DEPTH-ZERO SUPERCUSPIDAL L-PACKETS AND THEIR STABILITY

STEPHEN DEBACKER AND MARK REEDER

In this paper we verify the local Langlands correspondence for pure inner forms of unramified p-adic groups and tame Langlands parameters in “general position”. For each such parameter, we explicitly construct, in a natural way, a finite set (“L-packet”) of depth-zero supercuspidal representations of the appropriate p-adic group, and we verify some expected properties of this L-packet. In particular, we prove, with some conditions on the base field, that the appropriate sum of characters of the representations in our L-packet is stable; no proper subset of our Lpackets can form a stable combination. Our L-packets are also consistent with the conjectures of B. Gross and D. Prasad on restriction from SO_{2n+1} to SO_{2n} [24].

These L-packets are, in general, quite large. For example, Sp_{2n} has an L-packet containing 2^{n} representations, of which exactly two are generic. In fact, on a quasi-split form, each L-packet contains exactly one generic representation for every rational orbit of hyperspecial vertices in the reduced Bruhat-Tits building. When the group has connected center, every depth-zero generic supercuspidal representation appears in one of these L-packets.

We emphasize that there is nothing new about the representations we construct. They are induced from Deligne-Lusztig representations on subgroups of finite index in maximal compact mod-center subgroups, see [42], [44], [61]. The point here is to assemble these representations into L-packets in a natural and explicit way and to verify that these L-packets have the required properties.

To explain further, we need some notation. Let k be a p-adic field of characteristic zero, let K be a maximal unramified extension of k, let Γ = Gal(K/k), and let Frob ∈ Γ be a Frobenius element. Let W_{t}, I_{t} be the tame Weil group of k and its inertia subgroup. Let G be a connected reductive k-group which is K-split and k-quasi-split. To simplify the exposition, we assume in this introduction that G is semisimple. Let G := G(K), and let F be the action of Frob on G, arising from the given k-structure on G.

In the spirit of local class field theory, we construct both the “geometric” and “p-adic” sides of our local Langlands correspondence, and make an explicit connection between the two sides.

We start with the geometric side. The action of F on the root datum of G gives rise to an automorphism ϑ̂ of the Langlands dual group Ĝ. The Langlands parameters considered in this paper are continuous homomorphisms

φ : W_{t} ⟶ ⟨ϑ̂⟩ ⋉ Ĝ,

Date: February 2, 2007.

1991 Mathematics Subject Classification. Primary 22E50, 11S37.

Key words and phrases. Bruhat-Tits theory, characters, harmonic analysis, local Langlands correspondence, reductive p-adic group, stability.

(for the discrete topology on ⟨ϑ̂⟩ ⋉ Ĝ) whose centralizer in Ĝ is finite, and such that φ(Frob) is a semisimple element in ϑ̂Ĝ, and φ(I_{t}), a priori a finite cyclic group, is generated by a regular semisimple element in Ĝ. This latter condition is what we mean by “general position”. It implies that φ(I_{t}) is contained in a unique maximal torus T̂ ⊂ Ĝ. The element φ(Frob) normalizes T̂, acting via an element of the form ϑ̂ŵ, where ŵ belongs to the Weyl group of T̂ in Ĝ. Moreover, the centralizer of φ is the finite abelian group

C_{φ} := T̂^{ϑ̂ŵ}

of fixed-points of φ(Frob) in T̂.

For each irreducible character ρ ∈ Irr(C_{φ}), we will define a representation of the group of k-points of a certain inner form of G.

First, we parametrize Irr(C_{φ}) as follows. The automorphisms ϑ̂ and ŵ induce dual automorphisms ϑ and w of the character group X := X^{∗}(T̂), and each λ ∈ X determines a character ρ_{λ} ∈ Irr(C_{φ}) by restriction from T̂ to T̂^{ϑ̂ŵ}. Thus we have an isomorphism

X/(1 − wϑ)X ⟶^{∼} Irr(C_{φ}), λ ↦ ρ_{λ}.

Next, for each λ ∈ X we construct an unramified cocycle u_{λ} ∈ Z^{1}(Γ, G), hence an inner twist of G with Frobenius F_{λ} = Ad(u_{λ}) ∘ F, along with an irreducible depth-zero supercuspidal representation π_{λ} of GF_{λ}.

The cocycle u_{λ} is found as follows. Let W be the affine Weyl group of G, acting on the apartment A = ℝ ⊗ X in the Bruhat-Tits building B(G) of G. The character λ ∈ X determines a translation t_{λ} ∈ W. Since T̂^{ϑ̂ŵ} is finite, it follows that the operator t_{λ}wϑ has a unique fixedpoint x_{λ} ∈ A. If we choose an alcove C_{λ} ⊂ A containing x_{λ} in its closure, we can then uniquely write

(1)  t_{λ}wϑ = w_{λ}y_{λ}ϑ,

where w_{λ} belongs to the “parahoric subgroup” of W at x_{λ} and y_{λ} ∈ W satisfies y_{λ}ϑ · C_{λ} = C_{λ}. The cocycle u_{λ} : Γ ⟶ G sends Frob to an appropriately chosen representative of y_{λ} in G.

Now for the representation π_{λ}. The point x_{λ} is F_{λ}-stable, and is in fact a vertex in B(GF_{λ}). The parahoric subgroup G_{λ} of G at x_{λ} is F_{λ}-stable, and GF_{λ} ; λ is a maximal parahoric subgroup of GF_{λ}. The representation π_{λ} of GF_{λ} is compactly-induced from a representation κ_{λ} of GF_{λ} ; λ.

This κ_{λ} is obtained as follows. The element w_{λ} determines an F_{λ}-anisotropic torus T_{λ} of G with T F_{λ} ; λ ⊂ G_{λ}. By the depth-zero Langlands correspondence for tori (known, but reproved in Chapter 4.3 below), we can associate to (φ, λ) a depth-zero character χ_{λ} of T F_{λ} ; λ, whence a Deligne-Lusztig representation

κ_{λ} := ±R_{T_{λ}}^{G_{λ}}χ_{λ}.

Thus, for each λ ∈ X, we define

π_{λ} := ind_{G{F \atop λ} ; λ}^{G{F \atop λ}} κ_{λ},

using compact (equivalently, smooth) induction, and prove that π_{λ} is an irreducible representation of GF_{λ}. Of course, we now have infinitely many groups GF_{λ} and representations π_{λ}, whereas the L-packet Π(φ) should be parametrized by the finite set Irr(C_{φ}).

However, according to Vogan’s idea of “representations of pure-inner forms” [62], we must take into account the natural G-action on pairs (u, π_{u}), where u ∈ Z^{1}(Γ, G) and π_{u} is a representation of GF_{u} (here F_{u} = Ad(u) ∘ F). We prove that the G-orbit [u_{λ}, π_{λ}] is independent of all choices made in the construction, and that for λ, μ ∈ X, we have

[u_{λ}, π_{λ}] = [u_{μ}, π_{μ}] ⇔ ρ_{λ} = ρ_{μ} ∈ Irr(C_{φ}).

Thus, our construction leads to an L-packet Π(φ) in the form of equivalence classes:

Π(φ) = {[u_{λ}, π_{λ}] : ρ_{λ} ∈ Irr(C_{φ})}.

We have a partition

Π(φ) = ∐_{ω∈H^{1}(Γ,G)} Π(φ, ω),

where Π(φ, ω) consists of the classes [u_{λ}, π_{λ}] with u_{λ} ∈ ω. Let

Irr(C_{φ}) = ∐_{ω∈H^{1}(Γ,G)} Irr(C_{φ}, ω)

be the corresponding partition of Irr(C_{φ}).

The first expected property of Π(φ) is that Irr(C_{φ}, ω) should be the fiber over ω under the composition

(2)  Irr(C_{φ}) ⟶ Irr(Ẑϑ̂) ⟶^{∼} H^{1}(Γ, G),

where the first map is restriction, the second map is Kottwitz’ isomorphism [34], and Ẑ is the center of Ĝ. This amounts to proving that the map described in (2) sends ρ_{λ} ∈ Irr(C_{φ}) to the class of u_{λ} in H^{1}(Γ, G). For this, and other purposes, we need a very explicit description of Kottwitz’ isomorphism on the level of cocycles. Chapter 2 contains a simple proof of Kottwitz’ isomorphism in the form we need, along with related facts used in the proof of stability.

The second expected property of Π(φ) is that the ratio of formal degrees

\frac{deg(π_{λ})}{deg(St_{λ})},

where St_{λ} is the Steinberg representation of GF_{λ}, should be independent of λ ∈ X. This is proved by a direct calculation in Chapter 5.

The third expected property of Π(φ) is that π_{0} (here λ = 0) should be generic. This is true. In fact, we determine all generic representations in Π(φ), and show that they are in natural bijection with rational classes of hyperspecial vertices in the reduced building of G. En route, we classify all depth-zero supercuspidal generic representations of unramified groups, see Chapter 6. There is a more general conjecture, due to B. Gross and D. Prasad [23], about which Whittaker models are afforded by the generic representations in Π(φ). This conjecture is verified for Π(φ) in [19].

We illustrate the construction and above-mentioned properties, in Chapter 13, with a “canonical example” of L-packets arising from the opposition involution.

The rest of our paper is devoted to the fourth expected property, namely, the stability of Π(φ, ω).

We now consider L-packets from the p-adic side. Let G be any connected reductive K-split k-group with Frobenius automorphism F on G. Take a pair (S, θ), where S = S(K) is the group of K-points in an unramified k-anisotropic maximal torus S in G and θ is a depth-zero character of S^{F} = S(k). The group S^{F} has a unique fixed-point x ∈ B(G^{F}). We have a Deligne-Lusztig virtual character R_{S,θ}^{G_{x}} of the parahoric subgroup G_{x}^{F}, which we lift to a class function R(G, S, θ) on the set of regular semisimple elements of G^{F}, using Harish-Chandra’s character integral. One checks that R(G, S, θ) depends only on the G^{F} -orbit T̂ of the pair (S, θ). For (S, θ) ∈ T̂, we define

R(G, T̂) := R(G, S, θ).

We say that two pairs (S_{1}, θ_{1}), (S_{2}, θ_{2}) as above are G-stably-conjugate if there is g ∈ G such that Ad(g) sends (S_{1}^{F} , θ_{1}) to (S_{2}^{F} , θ_{2}). Each G-stable class T̂_{st} of pairs (S, θ) is a finite disjoint union

T̂_{st} = T̂_{1} ⊔ ⋯ ⊔ T̂_{n}

of G^{F} -orbits. We consider the function

R(G, T̂_{st}) := ∑_{i=1}^{n} R(G, T̂_{i}).

Our aim is to prove that R(G, T̂_{st}) is a stable class-function on the set of strongly regular semisimple elements in G^{F}.

But first, we relate R(G, T̂_{st}) to the L-packets constructed previously on the geometric side. To do this, we must put the representations in Π(φ) in “normal form”, as follows. We fix ω ∈ H^{1}(Γ, G), and choose u ∈ ω. For each λ ∈ X, with u_{λ} ∈ ω, there is m_{λ} in G such that Ad(m_{λ}) sends GF_{λ} to GF_{u}. For each ρ ∈ Irr(C_{φ}, ω), we define

π_{u}(φ, ρ) := Ad(m_{λ})_{∗}π_{λ}

where λ ∈ X is such that ρ_{λ} = ρ. Then π_{u}(φ, ρ) is a representation of GF_{u} whose isomorphism class is independent of the choices of λ and m_{λ}. The “normalized” L-packet is then defined as

Π_{u}(φ) := {π_{u}(φ, ρ) : ρ ∈ Irr(C_{φ}, ω)};

it consists of representations of the fixed group GF_{u}.

The comparison between the p-adic and geometric sides consists in proving that the sum of characters in Π_{u}(φ) is, up to a constant factor, a function of the form R(G, T̂_{st}) for an appropriate T̂_{st}. This involves an explicit parametrization of the G-stable classes of pairs (S, θ), in terms of characters λ ∈ X. This parametrization follows naturally from our study of Kottwitz’ isomorphism in Chapter 2.

Now, to prove stability for our L-packets, it remains to prove that the functions R(G, T̂_{st}) are stable. The first main step is a reduction formula, using the topological Jordan decomposition.

This reduction becomes trivial on the set of strongly regular topologically semisimple elements in G^{F}, proving stability there without any restrictions on the residue characteristic.

To prove stability everywhere, we must examine the restriction of R(G, T̂_{st}) to the topologically unipotent set. We are dealing here with a p-adic analogue of a Green function, so we write Q(G, T̂_{st}) for the restriction of R(G, T̂_{st}) to the topologically unipotent set in G^{F}.

To use the reduction formula, we must establish an identity between Q(G, T̂_{st}) and Q(G′, T̂′_{st}), where G′ is an inner form of G. To prove this identity, we use Murnaghan-Kirillov theory. The idea is to use a logarithm map and Kazhdan’s proof of the Springer Hypothesis [31] to express Q(G, T̂_{st}) as the Fourier transform of a stable orbital integral on the Lie algebra of G^{F}. We then invoke a deep result of Waldspurger [63], to the effect that the fundamental lemma is valid for inner forms, and this completes the proof.

