a Galois-stable maximal torus T and χ is a certain (carefully chosen) set of minimally ramified χ-data.We obtain the adjoint representation from ϕ = ϕ (S,θ) by composing it with the adjoint homomorphism L G → GL(V ), where V := ĝ/ẑ Γ k .The representation decomposes as a direct sum where V toral := t/ẑ Γ k and where ĝα .
Here ĝα is the usual α ∨ -eigenline for the action of S on ĝ, where α ∨ is interpreted, via  and the canonical identification X * ( T ) = X * (T ), as a root of X * ( T ).We call V toral the toral summand and V root the root summand.From our formula for ϕ we can work out the adjoint Weil actions on these summands.4.4.Toral summand.For the toral summand, it is useful to momentarily consider the vector space V toral := Lie( T ) equipped with the adjoint Weil action of ϕ, so that the projection V toral → V toral is Weil-equivariant.In general, for any complex torus T the natural inclusion X * (T ) ֒→ Lie(T ) gives rise to a canonical identification X * (T ) C ≃ Lie(T ).The representation V toral of the Weil group W k is therefore the complexification of the lattice Λ = X * ( T ), isomorphic to X * (S) by X * () and the canonical identification X * ( T ) = X * (T ).The Galois action on the lattice Λ is transferred via this chain of identifications from the Galois action on X * (S) arising from the structure of S as a torus over k.To summarize, there is an identification of representations V toral ≃ X * (S) C .Although X * (S a ) is a sublattice of X * (S), not a quotient, since X * (S a ) = X * (S) Γ k the smaller lattice becomes a canonical quotient of the larger after complexifying both.We thus have a second identification V toral ≃ X * (S a ) C .We can now compute the toral γ-factor using the lattice M := X * (S a ) I k , whose complexification is the vector space used to compute the L-factor of V toral .
Proof.We omit several details because the calculation closely follows [Kal15, Section 5.4].
It is easy to dispense with the L-factor at s = 0: where M Frob denotes the coinvariants of Frobenius.The L-factor at s = 1 is The determinantal factor in the last equation can be rewritten as Since S is tamely ramified, Equation (55) shows that the Artin conductor of the toral summand is just cond V toral = dim(V toral /V I k toral ) = dim S a − dim M. By our formula relating the Artin conductor and the ε-factor, Equation (54), |ε(V toral )| = exp q 1 2 (dim S a − dim M) .
Since θ is regular (Definition 8), the roots α with depth(θ kα/k • α ∨ ) ≤ 0 form a sub root system R 0 of R = R( T , G), and the action of inertia on R 0 preserves a set R + 0 of positive roots.Let H := Ad(ϕ(I k )), let Hα denote the H-orbit of α ∈ R, and let U α ⊂ G be the root group for α ∈ R. Since I k ∩ W α is the inertia group of k α and β∈Hα ĝβ is a monomial representation of I k induced from ψ α , the character ψ α is unramified if and only if the following three groups coincide: the stabilizer of U α in H, the centralizer of U α in H, and the centralizer of α in H.Moreover, ψ α is unramified if and only if ψ β is unramified for each β ∈ Hα.Assume α ∈ R + 0 satisfies these equivalent properties and has maximal length among all such roots.The proof works just as well if α ∈ R − 0 , so focus on the positive roots.First, suppose the roots in the H-orbit Hα of α are pairwise orthogonal.Choose a nontrivial element u α ∈ U α .For each β ∈ Hα, choose x ∈ H such that β = xα, and let u β := xu α .The element u β depends only on u α and β and not on x.Consider the product Then u is invariant under H, hence centralizes inertia.But at the same time u is not semisimple because the H-orbit of α consists of positive roots, contradicting regularity.
