aa r X i v : . [ m a t h . R T ] J a n FORMAL DEGREE OF REGULAR SUPERCUSPIDALS
DAVID SCHWEIN
Abstract.
Supercuspidal representations are usually infinite-dimensional, so the size ofsuch a representation cannot be measured by its dimension; the formal degree is a betteralternative. Hiraga, Ichino, and Ikeda conjectured a formula for the formal degree of asupercuspidal in terms of its L -parameter only. Our first main result is to compute theformal degrees of the supercuspidal representations constructed by Yu. Our second, usingthe 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 ofthe representation theory of the topological group G ( k ), the supercuspidal representationsare fundamental: there is a precise sense in which all irreducible (smooth or unitary) repre-sentations can be constructed from supercuspidals. Much recent work has thus focused onthe construction and study of supercuspidal representations.In 2001, Yu [Yu01], building on earlier work of Howe [How77] and Adler [Adl98], gave ageneral construction of supercuspidal representations when G splits over a tamely ramifiedextension of k . Six years later, Kim [Kim07] proved that Yu’s construction is exhaustivewhen k has characteristic zero and the residue characteristic p of k is larger than someineffective bound depending on G . Recently, Fintzen [Fin18] has improved Kim’s result toshow that Yu’s construction produces all supercuspidals when p does not divide the orderof the absolute Weyl group. Hence Yu’s construction produces all supercuspidals for manyreductive groups, though not all of them. Moreover, the explicit nature of Yu’s constructionmakes 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 iscanonical 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 asa point of the unitary dual. Since supercuspidal representations of semisimple groups arediscrete series, this point is isolated. We may thus ask for the measure of the point, aninteresting numerical invariant of π called the formal degree .When G is not semisimple it is no longer necessarily the case that all supercuspidal repre-sentations are unitary. Nonetheless, we can define the formal degree of an arbitrary supercus-pidal 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 forgetabout 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 represen-tations, Theorem A. The formula uses some notation that we must briefly recall for itsstatement to be intelligible. Yu’s construction takes as input a 5-tuple Ψ. The first mem-ber of Ψ is an increasing sequence ( G i ) ≤ 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 build-ing B ( G ). The third member is an increasing sequence ( r i ) ≤ i ≤ d of nonnegative real numbers.The fourth member is a certain irreducible representation ρ of the stabilizer G ( k ) [ y ] of theimage [ y ] of y in B red ( G ). We compute the formal degree with respect to a certain Haarmeasure µ constructed by Gan and Gross and discussed later in the introduction. Gan andGross’s measure depends on a choice of additive character and in the formula we choose alevel-zero character. For simplicity of exposition we assume in our discussion of the formulathat G is semisimple, though this assumption is relaxed in the paper proper. Finally, letexp q ( t ) := q t . Theorem A.
Let G be a semisimple k -group and let Ψ be a generic cuspidal G -datum withassociated supercuspidal representation π . Then deg( π, µ ) = dim ρ [ G ( k ) [ y ] : G ( k ) y, ] exp q (cid:18) dim G + dim G y, + d − X i =0 r i ( | R i +1 | − | R i | ) (cid:19) . 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-dimensionalirreducible representation τ of a certain compact-open subgroup K . There is a generalformula for the formal degree of a compact induction which specializes, in this case, to theratio 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 tocomputing the index. Using the theory of the Moy-Prasad filtration, we can translate thecomputation of this index into the computation of the length of a certain subquotient of theLie algebra. The length computation, Theorem 38, is our key technical result in the proofof 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 arithmeticparameterization of supercuspidals.It is expected that the set Π( G ) of smooth irreducible representations of G ( k ) is classifiedby certain homomorphisms ϕ : W ′ k → L G , called L -parameters . Here W k is the Weil groupof k , W ′ k := SL ( C ) × W k is the Weil-Deligne group of k , and L G := b G ⋊ W k is the (Weilform of the) L -group of G . The expected classification consists of a partitionΠ( G ) = G ϕ Π ϕ ( G )of Π( G ) into finite subsets Π ϕ ( G ), called L -packets , indexed by (equivalence classes of) L -parameters ϕ . The sets Π ϕ ( G ) are supposed to satisfy many compatibility conditions, thesimplest of which are summarized in Borel’s Corvallis article [Bor79, Section 10.3]. Theresulting partition of Π( G ) is called a local Langlands correspondence . Although a localLanglands correspondence has been constructed for many classes of groups and represen-tations, the full correspondence remains a conjecture. Even after fixing the group G , it isusually quite difficult to establish the correspondence for the entirety of Π( G ). Recent workhas thus focused on constructing the L -packets of particular L -parameters.Building on the outline of the correspondence, Langlands suggested [Lan, Chapitre IV] thatthe elements of the L -packet Π ϕ ( G ) are parameterized by representations of a certain finitegroup attached to ϕ . Refining Langlands’s proposal, Vogan enhanced [Vog93, Section 9] L -parameters to pairs ( ϕ, ρ ) consisting of an L -parameter ϕ and an irreducible representation ρ ORMAL DEGREE OF REGULAR SUPERCUSPIDALS 3 of the finite group π ( S ϕ ), where S ϕ is the preimage in b G sc of the centralizer in b G of ϕ . Unlikeordinary L -parameters, these enhanced parameters keep track of the inner class of G : oneimposes 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 thisformulation, it is expected [ABPS17, Section 1.2] that the local Langlands correspondencebecomes a bijection, in other words, that the irreducible representations ρ of S ϕ satisfyingthe 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 interms of its L -parameter. They verified the conjecture in many cases, in particular, for realreductive groups and for inner forms of SL n and GL n . We will state their conjecture in amoment after reviewing two of its inputs.First, in a paper attaching motives to reductive groups, Gross constructed [Gro97a, Sec-tion 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] thatconjecturally agrees with the original one. Hiraga, Ichino, and Ikeda originally predictedthat Gross’s measure was the right one for their conjecture, but realized later [HII08a] thatone should use Gross and Gan’s measure instead.Second, one can attach to the parameter ϕ and certain additional data – a finite-dimensionalrepresentation 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 wechoose the adjoint representation Ad of L G on b g / b z Γ k , where b g and b z are the Lie algebras of b G and of its center and where Γ k is the absolute Galois group of k . Division by b z Γ k ensures thatthe γ -factor is defined at s = 0. We call the factor γ (0 , ϕ, Ad , ψ ) appearing in the conjecturethe the adjoint γ -factor of ϕ .We can now state the conjecture [HII08b, Conjecture 1.4] of Hiraga, Ichino, and Ikeda onthe formal degree, referred to in this paper as the “formal degree conjecture” for the sakeof brevity. Let S ♮ϕ denote the centralizer of ϕ in c G a , where G a := G/A with A the maximalsplit central torus of G . Conjecture 1.
Let π be an essentially discrete series representation of G ( k ) with extendedparameter ( ϕ, ρ ) , let ψ be an additive character of k , and let µ ψ be the Gross-Gan measureon G ( k ) attached to ψ . Then deg( π, µ ψ ) = dim ρ | π ( S ♮ϕ ) | · | γ (0 , ϕ, Ad , ψ ) | . Hiraga, Ichino, and Ikeda verified their conjecture in several cases, building on the work ofmany 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]; forsome 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 theconjecture, Gan and Ichino [GI14] showed that it holds for U , Sp , and GSp ; 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 DAVID SCHWEIN that it holds for simple supercuspidals; Ichino, Lapid, and Mao [ILM17] showed that it holdsfor odd special orthogonal and metaplectic groups; and Beuzart-Plessis [BP20] showed thatit holds for unitary groups.The formal degree conjecture is a “meta-conjecture” in the sense that it depends itselfon a conjecture, the local Langlands correspondence. In order to verify the formal degreeconjecture one must first have access to a candidate local Langlands correspondence, or atleast, to candidate L -packets. Strictly speaking, the previous sentence is not entirely truebecause some groups admit an analytic construction of the γ -factor that bypasses the localLanglands correspondence, though the two are expected to be compatible. The main exampleis Godement-Jacquet’s [GJ72] construction of the L - and ε -factors for representations of thegeneral 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 groupusing work that predated the Henniart [Hen00] and Harris-Taylor [HT01] constructions ofthe local Langlands correspondence. Nonetheless, for the representations we consider in thispaper, 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 rep-resentations, the “regular supercuspidal representations”. His construction passes througha pair ( S, θ ) consisting of an elliptic maximal torus S of G and a character θ of S ( k ). Onthe Galois side, one uses the Langlands-Shelstad theory of χ -data and extensions of L -embeddings [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 supercuspidalrepresentation π of G ( k ). We can thus interpret π as a functorial lift of θ with respect tothe embedding S ֒ → G . The L -packet of ϕ consists, roughly, of all π produced in this wayas we pass through the various conjugacy classes of embeddings of S in G ; Section 2 reviewsthe construction in more detail.Since we can compute the formal degree of Yu’s representations, and Kaletha’s L -packetsconsist of such representations, it is natural to ask whether the L -packets satisfy the formaldegree 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 aregular supercuspidal parameter: it is a direct sum of the complexified character latticeof 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 forthe monomial representations, computing their γ -factors amounts to computing the depthsof the inducing characters. The inducing characters are very close to certain charactersnaturally constructed from θ , and whose depth is usually easy to understand; the difficultyin the proof is to quantify the difference between the two characters. To quantify it, weprove that χ -data satisfy a natural base change formula, Theorem 64, and that the inducingcharacters are ramified, Lemma 74.A refinement of the formal degree conjecture due to Gross and Reeder [GR10, Conjec-ture 8.3] predicts the root number of the adjoint representation. In future work, I hope todetermine whether Kaletha’s regular L -packets also satisfy this refined conjecture.The structure of this paper mirrors the formal degree conjecture. After two preliminarysections that fix notation and review the Langlands correspondence for regular supercusp-idals, we compute the formal degree of a Yu representation in Section 3, we compute the ORMAL DEGREE OF REGULAR SUPERCUSPIDALS 5
Galois side of the formal degree conjecture in Section 4, and we compare the two in the briefSection 5.
Contents
1. Notation 51.1. Sets 51.2. Filtrations 61.3. Fields 61.4. Groups 71.5. Root systems 81.6. Bruhat-Tits theory 81.7. Base change for groups and vector spaces 91.8. Base change for characters 92. Langlands correspondence for regular supercuspidals 102.1. Tame elliptic regular pairs 112.2. Regular representations 122.3. L -embeddings 142.4. Regular parameters 153. Automorphic side 163.1. Notation 173.2. Formal degree 173.3. Concave-function subgroups: split case 193.4. Concave-function subgroups: tame case 203.5. Yu’s groups 223.6. Length computation 243.7. Yu’s construction 263.8. Degree computation 273.9. Regular supercuspidals 304. Galois side 324.1. Review of L -, ε - and γ -factors 324.2. Base change for χ -data 344.3. Adjoint representation 374.4. Toral summand 384.5. Root summand 394.6. Summary 415. Comparison 42Acknowledgments 43References 431. Notation
Sets.