However, there are two difficulties with this argument, one pleasant, one not. The pleasant difficulty is about a certain sign in Waldspurger’s result. It is given in [63] as a ratio of gamma constants. For us, it is necessary that this ratio be equal to Kottwitz’ sign e(G) [33]. This equality of signs is a particular case of a conjecture of Kottwitz. Because of its importance, here and elsewhere, we give two proofs, the first using Shalika germs, the second continuing in the combinatorial spirit of [63].

The unpleasant difficulty is about the logarithm map, which is required to satisfy certain compatibility properties with respect to the Moy-Prasad filtrations on G and its Lie algebra. It is at this point that restrictions on k must be imposed. We require that p ≥ (2 + e)n, where p is the residual characteristic of k, e is the ramification degree of k/ℚ_{p}, and n is the dimension of a faithful algebraic representation of G over k.

Finally, some remarks about exhaustion. All depth-zero supercuspidal representations of G^{F} are constructed in [44]. Many of them do not appear in our L-packets Π(φ). They should appear in square-integrable L-packets where φ is tame, but has a nontrivial component on SL_{2}(ℂ) and therefore cannot be in general position. For groups with connected center, such L-packets have been found for unramified φ in [39], [40], [41], [48]. For groups with connected center, the L-packets constructed in this paper should be exactly those depth-zero L-packets which consist entirely of supercuspidal representations. See Chapter 3 for more discussion of this.

We thank Robert Kottwitz, Fiona Murnaghan, Dipendra Prasad, Gopal Prasad, Loren Spice, and Jiu-Kang Yu for helpful conversations. We thank Loren Spice for allowing us to use his proof of Lemma B.7.2. Part of this work was done in 2001 while one of us (MR) was visiting the École Normale Supérieure in Paris, and the Korteweg-de Vries Institute in Amsterdam. He thanks Anne-Marie Aubert and Eric Opdam, respectively, for their hospitality. Finally, we acknowledge the National Science Foundation for its support, via grants DMS-0200542 (SD) and DMS-0207231 (MR).

While we were writing the details of our stability proof, D. Kazhdan and Y. Varshavsky announced a similar stability result, also using Murhaghan-Kirillov theory. See [32].

Contents

1. Basic Notation 7 2. Remarks on Galois cohomology 10

2.1 Unramified cohomology 10 2.2 Steinberg’s vanishing theorem 11 2.3 Explicit cocycles 12 2.4 Kottwitz’ Theorem 14 2.5 The dual group 16 2.6 A commutative diagram 16 2.7 Fixed points and cocycles 17 2.8 A normal form for Frobenius endomorphisms 19 2.9 Conjugacy 20 2.10 Rational classes in a stable class 21 2.11 A partition of the rational classes in a stable class 22

3. The conjectural local Langlands correspondence 24

3.1 Frobenius endomorphisms and representations of p-adic groups 24 3.2 Dual group 24 3.3 Weil group 25 3.4 Elliptic Langlands parameters 25 3.5 The conjectures 25

4. From tame regular semisimple parameters to depth-zero supercuspidal L-packets 27

4.1 Tame regular semisimple parameters 27 4.2 Outline of the construction 28 4.3 Depth zero characters of unramified tori 29 4.4 From tame parameters to depth-zero types 31 4.5 Definition of the L-packets 34 4.6 Choosing representatives in an L-packet 36

5. Normalizations of measures and formal degrees 36

5.1 Haar measure 37 5.2 Formal degree of the Steinberg representations 38 5.3 Formal degrees in our L-packets 39

6. Generic Representations 40

6.1 Depth-zero generic characters and representations 40 6.2 Generic representations in our L-packets 42

7. Topological Jordan decomposition 45 8. Unramified and minisotropic maximal tori 47 9. Some character computations 48

9.1 Harish-Chandra’s character formula 49 9.2 The character integral 49 9.3 Relation to characters 49 9.4 Stable conjugacy of tori and their characters 50 9.5 The stable character integral 52 9.6 Relation to L-packets 52

10. Reduction formulae for character integrals 54

10.1 Characters in a simple case 58

11. Reduction formula for stable character integrals 59

11.1 A bijection between stable classes of unramified tori 61 11.2 Stable characters in a simple case 62

12. Transfer to the Lie algebra 62

12.1 Orbital Integrals 63 12.2 A result of Waldspurger 63 12.3 Another calculation of Waldspurger’s sign 67 12.4 Murnaghan-Kirillov theory 71 12.5 Completion of the proof of stability 72

13. L-packets arising from the opposition involution 73 Appendix A. Good bilinear forms and regular elements 74

A.1 Good bilinear forms 74 A.2 Regular elements 75

Appendix B. A logarithm mapping for G 76

B.1 The exponential map for the general linear group 76 B.2 The logarithmic mapping ψ 78 B.3 An extension of ψ 78 B.4 The adjoint representation and ψ 79 B.5 A brief introduction to the filtrations of Moy and Prasad 79 B.6 A logarithmic map for semisimple groups 83 B.7 An extension to reductive groups 91

Index of selected notation and terms 92 References 96

1. Basic Notation

The cardinality of a finite set X is denoted by |X|. We denote the action of a group G on a set X by g · x or ^{g}x, for g ∈ G, x ∈ X. The fixed point set of g in X is denoted by X^{g}, and X^{G} := ∩_{g∈G}X^{g}. The set of G-orbits in X is denoted by X/G. The centralizer of g ∈ G is denoted by C_{G}(g). The conjugation map g′ ↦ gg′g^{−1} on G is denoted by Ad(g). The normalizer of a subgroup S ⊂ G is denoted by N(G, S). In this paper, the phrase “representation of a group G” means “equivalence class of complex representations of G”. The set of irreducible representations of a finite group G is denoted by Irr(G).

In this paper, k is a field of characteristic zero with a nontrivial discrete valuation for which k is complete with finite residue field 𝔣. Let q = |𝔣|, and let p be the characteristic of 𝔣. We fix an algebraic closure k̄ of k. Let K be the maximal unramified extension of k in k̄, and let 𝔉 denote the residue field of K. Then 𝔉 is an algebraic closure of 𝔣. Until Section 12 there are no restrictions on p or q. We fix an element ϖ ∈ k of valuation equal to one.

Let I be the inertia subgroup of the Galois group Gal(k̄/k), and let Γ = Gal(k̄/k)/I. Then Γ is topologically generated by an element Frob whose inverse induces the automorphism x ↦ x^{q} on 𝔉. We let Frob, “the Frobenius”, denote both this automorphism of K/k and the automorphism of 𝔉/𝔣 which it induces. We have K = k̄^{I}, k = K^{Frob}.

We use the following conventions for algebraic groups and their groups of rational points. For any k-group G, we identify G with its group G(k̄) of k̄-rational points, and let G := G(K) = G^{I} denote the K-rational points of G. For most of our purposes, the group G will play the role of “algebraic group”. The given action of Gal(k̄/k) on G restricts to an action of Γ on G, which is completely determined by an automorphism F ∈ Aut(G) given by the action of Frob. We have G^{F} = G(k). Likewise, we identify 𝔣-groups G with their groups of 𝔉-rational points, and we have G^{F} = G(𝔣).

The set of irreducible admissible representations of G^{F} is denoted by Irr(G^{F}). The subset of square-integrable representations in Irr(G^{F}) is denoted by Irr^{2}(G^{F}).

If S is a k-torus in G, we say that a character θ ∈ Irr(S^{F}) is F -regular if θ has trivial stabilizer in [N(G, S)/S]^{F}.

Given an element γ in either G or G, we let G_{γ} or G_{γ} denote the identity component of the centralizer of γ in G or G, respectively. If γ ∈ G, then we set G_{γ} := G∩G_{γ}. We say the element γ in G or G is regular semisimple if G_{γ} or G_{γ} is a torus. We let G^{rss} denote the set of regular semisimple elements of G. We say that γ in G or G is strongly regular semisimple if C_{G}(γ) or C_{G}(γ) is a torus. We let G^{srss} denote the set of strongly regular semisimple elements of G. If S is a maximal k-torus in G, then by [8, 1.10] the set G^{srss} ∩ S^{F} is nonempty.

For two reductive groups G_{1}, G_{2} or G_{1}, G_{2} of respective ranks r_{1}, r_{2} over k or 𝔣, we let

ε(G_{1}, G_{2}) = (−1)^{r_{1}−r_{2}}, ε(G_{1}, G_{2}) = (−1)^{r_{1}−r_{2}},

respectively.

For any torus S or S, we let X_{∗}(S) or X_{∗}(S) denote the group of algebraic one-parameter subgroups of S or S. We say an 𝔣-torus S ⊂ G is F -minisotropic in G if every μ ∈ X_{∗}(S)^{F} has image contained in the center of G.

The analogous notion for tori in G has an extra condition: In this paper, an unramified torus is a group of the form S = S(K), where S is a k-torus which splits over K. These conditions mean that I acts trivially on X_{∗}(S), and the action of Gal(k̄/k) on X_{∗}(S) factors through Γ. An F -minisotropic torus in G is a group of the form S = S(K), where S is a k-torus in G such that S is split over K, and the Frobenius F, arising from the given k-structure on G, has the property that every μ ∈ X_{∗}(S)^{F} has image contained in the center of G.

If S is a K-split k-torus, we let ^{0}S denote the maximal bounded subgroup of the unramified torus S. We have an isomorphism

K^{×} ⊗ X_{∗}(S) ⟶^{∼} S

given by evaluation. This restricts to an isomorphism

R_{K}^{×} ⊗ X_{∗}(S) ⟶^{∼} ^{0}S,

where R_{K}^{×} is the group of units in the ring of integers of K.

For this paper, until the appendices, G denotes a connected reductive k-group which splits over K. Let F be the Frobenius automorphism of G arising from the given k-structure on G.

Let B(G), B(G^{F}) denote the Bruhat-Tits buildings of G, G^{F}, respectively. The Frobenius F acts naturally on B(G), and we have B(G^{F}) = B(G)^{F}.

Let j : G → G_{ad} denote the adjoint quotient. Following our conventions, we set G_{ad} := G_{ad}(K), and denote again by F the action of Frob on G_{ad}.

Via the map j, the group G acts on B(G_{ad}). The latter is sometimes referred to as the “reduced building” of G. Likewise, the reduced building of G^{F} is B(G_{ad}^{F}) = B(G_{ad})^{F}.

Each unramified torus S in G determines apartments A(S) ⊂ B(G) and A_{ad}(S) ⊂ B(G_{ad}); these apartments can be defined as the fixed-point sets of ^{0}S in B(G) and B(G_{ad}), respectively. The Euclidean closure of any subset J of an apartment is denoted by J̄.

If J is an F -stable subset of a facet in B(G) or B(G_{ad}), we let G_{J} denote the corresponding parahoric subgroup of G, and let G_{J}^{+} denote the pro-unipotent radical of G_{J}. The quotient G_{J} := G_{J}/G_{J}^{+} is the group of 𝔉-points of a connected reductive group over 𝔣. We have F (G_{J}) = G_{J}, F (G_{J}^{+}) = G_{J}^{+}, and the induced action of F on G_{J} agrees with the 𝔣-structure on G_{J}. We have G_{J}^{F} = G_{J}^{F} /G_{J}^{+F}.

Recall that G is split over K. By [10, 5.1.10], there exists a K-split maximal torus T ⊂ G which is defined over k and maximally k-split. We abbreviate X := X_{∗}(T), A := A(T). Let N be the normalizer of T in G. The affine Weyl group of T in G is the quotient

W := N/^{0}T .

We will use T = T(K) as a “platonic” unramified torus in G; various unramified tori S as above will arise from twisted embeddings of T in G.

Let T_{ad} = j(T) denote the image of T in G_{ad}, and abbreviate A_{ad} := A(T_{ad}), X_{ad} := X_{∗}(T_{ad}). Let W_{ad} be the affine Weyl group of T_{ad} in G_{ad}. Since T and T_{ad} are defined over k, the Frobenius F induces automorphisms of X, X_{ad}, A, A_{ad}, W, W_{ad}. We write also

j : X → X_{ad}, j : W ⟶ W_{ad}

for the maps induced by j. These maps are F -equivariant, since j is defined over k. The kernel and image of the latter map are given as follows.

We may identify X with the normal subgroup T/^{0}T ◁ W, via evaluation at ϖ. If λ ∈ X, we let t_{λ} := λ(ϖ) denote both the corresponding element of T and its image in W. There is a map W_{ad} ⟶ X_{ad}/jX, to be defined shortly, which fits into an exact sequence

(3)  1 ⟶ X^{W} ⟶ W ⟶^{j} W_{ad} ⟶ X_{ad}/jX ⟶ 1.

Note that the last group X_{ad}/jX is finite. The group X^{W} acts trivially on A_{ad}.

There exists an F -stable alcove C ⊂ A. Let W^{∘} be the subgroup of W generated by reflections in the walls of C, and let Ω_{C} := {ω ∈ W : ω · C = C}. The group Ω_{C} is abelian, isomorphic to the quotient of X by the co-root sublattice X^{∘} ⊂ X. The normal subgroup W^{∘} ◁ W acts simply-transitively on alcoves in A, so we have a semidirect product expression

W = Ω_{C}W^{∘}.