In the remaining case, when the roots in the H-orbit of α are not pairwise orthogonal, a slight elaboration of the previous argument yields a contradiction.In this case the H-orbit of α admits an involution β → β such that β, γ = 0 (for β, γ ∈ Hα) if and only if γ ∈ {β, β}.From each pair {β, β} with β ∈ Hα choose one element, including the element α, and let (Hα) + be the resulting set of orbit representatives, so that Hα = (Hα) + ⊔ (Hα) + .Choose a nontrivial element u α ∈ U α , choose x ∈ H such that xα = ᾱ, and define the element u ᾱ := xu α , independent of the choice of x.The commutator subgroup U α+ ᾱ = [U α , U ᾱ] is stabilized by x, and since we assumed that α had maximal length among the possible counterexamples to our theorem, it is not centralized by x.(In fact, x must act by inversion on this group because xu ᾱ = u α .)Hence there is an element u α+ ᾱ ∈ U α+ ᾱ with that is, u α+ ᾱu α u ᾱ = x(u α+ ᾱu α u ᾱ).
Since the action Ad • ϕ of wild inertia on G 0 is trivial, the group H acts on the factors of u through some abelian quotient.Hence u centralizes inertia but is not semisimple, contradicting regularity.
Remark 73.Kaletha defines an L-parameter to be torally wild if it takes wild inertia to a maximal torus of G, and shows that torally wild L-parameters factor through the L-group of a tame maximal torus [Kal19c].The proof of Lemma 72 shows that this larger class of parameters satisfies the conclusions of the lemma.
Proof.Recall from Section 2.1 that there is a twisted Levi subgroup G 0 of G such that α ∈ R(S, G 0 ) if and only if depth(θ kα/k • α ∨ ) ≤ 0, and that S is maximally unramified in G.
Lemma 72 handles the case where G = G 0 , so we assume that G 0 G.
To deal with this more general case we factor the L-embedding L S → L G through L G 0 .Kaletha showed [Kal19b, Lemmas 5.2.9, 5.2.8] that there is an L-embedding L j G 0 ,G : L G 0 → L G with the following property: the composite parameter W k → L S → L G is given by the formula w →  θb (w) θ(w)r χ (w) n(ω S,G (w))w where θ b : S(k) → C × is tamely ramified and Ω(S, G 0 ) Γ k -invariant.Furthermore, from the construction of L j G 0 ,G it is easy to see that the embedding g 0 ֒→ g is L j G 0 ,G -equivariant.In this way we reduce to the previous case of G = G 0 but with θ replaced by θ • θ −1 b .This replacement does not affect the validity of the reduction: since θ b is Ω(S, G 0 ) Γ k -invariant the character θ ′ = θ • θ −1 b is still regular [Kal19b, Fact 3.7.6],and since θ b is tamely ramified, we still have that depth(θ ′ kα/k • α ∨ ) ≤ 0. Remark 75.Unlike Corollary 71, Lemma 74 does not assert that ψ α and θ kα/k • α ∨ have the same depth when the latter has depth zero.I expect this stronger assertion to be true.It would be enough to prove that if θ is extra regular then depth(θ kα/k • α ∨ ) ≥ 0. But I was unable to prove the stronger assertion and a weaker statement sufficed.
In summary, the root summand decomposes as a direct sum where, for any α ∈ α( k), the representation V α is induced from a character ψ α of W α with known depth.We can now easily compute the γ-factor.Recall the Galois sets R i and depths r i ≥ 0 of Section 2.1.Proof.Suppose that α ∈ R i+1 , for 0 ≤ i ≤ d − 1. Lemmas 12 and 74 and Corollary 71 show that depth k ψ α = r i .Since L-factors are inductive L(s, V α ) = L(s, ψ α ), and since ψ α is ramified its L-factor is trivial.As for the absolute value of the ε-factor, since the extension k α ⊇ k is tame, Lemma 58 shows that cond V α = (1 + r i )|α( k)|, so that ε(V α ) = exp q 1 2 (1 + r i )|α( k)| .Summing over α ∈ R(G, S) finishes the proof.4.6.Summary.Let A be the maximal split central subtorus of G, let G a := G/A, let S a := S/A, and let M := X * (S a ) I k .Lemmas 69 and 76 show that the absolute value of the adjoint γ-factor is

Comparison
In this short final section we combine our work from Sections 3 and 4 with several results from the literature to show that the automorphic and Galois sides of the formal degree conjecture are equal, the following theorem.