Let | X | denote the cardinality of the set X . Given a subset Y ⊆ X , let Y : X → { , } denote the indicator function of Y .Many operations on sets are expressed by superscripts or subscripts. When we haveseveral operations denoted this way, say X X a and X X b , we use a comma to denote DAVID SCHWEIN the concatenation: X X a X a,b . This expression is notationally simpler than the longerform ( 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, I -indexed filtration of the set X . For i ∈ I , define(2) X i + := [ j>i X j . Let e I := I ∪ { i + : i ∈ I } ∪ {∞} denote Bruhat and Tits’s extension of I [BT72, 6.4.1]; theirdefinition is for I = R only, but it is clear how to extend it to arbitrary I . Equation (2)together with the convention X ∞ := \ i X i defines an extension of the given filtration to a e I -indexed filtration. If in addition X = G isa group and each X i = G i is a subgroup of G then define, for i < j in e I , G i : j := G i /G j . We apply this formalism to the Moy-Prasad filtration on a p -adic group G ( k ) and its Liealgebra 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 ≥ , and the filtrations on g are indexed by R . In Section 2.1 weconsider a filtration (of a root system) that is increasing, not decreasing. When the filtrationon X is increasing, its extension to e I is defined by X i + := \ j>i X j , X ∞ := [ i X i . 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 finitealgebraic 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 constructionof 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 . Weextend ord k to a valuation on the algebraic closure ¯ k and denote the extension by ord k aswell. Hence the value group for a finite extension ℓ is ord k ( ℓ × ) = e − 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 thanlarger than Z . Most of the computations in this paper can be modified to accommodate adifferent choice of value group because their defining object inherit that choice. Many ofthe 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 conductorcomputations of Section 4.1 could be modified to carry through because the upper numberingfiltration inherits its jumps from the value group. However, this modification would breakthe connection between the Artin conductor and the Artin representation. Moreover, sincethe ε -factor is defined independent of the value group, the relationship between the Artinconductor and the ε -factor, Equation (54), holds only for value group Z . ORMAL DEGREE OF REGULAR SUPERCUSPIDALS 7
Given a finite O -module M , let len M denote the length of M . When k has positivecharacteristic 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 insteadwith 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 equationexp 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 ≥ W rk ⊆ W k denote the r th subgroupof 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 asdepth π := inf { r ∈ R : π ( W rk ) = 1 } . In order to make the filtration on the Weil group compatible with the Moy-Prasad filtration,we need to modify the upper numbering filtration on W ℓ for a finite extension ℓ of k by usingthe valuation ord k to define it instead of the valuation ord ℓ . When the extension is tame, asis usually the case, this modification has the effect of scaling the indices of the filtration by e − ℓ/k . To make the dependence on k clear, let depth k denote the depth of a representationof W ℓ where its filtration is computed using ord k . The distinction is crucial for the proof ofLemma 58.1.4. Groups.
Let G be a reductive k -group, let Z be the center of G , and let A be themaximal split subtorus of Z . We reserve the symbols S and T for tori, often maximal toriof 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 re-ductive group and S and T denotes tori.First, we sometimes need to work with κ -groups instead of k -groups. This practice cannotbe avoided, but is so rare that we did not see the value in introducing a separate notationalconvention for κ -groups. So G , S , and T denote κ -groups in this setting, which takes placein 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 andthe 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 locallyprofinite group, the proper setting for that theory.Given a subgroup H of G , let H a := G/A . The letter a abbreviates “anisotropic”. Thenotation hides the dependence on the ambient group G , but the meaning should be clearfrom context: typically H is a maximal torus or (twisted) Levi subgroup of G .Let b G denote the Langlands dual group of G , a complex reductive group, and let L G = b 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 . DAVID SCHWEIN
Root systems.
Given a reductive k -group G and a maximal torus T of G , let R ( G, T )be the absolute root system of G with respect to T , that is, the root system of G ¯ k withrespect to T ¯ k , together with its natural Galois action. Let R ( G, T ) denote the set of Galoisorbits 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 := ¯ k Γ α denote the fixedfield of Γ α . Given α ∈ R , let k α denote a fixed field extension of k that is isomorphic, forsome 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 thestyle of Deligne and Lusztig [DL76, Section 1.1], but since Γ k is nonabelian, it is impossibleto canonically identify this limit with any one k α . The notation k α helps us avoid choosingsuch an α , and is reserved for expressions that depend only on the isomorphism class of theextension, 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 ex-tended and reduced Bruhat-Tits buildings of G . Let [ x ] denote the image of x under thecanonical 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 “apart-ment” refer to the extended forms.The apartment of a split maximal torus is a classical construction, defined in Bruhat andTits’s original papers on buildings [BT72, BT84]. More recently, in a paper establishingtame 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 notcanonical, the image of the embedding is canonical. We denote this image by A ( G, S ) andcall it the apartment of G in S , though that terminology is typically reserved for maximalsplit 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 thecenter of G is anisotropic the parahoric group G ( k ) x, is of finite index in the stabilizer G ( k ) x ;in general, A ( k ) G ( k ) x, is of finite index in G ( k ) x .For each x ∈ B ( G ), Moy and Prasad defined [MP94, Sections 2 and 3] a canonical de-creasing R ≥ -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 torusthe point x is irrelevant and we suppress it from the notation. Although the filtrations aredefined 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 muchmore 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 thesense that for r > k × r := { a ∈ k × : ord k ( a − ≥ 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 ORMAL DEGREE OF REGULAR SUPERCUSPIDALS 9 that the Moy-Prasad filtration is a topological invariant independent of the base field in thefollowing sense: the Moy-Prasad filtration on the group G ( ℓ ) agrees with the Moy-Prasadfiltration 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 , isthe smallest real number r such that for some x ∈ B ( G ), the representation π has a nonzerovector fixed by G ( k ) x,r + . The subscript k is a reminder that the depth, via the Moy-Prasadfiltration by which it is defined, depends on the base field k . It is not a priori clear that thisminimum is attained, but Moy and Prasad showed [MP96, Theorem 3.5] that the depth isa nonnegative rational number. This result makes the depth an indispensable tool in therepresentation theory of p -adic groups. Remark 6.
Most of Bruhat-Tits theory carries through when the field k is assumed only tobe 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, thepullback X × Y Z . When Z = Spec A is the spectrum of a field A , we write X A for X Z . Theschemes X that we base change are in practice always group schemes.Similar notation can be used for base change of modules. Given a B -algebra A and an A -module M , let M B := M ⊗ A B .We take the position that an algebraic group carries the information of its base scheme.This position forces our terminology to differ slightly from common practice in the literaturewhere a k -group is thought of as a ¯ k -group with a k -rational structure. In this commonlanguage one can speak, given two k -groups G and H , of morphisms G → H that are notdefined over k . For us, a morphism G → H is automatically defined over k . To speakof 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 extensionof 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 dualof this unit is a canonical map N ℓ/k : T → S , called the norm map ; we use the samename 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: θ ℓ/k := θ ◦ N ℓ/k , We call θ ℓ/k the base change of θ from k to ℓ . Contrary to the usual notation for base changeof schemes, the notation for base change of characters must include k , not just ℓ , becausethe 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 corre-spondence. For tori, this correspondence is a bijection between the complex character groupof S ( k ) and the Galois cohomology group H ( W k , b S ). On the Galois side, we can restrictthe L -parameter ˆ θ of a character θ to the Weil group W ℓ of a finite separable extension. Butby the local Langlands correspondence for S ℓ , this parameter ˆ θ | ℓ corresponds to a characterof S ( ℓ ). It is well known, and a formal consequence of the properties of the local Lang-lands correspondence for tori, that this character is precisely the base-changed character just defined: symbolically, ˆ θ | ℓ = d 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 ≥ or (2) ℓ/k tamely ramified and depth k θ > 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 sufficesto prove the lemma in the case where S = G m . The first case is an immediate consequenceof local class field. Using the first result, reduce the second to the case where ℓ is obtainedfrom k by adjoining a root of a uniformizer; from here, the result is a straightforwardcomputation. (cid:3) Langlands correspondence for regular supercuspidals
In this section we review the Langlands correspondence for regular supercuspidal represen-tations, following Kaletha’s article [Kal19b]. Many of the definitions, for instance, regularityof L -parameters, are rather technical, and instead of restating them, we point to their defi-nitions in the literature. The description of regular representations and the construction oftheir 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 verifythe formal degree conjecture, how these pairs interface with both sides of the Langlandscorrespondence.On the automorphic side, the pair ( S, θ ) produces an input to Yu’s construction [Yu01] ofsupercuspidals; we explain how this works in Section 2.2. In this way we produce a supercus-pidal representation π ( S,θ ) of G ( k ). When ( S, θ ) is “tame elliptic regular”, the representationsthat 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 W k L θ −→ L S L j χ −−→ L G. Here the first map corresponds to θ under the local Langlands correspondence for tori andthe second map is an extension of a given Galois-stable embedding ˆ : b S → b G . There is ageneral procedure, reviewed in Section 2.3, for extending ˆ j 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 thepair ( S, θ ), producing a canonical extension L j χ and thus a canonical L -parameter. Usingpairs ( S, θ ), we organize regular supercuspidal parameters into L -packets in Section 2.4. Tofirst approximation a regular supercuspidal L -packet consists of the regular supercuspidalrepresentations π ( jS,θ ◦ j − ) 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 − . ORMAL DEGREE OF REGULAR SUPERCUSPIDALS 11
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 correspondencefor regular supercuspidals. This subsection reviews these conditions, following the discussionin Kaletha’s article [Kal19b, Sections 3.6 and 3.7], and uses them to compute the depths ofcertain 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 splitsover a tamely ramified extension of k . In this paper S is always assumed tame, though wesometimes repeat the hypothesis for emphasis.All other conditions on our pair require that S be embedded as a maximal torus of areductive group G . This requirement is extremely natural on the automorphic side, but onthe 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 > R r := { α ∈ R | ( θ ℓ/k ◦ α ∨ )( ℓ × r ) = 1 } . The assignment r R r is an increasing, Galois-stable, R -indexed filtration of R . Let R r + := S s>r R s , let r d − > · · · > r > r − = 0 and r d = depth k θ . For each 0 ≤ i ≤ d , let R i := R ( r i − )+ . It turns out [Kal19b, Lemma 3.6.1] that R i is a Levi subsystem of R . Let G i be the connected reductive subgroup of G containing S whose root system with respect to S is R i ; the group G i can be constructed by Galois descent,for example. Let G − := S . Definition 8 ([Kal19b, Definition 3.7.5]) . A tame elliptic pair is regular if(1) the action of the inertia group on R preserves a set of positive roots, and(2) the stabilizer of the action of the group N ( G , S )( k ) /S ( k ) on θ | S ( k ) is trivial.It is extra regular if, in addition,(2 ′ ) the stabilizer of the action of the group Ω( G , S )( k ) on θ | S ( k ) is trivial.Here N ( G , S ) is the normalizer of G in S and Ω( G , S ) is the Weyl group.When we compute in Section 4 the formal degree of the regular parameter attachedto ( S, θ ), half of the problem (the “root summand” of Section 4.5) boils down to know-ing for each coroot α ∨ the depth of the character θ k α /k ◦ α ∨ ;here α ∨ is interpreted as a homomorphism k × α → S ( k α ). Therefore, our main goal in thissubsection is to compute the depth of this character. To carry out the computation, wesystematically decompose θ as a product of characters of known depth using Kaletha’s notionof 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 acharacter φ : H ( k ) → C × of a twisted Levi subgroup H of G to be G -generic of depth r . Weneed not concern ourselves with the precise definition of G -genericity, but we do need one ofits 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 φ ℓ/k ◦ α ∨ ℓ of ℓ × has depth r .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 usingthe naturality of the norm map, and from there, we can deduce the result in general usingnaturality and Lemma 7. (cid:3) Corollary 10.
In the setting of Lemma 9, depth k φ ∈ ord( k × α ) for each α ∈ R ( G, S ) \ R ( H, S ) . Regularity is much less restrictive than genericity, but we need to know something aboutgenericity in order to understand the depths of various auxiliary characters constructedfrom θ in Section 4.5. Roughly speaking, any character, regular or not, can be decom-posed as a product of generic characters related to the filtration of the root system. Thisdecomposition is called a Howe factorization. Definition 11. A Howe factorization of (
S, θ ) is a sequence ( φ i : G i ( k ) → C × ) − ≤ i ≤ d ofcharacters satisfying the following properties.(1) θ = d Y i = − φ i | S ( k ) .(2) The character φ i is trivial on G i sc ( k ) for 0 ≤ i ≤ d .(3) The character φ i is G i +1 -generic of depth r i for 0 ≤ i ≤ d −
1; the character φ d istrivial if r d = r d − and has depth r d otherwise; and the character φ − is trivial if G = S and otherwise satisfies φ − | S ( k ) = 1.It turns out [Kal19b, Proposition 3.6.7] that every tame pair admits a Howe factorization.When α / ∈ R , we can use this factorization to compute the depth of θ k α /k ◦ α ∨ . Lemma 12.
Let ( S, θ ) be a tame pair and let α ∈ R i , where ≤ i ≤ d . Then the character θ k α /k ◦ α ∨ : k × α → C × has depth r i − .Proof. Let ( φ j : G j ( k ) → C × ) − ≤ j ≤ d be a Howe factorization of ( S, θ ). Then θ k α /k ◦ α ∨ = d Y j = − φ j,k α /k ◦ α ∨ . Since α ∨ factors through G i , condition (2) of a Howe factorization implies that the factorsof this product are trivial for j ≥ i . By Lemma 9, the j th remaining factor has depth r j ,and since the sequence j r j is strictly increasing, the product has depth r i − . (cid:3) Lemma 12 conspicuously omits the case where α ∈ R . We have more to say about thisin Section 4.5, especially in Lemma 74 and Remark 75.2.2. Regular representations.