A similar discussion and decomposition holds for W_{ad}.

We have been using F to denote the Frobenius arising from an arbitrary K-split k-structure on G. When this k-structure is in fact k-quasi-split, we denote the Frobenius by F. The key difference in the quasi-split case is the existence of an F-fixed hyperspecial vertex o ∈ A_{ad}.

In the quasi-split case, we denote by ϑ the automorphisms of X, X_{ad}, A, A_{ad}, W, W_{ad} induced by F. Choose a ϑ-fixed hyperspecial vertex o ∈ A_{ad}. We let W_{o} be the image of N_{o} := N ∩ G_{o} in W. We may identify W_{o} = N/T via the natural maps

W_{o} ↪ W = N/^{0}T ⟶ N/T.

The map j is injective on W_{o} and we identify W_{o} with j(W_{o}). We have semidirect product decompositions

W = X ⋊ W_{o}, W_{ad} = X_{ad} ⋊ W_{o},

and all factors are preserved by ϑ. The map W_{ad} ⟶ X_{ad}/jX in the exact sequence (3) is induced by projection onto the X_{ad} factor in W_{ad}.

Finally, an inner twist of F by a cocycle u ∈ Z^{1}(F, G) (see Section 2) will be denoted by F_{u} := Ad(u) ∘ F.

2. Remarks on Galois cohomology

To state the Langlands conjectures at the level of refinement considered in this paper requires some notions from the Galois cohomology of reductive groups over local fields. The central results here are due to Kottwitz [34, 35], who computes H^{1}(k, G) in terms of the action of Gal(k̄/k) on the center of the dual group of G, and Bruhat-Tits [11], who compute H^{1}(k, G) in terms of the building of G. Here we give simple proofs of the above-mentioned results at the level of cocycles. This allows us to construct cocycles in G from fixed-points in A of elements in the affine Weyl group. Such fixed-points arise from the Langlands parameters we consider. Thus we can associate an explicit Frobenius to each Langlands parameter. We also use our cocycles to give representatives for various stable and rational classes of tori and semisimple elements in G. These will be used in the proof of stability.

2.1. Unramified cohomology. Let U be a group and let F be an endomorphism of U. For an integer d ≥ 1 and g ∈ U, define

N_{d}(F)(g) := gF (g) ⋯ F^{d−1}(g) ∈ U.

Note that

(4)  N_{dm}(F) = N_{m}(F^{d}) ∘ N_{d}(F).

Assume that every element of U is fixed by some power of F. Giving U the discrete topology, this means that the group ℤ̂ of profinite integers, with topological generator F, acts continuously on U. We denote by

H^{1}(F, U) = H^{1}(ℤ̂, U)

the continuous (nonabelian) cohomology of U. Any cocycle is determined by its value on F, which is an element of the set

Z^{1}(F, U) := {u ∈ U : N_{m}(u) = 1 for some m ≥ 1}.

Thus we view cocycles as elements of U, and H^{1}(F, U) is the quotient of Z^{1}(F, U) under the U-action: g ∗ u = guF (g)^{−1}. Note that if N_{m}(u) = 1 and F^{d}(g) = g, then N_{md}(F)(g ∗ u) = 1.

If U is nonabelian, the set Z^{1}(F, U) of cocycles is not closed under multiplication. However, if u, v ∈ Z^{1}(F, U) and d ≥ 1 we have

(5)  N_{d}(F)(vu) = N_{d}(F_{u})(v) · N_{d}(F)(u),

where

F_{u} := Ad(u) ∘ F ∈ End(U).

From Equations (4) and (5) we conclude:

Lemma 2.1.1. If two of the following hold, then so does the third:

(1) u ∈ Z^{1}(F, U),

(2) v ∈ Z^{1}(F_{u}, U),

(3) vu ∈ Z^{1}(F, U).

Lemma 2.1.2. If the fixed-point group U^{F^{d}} is finite for each d ≥ 1, then Z^{1}(F, U) = U.

Proof. Fix d ≥ 1 and suppose that g^{m} = 1 for each g ∈ U^{F^{d}}. From Equation (4), we have

N_{dm}(F)(g) = N_{m}(F^{d}) (N_{d}(F)(g)) = (N_{d}(F)(g))^{m} = 1.

□

Lemma 2.1.3. Suppose U is a compact group with endomorphism F and a decreasing filtration U = U_{0} ⊃ U_{1} ⊃ U_{2} ⊃ ⋯ by open normal F -stable subgroups U_{n} such that ⋂_{n} U_{n} = {1}. Assume that H^{1}(F, U_{n}/U_{n+1}) = 1 for all n ≥ 0. Then H^{1}(F, U) = 1.

Proof. Let u ∈ Z^{1}(F, U), so that u ∈ U_{n} for some n ≥ 0. By the vanishing assumption and normality, there are g_{0} ∈ U_{n} and u_{1} ∈ U_{n+1} such that u = g_{0}∗u_{1}. Then u_{1} = g_{0}^{−1}∗u ∈ Z^{1}(F, U). Repeating, we have elements g_{k}, u_{k} ∈ U_{n+k} for all k ≥ 1, such that u = (g_{0}g_{1} ⋯ g_{k−1}) ∗ u_{k}. Since U is compact, the limit g := lim_{k} g_{0}g_{1} ⋯ g_{k} exists, and u = g ∗ 1. □

2.2. Steinberg’s vanishing theorem. In this section, G is only required to be a connected kgroup, with Frobenius automorphism F on G. At several points we use the following consequence of a well-known result of Steinberg [56, Thm. 1.9]:

Theorem 2.2.1. H^{1}(K, G) = 1.

One consequence of Theorem 2.2.1 is that the natural surjection Gal(k̄/k) → Γ induces an isomorphism

H^{1}(F, G) ≃ H^{1}(k, G).

Each cocycle u ∈ Z^{1}(F, G) arises from a twisted k-structure on G, under which Frob acts on G via the automorphism

F_{u} := Ad(u) ∘ F ∈ Aut(G),

so that G^{F_{u}} is the group of k-rational points under this twisted k-structure [51, III.1.3]. Note that for g ∈ G, we have

Ad(g) ∘ F_{u} = F_{g∗u} ∘ Ad(g),

so Ad(g) induces an isomorphism

Ad(g) : G^{F_{u}} ⟶^{∼} G^{F_{g∗u}}.

Thus, the isomorphism class of G^{F_{u}} depends only on the class of u in H^{1}(F, G). However, the dependence is non-canonical, in the sense that a class in H^{1}(F, G) does not determine a unique twist of F; one must choose a cocycle in the class. We must therefore accept a wide range of Frobenius endomorphisms F_{u} giving rise to the same k-isomorphism class of groups.

2.3. Explicit cocycles. For the rest of this chapter, G is a connected reductive k-group with Frobenius automorphism F on G. To keep things as simple and clear as possible, we assume that G is K-split and k-quasi-split, even though these assumptions are not necessary until later in the paper. The following result is a special case of [64, Prop. 2.3]. We give a direct proof, in our context.

Lemma 2.3.1. For each x ∈ B(G)^{F} we have H^{1}(F, G_{x}) = 1, where G_{x} is the parahoric subgroup attached to x.

Proof. If u ∈ Z^{1}(F, G_{x}), then u ∈ G_{x}^{F^{d}} for some d ≥ 1. We want to apply 2.1.3 to the compact group U = G_{x}^{F^{d}}. Let G_{x,r}, r ∈ ℝ_{≥0}, be the Moy-Prasad-Yu filtration of G_{x} [65]. There is an increasing sequence {r_{n} : n = 0, 1, 2, … } ⊂ ℝ_{≥0} such that for every r ≥ 0 we have G_{x,r} = G_{x,r_{n}} for a unique n. These filtration subgroups are F -stable; we set U_{n} := G_{x,r_{n}}^{F^{d}}.

Each quotient group U_{n}/U_{n+1} is the group of 𝔣_{d}-rational points in a connected 𝔣-group U_{n}. Here 𝔣_{d} denotes the degree d extension of 𝔣. By the Lang-Steinberg theorem, we have H^{1}(𝔣, U_{n}) = 1 for all n ≥ 0. Since the natural map

H^{1}(𝔣_{d}/𝔣, U_{n}(𝔣_{d})) ⟶ H^{1}(𝔣, U_{n})

is injective [51, I.5.8], we have H^{1}(𝔣_{d}/𝔣, U_{n}(𝔣_{d})) = 1 for all n ≥ 0.

We have shown that the groups U_{n} satisfy the conditions of Lemma 2.1.3, which implies that the cocycle u is a coboundary in H^{1}(F, U_{0}), hence also in H^{1}(F, G_{x}). □

Recall that T is a K-split maximal k-torus in G, such that T contains a maximal k-split torus in G, and N is the normalizer of T in G. The affine Weyl group of T in G is the quotient W := N/^{0}T, where ^{0}T is the maximal bounded subgroup of T. The apartment of T in B(G) is denoted by A, and the N-action on A factors through a faithful action of W on A.

To describe H^{1}(F, G) on the level of cocycles, the first step is to reduce the group in which the cocycles live. Let C be an F-stable alcove in A (see [60, 3.4.3]). Let G_{C} be the Iwahori subgroup of G attached to C. The normalizer in G of G_{C} is the group

G_{C}^{⋆} := {g ∈ G : g · C = C}.

We have N ∩ G_{C} = ^{0}T, and we set

N_{C} := N ∩ G_{C}^{⋆}.

Then the group

Ω_{C} := {ω ∈ W : ω · C = C}

is the image of N_{C} in W. The inclusion N_{C} ↪ G_{C}^{⋆} induces an isomorphism

d : Ω_{C} ⟶^{∼} G_{C}^{⋆}/G_{C}.

Since F ·C = C, we have F(G_{C}^{⋆}) = G_{C}^{⋆}, so we may define H^{1}(F, G_{C}^{⋆}) as in 2.2, and similarly for H^{1}(F, N_{C}). The first reduction relies on the existence and conjugacy of rational alcoves, already used above.

Lemma 2.3.2. The inclusion G_{C}^{⋆} ↪ G induces an isomorphism

H^{1}(F, G_{C}^{⋆}) ⟶^{∼} H^{1}(F, G).

Proof. We first prove surjectivity. Let u ∈ Z^{1}(F, G). By [60, 1.10.3] there is an F_{u}-stable alcove C_{u} ⊂ B(G). We have g · C_{u} = C for some g ∈ G. Since F_{u} ·C_{u} = C_{u}, we have u F(g^{−1}) · C = g^{−1} · C, i.e., g ∗ u ∈ G_{C}^{⋆}.

For injectivity, suppose u, v ∈ Z^{1}(F, G_{C}^{⋆}), and g ∗ u = v for some g ∈ G. Then

F_{v} ·g · C = v F(g) · C = gu · C = g · C.

Thus g · C and C are two F_{v}-stable alcoves in B(G). By [60, §2.5] there is h ∈ G^{F^{v}} such that hg · C = C, so hg ∈ G_{C}^{⋆}. However, h = F_{v}(h) implies

(hg)u F(hg)^{−1} = hv F(h)^{−1} = v,

so [u] = [v] in H^{1}(F, G_{C}^{⋆}). □

To go further, we need another vanishing result. The image of ^{0}T in G_{C} is a maximal 𝔣-torus T in G_{C}. We let ^{0}T^{+} be the kernel of the natural map ^{0}T ⟶ T. Then ^{0}T^{+} is the pro-unipotent radical of^{0}T.

Recall that Γ = Gal(K/k). A topological Γ-module [50, XIII, p.188] is a Γ-module in which every element is fixed by some power of Frob.

Lemma 2.3.3. For any n ∈ N, letting Frob act on ^{0}T via F_{n} := Ad(n) ∘ F makes ^{0}T a topological Γ-module for which H^{2}(F_{n},^{0}T) = 0.

Proof. As endomorphisms of T, we have F_{n} = F_{w}, where w is the image of n in N/T. Now

F_{w}^{k} = Ad[w F(w) ⋯ F^{k−1}(w)] ∘ F^{k}, for all k ≥ 1.

Since N/T is finite, the term in brackets is 1 for some k (cf. Lemma 2.1.2) and multiples thereof. Also every t ∈ T is fixed by some F^{m} and multiples thereof. The first assertion now follows.

The exact sequence of topological Γ-modules

1 ⟶ ^{0}T^{+} ⟶ ^{0}T ⟶ T ⟶ 1

gives an exact sequence [51, §2.2, p.10] in Galois cohomology

⋯ ⟶ H^{2}(F_{w},^{0}T^{+}) ⟶ H^{2}(F_{w},^{0}T) ⟶ H^{2}(F_{w}, T) ⟶ · · · .

Since T is a torsion group, we have H^{2}(F_{w}, T) = 0 by [50, Proposition 2, p.189]. Since 0T^{+} is the union of an inverse limit of torsion groups, from [50, Lemma 3, p.185] we have H^{2}(F_{w},^{0}T^{+}) = 0. □

Consider now the following commutative diagram, where the horizontal maps are inclusions, and the vertical maps are the natural projections.