Let (S, θ) be a tame elliptic regular pair and let ϕ = ϕ (S,θ) be the L-parameter attached to this pair by the constructions of Section 2.4.The Galois side of the formal degree conjecture for ϕ is expressed in Equation (77).Recall the notation of Section 4.6.
The supercuspidal representations in the L-packet for ϕ are of the form π (jS,jθ ′ ) as described in Section 2.4, where j ranges over conjugacy classes of admissible embeddings j : S ֒→ G. Since jθ ′ and θ • j −1 differ by a tamely ramified character, the formal degrees of π (S,θ) and π (jS,jθ ′ ) , as expressed in Equation (53), agree.So on the automorphic side, we can assume for the purpose of computing the formal degree that the relevant pair is (S, θ), even though it is actually (jS, jθ ′ ).
To compute the dimension of the lattice M := X * (S a ) I k from Section 4.6, we prove an analogue for tori of the Néron-Ogg-Shafarevich criterion for abelian varieties.
Lemma 78.Let k be a Henselian, discretely-valued field with residue field κ and let T be a tame k-torus.Then there is a canonical identification X * (T ) I k = X * (T 0:0+ ).
Proof.Since X * (T ) I k = X * (T ) k nr where k nr is the maximal unramified extension of k, we can use étale descent for the Moy-Prasad filtration [Yu02, 9.1] to reduce the proof to the case where κ is separably closed.Let T s ⊆ T be the maximal split subtorus, so that X * (T ) I k = X * (T s ) = X * (T s 0:0+ ) since now I k = Γ k .It suffices to prove that the canonical inclusion T s 0:0+ ֒→ T 0:0+ is an isomorphism.The proof rests on two facts from SGA 3. Since S 0 is smooth and affine, the moduli space of its maximal tori is represented by a smooth O-scheme [DG11, Exposé XII, Corollaire 1.10].By Hensel's Lemma [DG11, Exposé XI, Corollaire 1.11], every κ-point of this moduli space lifts to a O-point.
At this point, we know that the exp q factors in Eq. (53) and Eq.(77) are equal.
Lemma 79 ([Kal15, Lemma 5.17]).[S a (k) : S a (k Let's now compare the remaining factors outside of exp q .On the automorphic side we have [S a (k) : S a (k) 0+ ] −1 ; on the Galois side we have .
−1 and has depth r d otherwise; and the character φ −1 is trivial if G 0 = S and otherwise satisfies φ −1 | S(k) 0+ = 1.It turns out [Kal19b, Proposition 3.6.7]that every tame pair admits a Howe factorization.When α / ∈ R 0 , we can use this factorization to compute the depth of θ kα/k • α ∨ .
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, θ, j) a certain minimally ramified set χ of χ-data, which appears, for one, in the definition of ε.Then the L-packet corresponding to (S, , χ, θ) consists of the set of regular supercuspidal representations π (jS,jθ ′ )

Proof.
It suffices to prove the corollary in the case where R ′ = {α}.Then both sides equal t∈ordα x 0<t<f (α) len g α x,t:t+ by the third part of Lemma 35.
The ratio of one to the other is|X * (S a ) I k Frob | • |X * (S a ) Frob I k | |X * (S a ) Γ k | ,using that M = X * (S a ) I k .This ratio equals 1 [Kal15, Lemma 5.18].
[Yu01]'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.
and Lusztig computed the dimension of the virtual representation R (S,θ) in their original paper [DL76, Corollary 7.2]; it is where 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