Yu’s seminal paper [Yu01] constructs a broad class of su-percuspidal representations starting from a certain triple ( ~G, π − , ~φ ), which we call, for ref-erence, a Yu datum . This subsection reviews these triples and explains how a tame ellipticregular pair gives rise to a Yu datum. Later, Section 3.7 explains in more detail how toconstruct supercuspidal representations from Yu data.
ORMAL DEGREE OF REGULAR SUPERCUSPIDALS 13
There are three stages of representations used in in Yu’s construction, each informingthe previous one: representations of finite groups of Lie type; depth-zero supercuspidalrepresentations; and supercuspidal representations of arbitrary depth. Since the definitionof regular supercuspidal passes through each of these stages, we start by reviewing each stagein turn.The first stage is finite groups of Lie type. In this paragraph only, let G be a reductive κ -group. 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 torusof 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-Lusztigrepresentation.Passing to the second stage, depth-zero supercuspidals, the result that initiated generalconstructions of supercuspidals was the following classification theorem [MP96, Proposi-tion 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, contains the inflation to G ( k ) x, of an irre-ducible cuspidal representation e ρ of G ( k ) x, . Furthermore, we can recover π by compactinduction: there is some representation ρ of G ( k ) x such that ρ | G ( k ) x, contains the inflationof e ρ and such that π = c-Ind G ( k ) G ( k ) x ρ . The depth-zero supercuspidal representation π is regular if the representation ρ of G ( k ) x, is regular. Regular depth-zero supercuspidals π enjoytwo pleasant properties.First, there is a canonical bijection between regular depth-zero supercuspidals and conju-gacy classes of regular tame elliptic pairs ( S, θ ) of depth zero in which the torus S is maximallyunramified in G [Kal19b, Definition 3.4.1]. In particular, we can recover S from π .Second, it turns out [Kal19b, Sections 3.4.4 and 3.4.5] that π ( S,θ ) can be compactly inducedfrom a representation η ( S,θ ) of the group S ( k ) G ( k ) x, . This is an improvement over Moy andPrasad’s theorem, which uses the larger stabilizer group G ( k ) x instead; generally S ( k ) G ( k ) x, is easier to understand than G ( k ) x . The fact that π is compactly induced from this smallergroup plays a crucial role in our final computation, in Section 3.9, of the formal degree of aregular 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 Levisubgroup of G ℓ , and H is tame if ℓ can be taken to be a tame extension of k . A twisted Levisequence in G is an increasing sequence ~G = ( G ( G ( · · · ( G d )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 − of a Yu datumis a depth-zero supercuspidal representation of G ( k ); and the third component ~φ of a Yudatum is a sequence of characters ~φ = ( φ i : G i ( k ) → C × ) ≤ i ≤ d . These three objects are required to satisfy certain conditions that Section 3.7 spells out indetail. In fact, in that section we work with a certain five-tuple instead of a Yu datum, butthe 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 notdivide 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 − , ~φ ) is regular if π − is a regular depth-zero supercuspidalrepresentation. A supercuspidal representation is regular if it is isomorphic to a supercuspidalrepresentation constructed from a regular Yu datum.We have thus defined the supercuspidal representations of interest to us; the next matteris to connect them to torus-character pairs.Given a Yu datum ( ~G, π − , ~θ ), we can find a maximally unramified torus S of G and aregular depth-zero character φ − of S ( k ) such that π − = π ( S,φ − ) . Setting θ = d Y i = − φ i | S ( k ) then produces a tame elliptic regular pair ( S, θ ). Conversely, given such a pair (
S, θ ), withHowe factorization ( φ i : G i ( k ) → C × ) − ≤ i ≤ d , the triple( ~G = ( G i ) ≤ i ≤ d , π − = π ( S,φ − ) , ~φ = ( φ i ) ≤ i ≤ d )is a Yu datum. It turns out [Kal19b, Proposition 3.7.8] that these assignments are bijectionsmodulo the appropriate equivalences. In this way, we can form a regular supercuspidalrepresentation π ( S,θ ) from a tame elliptic regular pair ( S, θ ).2.3. L -embeddings. In this subsection we explain and study a formalism of Langlands andShelstad for extending an embedding ˆ : b S → b G with Galois-stable b G -conjugacy class to an L -embedding L j : L S → L G . Fix a Γ k -pinning of b G with maximal torus b T .The first difficulty in extending ˆ to L j is to reconcile the Galois actions on b S and b G .Specifically, let τ G denote the action (homomorphism) of W k on b T through its action on b G ;let τ S denote the action of W k on b T through its action on b S , transferred using ˆ ; let N bethe normalizer of b T in b G ; and let Ω = Ω( b G, b 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 ) − ∈ Ω , so that nw acts on b T by the b S -action. The lift exists precisely because the b G -conjugacy classof ˆ 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 liftfundamental 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 soimportant for us.Since the Tits lift is not a homomorphism, the candidate formula w n ( ω S,G ( w )) w for L j | W k does not define a homomorphism W k → L G . To get around this problem, Langlandsand Shelstad studied the failure of this formula to define a homomorphism as measured by ORMAL DEGREE OF REGULAR SUPERCUSPIDALS 15 the function t : W k × W k → b T given by t ( w , w ) := ( n ( w ) w )( n ( w ) w )( n ( w w ) w w ) − . Using an object χ called a set of χ -data, whose definition we review momentarily, theyconstructed a function r χ : W k → b S that negates this failure in the sense that ∂ (ˆ ◦ r χ ) = t − ,where ∂ is the coboundary operator. Hence the modified formula w ˆ ( r χ ( w )) n ( ω S,G ( w )) w does define a homomorphism W k → L G , and in total, the extension L j χ of ˆ is given by theformula L j χ ( sw ) = ˆ ( s r χ ( w )) n ( ω S,G ( w )) w. This concludes our outline of the Langlands-Shelstad procedure for extending an embed-ding of a torus to an L -embedding. Later, in Section 4.2, we recall the definitions of χ -dataand the function r χ .2.4. Regular parameters.
This subsection is largely an expository account of Sections 5.2and 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 importantfor us, so we omit the precise statement: roughly speaking, a parameter ϕ is regular if ittakes the wild subgroup to a torus and the centralizer of the inertia subgroup is abelian (infact, 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 theirirreducible representations are one-dimensional. This means that we can ignore the factordim 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 super-cuspidal L -packet data whose objects are quadruples ( S, ˆ , χ, θ ) consisting of • a tame torus S of dimension the absolute rank of G , • an embedding ˆ : b S → b G of complex reductive groups whose b G -conjugacy class isGalois stable, • a minimally ramified set of χ -data for R ( G, S ), and • a character θ : S ( k ) → C × .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 donot concern us here. One can also define a morphism of such data, organizing them into acategory 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 b G -conjugacy class of ˆ gives rise to a Galois-stable G (¯ k )-conjugacy class of embeddings S ¯ k ֒ → G ¯ k whose elements are called admissible (withrespect to ˆ ) [Kal19b, Section 5.1]. Since S is elliptic, this G (¯ k )-conjugacy class containsembeddings 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 supercuspidalrepresentations into L -packets. The key property of the category of regular supercuspidal L -packet data is that the iso-morphism classes of its objects are in natural bijection with equivalence classes of regularsupercuspidal parameters. Given a regular supercuspidal L -packet datum ( S, ˆ , θ, χ ), itsparameter is the composition L j χ ◦ L θ, where L j χ is the L -embedding of Section 2.3; this is the direction of the correspondence thatwe 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 admissibleembedding 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 jθ ′ := θ ◦ j − · ε where ε = ε f, ram · ε ram is a certain tamely-ramified Weyl-invariant character of jS [Kal19b,Section 5.3]. Kaletha used the character formula of Adler, DeBacker, and Spice [AS08, AS09,DS18] to construct from ( S, 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, ˆ , χ, θ ) consists of theset of regular supercuspidal representations π ( jS,jθ ′ ) where j : S ֒ → G ranges over the G ( k )-conjugacy classes of admissible embeddings.3. Automorphic side
In Section 2.2, we outlined Yu’s construction of supercuspidal representations. In thissection we calculate the formal degree of such a representation. This result is of independentinterest, and could be used to verify the formal degree conjecture for broader classes ofsupercuspidal representations than those considered in this paper.The basic idea of the computation is quite simple, but various technical complicationsarise in the process. As Section 3.7 explains, Yu’s representations are obtained by compact-induction of a finite-dimensional representation of a compact-open (or really, compact-mod-center) subgroup of G ( k ). There is a general formula for the formal degree of such a rep-resentation in terms of the dimension of the starting representation and the volume of thesubgroup. Section 3.2 explains this formula and reviews the notion of formal degree. To com-pute the formal degree of a Yu representation, then, one need only compute two numbers, adimension and a volume.The dimension comes from Deligne-Lusztig theory and is straightforward to compute inour case. We work it out in Section 3.9, where we specialize the formal degree computationto 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 tocomputing its index in a larger group of known volume, so the volume computation boilsdown 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 onthe Lie algebra.The groups used in Yu’s construction generalize the subgroups of the Moy-Prasad filtra-tion. In Sections 3.3 to 3.5 we review their construction and explain various ways in which ORMAL DEGREE OF REGULAR SUPERCUSPIDALS 17
Yu’s theory is an elaboration of the theory of Moy and Prasad [MP94, MP96], or going backeven farther, the theory of Bruhat and Tits [BT72, BT84], Our goal in that lengthy sectionis to generalize the Moy-Prasad isomorphism to Yu’s groups, and to understand the extentto 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 thatwe need to compute. At this point the main steps are in place, and in Section 3.8 we walkup the staircase and finish the computation.To make statements like Lemma 48 easier to read, in Section 3.1 we introduce notationparticular 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 justan abstract group, a topological group if k has a topology. Many constructions are easier onthe level of schemes, but for the kind of representation theory we consider in this paper, onecan 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 itsrational points by G ( k ). Following the convention in this section, however, would create aconfusing 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, . . . : G := G ( k ) . In particular, we write ~G for a twisted Levi sequence and ~G for the sequence of topologicalgroups obtained from it by taking k -points. The convention extends to O - and κ -schemesas 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, denotesthe maximal reductive quotient of the special fiber of G x , a κ -group.3.2. Formal degree.
Here we review the formal degree, following the relevant section ofRenard’s monograph on representations of p -adic groups [Ren10, IV.3], then calculate theformal degree of a compactly induced representation. Contrary to the conventions of therest 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 bea 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 V ∨ is the smoothdual of V ) is the function π v,v ∨ : G → C defined by π v,v ∨ ( x ) = h π ( x ) v, v ∨ i . Since π isirreducible, there is a character χ of Z , called the central character of π , such that π ( z ) = χ ( z )for all z ∈ Z . Assume that the central character is unitary. In this case, the function x
7→ | π v,v ∨ ( x ) | is constant on cosets of A , and hence defines a function on G/A . We write | π v,v ∨ | L ( G/A,µ ) for the integral of this function, and say that ( π, V ) is discrete series (withrespect to A ) if | π v,v ∨ | L ( G/A,µ ) < ∞ for all v ∈ V and v ∨ ∈ V ∨ . This condition is independentof 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 seriesafter twisting by some character of the group.