\xymatrix{N_{C} \ar[r]^{b} \ar[d]_{a} & G_{C}^{⋆} \ar[d]^{c} \\ Ω_{C} \ar[r]^{d} & G_{C}^{⋆}/G_{C}}

Lemma 2.3.4. The maps a, b, c, d in the above diagram induce isomorphisms a_{∗}, b_{∗}, c_{∗}, d_{∗} on H^{1}(F, ·).

Proof. The map d is already an isomorphism. The map a_{∗} is surjective by Lemma 2.3.3 and [51, Corollary, p.54]. Since the induced diagram on cohomology is commutative, the map c_{∗} is also surjective.

If u ∈ Z^{1}(F, G_{C}^{⋆}), then from [51, Corollaries 1 and 2, p.52] the fiber of c_{∗} through [u] is in bijection with ker[H^{1}(F_{u}, G_{C}^{⋆}) → H^{1}(F_{u}, G_{C}^{⋆}/G_{C})]. By the exact sequence

⋯ ⟶ H^{1}(F_{u}, G_{C}) ⟶ H^{1}(F_{u}, G_{C}^{⋆}) ⟶ H^{1}(F_{u}, G_{C}^{⋆}/G_{C})

in nonabelian cohomology [51, Proposition 38, p.51] and the vanishing of H^{1}(F_{u}, G_{C}) by Lemma 2.3.1, the above kernel is trivial. Hence c_{∗} is injective. A similar argument shows that a_{∗} is injective, which completes the proof. □

2.4. Kottwitz’ Theorem. In this section we will recover Kottwitz’ theorem on the level of cocycles. First we need an elementary result.

Lemma 2.4.1. Let A be a finitely generated abelian group, and let σ ∈ Aut(A) be an automorphism of finite order. Define

A_{1} := {a ∈ A : (1 + σ + ⋯ + σ^{n−1})a = 0 for some n ≥ 1},
A_{2} := {a ∈ A : ma ∈ (1 − σ)A for some m ≥ 1}.

Then A_{1} = A_{2}.

Proof. For p ≥ 1 let N_{p} = 1 + σ + ⋯ + σ^{p−1} ∈ End(A). Then

N_{pq} = N_{p} + σ^{p}N_{p} + ⋯ + σ^{p(q−1)}N_{p} = (1 + σ^{p} + ⋯ + σ^{p(q−1)})N_{p}.

Hence if N_{p}(a) = N_{q}(b) = 0, then N_{pq}(a + b) = 0. That is, A_{1} is a subgroup of A. Also, since A_{tor} is finite, every element of A_{tor} is fixed by some power of σ. If qa = 0, then σ^{p}(a) = a for some p ≥ 1, so N_{pq}(a) = qN_{p}(a) = N_{p}(qa) = 0. Thus, A_{tor} ⊆ A_{1}.

Set Ā = A_{1}/A_{tor}, V = ℚ ⊗ Ā. The latter is a finite-dimensional ℚ-vector space, to which σ and N_{p} extend for all p. We claim that V^{σ} = {0}. If 0 ≠ v ∈ V^{σ}, we may assume, by clearing denominators, that v ∈ Ā. Then N_{p}(v) = pv ≠ 0 for all p ≥ 1, a contradiction. Hence 1 − σ is invertible on V. Let a ∈ A_{1} have image ā ∈ Ā. Write ā = (1 − σ)b̄, for some b̄ ∈ V. Clearing denominators, we have mā = (1 − σ)c̄ for some m ∈ ℤ, c ∈ A_{1}. So ma = (1 − σ)c + z, where z ∈ A_{tor}. Say qz = 0. Then qma = (1 − σ)qc ∈ (1 − σ)A, showing that A_{1} ⊆ A_{2}.

The other containment is easy: If qa = (1 − σ)b, and p is the order of σ, then

N_{pq}(a) = qN_{p}(a) = N_{p}(qa) = N_{p}(1 − σ)b = (1 − σ^{p})b = 0.
□

Let X = X_{∗}(T), and let W^{∘} be the subgroup of W generated by reflections in the walls of an alcove in A. Evaluation at ϖ identifies λ ∈ X with the operator t_{λ} ∈ W of translation by λ on A. Under this identification, X ∩ W^{∘} =: X^{∘} is the co-root lattice of T. We set X̄ := X/X^{∘}. The group W^{∘} acts simply-transitively on alcoves, hence we have the semidirect product decomposition

W = W^{∘} ⋊ Ω_{C}.

The automorphism F preserves T, hence induces an automorphism ϑ of W, which preserves X, W^{∘}, Ω_{C}. If G is actually k-split then ϑ is trivial. In general, ϑ has finite order.

For λ ∈ X, let ω_{λ} be the unique element of t_{λ}W^{∘} ∩ Ω_{C}. Then ω_{λ} = 1 exactly when λ belongs to X^{∘}; the map λ ↦ ω_{λ} induces a ϑ-equivariant group isomorphism X̄ ⟶^{∼} Ω_{C}.

Corollary 2.4.2. The map λ ↦ ω_{λ} induces an isomorphism

[X̄/(1 − ϑ)X̄]_{tor} ⟶^{∼} H^{1}(F, Ω_{C}).

Proof. Apply Lemma 2.4.1 to the abelian group A = Ω_{C}, σ = ϑ. Since (1 − ϑ)Ω_{C} ⊂ (Ω_{C})_{2}, we have

H^{1}(F, Ω_{C}) = (Ω_{C})_{1}/(1 − ϑ)Ω_{C} = (Ω_{C})_{2}/(1 − ϑ)Ω_{C} = [Ω_{C}/(1 − ϑ)Ω_{C}]_{tor}.

The isomorphism X̄ ≃ Ω_{C} finishes the proof. □

Combining 2.3.4 and 2.4.2, we can express Kottwitz’ isomorphism in the following form.

Corollary 2.4.3. The composition

[X̄/(1 − ϑ)X̄]_{tor} ⟶^{∼} H^{1}(F, Ω_{C}) ⟶^{a_{∗}^{−1}} H^{1}(F, N_{C}) ⟶^{b_{∗}} H^{1}(F, G)

is a bijection. A class [λ] ∈ [X̄/(1 − ϑ)X̄]_{tor}, represented by λ ∈ X, corresponds to the class [ω̇_{λ}] ∈ H^{1}(F, G), where ω̇_{λ} ∈ Z^{1}(F, N_{C}) is any element whose image in W is the unique element ω_{λ} of t_{λ}W^{∘} ∩ Ω_{C}.

2.5. The dual group. Corollary 2.4.3 is usually expressed in terms of the dual group Ĝ of G. Let Y := X^{∗}(T) be the algebraic character group of T, and let ⟨ , ⟩ : X × Y → ℤ be the natural pairing. The dual group of T is the complex torus T̂ := Y ⊗ ℂ^{×}; it is a maximal torus in Ĝ. Let Ẑ denote the center of Ĝ.

For any σ ∈ Aut(X), let σ̂ ∈ Aut(Y) be defined by

⟨σλ, η⟩ = ⟨λ, σ̂η⟩, λ ∈ X, η ∈ Y.

The action of ϑ̂ on Y extends to the automorphism ϑ̂ ⊗ 1 of T̂, thence by restriction to an automorphism of Ẑ.

We may identify

X̄ = Hom(Ẑ, ℂ^{×}),

via restriction of characters. Restricting further to Ẑϑ̂, we may identify

X̄/(1 − ϑ)X̄ = Hom(Ẑϑ̂, ℂ^{×}).

The elements in X̄/(1 − ϑ)X̄ vanishing on the identity component of Ẑϑ̂ are exactly the torsion elements in X̄/(1 − ϑ)X̄. Hence we may identify

[X̄/(1 − ϑ)X̄]_{tor} = Irr[π_{0}(Ẑϑ̂)].

With these identifications, Corollary 2.4.3 becomes the usual expression of Kottwitz’ isomorphism.

2.6. A commutative diagram. It is at this point that we first use seriously the assumption that F arises from a quasi-split k-structure on G which is K-split. Such an assumption ensures the existence of a F-fixed hyperspecial vertex o ∈ j(C̄).

Let W_{o} be the image of N_{o} = N ∩ G_{o} in W. The latter has another semidirect product expression

W = X ⋊ W_{o},

and both factors are preserved by ϑ.

Since G is K-split and k-quasi-split, there is a Gal(k̄/k)-invariant pinning in G. Applying Prop. 3 of [59] to this pinning, we see that there is an F-stable finite subgroup Ẇ_{o} ⊂ N_{o} projecting onto W_{o}.

Let w ∈ W_{o}, choose a lift ẇ ∈ Ẇ_{o} of w, and set F_{w} := Ad(ẇ) ∘ F. Applying Lemmas 2.1.2 and 2.3.1 to the groups Ẇ_{o} and G_{o}, respectively, there exists p_{0} ∈ G_{o} such that

ẇ = p_{0}^{−1} F(p_{0}).

The map Ad(p_{0}) : T ⟶ G intertwines the pairs (T, F_{w}), (G, F). Let

(6)  r : H^{1}(F_{w}, T) ⟶ H^{1}(F, G),

be the map induced by Ad(p_{0}).

A version of the following result was proved by Kottwitz [35, Thm. 1.2].

Lemma 2.6.1. We have a commutative diagram

[X/(1 − wϑ)X]_{tor} ⟶ [X̄/(1 − ϑ)X̄]_{tor}
≃↓  ↓≃
H^{1}(F_{w}, T)  ⟶^{r}  H^{1}(F, G)

where the vertical maps are from 2.4.3 applied to T and G, the top row is the natural projection and the map r is defined in Equation (6).

Proof. Starting at [X/(1 − wϑ)X]_{tor} and going down the left side, then over on the bottom row, the class of λ ∈ [X/(1 − wϑ)X]_{tor} goes to the class

[p_{0}t_{λ}p_{0}^{−1}] = [t_{λ}p_{0}^{−1} F(p_{0})] = [t_{λ}ẇ] ∈ H^{1}(F, G).

Equation (9) below shows that [t_{λ}ẇ] = [ω̇_{λ}], which is the result of the other route, by Corollary 2.4.3. □

2.7. Fixed points and cocycles. We continue in the set-up of Section 2.4. In this section we show how cocycles in Z^{1}(F, G) arise from fixed-points in A of elements in the affine Weyl group W. This will be used to associate Frobenius endomorphisms on G to Langlands parameters in LG.

Let w ∈ W_{o}, and let X_{w} be the preimage in X of [X/(1 − wϑ)X]_{tor}. For λ ∈ X_{w}, we define

σ_{λ} := t_{λ}wϑ ∈ W ⋊ ⟨ϑ⟩.

Lemma 2.7.1. The element σ_{λ} ∈ W ⋊ ⟨ϑ⟩ has finite order.

Proof. The element wϑ has finite order, say n, since it belongs to the finite group W_{o} ⋊ ⟨ϑ⟩. We let N_{wϑ} = 1 + wϑ + ⋯ + (wϑ)^{n−1} ∈ End(X) be the associated norm mapping. Since λ ∈ X_{w}, there is m ≥ 1 such that mλ = (1 − wϑ)ν, for some ν ∈ X. Then

σ_{λ}^{nm} = N_{wϑ}(t_{mλ}) = N_{wϑ}(1 − wϑ)(t_{ν}) = 1.
□

By Lemma 2.7.1, σ_{λ} preserves a facet J_{λ} in A. Choose an alcove C_{λ} in A containing J_{λ} in its closure. Let W_{λ} be the subgroup of W^{∘} generated by reflections in the hyperplanes containing J_{λ}. The group W_{λ} acts simply-transitively on alcoves in A containing J_{λ} in their closure. Hence there is a unique element w_{λ} ∈ W_{λ} such that

σ_{λ} · C_{λ} = w_{λ} · C_{λ}.

Set y_{λ} := w_{λ}^{−1}t_{λ}w. Thus we have two expressions for σ_{λ}:

(7)  t_{λ}wϑ = σ_{λ} = w_{λ}y_{λ}ϑ,

and the latter is characterized as the unique factorization of σ_{λ} such that w_{λ} ∈ W_{λ} and y_{λ} ∈ W satisfies y_{λ}ϑ · C_{λ} = C_{λ}. Since w_{λ} fixes J_{λ} pointwise, we also have y_{λ}ϑ · J_{λ} = J_{λ}; indeed, σ_{λ} and y_{λ}ϑ have the same action on J_{λ}.

To briefly look ahead: Equation (7) is the essence of our Langlands correspondence. The expression t_{λ}wϑ will arise from a certain kind of Langlands parameter, that is, t_{λ}wϑ is an object on the “geometric side”. On the other hand, y_{λ} and w_{λ} will determine a twisted Frobenius F_{λ} and an unramified torus in GF_{λ}, respectively, so y_{λ} and w_{λ} are objects on the “p-adic side”. The next result leads us to F_{λ}.

Lemma 2.7.2. There exists a lift u_{λ} ∈ N of y_{λ} such that u_{λ} ∈ Z^{1}(F, N).

Proof. If j is the order of σ_{λ} (see Lemma 2.7.1), then

1 = (w_{λ}y_{λ}ϑ)^{j} = w′_{λ}(y_{λ}ϑ)^{j},

for some w′_{λ} ∈ W^{∘}. Since W^{∘} acts simply-transitively on alcoves in A, we can decompose

W ⋊ ⟨ϑ⟩ = W^{∘} ⋊ Ω̃_{C_{λ}},

where Ω̃_{C_{λ}} is the stabilizer of C_{λ} in W ⋊ ⟨ϑ⟩. It follows that (y_{λ}ϑ)^{j} = 1. Let k be the order of ϑ. Then

1 = (y_{λ}ϑ)^{jk} = [y_{λ}ϑ(y_{λ}) ⋯ ϑ^{jk−1}(y_{λ})]ϑ^{jk} = y_{λ}ϑ(y_{λ}) ⋯ ϑ^{jk−1}(y_{λ}).

That is, y_{λ} ∈ Z^{1}(F, W). Hence, for all x ∈ W, we have x^{−1}y_{λ}ϑ(x) ∈ Z^{1}(F, W).

Recall that y_{λ}ϑ · C_{λ} = C_{λ}. Let x ∈ W^{∘} be the element such that C_{λ} = x · C. Then y_{λ}ϑ · x · C = x · C, so x^{−1}y_{λ}ϑ(x) ∈ Ω_{C} (recall C is ϑ-stable). By the previous paragraph, we have in fact x^{−1}y_{λ}ϑ(x) ∈ Z^{1}(F, Ω_{C}).

By Lemma 2.3.4 there is a lift n ∈ Z^{1}(F, N) of x^{−1}y_{λ}ϑ(x) such that n · C = C. Choose a lift ẋ ∈ N of x. Then the element

u_{λ} := ẋn F(ẋ)^{−1} ∈ Z^{1}(F, N)

is a lift of y_{λ} as claimed. □

Lemma 2.7.3. The class of u_{λ} in H^{1}(F, G) is equal to that of ω̇_{λ} ∈ Z^{1}(F, G). (See 2.4.3.)

Proof. By the construction of u_{λ} in Lemma 2.7.2, we have [u_{λ}] = [n], where n ∈ Z^{1}(F, N) is a lift of x^{−1}y_{λ}ϑ(x) ∈ Ω_{C}, and x is a certain element of W^{∘}. By Corollary 2.4.3, it suffices to show that

x^{−1}y_{λ}ϑ(x) ∈ t_{λ}W^{∘}.

First note that t_{λ}W^{∘} is preserved under conjugation by W^{∘}. The equation t_{λ}w = w_{λ}y_{λ} then implies y_{λ} ∈ t_{λ}W^{∘}. Since x ∈ W^{∘} as well, it follows that x^{−1}y_{λ}ϑ(x) ∈ t_{λ}W^{∘}. □

Fix once and for all a lift ẇ of w in Ẇ_{o}. Since t_{λ}wy_{λ}^{−1} = w_{λ} ∈ W^{∘}, there exists a unique lift ẇ_{λ} ∈ N of w_{λ} satisfying

t_{λ}ẇ = ẇ_{λ}u_{λ}.

Set

G_{λ} := G_{J_{λ}}, F_{λ} := Ad(u_{λ}) ∘ F .

Since y_{λ}ϑ · C_{λ} = C_{λ}, we have

F_{λ} ·C_{λ} = C_{λ}.

We have

t_{λ} ∈ Z^{1}(F_{w}, G),  ẇ ∈ Z^{1}(F, G),

the first by the definition of X_{w} and the second by Lemma 2.1.2 applied to the group Ẇ_{o}. From Lemma 2.1.1 we conclude that

t_{λ}ẇ ∈ Z^{1}(F, G).

But also u_{λ} ∈ Z^{1}(F, G), so using Lemma 2.1.1 again, we conclude that

ẇ_{λ} ∈ Z^{1}(F_{λ}, G_{λ}).

By Lemma 2.3.1 there is an element p_{λ} ∈ G_{λ} such that

p_{λ}^{−1} F_{λ}(p_{λ}) = ẇ_{λ}.

This equation can be written as

(8)  t_{λ}ẇ = p_{λ}^{−1}u_{λ} F(p_{λ}).

It follows that t_{λ}ẇ ∈ Z^{1}(F, G), and, in view of Lemma 2.7.3, we have

(9)  [t_{λ}ẇ] = [u_{λ}] = [ω̇_{λ}] ∈ H^{1}(F, G),

as claimed in the proof of Lemma 2.6.1.

2.8. A normal form for Frobenius endomorphisms. Keep the set-up of Section 2.7. For each λ ∈ X_{w} we have defined a Frobenius automorphism F_{λ} and an F_{λ}-stable alcove C_{λ} in A. For certain w ∈ W_{o}, we will eventually associate to λ, and some additional data, a representation π_{λ} ∈ Irr(GF_{λ}). This association will be quite natural, but it will leave us with infinitely many pairs (GF_{λ}, π_{λ}), which are almost all conjugate to one another in some sense, and we will need to compare them. To do this, we seek a normal form for our Frobenius endomorphisms F_{λ}.

Fix a class ω ∈ H^{1}(F, G), along with a representative u ∈ ω ∩ N, such that u · C = C. This is possible by Lemma 2.3.4. In this section we will gather together all of the F_{λ} for which u_{λ} ∈ ω. We will then use our explicit cohomology picture to keep track of conjugacy classes of tori and certain semisimple elements in a fixed group GF_{u}. This, in turn, will be used in our stability calculations.

From Lemma 2.6.1, we have a map

r : X_{w} → H^{1}(F, G)

sending λ ↦ [ω̇_{λ}]. For λ ∈ r^{−1}(ω), define σ_{λ} = t_{λ}wϑ, and choose J_{λ}, C_{λ}, u_{λ} as in Section 2.7. Recall that the Frobenius F_{λ} = Ad(u_{λ}) ∘ F stabilizes the alcove C_{λ}.

Lemma 2.8.1. For each λ ∈ r^{−1}(ω), there exists m_{λ} ∈ N such that

m_{λ} ∗ u_{λ} = u, m_{λ} · C_{λ} = C.

Proof. Choose k_{λ} ∈ N such that k_{λ} · C_{λ} = C. Since F_{λ} ·C_{λ} = C_{λ}, it follows that k_{λ} ∗ u_{λ} ∈ N_{C}. In Lemma 2.7.3 we proved that [u] = [u_{λ}] in H^{1}(F, G). Therefore [u] = [k_{λ} ∗ u_{λ}] in H^{1}(F, G). Since u and k_{λ} ∗ u_{λ} belong to N_{C}, and H^{1}(F, N_{C}) → H^{1}(F, G) is injective (see Lemma 2.3.4), we have [u] = [k_{λ} ∗ u_{λ}] in H^{1}(F, N_{C}). Hence there is ℓ_{λ} ∈ N_{C} such that u = (ℓ_{λ}k_{λ}) ∗ u_{λ}. Then m_{λ} := ℓ_{λ}k_{λ} has the required properties. □

As in Section 2.7, we have the alternative expression σ_{λ} = w_{λ}y_{λ}ϑ, where w_{λ} ∈ W_{λ} and y_{λ} is the image of u_{λ} in W. Recall that we have fixed a lift ẇ ∈ Ẇ_{o} of w, which determines a lift ẇ_{λ} ∈ N ∩ G_{λ} by the equation t_{λ}ẇ = ẇ_{λ}u_{λ}, and we have an element p_{λ} ∈ G_{λ} such that p_{λ}^{−1} F_{λ}(p_{λ}) = ẇ_{λ}. Choose m_{λ} as in Lemma 2.8.1 and set

q_{λ} := m_{λ}p_{λ} ∈ G, S_{λ} := Ad(q_{λ})T.

Then in G we have, using equation (8),

q_{λ}^{−1} F_{u}(q_{λ}) = p_{λ}^{−1} · m_{λ}^{−1}u F(m_{λ} · p_{λ})u^{−1} = p_{λ}^{−1}u_{λ} F(p_{λ})u^{−1} = t_{λ}ẇu^{−1}.

Thus, we have the analogue of equation (8) for q_{λ}:

(10)  t_{λ}ẇ = q_{λ}^{−1}u F(q_{λ}).

Equation (10) will be used repeatedly in future calculations. It implies that the map Ad(q_{λ}) : T ⟶ S_{λ} satisfies

F_{u} ∘ Ad(q_{λ}) = Ad(q_{λ}) ∘ F_{w} .

In particular, S_{λ} is an F_{u}-stable unramified maximal torus in G, whose underlying algebraic group S_{λ} is k-isomorphic to the twist of T by w.

In this section we have constructed an infinite family {S_{λ} : λ ∈ r^{−1}(ω)} of such tori, and our next task is to group these tori, and their strongly regular elements, into GF_{u}-conjugacy classes.

2.9. Conjugacy. We will use several times another consequence of Steinberg’s vanishing result, Theorem 2.2.1.

Lemma 2.9.1. Let G_{ad} be the adjoint group of G, and let G_{ad} = G_{ad}(K). Suppose G_{ad} acts on a k-variety X, with connected stabilizers. For x, y ∈ X(K), the following are equivalent:

(1) x and y are in the same G-orbit

(2) x and y are in the same G_{ad}-orbit

(3) x and y are in the same G-orbit. Here G acts on X via the canonical map j : G ⟶ G_{ad}.

Proof. Implication 1 ⇒ 2 is clear. Since G ⟶ G_{ad} is surjective, 2 ⇒ 3 is also clear. Assume 3 holds, so there is g ∈ G such that g · x = y. Since x, y ∈ X(K) = X^{I}, the map sending σ ∈ I to g^{−1}σ(g) ∈ G is a cocycle in Z^{1}(K, G_{x}). By hypothesis, G_{x} is the full stabilizer of x in G. By Theorem 2.2.1 we have H^{1}(K, G_{x}) = 1, so there is h ∈ G_{x} such that (gh)^{−1}σ(gh) = 1 for all σ ∈ I. Hence gh ∈ G, and 1 follows. □

We say that γ ∈ G^{F} is strongly regular semisimple in G if the centralizer of γ in G is a torus. By 2.9.1 we have for such γ the equalities

[Ad(G)γ]^{Gal(k̄/k)} = [Ad(G_{ad})γ]^{F} = [Ad(G)γ]^{F} .

Such a set is called the G-stable conjugacy-class of γ. It is a finite union of Ad(G^{F})-orbits, which are called the rational classes in the stable class.

2.10. Rational classes in a stable class. We continue with the setup of Section 2.8. Our aim is to explicitly parametrize the rational classes in the stable classes of certain elements γ ∈ G.

Recall that the map r : X_{w} ⟶ H^{1}(F, G) is defined as a composition

r : X_{w} ⟶ [X/(1 − wϑ)X]_{tor} ⟶ H^{1}(F, G).

We have fixed ω ∈ H^{1}(F, G), and have considered the fiber r^{−1}(ω) ⊂ X_{w}. Now let [r^{−1}(ω)] denote the image of r^{−1}(ω) in [X/(1 − wϑ)X]_{tor}. In other words, [r^{−1}(ω)] is the fiber over ω in the second map in the above composition. By Lemma 2.6.1, we may identify [r^{−1}(ω)] with the fiber over ω of the natural map H^{1}(F_{w}, T) ⟶ H^{1}(F, G).

Let γ ∈ T F_{w} be a strongly regular element of G. For λ ∈ r^{−1}(ω), we set

γ_{λ} := q_{λ}γq_{λ}^{−1} ∈ SF_{u} ; λ.

Lemma 2.10.1. For λ, μ ∈ r^{−1}(ω), the elements γ_{λ} and γ_{μ} are GF_{u}-conjugate if and only if λ ≡ μ mod (1 − wϑ)X. Thus, sending λ ↦ γ_{λ} defines a bijection

[r^{−1}(ω)] ⟶^{∼} [Ad(G)γ]F_{u}/GF_{u}.

Proof. Since S_{λ} = G_{γ_{λ}}, this is almost obvious from Lemma 2.6.1. However, we will give a direct proof which produces the conjugation from elements already in play.

By Equation (10) we have

q_{λ}^{−1}u F(q_{λ}) = t_{λ}ẇ, q_{μ}^{−1}u F(q_{μ}) = t_{μ}ẇ.

Let h = q_{μ}q_{λ}^{−1}, so that Ad(h)γ_{λ} = γ_{μ}. Then h^{−1} F_{u}(h) ∈ S_{λ} since γ is strongly regular. Moreover, γ_{μ} ∈ Ad(GF_{u})γ_{λ} if and only if the class [h^{−1} F_{u}(h)] in H^{1}(F_{u}, S_{λ}) is the identity element. We have

h^{−1} F_{u}(h) = q_{λ} · q_{μ}^{−1}u F(q_{μ}) · F(q_{λ})^{−1}u^{−1}
= q_{λ}t_{μ}ẇ F(q_{λ})^{−1}u^{−1}
= q_{λ}t_{μ}ẇẇ^{−1}t_{−λ}q_{λ}^{−1}
= q_{λ}t_{μ−λ}q_{λ}^{−1},

so

[h^{−1} F_{u}(h)] = [q_{λ}t_{μ−λ}q_{λ}^{−1}] ∈ H^{1}(F_{u}, S_{λ}).