It can be shown that every discrete series representation is unitary , not in the sense ofHilbert space representations, but in the sense that it admits a positive-definite G -invariantHermitian product. The resulting isomorphism between V and V ∨ defines a matrix coeffi-cient 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 thenorm of a vector and the L -norm of its matrix coefficient. Although these norms are notequal in general, it turns out that they differ by a multiplicative constant depending onlyon π and the Haar measure on G/A , not on the vector. This constant is called the formaldegree . Definition 15 ([Ren10, IV.3.3]) . Let ( π, V ) be an essentially discrete series representationand let µ be a Haar measure on G/A . If π is in addition discrete series then there exists apositive real constant deg( π, µ ), called the formal degree of π , such that for all v ∈ V , | π v | L ( G/A,µ ) = | v | deg( π, µ ) . In general, we define the formal degree of π as the formal degree of any discrete seriesrepresentation 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:deg( π, cµ ) = c − deg( π, µ ) . 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 certaintopology 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 measureis the formal degree of π with respect to µ . In particular, if G is compact and we choosethe Haar measure on G giving it volume one then the formal degree of ( π, V ) is just thedimension of V .Discrete series representations are closely related to supercuspidal representations, butneither notion implies the other. Recall that a smooth irreducible representation is super-cuspidal 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 cer-tainly discrete series, since compactly supported functions are square-integrable; but sincenot every compactly supported function is square-integrable, in general, there are discreteseries representations that are not supercuspidal. At the same time, since supercuspidal rep-resentations need not have unitary central character, not all supercuspidal representationsare discrete series. However, when G is a p -adic reductive group, every supercuspidal repre-sentation of G is essentially discrete series, and we may thus speak of the representation’sformal degree.Since Yu’s supercuspidal representations are compactly induced, we compute their formaldegree using a general formula for the formal degree of a compactly induced representa-tion. As a preliminary step, we define a natural Hermitian product on a compactly inducedrepresentation.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 repre-sentation of G compactly induced from ( ρ, W ). That is, V is the space of smooth functions ORMAL DEGREE OF REGULAR SUPERCUSPIDALS 19 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 productis given by the formula(17) h f , f i = Z K \ G h f ( x ) , f ( x ) i d µ K \ G ( x ) , where µ K \ G is any positive G -invariant Radon measure on K \ G , for instance, the countingmeasure. Lemma 18.
Let ( ρ, W ) be a finite-dimensional unitary representation of K , let ( π, V ) bethe compact induction of W to G , and let µ be a Haar measure on G/A . If π is irreduciblethen deg( π, µ ) = dim ρ vol( K/ ( K ∩ A ) , µ ) . Proof.
We start by defining an isometric embedding
W ֒ → V . Given a vector w ∈ W , define˙ w ∈ V by ˙ w ( x ) = H ( x ) ρ ( x ) w. 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 ˙ w is anisometric embedding. It follows that the matrix coefficient of ˙ w is the extension by zero ofthe matrix coefficient of w , and that their L -norms coincide provided that we take the Haarmeasure on K/ ( K ∩ A ) to be the restriction of µ , denoted also by µ . Then for any nonzero w ∈ W , deg( π, µ ) = | ˙ w | · | π ˙ w | − L ( G/A,µ ) = | w | · | ρ w | − L ( K/ ( K ∩ A ) ,µ ) = deg( ρ, µ ) . Finally, since the formal degree scales inversely to the Haar measure used to define it,deg( ρ, µ ) = vol( K/ ( K ∩ A ) , µ ) − deg( ρ, µ )where µ is the measure on K/ ( K ∩ A ) assigning it total volume one. The degree of ρ withrespect to this measure is just the dimension of W . (cid:3) Concave-function subgroups: split case.
Suppose G is split with split maximaltorus T and root system R = R ( G, T ), so that R = R . In this setting, Bruhat and Titsshowed [BT72, Section 6.4] how to construct from a function f : R ∪ { } → e 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 Bruhatand Tits’s construction. The eventual goal, in later subsections, is to explain how theirconstruction generalizes to the construction of subgroups that appear in Yu’s constructionof 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”, dependingon one’s point of view, a family of additive valuations ( v αx : U α ( k ) → R ) α ∈ R , where U α is theroot group of α . Since U α is canonically isomorphic to the root line g α , we may also thinkof v αx as a valuation g α → R . Now let U αx,r := { u ∈ U α : v αx ( u ) ≥ r } , g αx,r := { X ∈ g α : v αx ( X ) ≥ r } . As for the point α = 0, we can think of T as the root group U and its Lie algebra t as theroot space g . These objects carry their own filtrations: let T r := { t ∈ T : ∀ χ ∈ X ∗ ( T ) , ord( χ ( t ) − ≥ r } and let t r := { X ∈ T : ∀ χ ∈ X ∗ ( T ) , ord(d χ ( 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 written here is unrelated to the zeroth group in a twisted Levisequence, even though the notation for the two is the same. The two notations never appearin the same subsection, however, so there is little risk of confusion.Given a function f : R ∪ { } → e R , let U αx,f := U αx,f ( α ) , g αx,f := g αx,f ( α ) for any α ∈ R ∪ { } . The group G x,f is then defined as the subgroup of G generated by thesubgroups U αx,f with α ∈ R ∪ { } , and the lattice g x,f is defined as the subgroup of g spannedby the subgroups g αx,f with α ∈ R ∪ { } . Remark 20.
When r = ∞ the group U αx, ∞ is trivial, and when in addition α = 0 we canrecover the filtrations on the root groups and root lines as g αx,r = g x,f and U αx,r = G x,f where f ( β ) = ( r if β = α ∞ if not.In order for the construction of G x,f to behave nicely we must assume that f is nonnegativeand concave , that is, that for all finite families ( α i ) i ∈ I of elements of R ∪ { } , f (cid:16)X i ∈ I α i (cid:17) ≤ X i ∈ I f ( α i )whenever P i ∈ I α i ∈ R ∪{ } . 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 thetame case, a simple exercise in Galois descent. Remark 21.
For simplicity of narrative we attributed the construction of G x,f to Bruhatand Tits, but Yu is partially responsible for the construction, even in the split case. Bruhatand Tits worked only with functions f such that f (0) = 0, but Yu extended their theory toall 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 constructionsof the previous subsection, he constructed for each function f : R ∪ { } → e R and point x ∈ A ( G, T ), a subgroup G x,f . In fact, Yu defined his construction only for a special class offunctions, which we review in the next subsection. However, for aesthetic reasons, we foundit 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 andlet Γ ℓ/k be the Galois group of ℓ over k . A function f : R ∪ { } → e R can be interpretedas a Galois-invariant function R ∪ { } → e R , and we say that f is concave if the associatedGalois-invariant function is concave. ORMAL DEGREE OF REGULAR SUPERCUSPIDALS 21
Since f is Galois-invariant, the subgroups ( g ℓ ) x,f and G ( ℓ ) x,f are Galois-invariant and wedefine g x,f := ( g ℓ ) Γ ℓ/k x,f , G x,f := G ( ℓ ) Γ ℓ/k x,f . These groups do not depend on the choice of ℓ . We need not assume that f is concave todefine these objects, although they are best-behaved in that case. We can also construct foreach α in R ∪ { } a root space g α := (cid:16) M α ∈ α (¯ k ) g αℓ (cid:17) Γ ℓ/k . When α = 0, the root space g is just the Lie algebra of T . Combining this constructionwith Remark 20, we see that each root space g α admits a natural filtration: define g αx,f = (cid:16) M α ∈ α (¯ k ) ( g αℓ ) x,f (cid:17) Γ ℓ/k . It follows immediately from the definitions that g x,f = M α ∈ R ∪{ } g αx,f . More generally, given a subset R ′ ⊆ R ∪ { } , let g R ′ := M α ∈ R ′ g α , g R ′ x,f := M α ∈ R ′ g αx,f . Warning 22.
In comparison to the function of Remark 20, the function f : R ∪ { } → e R defined, for a fixed α ∈ R , by f ( β ) = ( r if β = α ∞ if notis 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 valuesin 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 G x,f ( O ) = G x,f as subgroups of G . The construction is due to Yu [Yu02, Section 8].Having defined the group G x,f and Lie algebra g x,f , we now study them. There are threeareas of interest for our later applications.First, in our calculation of the formal degree, it greatly simplifies notation to reduce tothe 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 ) → e 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 followsimmediately from the split case by taking Galois invariants. (cid:3)
Second, when we compute certain subgroup indices in Section 3.8, we need to understandhow to intersect groups of the form G x,f , for fixed x . Lemma 25.
Let f, g : R ∪ { } → e R be positive concave functions. Then G x,f ∩ G x,g = G x, max( f,g ) . Proof.
In the case where G is split, a classical result of Bruhat and Tits [BT72, (6.4.48)] canbe used to show [Yu02, 8.3.1] that the natural multiplication map Y α ∈ R ∪{ } U αx,f ( α ) → G x,f is a bijection for a certain ordering of the factors. Using tame descent, this observationreduces the proof to the obvious fact (still in the split case) that U αx,r ∩ U αx,s = U αx, max( r,s ) . (cid:3) 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 alength computation, which we carry out in Section 3.6. Lemma 26.
Let f, g : R ∪ { } → e R be positive concave functions such that f ≤ g . Assumein addition that the following condition is satisfied: g ( a ) ≤ X i ∈ I f ( a i ) + X j ∈ J f ( b j ) for all non-empty finite sequences ( a i ) i ∈ I and ( b j ) j ∈ J of elements of R ∪ { } such that a := P i ∈ I a i + P j ∈ J b j ∈ R ∪ { } . Then (1) [ G x,f , G x,f ] ⊆ G x,g , 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.Proof. This follows from [BT72, 6.4.44] and [BT72, 6.4.48] in the split case and [Yu02,Section 2] in general. (cid:3)
Remark 27.
In Lemma 26, it is tempting to instead impose the simpler condition that g ( a ) ≤ P i ∈ I f ( a i ) for all non-empty finite sequences ( a i ) i ∈ I such that P i ∈ I a i = a . However,this stronger condition would significantly weaken the conclusion: the condition implies, bytaking the constant sequence, that g ≤ f , so that g = f . Corollary 28.
Let f, g : R ∪ { } → R be positive concave functions such that f ≤ g , andsuppose there is a chain of concave functions f = f ≤ f ≤ · · · ≤ f n = g such that for each i with ≤ i ≤ n , the pair ( f i − , f i ) satisfies the conditions of Lemma 26.Then | G x,f : g | = | g x,f : g | . Yu’s groups. , In our application, it is enough to work with the subgroups constructedfrom a certain restricted class of concave functions, those constructed from admissible se-quences and tame twisted Levi sequences. The construction specializes that of Section 3.4, sowe work in the same setting. After reviewing Yu’s construction of these groups, we specializethe theory of Section 3.4 to show that they admit a Moy-Prasad isomorphism.
ORMAL DEGREE OF REGULAR SUPERCUSPIDALS 23
A sequence ~r = ( r i ) di =0 in e R is admissible if there is some j with 0 ≤ j ≤ d such that0 ≤ r = · · · = r j and r j ≤ r j +1 ≤ · · · r d . The admissible sequence is weakly increasing if,in addition, r i ≤ r i +1 for all i with 0 ≤ i ≤ d − ~G from Section 2.2. We assumethat the tame maximal torus T of G is contained in G , so that x ∈ B ( G i ) for each i . Foreach i with 0 ≤ i ≤ d , let R i := R ( G i , T ).It is sometimes necessary to work with truncated twisted Levi sequences. Given integers a and b such that 0 ≤ a ≤ b ≤ d , let ~G [ a,b ] := ( G a ( · · · ( G b );given an integer i with 0 ≤ i ≤ d , define ~G ( i ) := ~G [0 ,i ] .Given an admissible sequence ~r and a twisted Levi sequence ~G of the same length d , definethe function f ~r : R → e R by f ~r ( α ) := ( r if α ∈ R ∪ { } r i if α ∈ R i \ R i − , ≤ i ≤ d. Since ~r is admissible, the function f ~r is concave [Yu01, Lemma 1.2]. Hence we may define ~G x,~r := G x,f ~r , ~ g x,~r := g x,f ~r . Given a second admissible sequence ~s of length d with r i ≤ s i for all i , define ~G x,~r : ~s := G x,f ~r : f ~s , ~ g x,~r : ~s := g x,f ~r : f ~s . Remark 29.
The group ~G x,~r depends only on ~G , x , and ~r ; in particular, it is independent ofthe choice of torus T , provided that x ∈ B ( T ) [Yu01, Section 1]. The same cannot necessarilybe 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. ThisGalois action can vary among maximal tori whose buildings are identical subsets of B ( G ).In this setting, Yu generalized [Yu01, Lemma 1.3 and Corollary 2.4] the Moy-Prasadisomorphism. Lemma 30.