On the other hand, we have isomorphisms

[X/(1 − wϑ)X]_{tor} ⟶^{∼} H^{1}(F_{w}, T) ⟶^{Ad(q_{λ})} H^{1}(F_{u}, S_{λ}).

The first isomorphism is Corollary 2.4.3 applied to T, F_{w}; for ν ∈ X_{w}, it sends the class of ν mod (1 − wϑ)X to the class of t_{ν} in H^{1}(F_{w}, T). Thus, [q_{λ}t_{μ−λ}q_{λ}^{−1}] is trivial in H^{1}(F_{u}, S_{λ}) if and only if λ − μ ∈ (1 − wϑ)X. □

2.11. A partition of the rational classes in a stable class. We have seen in Lemma 2.10.1 that the fiber [r^{−1}(ω)] parametrizes the GF_{u}-conjugacy classes in the stable class of γ_{λ}, for λ ∈ r^{−1}(ω). In this section we study an additional structure on this fiber. Namely, the group

W_{o}^{wϑ} := {z_{o} ∈ W_{o} : wϑ(z_{o})w^{−1} = z_{o}}

acts naturally on X_{w}, [X/(1 − wϑ)X]_{tor}, and [X̄/(1 − ϑ)X̄]_{tor}, and W_{o}^{wϑ} acts trivially on the latter. Hence there is a natural W_{o}^{wϑ}-action on the fiber [r^{−1}(ω)]. This action in fact corresponds to GF_{u}-conjugacy among the family of tori {S_{λ} : λ ∈ r^{−1}(ω)}, as follows.

Lemma 2.11.1. For λ, μ ∈ r^{−1}(ω) the following are equivalent.

(1) There is z_{o} ∈ W_{o}^{wϑ} such that z_{o}μ ≡ λ mod (1 − wϑ)X.

(2) There is g ∈ GF_{u} such that^{g}γ_{μ} ∈ S_{λ}.

(3) There is g ∈ GF_{u} such that^{g}S_{μ} = S_{λ}.

Proof. Assertions 2 and 3 are equivalent because S_{λ} = G_{γ_{λ}} for all λ ∈ r^{−1}(ω). (We have made them separate statements for later convenience.)

Assume 3 holds. Then q_{λ}^{−1}gq_{μ} ∈ N. Applying Equation (10) for μ and λ, we find that

\begin{aligned}q_{λ}^{−1}gq_{μ} · t_{μ}ẇ · F(q_{μ}^{−1}g^{−1}q_{λ}) & = q_{λ}^{−1}gu F(g)^{−1} F(q_{λ}) \\ & = q_{λ}^{−1}g · u F(g)^{−1}u^{−1} · q_{λ}t_{λ}ẇ \\ & = q_{λ}^{−1}g · F_{u}(g)^{−1} · q_{λ}t_{λ}ẇ \\ & = t_{λ}ẇ. \end{aligned} ; (11)

Let z ∈ W be the image of q_{λ}^{−1}gq_{μ}, and write z = t_{ν}z_{o} with ν ∈ X, z_{o} ∈ W_{o}. Mapping the first and last terms of Equation (11) to W, we have

t_{ν}z_{o} · t_{μ}w · ϑ(z_{o}^{−1}t_{−ν}) = t_{λ}w.

This shows that z_{o} ∈ W_{o}^{wϑ}, and then projection onto W_{o} yields

λ = z_{o}μ + (1 − wϑ)ν,

so 1 holds.

Conversely, if 1 holds, then λ = z_{o}μ + (1 − wϑ)ν for some ν ∈ X, and we set z = t_{ν}z_{o}. Since H^{1}(F_{w},^{0} T) = 1, there is a lift ż_{o} ∈ N F_{w} of z_{o}, and we set ż = t_{ν}ż_{o}. Then

F(ż) = t_{ϑν}ẇ^{−1}ż_{o}ẇ.

Set g = q_{λ}żq_{μ}^{−1}. It is clear that ^{g}S_{μ} = S_{λ}. To prove 3, it remains to show that g ∈ GF_{u}. Using equation (10) again, we compute

F_{u}(g) = u F(q_{λ}) · F(ż) · F(q_{μ})^{−1}u^{−1}
= q_{λ}t_{λ}ẇ · t_{ϑν}ẇ^{−1}ż_{o}ẇ · ẇ^{−1}t_{−μ}q_{μ}^{−1}
= q_{λ} · t_{λ+wϑν−z_{o}μ} · ż_{o}q_{μ}^{−1}
= q_{λ}t_{ν}ż_{o}q_{μ}^{−1}
= g,

as desired. □

Let

W_{o,λ}^{wϑ} := {z ∈ W_{o}^{wϑ} : zλ ≡ λ mod (1 − wϑ)X}

be the stabilizer in W_{o}^{wϑ} of the class of λ in [r^{−1}(ω)]. The next result interprets W_{o}^{wϑ} and W_{o,λ}^{wϑ} as “large” and “small” Weyl groups of S_{λ}, respectively. This will be used to relate L-packets to stable conjugacy classes of tori.

Lemma 2.11.2. For λ ∈ r^{−1}(ω), the map Ad(q_{λ}) induces isomorphisms

W_{o}^{wϑ} ⟶^{∼} N(G, SF_{u} ; λ)/S_{λ}, W_{o,λ}^{wϑ} ⟶^{∼} N(G, S_{λ})F_{u}/SF_{u} ; λ.

Proof. First, a remark about normalizers of tori. Let F be a Frobenius on G arising from some k-structure, and let S be the group of K-points of a maximal k-torus S ⊂ G. We claim that

(12)  [N(G, S)/S]^{F} = N(G, S^{F})/S.

For ⊆: Let n ∈ N(G, S) be such that F (n) = ns for some s ∈ S. Then on S we have Ad(n) ∘ F = F ∘ Ad(n), implying that n ∈ N(G, S^{F}). For ⊇: Choose s_{0} ∈ S^{F} ∩ G^{srss}. For n ∈ N(G, S^{F}), and s ∈ S, the element nsn^{−1} centralizes s_{0}, hence lies in S. This shows that N(G, S^{F}) ⊆ N(G, S). Moreover, we have Ad(n)s_{0} ∈ S^{F}, implying that Ad(n^{−1}F (n))s_{0} = s_{0}, hence F (n) ∈ nS, as desired.

This remark shows that W_{o}^{wϑ} = N(G, T F_{w})/T, and the first isomorphism follows. The second isomorphism amounts to showing that the projections N → W → W_{o} induces an isomorphism

(13)  N F_{t_{λ}w}/T F_{w} ⟶ W_{o,λ}^{wϑ}.

Let n ∈ N F_{t_{λ}w}, and let t_{ν}z be the image of n in W, where ν ∈ X and z ∈ W_{o}. We want to show that z ∈ W_{o,λ}^{wϑ}. From the equation Ad(t_{λ}) F_{w}(n) = n, we get

Ad(t_{λ}w)ϑ(t_{ν}z) = t_{ν}z,

which leads to

t_{λ+(wϑ−1)ν} Ad(w)ϑ(z) = t_{zλ}z,

hence z ∈ W_{o}^{wϑ} and zλ = λ + (wϑ − 1)ν, as desired. This shows also that (13) is injective.

To see that (13) is surjective, let z ∈ W_{o,λ}^{wϑ}, and choose a lift ż ∈ N_{o}. Since Ad(t_{λ}w)ϑ(z) = z in W_{o} = N_{o}/^{0}T, we have

Ad(t_{λ}) F_{w}(ż) = żt,

for some t ∈ ^{0}T. Since H^{1}(F_{w},^{0}T) = 1 we can write t = s F_{w}(s^{−1}) for some s ∈ ^{0}T. Then żs is a lift of z in N F_{t_{λ}w}. □

3. The conjectural local Langlands correspondence

Very roughly speaking, the conjectural local Langlands correspondence predicts a relationship between representations of a p-adic group and certain maps from the Weil group into the dual group. The latter maps are called “Langlands parameters”; they should partition the representations of the p-adic group into finite sets, called “L-packets”, and it is conjectured that these L-packets have many nice properties. We now make these statements more precise.

3.1. Frobenius endomorphisms and representations of p-adic groups. Continue with the setup of section 2.3: G is a connected reductive k-group which is k-quasi-split and K-split, with Frobenius automorphism F on the group G = G(K).

For each cocycle u ∈ Z^{1}(F, G), we have a twisted Frobenius

F_{u} := Ad(u) ∘ F

on G, and for g ∈ G, we have

Ad(g) ∘ F_{u} ∘ Ad(g)^{−1} = F_{g∗u} .

Therefore Ad(g) is an isomorphism

Ad(g) : GF_{u} ⟶ G^{F^{g∗u}},

which induces a bijection on irreducible representations, denoted by

Ad(g)_{∗} : Irr(GF_{u}) ⟶ Irr(G^{F^{g∗u}}).

This bijection preserves the sets Irr^{2}(·) of square-integrable representations.

Thus we have a G-action on the set

R^{2}(F, G) := {(u, π) : u ∈ Z^{1}(F, G), π ∈ Irr^{2}(GF_{u})}.

Considering the u-coordinate, we can partition R^{2}(F, G) into G-stable subsets

R^{2}(F, G) = ∐_{ω∈H^{1}(F,G)} R^{2}(F, G, ω),

where R^{2}(F, G, ω) consists of the pairs (u, π) ∈ R^{2}(F, G) with u ∈ ω.

3.2. Dual group. Let Ĝ be the dual group of G. By definition, the dual torus

T̂ := Y ⊗ ℂ^{×}

is a maximal torus in Ĝ. The operator ϑ̂ ∈ Aut(Y) dual to ϑ extends to an automorphism of the torus T̂, with trivial action on ℂ^{×}.

We choose, once and for all, a pinning (T̂, B̂, {x_{α}}) where B̂ is a Borel subgroup of Ĝ containing T̂ and the x_{α} are non-trivial elements in the simple root groups of T̂ in B̂. There is a unique extension of ϑ̂ to an automorphism of Ĝ, satisfying ϑ̂(x_{α}) = x_{ϑ·α} (see [7]). We can then form the semidirect product

LG := ⟨ϑ̂⟩ ⋉ Ĝ.

3.3. Weil group. Recall that the inertia subgroup I ≤ Gal(k̄/k) is the kernel of the natural map

Gal(k̄/k) ⟶ Gal(K/k).

The Weil group W is the subgroup of Gal(k̄/k) generated by I and the Frobenius Frob. The wild inertia subgroup I^{+} ◁ I is the maximal pro-p subgroup of I. The tame inertia group is the quotient I_{t} := I/I^{+}, and the tame Weil group is the quotient W_{t} := W/I^{+}. We will have more to say about these groups in Section 4.3.

3.4. Elliptic Langlands parameters. An elliptic Langlands parameter is a homomorphism

φ : W × SL_{2}(ℂ) ⟶ ^{L}G

with the following properties:

• φ(I) is a finite subgroup of Ĝ,

• φ(Frob) = ϑ̂f, where f ∈ Ĝ is semisimple,

• the restriction of φ to SL_{2}(ℂ) is algebraic,

• The identity component CĜ(φ)^{∘} of CĜ(φ) is equal to the identity component (Ẑϑ̂)^{∘} of

Ẑϑ̂. The last condition expresses the “ellipticity” of φ; it is equivalent to requiring that the image of φ is not contained in a proper Levi subgroup of^{L}G, where the meaning of “Levi subgroup” is as in [7, 3.4].

We let C_{φ} denote the component group of CĜ(φ). Since Ẑϑ̂ is contained in the center of CĜ(φ), each ρ ∈ Irr(C_{φ}) determines a central character on Ẑϑ̂ hence, via Kottwitz’ isomorphism (Corollary 2.4.3), a class ω_{ρ} ∈ H^{1}(F, G). Thus we may partition

Irr(C_{φ}) = ∐_{ω∈H^{1}(F,G)} Irr(C_{φ}, ω),

where Irr(C_{φ}, ω) consists of the representations ρ ∈ Irr(C_{φ}) with ω_{ρ} = ω.

3.5. The conjectures. The version of the Langlands conjectures stated here is the product of many refinements, by Deligne, Lusztig, Vogan and others. The local Langlands correspondence for G is a conjectural bijection between the set of Ĝ-orbits of pairs (φ, ρ), where φ is an elliptic Langlands parameter and ρ ∈ Irr(C_{φ}), and the set of G-orbits in R^{2}(F, G). Among many other expected properties, the G-orbit corresponding to (φ, ρ) should lie in R^{2}(F, G, ω) precisely when ω_{ρ} = ω.

Thus, we expect to have, for each Ĝ-conjugacy class of elliptic Langlands parameters φ, a finite set

Π(φ) = ∐_{ω∈H^{1}(F,G)} Π(φ, ω),

where

(14)  Π(φ, ω) := {[π(φ, ρ)] : ρ ∈ Irr(C_{φ}, ω)},

and [π(φ, ρ)] = {(u, π_{u}(φ, ρ)) : u ∈ ω} is a G-orbit in R^{2}(F, G, ω).