Let ~G be a tame Levi sequence of length d and let ~r and ~s be admissible sequencesof 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. (cid:3) Corollary 32.
Let ~G be a tame Levi sequence of length d and let ~r and ~s be weakly increasingadmissible sequences of length d such that < r i ≤ s i < ∞ for all i . Then | ~G x,~r : ~s | = | ~ g x,~r : ~s | . Proof.
Since ~r is weakly increasing, Condition 31 simplifies to the condition that(33) s i ∈ [ r i , r i + r ] for all i .It is now an elementary but tedious exercise to construct a chain ( ~s ( j ) ) ≤ j ≤ N of weaklyincreasing admissible sequences ~s ( j ) = ( s ( j )0 ≤ · · · ≤ s ( j ) d ), where N ≫
0, such that s (0) = ~r and s ( d ) = ~s , such that s ( j ) i ≤ s ( j +1) i for all i and j with 0 ≤ j ≤ N −
1, and such thateach pair ( ~s ( j − , ~s ( j ) ) satisfies Condition 31 of Lemma 30. After completing this exercise,we invoke Corollary 28. (cid:3) 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 increasingadmissible sequence of length d . Then | ~G x, ~r | = | ~ g x, ~r | . Length computation.
Retaining the notation of Section 3.4, let f : R → R be apositive function and let R ′ ⊆ R be a subset. Our goal in this subsection is to compute thelength of the O -module g R ′ x, f , culminating in Theorem 38.We start by studying the jumps in the filtration on g αx . For α ∈ R , consider the setord x α := { t ∈ R : g αx,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 forus it is enough to know several weak properties of these sets. Lemma 35.
Let T be a tame maximal torus of G , let x ∈ A ( G, T ) , and let α ∈ R . Then (1) ord x ( − α ) = − ord x α , (2) ord x α is an ord k ( k × α ) -torsor, and (3) len g αx,t : t + = [ κ α : κ ] ord x α ( t ) .Proof. Property (1) follows from a PGL calculation [DS18, Corollary 3.11]. As for theother properties, earlier we mentioned that g α was isomorphic to k α , though not canonically.Choose one such isomorphism φ : k α → g α . This isomorphism is compatible with the Moy-Prasad filtration in the following sense: there is a real number r such that for all r ∈ R ,the isomorphism φ restricts to an isomorphism k α,r + r ≃ g α,r of O -modules. Properties (2)and (3) now follow immediately. (cid:3) Corollary 36.
Let T be a tame maximal torus of G , let x ∈ A ( G, T ) , let f : R ∪ { } → R be a positive function, and let R ′ ⊆ R . Then len g R ′ x, f = X α ∈ R ′ X It suffices to prove the corollary in the case where R ′ = { α } . Then both sides equal X t ∈ ord α x Let Λ ⊂ R be a free abelian group of rank one, let λ := min(Λ ∩ R > ) , let h : R → N be a discretely supported function, and let H ( s ) := P ′ ≤ t ≤ s h ( t ) for s > . Suppose h is even and Λ -periodic. Then for all s ∈ Λ ∩ R > , H ( s ) = sλ H ( λ ) . Proof. Since H ( s + λ ) = H ( s ) + H ( λ ), induction reduces the proof to the case where s = λ or s = λ . The first case is obvious; for the second, use that h ( t ) = h ( λ − t ). (cid:3) 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 ∪ { } → R be a positive even function, and let R ′ ⊆ R be a subset closed under negation. Suppose that f ( α ) ∈ ord( k × α ) for all α ∈ R . Then len( g R ′ x, f ) + len( g R ′ x, ) + len( g R ′ x,f : f + ) = X α ∈ R ′ [ k α : k ] f ( α ) . Proof. By Corollary 36, the lefthand side of the theorem islen( g R ′ x, f ) + len( g R ′ x, ) + len( g R ′ x,f : f + ) = X α ∈ R ′ X ′ ≤ t ≤ f ( α ) [ κ α : κ ] ord x α ( t ) . Since f is even and κ α = κ − α , the righthand side above is X α ∈ R ′ X ′ ≤ t ≤ f ( α ) 12 [ κ α : κ ] (cid:0) ord x α ( t ) + ord x ( − α ) ( t ) (cid:1) . By Lemma 35, the function ord x α + ord x ( − α ) is even and ord( k × α )-periodic. Hence we mayapply Lemma 37 to conclude that X ′ ≤ t ≤ f ( α ) 12 [ κ α : κ ] (cid:0) ord x α ( t ) + ord x ( − α ) ( t ) (cid:1) = [ k α : k ] f ( α ) , using the fact that X ′ ≤ t ≤ [ κ α : κ ] ord x α ( t ) = X ≤ t< [ κ α : κ ] ord x α ( t ) = [ κ α : κ ] ord k ( k × α ) = [ k α : k ] . (cid:3) Yu’s construction. In this subsection we describe Yu’s supercuspidal representations,following Hakim and Murnaghan’s expanded exposition [HM08, Section 3] of Yu’s originalpaper [Yu01]. Yu’s full construction is quite elaborate, but fortunately, it is enough for usto understand only the parts of the construction needed to calculate the formal degree.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 /Z is anisotropic, where Z is the centerof G and Z is the center of G ; • a point y in the apartment of a tame maximal torus of G ; • an increasing sequence ~r = (0 < r < r < · · · < r d − ≤ r d ) of real numbers (if d = 0then we only require that 0 ≤ r ); • an irreducible representation ρ of G y ] whose restriction to G y, is 1-isotypic and forwhom the compact induction c-Ind G K ρ is irreducible (hence supercuspidal); • a sequence ~φ = ( φ , . . . , φ d ) of characters, with φ i a character of G i , such that φ d = 1if r d = r d − 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 ofSection 2.1, and the datum is regular if, in addition, the depth-zero supercuspidal represen-tation c-Ind G K ρ is regular.Many of the objects used in Yu’s construction and built from a cuspidal G -datum donot depend on the representations ρ and ~φ . To make this independence explicit, we definea cuspidal G -datum without representations to be a 3-tuple ( ~G, y, ~r ) consisting of the firstthree components of a cuspidal G -datum.Let Ψ = ( ~G, ~r, y ) be a cuspidal G -datum without representations. From Ψ we can con-struct the following subgroups: K = G y ] K = G y, K i +1 = K ~G ( i +1) y, (0+ ,s ,...,s i ) K i +1+ = ~G ( i +1) y, (0+ ,s + ,...,s i +) J i +1 = ( G i , G i +1 ) y, ( r i ,s i ) J i +1+ = ( G i , G i +1 ) y, ( r i ,s i +) K := K d := K d +1 K + := K d + := K d +1+ . Here 0 ≤ i ≤ d − 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 . The groups K i +1 and K i +1+ are particularlyimportant; later on, we need to express them in the following alternative ways. K i +1 = K ~G [1 ,i +1] y,s ,...,s i K i +1+ = K ~G [1 ,i +1] y,s + ,...,s i + (39) K i +1 = K i J i +1 K i +1+ = K i + J i +1 . (40)We can now outline Yu’s construction. Let Ψ = ( ~G, y, ~r, ρ, ~φ ) be a generic cuspidal G -datum. For each 0 ≤ i ≤ d − 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 thatdim ρ i = [ J i +1 : J i +1+ ] / . Our ρ i is Hakim and Murnaghan’s φ ′ i . ORMAL DEGREE OF REGULAR SUPERCUSPIDALS 27 The representation ρ i is then inflated to a representation τ i of K . Section 3.9 explains inmore detail how the inflation procedure works, but at the moment, it is enough to knowthat the inflation procedure preserves dimension. In the edge case i = − τ − to bethe inflation (by the same procedure) of ρ to K , and in the edge case i = d take τ d to bethe restriction of φ d to K ; we could also handle the case i = − ≤ i ≤ d − ρ − := ρ . Finally, define the supercuspidal representation π attached to Ψ as the compact induction π = c-Ind GK τ, τ := τ − ⊗ τ ⊗ · · · ⊗ τ d . 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(41) deg( π Ψ , µ ) = dim ρ · Q d − i =0 [ J i +1 : J i +1+ ] / vol( K/A, µ )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 tothe case where the center of G is anisotropic, that is, A = 1. Given a generic cuspidal G -datum without representations Ψ = ( ~G, y, ~r ), let Ψ = ( −→ G a , y, ~r ) denote the reduction of Ψmodulo A : that is, G a ,i := G i, a = G i /A . Lemma 42. Let Ψ be a generic cuspidal G -datum without representations and let Ψ be thereduction of Ψ modulo A . Then (1) the groups K Ψ and K Ψ /A are identical as subgroups of G/A , and (2) [ J i +1Ψ : J i +1Ψ , + ] = [ J i +1Ψ : J i +1Ψ , + ] for all ≤ i ≤ d − .Proof. For the first part, since K Ψ A = G y ] A · A ~G ( d ) y, (0+ ,~s ) ~G ( d ) y, (0+ ,~s ) = G y ] A · ~G ( d ) y, (0+ ,~s ) ~G ( d ) y, (0+ ,~s ) ∩ A , it suffices by Lemma 24 to show that G y ] /A = ( G /A ) [ y ] , and this follows immediately fromthe surjectivity of the map G → ( G/A )( k ) and the natural identification of the reducedbuildings of G and G/A .The second part follows by an argument similar to the proof of Lemma 24. (cid:3) Degree computation. In this subsection we compute the formal degree of Yu’s su-percuspidal 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 measureon 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 tothe problem of computing certain subgroup indices. We will then be in a position to applyTheorem 38, finishing the calculation. By Lemma 42, we may assume for now that thecenter of G is anisotropic, though of course this restriction will have to be relaxed in thefinal formula.We will start by simplifying the volume of K . To begin with,(43) vol( K, µ ) − = vol( G y, , µ ) − [ KG y, : K ][ KG y, : G y, ] . Our τ i is Hakim and Murnaghan’s κ i . 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 . Then [ M H : N H ] = [ M : N ][ M ∩ H : N ∩ H ] , 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 wecan use to interpret the former as a subgroup of the latter. The group M/N acts transitivelyby left multiplication on the coset space M H/N H . Consider the stabilizer of the identitycoset under this action. Clearly M ∩ H/N ∩ H lies in the stabilizer, and conversely, it iseasy to see that any representative of an element of the stabilizer can be translated by anelement 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. (cid:3) By Equation (39) and Lemmas 25 and 44,(45) [ KG y, : K ] = [ K G y, : K ~G [1 ,d ] y, ( s ,...,s d − ) ] = [ G y, : ~G [1 ,d ] y, ( s ,...,s d − ) ][ G y, : G y,s ] . It follows from Lemma 25 again that [ KG y, : G y, ] = [ K : K ]. Combining this calcula-tion with Eqs. (39), (43), and (45) yieldsvol( K, µ ) − = vol( G y, , µ ) − [ KG y, : K ][ KG y, : G y, ] = vol( G y, , µ ) − [ G y, : ~G [1 ,d ] y, ( s ,...,s d − ) ][ G y ] : G y,s ] . The volume of G y, is known from the literature. Lemma 46. Let x ∈ B ( G ) . Then vol( G ( k ) x, , µ ) − = q (dim G ) / | g x, | / .Proof. DeBacker and Reeder [DR09, Section 5.1] defined a Haar measure ν on G ( k ) suchthat for any x ∈ B ( G ), vol( G ( k ) x, , ν ) = | G x, || g x, | / ;in particular, ν does not depend on the choice of x . Kaletha showed [Kal15, Lemma 5.15]that ν = q (dim G ) / µ . Combining these results proves the lemma. (cid:3) At this point, we can say that(47) deg( π, µ ) = dim ρ [ G y ] : G y, ] q (dim G ) / | g y, | / [ G y, : ~G [1 ,d ] y, ( s ,...,s d − ) ][ G y ] : G y,s ] d − Y i =0 [ J i +1 : J i +1+ ] / . We can now simplify Eq. (47) using our earlier results on concave functions. Lemma 48. | g y, | / [ G y, : ~G [1 ,d ] y, ( s ,...,s d − ) ][ G y, : G y,s ] d − Y i =0 [ J i +1 : J i +1+ ] / = exp q (cid:18) len g y, + d − X i =0 r i ( | R i +1 |−| R i | ) (cid:19) This condition is needed for M H and N H to be groups; it might be possible to weaken it. ORMAL DEGREE OF REGULAR SUPERCUSPIDALS 29 Proof. Let f = f ( s ,...,s d − ) for the twisted Levi sequence ~G [1 ,d ] . By Lemma 30,[ J i +1 : J i +1+ ] = | ( g i , g i +1 ) y, ( r i ,s i ):( r i ,s i +) | = (cid:12)(cid:12) ( g i +1 ) R i +1 \ R i y,s i : s i + (cid:12)(cid:12) = exp q (cid:16) X α ∈ R i +1 \ R i len g αy,f : f + (cid:17) . By Corollary 34,[ G y, : ~G [1 ,d ] y, ( s ,...,s d − ) ] = | ~G [1 ,d ] y, (0+ ,..., s ,...,s d − ) | = exp q (cid:16) len g y, s + d − X i =0 X R i +1 \ R i len g αy, f (cid:17) and [ G y, : G y,s ] = exp q len g y, s , so that[ G y, : ~G [1 ,d ] y, ( s ,...,s d − ) ][ G y, : G y,s ] = exp q (cid:16) d − X i =0 X R i +1 \ R i len g αy, f (cid:17) = exp q (cid:16) X α ∈ R \ R len g αy, f (cid:17) . We now recognize that | g y, | / | g y, | / · [ G y, : ~G [1 ,d ] y, ( s ,...,s d − ) ][ G y, : G y,s ] d − Y i =0 [ J i +1 : J i +1+ ] / equals exp q of X α ∈ R \ R X ′ ≤ t ≤ f ( α ) len g αy,t : t + . By Corollary 10 the hypotheses of Theorem 38 are satisfied, and the expression above be-comes d − X i =0 r i ( | R i +1 | − | R i | )The proof is finished by recalling that | g y, | / = exp q (cid:0) len( g y, ) (cid:1) . (cid:3) We now make the reduction promised in Lemma 42. Recall the notation for Ψ definedimmediately before Lemma 42. Theorem A. Let G be a reductive k -group and let Ψ be a generic cuspidal G -datum withassociated supercuspidal representation π . Then deg( π, µ ) = dim ρ [ G a , y ] : G a , y, ] exp q (cid:18) dim G a + dim G a , y, + d − X i =0 r i ( | R i +1 | − | R i | ) (cid:19) . Proof. When A = 1, the formula follows from Eq. (47), Lemma 48, and the fact that g y, is the Lie algebra of G a , y, . In general, according to Eq. (41), with the exception of thefactor dim ρ , the formal degree depends only on the underlying G -datum without represen-tations Ψ = ( ~G, y, ~r ). It now suffices to observe that by Lemma 42, the expression Q d − i =0 [ J i +1 : J i +1+ ] / vol( K/A, µ )is the same for both Ψ and the reduced datum Ψ := ( −→ G a , y, ~r ). (cid:3) Regular supercuspidals. In the special case where the supercuspidal representationis 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 iscompactly induced from a finite-dimensional representation ρ of the group G [ y ] , but whenthe supercuspidal π is regular, we can recover a maximal torus S from π , and π is inducedfrom a finite-dimensional representation η of the smaller group SG y, . In fact, the formerrepresentation is induced from the latter:(49) ρ = Ind G [ y ] SG y, η. In the depth zero case, replacing ρ by η in the formula for the formal degree thus mul-tiplies the rest of the formula by the index [ G [ y ] : SG y, ]. And since η is an extension of aDeligne-Lusztig representation, the literature provides a formula for its dimension. Theseobservations 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 to K . And to propagate the formula, weneed 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, theeffect 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 arepresentation 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 Con-dition 50. Hence ρ may be interpreted as (the inflation of) a representation of the quotientgroup K i / ( K i ∩ J i +1 ). Since K i normalizes J i +1 , the decomposition of Equation (40) becomesa semidirect product after dividing by K i ∩ J i +1 . We can now inflate this representation ofthe 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. Byinduction, we can therefore define for any j ≥ i + 1, and in particular for j = d (if i < d ),the inflated representation Inf K j K i ρ := Inf K j K j − · · · Inf K i +1 K i ρ. Lemma 51. Let Ψ be a cuspidal G -datum; recall the notations of Section 3.7 for the variousobjects attached to Ψ . Let H i := SG y, K i + and let H := H d . Suppose that there is a(necessarily irreducible) representation η of H such that ρ := Ind K H η . Then σ − := Inf HH η is defined and there is a canonical identification τ = Ind KH ( σ − ⊗ τ | H ⊗ · · · ⊗ τ d | H ) of representations of K .Proof. The notation inf H j H i mimics the notation inf K j K i : the same construction works if thesymbol K is replaced everywhere by H . Since ρ is 1-isotypic on J , so is η ; hence σ − := ORMAL DEGREE OF REGULAR SUPERCUSPIDALS 31 inf HH η is defined. There is a canonical identificationInd KH Inf HH η = Inf KK Ind K H η, so that τ = (Ind KH σ − ) ⊗ τ ⊗ · · · ⊗ τ d . The result now follows from the well-known [Ren10,Section III.2.11] formula (Ind KH π ) ⊗ π = Ind KH ( π ⊗ ( π | H )), where π is a representationof H and π is a representation of K . (cid:3) Kaletha showed [Kal19b, Section 3.4.4, proof of Lemma 3.4.20] that for a regular Yudatum, the finite-dimensional representation ρ has the property that ρ = Ind K SG y, η where η extends the inflation to G y, of a Deligne-Lusztig induced representation of the group G y, .We can thus compute dim η using Deligne-Lusztig theory.Let’s briefly recall the dimension formula for a Deligne-Lusztig induction. In this para-graph 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 and Lusztig computed the dimension of the virtualrepresentation R ( S,θ ) in their original paper [DL76, Corollary 7.2]; it isdim R ( S,θ ) = [ G : S ]dim St G where St G is the Steinberg representation of G . In his classic book on representations offinite groups of Lie type, Carter computed [Car85, Corollary 6.4.3] the dimension of theSteinberg representation; it islog q dim St G = (dim G − dim S ) . We can now assemble these results to make our final formula. Recall the notation of Theo-rem A. Corollary 52. Let Ψ be a regular generic cuspidal G -datum with resulting supercuspidalrepresentation π and let S be the maximally unramified maximal torus of G resulting from Ψ ,as explained in Section 2.2. Then (53) deg( π, µ ) = | S a0:0+ | − exp q (cid:16) dim G a + rank G a , y, + d − X i =0 s i ( | R i +1 | − | R i | ) (cid:17) . Proof. By Lemma 51, Lemma 18, and Kaletha’s description of ρ , the formula of Theorem Aremains true if dim ρ is replaced by[ K : SG y, ] dim η = [ G a[ y ] : S a G a , y, ] dim η. The dimension formula for η discussed in the paragraph above shows thatdim η = [ G y, : S G y, ] exp q (cid:0) (dim G y, − dim S ) (cid:1) − = [ G a , y, : S a0 G a , y, ] exp q (cid:0) (dim G a , y, − rank G a , y, ) (cid:1) − . The formula now follows. (cid:3) Galois side Let ( S, θ ) be a tame elliptic pair. We saw in Section 2.4 that when θ is extra regular, sucha pair can be extended to a regular supercuspidal L -packet datum ( S, ˆ , χ, θ ), and that theresulting 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 sectionis 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-dimensionalirreducible representations, so that the Galois side of the conjecture simplifies to | γ (0 , ϕ, Ad , ψ ) || π ( S ♮ϕ ) | . Moreover, the factor | π ( S ♮ϕ ) | has been computed in the literature. So our task is to computethe 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 secondpreliminary step, we work out in Section 4.2 how to base change the function r χ used tosolve the extension problem of Section 2.3.To compute the adjoint γ -factor, we give an explicit description of the adjoint represen-tation attached to the L -parameter ϕ . It turns out that this representation decomposes asa direct sum of two representations, one coming from the maximal torus of the dual groupand 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 ˆ and χ in the L -packet datum ( S, ˆ , χ, θ ) extending ( S, θ ).4.1. Review of L -, ε - and γ -factors. The γ -factor of a representation ( π, V ) of the Weilgroup W k is defined by the formula γ ( s, π, ψ, µ ) := ε ( s, π, ψ, µ ) L (1 − s, π ∨ ) L ( s, π )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, followingTate’s Corvallis notes [Tat79]. Roughly speaking, the L -factor carries information about theunramified part of the representation and the ε -factor carries information about the ramifiedpart of the representation. Since computing the absolute value of an ε -factor amountsto computing an Artin conductor, we also explain how to compute this quantity in ourapplication, following Chapter VI of Serre’s Local Fields [Ser79].The L -factor of π is the holomorphic function L ( s, π ) := det (cid:0) − q − s π (Frob) | V I k (cid:1) − where I k ⊂ W k is the inertia group and Frob ∈ W k is a Frobenius element. Later, we usethe fact that the L -factor is inductive : if ℓ ⊇ k is a field extension of k and ( π, V ) is afinite-dimensional complex representation of W k then L ( s, Ind ℓ/k ( π )) = L ( s, π ) , Ind ℓ/k := Ind W k W ℓ . ORMAL DEGREE OF REGULAR SUPERCUSPIDALS 33 The ε -factor is more subtle to define than the L -factor, and most of the subtlety resides inits 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 absolutevalue of the ε -factor instead.Changing s , ψ , or µ scales the ε -factor by a known quantity. We may thus define with noloss 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(54) | ε ( π ) | = q cond π where cond π is the Artin conductor of π . So computing the absolute value of the ε -factoramounts to computing the Artin conductor.The Artin conductor is defined by the following procedure. Given a Galois representa-tion ( π, V ), choose a Galois extension ℓ of k such that π | W ℓ is trivial, and let Γ ℓ/k be theGalois group of ℓ over k . Then the Artin conductor satisfies the formulacond π = X i ≥ codim( V Γ ℓ/k,i )[Γ ℓ/k, : Γ ℓ/k,i ] , where i G ℓ/k,i is the lower numbering filtration. This formula is independent of the choiceof ℓ . We can now extend the definition to all complex representations of the Weil group, notnecessarily those of Galois type, by stipulating that the Artin conductor be unchanged byunramified 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 ofEquation (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 onlyunderstand how to compute the Artin conductor of a tamely induced representation. Unlikethe L -factor, the Artin conductor is not invariant under induction. The best we can sayin general is that given a finite extension ℓ of k and a representation π of W ℓ , the inducedrepresentation has conductor(57) cond Ind ℓ/k π = ord k (disc ℓ/k ) dim π + f ℓ/k cond π, 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), theinduced representation is reducible. Lemma 58. Let ℓ ⊇ k be a finite tame extension and let χ : W ℓ → C × be a character. Then cond Ind ℓ/k χ = [ ℓ : k ](1 + depth k χ ) . Proof. Check, using the tameness of the extension, that ord k disc ℓ/k = [ ℓ : k ] − f ℓ/k . Thiscomputation together with Equations (56) and (57) yieldscond Ind ℓ/k χ = [ ℓ : k ] + f ℓ/k depth ℓ χ. Now use that depth ℓ χ = e ℓ/k depth k χ . (cid:3) The L -factor, ε -factor, and Artin conductor are additive in the sense that L ( s, π ) = L ( s, π ) L ( s, π ) , ε ( π ) = ε ( π ) ε ( π ) , cond π = cond π + cond π where π = π ⊕ π . Hence the γ -factor is additive as well. This simple but crucial fact allowsus 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 χ -data behave under base change, that is, restriction to the Weil group of a finite separableextension of k . Once we understand the effect of base change for arbitrary χ -data, we studyits effect on minimally ramified χ -data.Most of the definitions of this subsection are due to Langlands and Shelstad [LS87, Sec-tion 2.5], but our treatment is also influenced by Kaletha’s recent reinterpretation of Lang-lands and Shelstad’s formalism [Kal19a].Let ℓ be a separable quadratic extension of k . Local class field theory shows that thequotient k × /N ℓ/k ( ℓ × ) is cyclic of order two. The quadratic sign character of the extension ℓ ⊇ k is the character k × → {± } given by projection onto this quotient.A root α ∈ R ( G, S ) is symmetric if it is Galois-conjugate to − α , and is asymmetric oth-erwise. A symmetric root α is unramified if the quadratic extension k α ⊃ k ± α is unramifiedand 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. Definition 59. Let R = R ( G, S ). A set of χ -data for ( S, G ) (or just S if G is understood)is a collection χ = ( χ α : k × α → C × ) α ∈ R of characters satisfying the following properties.(1) χ − α = χ − α .(2) χ σα = χ α ◦ σ − for all σ ∈ Γ k .(3) If α is symmetric then χ α extends the quadratic sign character of k α ⊃ k ± α .Kaletha has interpreted a set of χ -data as giving rise to a character of a certain doublecover 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 discussionfrom 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,ℓ α /k α .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 theeasily verified formula σ − ◦ N ℓ σα /k σα = N ℓ α /k α ◦ σ − . ORMAL DEGREE OF REGULAR SUPERCUSPIDALS 35 As for the third property, if α is symmetric over ℓ then it is also symmetric over k andthe canonical map Γ ℓ α /ℓ ± α → Γ k α /k ± α is an isomorphism. Now recall [Tat79, (1.2.2)] thatthe local class field theory homomorphism W k → k × (whose abelianization is the Artinreciprocity isomorphism) intertwines the inclusion W ℓ ֒ → W k with the norm map ℓ × → k × .It follows that the canonical map N ℓ α /k α ( ℓ × α ) /N ℓ α /k α ( ℓ ×± α ) → k × α /k ×± α is an isomorphism, and hence that χ ℓ,α extends the quadratic sign character of ℓ α ⊃ ℓ ± α . (cid:3) It is time to start defining the function r χ . The definition requires a brief preliminarydiscussion of abstract group theory. Let G be a group and K a subgroup. A section u : K \ G → G of the projection G → K \ G – in other words, a choice of coset representatives– gives rise to a K \ G -indexed family of set maps u x : G → K , for x ∈ K \ G . To define themaps u x , we write down the element u ( x ) g and decompose it as a product of an element of K followed by its coset representative in K \ G ; that is, u x is defined by the following equation: u x ( g ) u ( xg ) = u ( x ) g. Before defining r χ , a word is on order on the exact nature of the object we are defining,since that object depends on many arbitrary choices. A gauge is a function p : R → {± } such that p ( − α ) = − p ( α ) for all α ∈ R . Our construction produces for each gauge p a cohomology class r χ,p of 1-cochains W k → b S . As the gauge p varies the cohomologyclasses r χ,p are not identical. However, there is a canonical means of relating one class tothe other. For any two gauges p and q , Langlands and Shelstad constructed a canonical1-cochain s q/p , depending only on R and not on the choice of χ -data. By definition, thecohomology classes r χ,p , are related by the equation(62) r χ,q = s q/p r χ,p , and the 1-cochains s q/p satisfy the right compatibility conditions to make these equationsconsistent [LS87, Corollary 2.4.B]. In the construction that follows we therefore define r χ,p for a particular choice of gauge, and Equation (62) then defines r χ,q for every other q .We can now write down the formula defining r χ , making several arbitrary choices alongthe way. First, choose(1) a section [ α ] α of the orbit map R → R/ {± } .Each [ α ] ∈ R/ {± } thus gives rise to two subgroups W α and W ± α , the stabilizers of α and {± α } in W k . For each [ α ], choose in addition(2) a section u α : W ± α \ W k → W k , and(3) a section v α : W α \ W ± α → W ± α .What we call “choosing a section” is more commonly called “choosing coset representatives”.Using these choices, define the element r χ ( w ) of b S = X ∗ ( S ) C by(63) r χ ( w ) = Y [ α ] ,x χ α ( v α ( u αx ( w ))) u α ( x ) − α , [ α ] ∈ R/ {± } , x ∈ W ± α \ W k . We still have to explain the dependence on the gauge. Use choices (1) and (2) above todefine the gauge p : R → {± } by setting p ( β ) = 1 if and only if β = u α ( x ) − α for some x ∈ W ± α \ W k . Then Equation (63) defines r χ,p := r χ . Now use Equation (62) to extend thedefinition to all gauges. Theorem 64. Let p : R → {± } be a gauge, let χ be a set of χ -data for S , and let ℓ be afinite separable extension of k . Then r χ,p | ℓ = r χ ℓ ,p for some set of auxiliary choices ((1), (2), and (3) above) in the definition of r χ and r χ ℓ .Proof. First, some preliminary notation on group actions. Given a right G -set X and ele-ments x, y ∈ X that lie in the same G -orbit, let x − y , the transporter from x to y , denotethe set of elements of G taking x to y . If y = gx then x − y = G x g where G x is the stabilizerin G of x .Let W := W k and W ′ := W ℓ . It suffices to consider the case where R is a transitive(Γ k × {± } )-set. The plan of the proof is to make choices (1), (2), and (3) for r χ and for r χ ℓ so that the equation r χ,p | ℓ = r χ ℓ ,p holds on the nose, not up to cohomology. To get equality,not just cohomology, some of our choices depend other choices.Fix α ∈ R (choice (1) for r χ ). The double cosets z ∈ W ± α \ W/W ′ index the W ′ -orbitsof R/ {± } . Choose a section c : W ± α \ W/W ′ → W , and for each double coset z let α z := c ( z ) − α (choice (1) for r χ ℓ ), so that the α z form a set of representatives for the W ′ -orbits of R/ {± } . Choose sections u z : W ′± α \ W ′ → W ′ (choice (2) for r χ ℓ ) and v : W α \ W ± α → W ± α (choice (3) for r χ ), and use them to define sections v z : W ′ α z \ W ′± α z → W ′± α z (choice (3) for r χ ℓ ) by the formula v z ( y ) = c ( z ) − v ( c ( z ) yc ( z ) − ) c ( z ) . Define the section u : W ± α \ W → W (choice (2) for r χ ) by u ( x ) := c ( z ) u z ( y ) where z = xW ′ and y = ( Kc ( z )) − x .We have now made all necessary choices to define r χ and r χ ℓ . It remains to check thatthese choices define the same gauge p and that r χ,p | ℓ = r χ ℓ ,p .To check that the gauges agree, first check that the assignment x ( z, y ) is a bijectionfrom W ± α \ W to the set of pairs ( z, y ) with z ∈ W ± α \ W/W ′ and y ∈ W ′± α z \ W ′ . In the rest ofthe proof, we assume that x is related to ( y, z ) by this bijection. Hence a root is of the form u ( x ) − α with x ∈ W ± α \ W if and only if it is of the form u z ( y ) − α z with z ∈ W ± α \ W/W ′ and y ∈ W ′± α z \ W ′ .Recall that for each x ∈ W ± α \ W , there is a function u x : W → W ± α obtained from u bythe equation u ( x ) w = u x ( w ) u ( xw );similarly, for each z ∈ W ± α \ W/W ′ and and y ∈ W ′± α z \ W ′ , there is a function u zy : W ′ → W ′± α z obtained from u z by the equation u z ( y ) w ′ = u zy ( w ′ ) u z ( yw ′ ) . These two constructions are related in the following way. Claim 65. Let w ∈ W ′ . Then u x ( w ′ ) = c ( z ) u zy ( w ′ ) c ( z ) − . Proof. Let x ′ = xw ′ , let z ′ = x ′ W ′ , and let y ′ = ( Kc ( z ′ )) − x ′ , so that ( z ′ , y ′ ) is obtainedfrom x ′ in the same way that ( z, y ) was obtained from x . Expand the defining equation of u x ( w ′ ): c ( z ) u z ( y ) w ′ = u x ( w ′ ) c ( z ′ ) u z ′ ( y ′ ) . Since z = z ′ and y ′ = yw ′ , c ( z ) − u x ( y ) c ( z ) u z ( yw ′ ) = u z ( y ) w ′ . (cid:3) ORMAL DEGREE OF REGULAR SUPERCUSPIDALS 37 Use the sections u and v to compute the L -parameter of χ : r χ ( w ) = Y x χ α ( v ( u x ( w ))) u ( w ) − α , x ∈ W ± α \ W. Assume now that w = w ′ ∈ W ′ . Use Claim 65 to simplify the expression to r χ ( w ′ ) = Y z,y χ α ( v ( c ( z ) u zy ( w ′ ) c ( z ) − )) u ( w ′ ) − α , z ∈ W ± α \ W/W ′ , y ∈ W ′± α z \ W ′ = Y z,y χ α ( c ( z ) − v z ( u zy ( w ′ )) c ( z )) u ( w ′ ) − α = Y z,y χ α z ( v z ( u zy ( w ′ ))) u z ( w ′ ) − α z . To complete the argument, use the property cited above that the local class field theoryhomomorphism intertwines inclusion with the norm map. (cid:3) Remark 66. Unlike most of the other parts of this paper, Theorem 64 and the definitionspreceding it do not require the torus S to split over a tamely ramified extension; they holdfor 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 tamelyramified 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 64and is therefore tamely ramified of finite order. (cid:3) 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 ramifiedset of χ -data satisfies the hypotheses, and thus the conclusion, of Corollary 67; we use thisobservation 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 forramified symmetric α .4.3. Adjoint representation. Our goal in this subsection is to describe the adjoint rep-resentation 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 ). Insubsequent 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 ϕ ( S,θ ) ( w ) = ˆ (cid:0) ˆ θ ( w ) r χ ( w ) (cid:1) n ( ω S,G ( w )) w where ˆ : b S → b G is an admissible embedding with image a Galois-stable maximal torus b T and χ is a certain (carefully chosen) set of minimally ramified χ -data. We obtain the adjointrepresentation from ϕ = ϕ ( S,θ ) by composing it with the adjoint homomorphism L G → GL( V ), where V := ˆ g / ˆ z Γ k . The representation decomposes as a direct sum V = V toral ⊕ V root where V toral := ˆ t / ˆ z Γ k and where V root := M α ∈ R ( G,S ) ˆ g α . Here ˆ g α is the usual α ∨ -eigenline for the action of b S on ˆ g , where α ∨ is interpreted, via ˆ and the canonical identification X ∗ ( b T ) = X ∗ ( T ), as a root of X ∗ ( b T ). We call V toral the toralsummand and V root the root summand . From our formula for ϕ we can work out the adjointWeil actions on these summands.4.4. Toral summand. For the toral summand, it is useful to momentarily consider thevector space e V toral := Lie( b T ) equipped with the adjoint Weil action of ϕ , so that the projection e 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 e V toral of the Weil group W k is therefore the complexification of the lattice Λ = X ∗ ( b T ),isomorphic to X ∗ ( S ) by X ∗ (ˆ ) and the canonical identification X ∗ ( b T ) = X ∗ ( T ). The Galoisaction on the lattice Λ is transferred via this chain of identifications from the Galois actionon X ∗ ( S ) arising from the structure of S as a torus over k . To summarize, there is anidentification of representations e 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 thesmaller lattice becomes a canonical quotient of the larger after complexifying both. We thushave 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 . Lemma 69. | γ (0 , V toral ) | = exp q (cid:0) (dim S a + dim M ) (cid:1) | M Frob || ( κ × ⊗ M ∨ ) Frob | .