These putative sets Π(φ) are known as “L-packets”. These L-packets should form partitions

R^{2}(F, G)/G = ∐_{{φ}/Ĝ} Π(φ), R^{2}(F, G, ω)/G = ∐_{{φ}/Ĝ} Π(φ, ω).

To describe the properties we expect of an L-packet, we fix a representative u ∈ Z^{1}(F, N) of each class ω ∈ H^{1}(F, G). We represent the trivial class by u = 1, recalling that F_{1} = F. Then {π_{u}(φ, ρ) : ρ ∈ Irr(C_{φ}, ω)} is a set of representatives for the G-orbits comprising Π(φ, ω).

We expect L-packets to have the following properties.

(i) The representation π_{u}(φ, ρ) is unipotent [39] if and only if φ is unramified, (that is, if φ is trivial on the inertia subgroup I of W). For G with connected center, Lusztig has constructed unipotent L-packets corresponding to unramified φ [39], [40]. See also [41] and [48] and for orthogonal and split adjoint exceptional groups, respectively.

(ii) π_{u}(φ, ρ) has depth-zero (that is, has nonzero vectors fixed under the pro-unipotent radical of some parahoric subgroup in GF_{u}) if and only if φ is tame (that is, φ is trivial on the wild inertia subgroup I^{+} of I).

(iii) π_{1}(φ, 1) should be generic (that is, has a Whittaker model). If G has connected center, then π_{1}(φ, 1) should be the unique generic representation in Π(φ).

(iv) Let^{L}M be a minimal Levi subgroup of^{L}G containing φ(W). (It is unique up to conjugacy by the connected centralizer of φ(W) [7].) If^{L}M = ^{L}G, then every class in Π(φ) should consist of supercuspidal representations. In this case, we say that Π(φ) itself is “supercuspidal”. The L-packets in this paper are all supercuspidal.

If ^{L}M ≠ ^{L}G, then ^{L}M corresponds to an F-stable Levi subgroup M ⊂ G contained in an F-stable proper parabolic subgroup P ⊂ G. The restriction φ : W ⟶ ^{L}M inductively corresponds to a generic supercuspidal representation π_{1}^{M}(φ, 1) of M^{F}, and π_{1}(φ, 1) should be a generic constituent of the smoothly-induced representation Ind_{P^{F}}^{G^{F}} π_{1}^{M}(φ, 1). For (u, ρ) ≠ (1, 1), the representation π_{u}(φ, ρ) should be supported on Levi subgroups of GF_{u} whose center has k-rank no larger than that of M^{F}.

(v) For each u, normalize Haar measure on GF_{u} so that the formal degree of the Steinberg representation of GF_{u} is independent of u. (For example, one could make all Steinberg formal degrees equal to one, but we will choose a different normalization.) Let Deg denote formal degree with respect to these measures. Then we should have

Deg[π_{u}(φ, ρ)] = dim ρ · Deg[π_{1}(φ, 1)].

Recall that π_{u}(φ, ρ) and π_{1}(φ, 1) may be representations of non-isomorphic groups.

Properties (i-v) were verified in [48] for unipotent L-packets of split adjoint exceptional groups (see (i) above).

(vi) Fix u and φ, and let Θ_{ρ} be the character of π_{u}(φ, ρ), viewed as a function on the set (G^{rss})F_{u} of regular semisimple elements of GF_{u}. The function

∑_{ρ∈Irr(C_{φ},ω)} dim ρ · Θ_{ρ}

should be stable. That is, if γ, γ′ ∈ (G^{srss})F_{u} are G-conjugate^{1} strongly regular elements (see Section 1), then we should have

∑_{ρ∈Irr(C_{φ},ω)} dim ρ · Θ_{ρ}(γ) = ∑_{ρ∈Irr(C_{φ},ω)} dim ρ · Θ_{ρ}(γ′).

This was verified in [41] for unipotent L-packets for inner forms of SO(2n + 1) (see (i) above).

4. From tame regular semisimple parameters to depth-zero supercuspidal L-packets

We shall construct L-packets satisfying (ii)-(vi) above, for tame parameters φ in “general position”. We will first make this condition precise, and outline the construction.

Our construction relies on the tame Langlands correspondence for tori. A general Langlands correspondence for tori was proved by Langlands [37] but it seems more difficult to extract the depth-zero correspondence from [37] than to re-prove it from scratch, so we give a short self-contained account of the tame Langlands correspondence for tori. Then we construct our L-packets, using the material from Section 2.7.

4.1. Tame regular semisimple parameters. We say that a Langlands parameter φ is tame regular semisimple if it is trivial on the wild inertia subgroup I^{+} and the centralizer of φ(I) in Ĝ is a torus. The latter condition is what we mean by “general position”. This forces φ to be trivial on SL_{2}(ℂ). (There is a more general notion of “tame regular” parameter which we will consider elsewhere.)

Recall that W_{t} = W/I^{+} and I_{t} = I/I^{+}. Our choice of inverse Frobenius determines a splitting

W_{t} = ⟨Frob⟩ ⋉ I_{t},

where Frob^{−1} x Frob = x^{q} for x ∈ I_{t}.

Recall that the Weyl group N/T is identified with W_{o}, the image of N_{o} in W. We let Ŵ_{o} denote the Weyl group N̂/T̂ where N̂ is the normalizer of T̂ in Ĝ. The restriction of the duality map

Aut(X) ⟶^{σ↦σ̂} Aut(Y)

defines an anti-isomorphism w ↦ ŵ from W_{o} to Ŵ_{o}.

1It is customary to require the elements to be G-conjugate, but we have seen in Lemma 2.9.1 that two strongly regular semisimple elements of G are G-conjugate if and only if they are G-conjugate.

After conjugating by Ĝ, we may assume that φ(I_{t}) ⊂ T̂ and φ(Frob) = ϑ̂f, where f ∈ N̂. Let ŵ be the image of f in Ŵ_{o}, corresponding to w ∈ W_{o} via the above anti-isomorphism.

Then

CĜ(φ) = T̂ ŵϑ,

which implies that the restriction map X → Hom(T̂ ŵϑ, ℂ^{×}) induces an isomorphism

(15)  [X/(1 − wϑ)X]_{tor} ⟶^{∼} Irr(C_{φ}),  λ ↦ ρ_{λ}.

Moreover, φ is elliptic if and only if

(T̂ ŵϑ)^{∘} = (Ẑϑ̂)^{∘}.

To summarize: A tame regular semisimple elliptic Langlands parameter (TRSELP) is given by two objects:

• a continuous homomorphism s : I_{t} ⟶ T̂, with CĜ(s) = T̂, and

• an element f ∈ N̂ satisfying the two conditions

ϑ̂ ∘ Ad(f) ∘ s^{q} = s, (T̂ ŵϑ)^{∘} = (Ẑ^{ϑ})^{∘},

where w ∈ W_{o} arises from f as above.

Remark 4.1.1. If Ĝ is semisimple, then the ellipticity condition on φ is that T̂ ŵϑ be finite. In this case, the map T̂ ⟶ T̂ given by t ↦ t^{−1}ŵϑ(t) has finite fibers, hence is surjective. Hence, if we conjugate ϑ̂f by elements of T̂, we can change f to any other representative of ŵ. This means the T̂-conjugacy class of ϑ̂f is determined by the image ŵ of f in Ŵ_{o}, so the Ĝ-conjugacy classes of TRSELPs are in bijection with Ŵ_{o}-conjugacy classes of pairs (s, ŵ), where s : I_{t} ⟶ T̂ is continuous, with CĜ(s) = T̂, and ŵ ∈ Ŵ_{o} satisfies

ŵϑ ∘ s^{q} = s,  T̂ ŵϑ is finite.

4.2. Outline of the construction. Suppose we have a TRSELP φ, with s, f, ŵ as above. Recall from Section 2.7 that X_{w} denotes the preimage in X of [X/(1 − wϑ)X]_{tor}. For λ ∈ X_{w}, let ρ_{λ} be as in (15). In Section 2.7 we associated to λ a cocycle u_{λ} whose class in H^{1}(F, G) is ω_{ρ_{λ}}. The twisted Frobenius F_{λ} = F_{u_{λ}} stabilizes a facet J_{λ} ⊂ A with corresponding parahoric subgroup G_{λ}. Ellipticity will imply that the facet J_{λ} is in fact a minimal F_{λ}-stable facet in A, so GF_{λ} ; λ is a maximal parahoric subgroup of GF_{λ}.

To (φ, λ) we will further associate an F_{λ}-minisotropic torus T_{λ}, a depth-zero character χ_{λ} of T_{λ}, whence an irreducible cuspidal representation κ_{λ}^{0} of GF_{λ} ; λ := (G_{λ}/G_{λ}^{+})F_{λ} (viewed as a representation of GF_{λ} ; λ), via the Deligne-Lusztig construction. In fact, χ_{λ} will define an extension κ_{λ} of κ_{λ}^{0} to Z^{F}GF_{λ} ; λ such that the smoothly-induced representation

π_{λ} := Ind_{Z^{F}G{F \atop λ} ; λ}^{G{F \atop λ}} κ_{λ}

is irreducible. Here Z denotes the group of K-rational points of the maximal k-split torus in the center of G. An exercise shows that the functions in π_{λ} necessarily have compact support modulo Z^{F}, so we could just as well define π_{λ} using the compact induction functor ind.

In our construction, u_{λ} and J_{λ} are not uniquely defined, but the G-orbit [u_{λ}, π_{λ}] ∈ R^{2}(F, G) will be independent of the choices of u_{λ} and J_{λ}. Moreover, for λ, μ ∈ X_{w}, we will have

[u_{λ}, π_{λ}] = [u_{μ}, π_{μ}] ⇔ ρ_{λ} = ρ_{μ}.

Thus, to (φ, ρ) we will associate the G-orbit [u_{λ}, π_{λ}] ∈ R^{2}(F, G, ω_{ρ}), where λ ∈ X_{w} is any character of T̂ restricting to ρ.

The L-packets thus defined are the “natural” ones: All choices involved in the construction are rendered equivalent by taking G-orbits. However, to make the stability calculations, we need representations of a fixed group. Using Section 2.8, we will choose a representative (u, π) in each G-orbit [u_{λ}, π_{λ}], so as to have all representations in the “unnatural” L-packet living on the single group GF_{u}.

4.3. Depth zero characters of unramified tori. Recall that X = X_{∗}(T), Y = X^{∗}(T). Let σ ∈ Aut(X) be an automorphism of X of order n, and let F_{σ} = σ⊗Frob^{−1} be the corresponding twisted Frobenius of both T = X ⊗ K^{×} and T = X ⊗ 𝔉^{×}. (Recall that Frob^{−1} is the q-power map on 𝔉.) Let 𝔣_{n} be the degree n extension of 𝔣 contained in 𝔉. Since σ has order n, the torus T with Frobenius F_{σ} splits over 𝔣_{n}, and T^{F^{n}^{σ}} = X ⊗ 𝔣_{n}^{×}.

Given automorphisms α, β of abelian groups A, B, respectively, let

Hom_{α,β}(A, B)

denote the set of homomorphisms f : A ⟶ B such that f ∘ α = β ∘ f.

We have an exact sequence

1 ⟶ TF_{σ} ⟶ T^{F^{n}^{σ}} ⟶^{1−F_{σ}} T^{F^{n}^{σ}} ⟶^{N_{σ}} TF_{σ} ⟶ 1,

where N_{σ}(t) = t F_{σ}(t) F_{σ}^{2}(t) ⋯ F_{σ}^{n−1}(t). So N_{σ} induces an isomorphism

Hom(TF_{σ}, ℂ^{×}) ⟶^{∼} Hom_{F_{σ},Id}(T^{F^{n}^{σ}}, ℂ^{×}) = Hom_{F_{σ},Id}(X ⊗ 𝔣_{n}^{×}, ℂ^{×}).

There is also an isomorphism

Hom(𝔣_{n}^{×}, T̂) ⟶^{∼} Hom(X ⊗ 𝔣_{n}^{×}, ℂ^{×}), s ↦ χ_{s},

where χ_{s}(λ ⊗ a) = λ(s(a)), for λ ∈ X, a ∈ 𝔣_{n}^{×}. One checks that

χ_{s} ∈ Hom_{F_{σ},Id}(X ⊗ 𝔣_{n}^{×}, ℂ^{×}) ⇔ σ̂ ∘ s = s ∘ Frob,

where σ̂ ∈ Aut(Y) is dual to σ. (The action of σ̂ on T̂ is such that σ · λ = λ ∘ σ̂ for all λ ∈ X.) Hence s ↦ χ_{s} is an isomorphism

Hom_{Frob,σ̂}(𝔣_{n}^{×}, T̂) ⟶^{∼} Hom_{F_{σ},Id}(X ⊗ 𝔣_{n}^{×}, ℂ^{×}).

The tame inertia group I_{t} is identified with the projective limit

I_{t} = lim_{m^{←}} 𝔣_{m}^{×},

with respect to the norm mappings on the finite fields 𝔣_{m}. The canonical projection

I_{t} ⟶ 𝔣_{m}^{×}

induces an isomorphism as Frob-modules

I_{t}/(1 − Ad Frob^{m})I_{t} ⟶^{∼} 𝔣_{m}^{×}.