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: L (0 , V toral ) − = det(1 − Frob | M C ) = | M Frob | , where M Frob denotes the coinvariants of Frobenius. The L -factor at s = 1 is L (1 , V toral ) − = det(1 − q − Frob | M C ) = ( − q ) − dim M det(1 − q Frob − | M C ) . The determinantal factor in the last equation can be rewritten asdet(1 − q Frob − | M C ) = det(1 − q Frob | M ∨ C ) = (cid:12)(cid:12) ( κ × ⊗ M ∨ ) Frob (cid:12)(cid:12) , meaning that L (1 , V toral ) − = q − dim M · (cid:12)(cid:12) ( κ × ⊗ M ∨ ) Frob (cid:12)(cid:12) . Collecting the two L -factors gives (cid:12)(cid:12)(cid:12)(cid:12) L (1 , V toral ) L (0 , V toral ) (cid:12)(cid:12)(cid:12)(cid:12) = q dim M | M Frob || ( κ × ⊗ M ∨ ) Frob | . Since S is tamely ramified, Equation (55) shows that the Artin conductor of the toralsummand 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 (cid:0) (dim S a − dim M ) (cid:1) . (cid:3) ORMAL DEGREE OF REGULAR SUPERCUSPIDALS 39 Root summand. The root summand is a direct sum of representations induced fromcharacters of closed, finite-index subgroups of W k . An element w ∈ W k acts on V root through ϕ as follows. First, the action of W k on X ∗ ( b S )(= X ∗ ( S )) induces an action onthe root system R = R ( G, S ), and the element w permutes the root lines by this action.Second, the toral element t w := ˆ (ˆ θ ( w ) r χ ( w )) ∈ b T scales each root line b g α by α ∨ ( t w ), where α ∨ ∈ R ∨ ( G, S ) is interpreted as a character of b T using ˆ . It follows that V root is a direct sum of monomial representations. That is, for eachGalois orbit α ∈ R ( G, S ) the subrepresentation V α := M α ∈ α (¯ k ) b g α is monomial and V root is the direct sum (over R ( G, S )) of these representations. Further,after choosing a representative α ∈ α (¯ k ), we can identify V α with the representation inducedfrom the action of W α on ˆ g α , a certain character ψ α of W α . The essential matter, then, is tounderstand these characters ψ α , and specifically, as it turns out, their depths.Although the factor n ( ω S,G ( w )) w stabilizes the line ˆ g α , it may fail to centralize it: instead,the factor scales the line by a certain sign d α ( w ) ∈ {± } . It follows that ψ α is the productof three characters: ψ α ( w ) = d α ( w ) · h α ∨ , ˆ ( r χ ( w )) i · (cid:10) α ∨ , (cid:0) ˆ ◦ ˆ θ (cid:1) ( w ) (cid:11) where h− , −i denotes the evaluation pairing X ∗ ( b T ) ⊗ b T → C × . There are two essential casesin the analysis of this character, depending on whether or not the character h α ∨ , (ˆ ◦ ˆ θ (cid:1)(cid:11) | W α has positive depth. By the local Langlands correspondence the depth of this character is thesame as the depth of the character θ k α /k ◦ α ∨ : k × α → C × , and we know something aboutthese depths from Section 2.1. Lemma 70. The character h α ∨ , ˆ ◦ r χ i| W α of W α is tamely ramified.Proof. This is an immediate corollary of Corollary 67. (cid:3) Since p is odd (see Remark 3) the character d α is also tamely ramified, as it takes valuesin {± } . So h α ∨ , (cid:0) ˆ ◦ ˆ θ (cid:1)(cid:11) | W α differs from ψ α by a tamely ramified character. From this wecan immediately deduce the following corollary. Corollary 71. If depth( θ k α /k ◦ α ∨ ) > then depth( θ k α /k ◦ α ∨ ) = depth( ψ α ) . It remains to analyze the case where the depth of θ k α /k ◦ α ∨ is not positive. We first assumethat S is maximally unramified in G , then remove this assumption. Lemma 72. Suppose S is maximally unramified. If depth( θ k α /k ◦ α ∨ ) ≤ then depth ψ α = 0 .Proof. It is clear from Lemma 70 that depth ψ α ≤ 0, so we need only show that ψ α is ramified.Using the assumption that θ is extra regular, Kaletha proved [Kal19b, Proposition 5.2.7]that the parameter ϕ = L j χ ◦ L θ is regular [Kal19b, Definition 5.2.3], meaning in particularthat the connected centralizer of the inertia subgroup I k in b G is abelian. So although thefull centralizer of inertia may not be abelian, it does at least have the property that allof its elements are semisimple. Our proof proceeds by contradiction: assuming that ψ α isunramified, we show that the centralizer of inertia contains a nontrivial unipotent elementand is therefore nonabelian, a contradiction. Since θ is regular (Definition 8), the roots α with depth( θ k α /k ◦ α ∨ ) ≤ R of R = R ( b T , b G ), and the action of inertia on R preserves a set R +0 of positiveroots. Let H := Ad( ϕ ( I k )), let Hα denote the H -orbit of α ∈ R , and let U α ⊂ b G bethe root group for α ∈ R . Since I k ∩ W α is the inertia group of k α and L β ∈ Hα ˆ g β is amonomial representation of I k induced from ψ α , the character ψ α is unramified if and onlyif 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 foreach β ∈ Hα . Assume α ∈ R +0 satisfies these equivalent properties and has maximal lengthamong all such roots. The proof works just as well if α ∈ R − , so focus on the positive roots.First, suppose the roots in the H -orbit Hα of α are pairwise orthogonal. Choose a nontriv-ial 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 u = Y β ∈ Hα u β . Then u is invariant under H , hence centralizes inertia. But at the same time u is notsemisimple 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, aslight elaboration of the previous argument yields a contradiction. In this case the H -orbitof α admits an involution β ¯ β such that h β, γ i 6 = 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 ) + . Choosea nontrivial element u α ∈ U α , choose x ∈ H such that xα = ¯ α , and define the element u ¯ α := xu α , independent of the choice of x . The commutator subgroup U α +¯ α = [ U α , U ¯ α ]is stabilized by x , and since we assumed that α had maximal length among the possiblecounterexamples to our theorem, it is not centralized by x . (In fact, x must act by inversionon this group because xu ¯ α = u α .) Hence there is an element u α +¯ α ∈ U α +¯ α with u − α +¯ α · xu α +¯ α = [ u α , u ¯ α ] , that is, u α +¯ α u α u ¯ α = x ( u α +¯ α u α u ¯ α ) . For each β ∈ ( Hα ) + choose x ∈ H such that β = xα , let u β := xu α , and let u ¯ β := xu ¯ α ; theseelements are independent of the choice of x . As before, define the element u = Y β ∈ ( Hα ) + u β + ¯ β u β u ¯ β . Since the action Ad ◦ ϕ of wild inertia on b G is trivial, the group H acts on the factorsof u through some abelian quotient. Hence u centralizes inertia but is not semisimple,contradicting regularity. (cid:3) Remark 73. Kaletha defines an L -parameter to be torally wild if it takes wild inertia to amaximal torus of b G , and shows that torally wild L -parameters factor through the L -groupof a tame maximal torus [Kal19c]. The proof of Lemma 72 shows that this larger class ofparameters satisfies the conclusions of the lemma. Lemma 74. If depth( θ k α /k ◦ α ∨ ) ≤ then depth ψ α = 0 .Proof. Recall from Section 2.1 that there is a twisted Levi subgroup G of G such that α ∈ R ( S, G ) if and only if depth( θ k α /k ◦ α ∨ ) ≤ 0, and that S is maximally unramified in G .Lemma 72 handles the case where G = G , so we assume that G ( G . ORMAL DEGREE OF REGULAR SUPERCUSPIDALS 41 To deal with this more general case we factor the L -embedding L S → L G through L G .Kaletha showed [Kal19b, Lemmas 5.2.9, 5.2.8] that there is an L -embedding L j G ,G : L G → L G with the following property: the composite parameter W k → L S → L G is given by theformula w ˆ (cid:0) ˆ θ b ( w )ˆ θ ( w ) r χ ( w ) (cid:1) n ( ω S,G ( w )) w where θ b : S ( k ) → C × is tamely ramified and Ω( S, G ) Γ k -invariant. Furthermore, from theconstruction of L j G ,G it is easy to see that the embedding b g ֒ → b g is L j G ,G -equivariant. Inthis way we reduce to the previous case of G = G but with θ replaced by θ · θ − b . Thisreplacement does not affect the validity of the reduction: since θ b is Ω( S, G ) Γ k -invariant thecharacter θ ′ = θ · θ − b is still regular [Kal19b, Fact 3.7.6], and since θ b is tamely ramified, westill have that depth( θ ′ k α /k ◦ α ∨ ) ≤ (cid:3) Remark 75. Unlike Corollary 71, Lemma 74 does not assert that ψ α and θ k α /k ◦ α ∨ havethe 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 k α /k ◦ α ∨ ) ≥ 0. But I wasunable to prove the stronger assertion and a weaker statement sufficed.In summary, the root summand decomposes as a direct sum V root = M α ∈ R ( G,S ) V α , where, for any α ∈ α (¯ k ), the representation V α is induced from a character ψ α of W α withknown depth. We can now easily compute the γ -factor. Recall the Galois sets R i anddepths r i ≥ Lemma 76. | γ (0 , V root ) | = exp q (cid:16) | R | + d − X i =0 r i ( | R i +1 | − | R i | ) (cid:17) .Proof. Suppose that α ∈ R i +1 , for 0 ≤ i ≤ d − 1. Lemmas 12 and 74 and Corollary 71show that depth k ψ α = r i . Since L -factors are inductive L ( s, V α ) = L ( s, ψ α ), and since ψ α isramified 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 (cid:0) (1 + r i ) | α (¯ k ) | (cid:1) . Summing over α ∈ R ( G, S ) finishes the proof. (cid:3) 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 theadjoint γ -factor is | γ (0 , V ) | = | M Frob || ( κ × ⊗ M ∨ ) Frob | exp q (cid:16) dim G a + dim M + d − X i =0 r i ( | R i +1 | − | R i | ) (cid:17) . Finally, since | π ( S ♮ϕ ) | = | X ∗ ( S ) Γ k | [Kal15, Lemma 5.13], the Galois side of the formal degreeconjecture is(77) | M Frob || X ∗ ( S a ) Γ k | · | ( κ × ⊗ M ∨ ) Frob | exp q (cid:16) dim G a + dim M + d − X i =0 r i ( | R i +1 | − | R i | ) (cid:17) . Comparison In this short final section we combine our work from Sections 3 and 4 with several resultsfrom the literature to show that the automorphic and Galois sides of the formal degreeconjecture are equal, the following theorem. Theorem B. Kaletha’s regular L -packets satisfy the formal degree conjecture, Conjecture 1. Let ( S, θ ) be a tame elliptic regular pair and let ϕ = ϕ ( S,θ ) be the L -parameter attached tothis pair by the constructions of Section 2.4. The Galois side of the formal degree conjecturefor 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 describedin Section 2.4, where j ranges over conjugacy classes of admissible embeddings j : S ֒ → G .Since jθ ′ and θ ◦ j − 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 assumefor the purpose of computing the formal degree that the relevant pair is ( S, θ ), even thoughit is actually ( jS, jθ ′ ).To compute the dimension of the lattice M := X ∗ ( S a ) I k from Section 4.6, we prove ananalogue for tori of the N´eron-Ogg-Shafarevich criterion for abelian varieties. Lemma 78. Let k be a Henselian, discretely-valued field with residue field κ and let T be atame k -torus. Then there is a canonical identification X ∗ ( T ) I k = X ∗ ( T ) .Proof. Since X ∗ ( T ) I k = X ∗ ( T ) k nr where k nr is the maximal unramified extension of k , wecan use ´etale descent for the Moy-Prasad filtration [Yu02, 9.1] to reduce the proof to thecase 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 s0:0+ ) since now I k = Γ k .It suffices to prove that the canonical inclusion T s0:0+ ֒ → T is an isomorphism. Theproof rests on two facts from SGA 3. Since S is smooth and affine, the moduli space of itsmaximal tori is represented by a smooth O -scheme [DG11, Expos´e XII, Corollaire 1.10]. ByHensel’s Lemma [DG11, Expos´e XI, Corollaire 1.11], every κ -point of this moduli space liftsto a O -point. (cid:3) 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 ) ] = | X ∗ ( S a ) Frob I k | · | ( κ × ⊗ M ∨ ) Frob | . Let’s now compare the remaining factors outside of exp q . On the automorphic side wehave [ S a ( k ) : S a ( k ) ] − ; on the Galois side we have | M Frob || X ∗ ( S a ) Γ k | · | ( κ × ⊗ M ∨ ) Frob | . 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]. ORMAL DEGREE OF REGULAR SUPERCUSPIDALS 43 Acknowledgments I would like to thank my advisor Tasho Kaletha for proposing the problem of calculatingthe formal degree, sharing his wealth of knowledge in the Langlands program, and encour-aging me when I felt hopelessly stuck. I’m also grateful to Atsushi Ichino for notifying meof other cases where the formal degree conjecture has been proved, to Stephen DeBacker forexplaining how to compute the apartment of an elliptic maximal torus, and to Karol Koziolfor discussing the problem and encouraging me to work toward the proof of Lemma 74.This research was supported by the National Science Foundation RTG grant DMS 1840234. 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