Since σ̂ has order n, any s ∈ Hom_{Ad Frob,σ̂}(I_{t}, T̂) is trivial on (1 − Ad Frob^{n})I_{t}. It follows that

Hom_{Frob,σ̂}(𝔣_{n}^{×}, T̂) ≃ Hom_{Ad Frob,σ̂}(I_{t}, T̂).

Thus the map s ↦ χ_{s} is a canonical bijection

Hom_{Ad Frob,σ̂}(I_{t}, T̂) ⟶^{∼} Hom(TF_{σ}, ℂ^{×}).

Now s∘Ad Frob = σ̂ ∘s iff for some (equivalently, any) τ ∈ T̂, the assignment Frob ↦ σ̂ ⋉τ extends s to a homomorphism

φ : W_{t} ⟶ ^{L}T_{σ},

where ^{L}T_{σ} = ⟨σ̂⟩ ⋉ T̂ is the L-group of the torus T with Frobenius F_{σ}. The T̂-conjugacy class of the extension φ is uniquely determined by the image of τ in T̂/(1 − σ̂)T̂. The latter group is identified with the character group of X^{σ}, whereby τ corresponds to

χ_{τ} ∈ Hom(X^{σ}, ℂ^{×}), χ_{τ}(λ) = λ(τ).

Our choice of uniformizer in k gives an isomorphism

T F_{σ} ≃ ^{0}T F_{σ} × X^{σ},

where ^{0}T is the group of R_{K}-points of T. Hence the above isomorphisms give a canonical bijection between T̂-conjugacy classes of admissible homomorphisms φ : W_{t} ⟶ ^{L}T_{σ} and depth-zero characters

χ_{φ} := χ_{s} ⊗ χ_{τ} ∈ Irr(T F_{σ}),

where s = φ|_{I_{t}} and φ(Frob) = σ̂ ⋉ τ. This bijection has the following naturality property.

Lemma 4.3.1. Let α be an algebraic automorphism of T commuting with F_{σ}, so α ∈ Aut(X) and α̂ ∈ Aut(Y). Then χ_{φ} ∘ α = χ_{α̂∘φ}.

Proof. We check it first on X^{σ}. Since χ_{φ}(μ) = μ(τ), for μ ∈ X^{σ}, we have

χ_{φ}(α · μ) = (α · μ)(τ) = μ(α̂ · τ) = χ_{α̂∘φ}(μ).

Now on TF_{σ} we have χ_{φ} = χ_{s}, where χ_{s} ∈ Hom_{F_{σ},Id}(X ⊗ 𝔣_{n}^{×}, ℂ^{×}). For λ ∈ X, a ∈ 𝔣_{n}^{×}, we have

χ_{α̂∘φ}(λ⊗a) = χ_{α̂∘s}(λ⊗a) = λ(α̂·(s(a))) = (α·λ)(s(a)) = (χ_{s} ∘α)(λ⊗a) = (χ_{φ} ∘α)(λ⊗a).

□

Now let φ : W_{t} → ^{L}G be a TRSELP, with associated w ∈ W_{o} and set σ = wϑ. We want to construct from φ a T̂-conjugacy class of Langlands parameters

φ_{T} : W_{t} ⟶ ^{L}T_{σ},

such that φ_{T} = φ on I, and such that φ_{T} (Frob) and φ(Frob) have the same action on T̂. We will have

φ_{T} (Frob) = σ̂ ⋉ τ

for some τ ∈ T̂, which is only defined up to σ̂-twisted conjugacy. That is, we need only define the coset of τ in T̂/(1 − σ̂)T̂.

We define the coset of τ as follows. Let Ĝ′ be the derived group of Ĝ, and let T̂′ = T̂ ∩ Ĝ′. Ellipticity implies that the map τ ↦ τσ̂(τ)^{−1} has finite kernel on T̂′, which means that

(1 − σ̂)T̂′ = T̂′,

so the inclusion T̂ ↪ Ĝ induces a bijection

T̂/(1 − σ̂)T̂′ ⟶^{∼} Ĝ/Ĝ′ =: Ĝ_{ab}.

It follows that T̂ ↪ Ĝ induces a bijection

(16)  T̂/(1 − σ̂)T̂ ⟶^{∼} Ĝ_{ab}/(1 − ϑ̂)Ĝ_{ab}

between the the set of σ̂-twisted conjugacy classes in T̂ and the set of ϑ̂-twisted conjugacy classes in the abelianization Ĝ_{ab}. Now, if φ(Frob) = ϑ̂ ⋉ f, we take any τ ∈ T̂ whose class in T̂/(1 − σ̂)T̂ corresponds under (16) to the image of f in Ĝ_{ab}/(1 − ϑ̂)Ĝ_{ab}.

Hence, from the TRSELP φ we get a character χ_{φ_{T}} ∈ Irr(T F_{σ}). We will abuse notation slightly and again denote this character by χ_{φ}.

4.4. From tame parameters to depth-zero types. Let φ : W_{t} ⟶ ^{L}G be a TRSELP with φ(Frob) = ϑ̂f as in Section 4.1. Let w ∈ W_{o} be the element such that ŵ is the image of f in Ŵ_{o}. Since φ is elliptic, we have

(17)  X^{wϑ} = X_{∗}(Z^{∘})^{ϑ},  X_{ad}^{wϑ} = {0},

where Z^{∘} is the identity component of the center Z of G.

Let λ ∈ X_{w}, and set

σ_{λ} = t_{λ}wϑ ∈ W ⋊ ⟨ϑ⟩,

as in Section 2.7. By the second equation in (17), the operator I − wϑ acts invertibly on A_{ad}, so σ_{λ} has a unique fixed-point x_{λ} ∈ A_{ad}, given by

x_{λ} = (I − wϑ)^{−1}t_{jλ} · o.

Let x̃_{λ} be the pre-image of x_{λ} in A^{σ_{λ}}.

The facet J_{λ} from Section 2.7 is the unique facet in A containing x̃_{λ}. As in Section 2.7, we choose an alcove C_{λ} in A containing J_{λ} in its closure, and write

G_{λ} := G_{J_{λ}},  W_{λ} := [N ∩ G_{λ}]/^{0}T ,  G_{λ} := G_{λ}/G_{λ}^{+}.

We choose u_{λ} ∈ Z^{1}(F, N) as in Lemma 2.7.2, and define

F_{λ} := Ad(u_{λ}) ∘ F .

Then F_{λ} is a Frobenius endomorphism of G for some k-rational structure on G which is inner to the quasi-split structure on G given by F. Recall also that F_{λ} stabilizes the apartment A, the alcove C_{λ}, and the facet J_{λ}.

Lemma 4.4.1. We have

A^{σ_{λ}} = J_{λ}^{σ_{λ}} = J F_{λ} ; λ = x̃_{λ}.

In particular, the point x_{λ} is a vertex in B(G_{ad})F_{λ}.

Proof. From (7) of Section 2.7 we may decompose σ_{λ} in two ways:

σ_{λ} = t_{λ}wϑ = w_{λ}y_{λ}ϑ.

Since w_{λ} fixes J_{λ} pointwise, we have

J F_{λ} ; λ = J_{λ}^{y_{λ}ϑ} = J_{λ}^{σ_{λ}} = x̃_{λ}.

Also, A^{σ_{λ}} = x̃_{λ} ⊆ J_{λ}, implying that A^{σ_{λ}} = J_{λ}^{σ_{λ}}. □

Since F_{λ} ·J_{λ} = J_{λ}, F_{λ} induces a Frobenius endomorphism of G_{λ}, preserving T. Since F_{λ} ·C_{λ} = C_{λ}, the Frobenius F_{λ} also preserves a Borel subgroup of G_{λ} containing T. It follows [6, 20.6] that T is a maximally 𝔣-split torus in G_{λ} with respect to F_{λ}.

From Section 2.7 we have the alternative expression

σ_{λ} = w_{λ}y_{λ}ϑ,

where w_{λ} ∈ W_{λ} and y_{λ} is the image of u_{λ} in W. Moreover, our fixed choice of lift ẇ of w defines a lift ẇ_{λ} ∈ N ∩ G_{λ} of w_{λ}, via the equation

t_{λ}ẇ = ẇ_{λ}u_{λ}.

Recall we can then choose an element p_{λ} ∈ G_{λ} such that

p_{λ}^{−1} F_{λ}(p_{λ}) = ẇ_{λ}.

Note that

F_{λ} ∘ Ad(p_{λ}) = Ad(p_{λ}) ∘ Ad(ẇ_{λ}u_{λ}) ∘ F .

Define

T_{λ} := Ad(p_{λ})T.

Then T_{λ} is an F_{λ}-stable unramified torus in G. On T, we have Ad(ẇ_{λ}u_{λ}) = Ad(w), so Ad(p_{λ}) : T ⟶ T_{λ} satisfies

F_{λ} ∘ Ad(p_{λ}) = Ad(p_{λ}) ∘ F_{w},

where F_{w} = Ad(ẇ) ∘ F.

By ellipticity, we have

T F_{w} = X^{wϑ} ×^{0}T F_{w} = X_{∗}(Z^{∘})^{ϑ} ×^{0}T F_{w} = Z^{F} ·^{0}T F_{w}.

This implies that T_{λ} is F_{λ}-minisotropic. Moreover, we have^{0}T_{λ} = T_{λ} ∩ G_{λ}, and^{0}T_{λ} projects to an F_{λ}-minisotropic maximal torus T_{λ} in G_{λ}.

On A and A_{ad} we have Ad(ẇ_{λ}u_{λ}) F = σ_{λ}. By Lemma 4.4.1 the unique fixed-point of T_{λ} in

B(G_{ad})F_{λ} is

[p_{λ} · A_{ad}]F_{λ} = p_{λ} · A_{ad}^{σ_{λ}} = p_{λ} · x_{λ} = x_{λ}.

As in Section 4.3, we have a depth-zero character χ = χ_{φ} of T F_{w}. Since φ is in general position, Lemma 4.3.1 implies that χ is F_{w}-regular.

This character χ transports to a depth-zero F_{λ}-regular character

χ_{λ} := Ad(p_{λ})_{∗}χ ∈ Irr(T F_{λ} ; λ).

The restriction of χ_{λ} to ^{0}T F_{λ} ; λ factors through a character χ_{λ}^{0} ∈ Irr(TF_{λ} ; λ), which is in “general position” with respect to F_{λ}, in the sense of [20, 5.16]. By [20, 8.3], Deligne-Lusztig induction then gives an irreducible cuspidal representation

κ_{λ}^{0} := ϵ(G_{λ}, T_{λ}) · R_{T_{λ},χ_{λ}^{0}}^{G_{λ}} ∈ Irr(GF_{λ} ; λ).

Inflate κ_{λ}^{0} to a representation of GF_{λ} ; λ and define an extension to Z^{F}GF_{λ} ; λ by

κ_{λ} := χ_{λ} ⊗ κ_{λ}^{0}.

This makes sense since (Z ∩ G_{λ})F_{λ} acts on κ_{λ}^{0} via the restriction of the scalar character χ_{λ}^{0}.

So far, to the TRSELP φ and λ ∈ X_{w}, we have associated a Frobenius F_{λ}, an F_{λ}-stable parahoric subgroup G_{λ}, and an irreducible representation κ_{λ} of Z^{F}GF_{λ} ; λ. In the process we made choices of ẇ, C_{λ}, u_{λ}, p_{λ}.

Lemma 4.4.2. Given a TRSELP φ and λ ∈ X_{w}, both fixed, suppose we make two sets of choices (ẇ, C_{λ}, u_{λ}, p_{λ}) and (ẇ′, C′_{λ}, u′_{λ}, p′_{λ}) as above, giving rise to (F_{λ}, T_{λ}, χ_{λ}, κ_{λ}) and (F′_{λ}, T′_{λ}, χ′_{λ}, κ′_{λ}) as above. Then there is h ∈ G_{λ} such that

(1) h ∗ u′_{λ} = u_{λ};

(2) Ad(h)_{∗}(T′_{λ}, χ′_{λ}, κ′_{λ}) = (T_{λ}, χ_{λ}, κ_{λ}).

Proof. Note that (1) implies that Ad(h)(G_{λ}^{F′_{λ}}) = GF_{λ} ; λ, so (2) makes sense. Since σ_{λ} is defined before the choices are made, we have

w_{λ}y_{λ} = t_{λ}w = w′_{λ}y′_{λ},

so there is t ∈ ^{0}T such that

ẇ_{λ}u_{λ} = tẇ′_{λ}u′_{λ}.

Here, both sides belong to Z^{1}(F, N) and act on T via w. Lemma 2.1.1 implies that t ∈ Z^{1}(F_{w},^{0}T). By Lemma 2.3.1 for^{0}T, there is s ∈ ^{0}T such that

(18)  s F_{w}(s)^{−1} = t.

Since Ad(w) = Ad(ẇ′_{λ}u′_{λ}) on T, equation (18) can be written

(19)  sẇ′_{λ}u′_{λ} = tẇ′_{λ}u′_{λ} F(s) = ẇ_{λ}u_{λ} F(s).

Recall our equations characterizing p_{λ} and p′_{λ}:

(20)  p_{λ}^{−1} F_{λ}(p_{λ}) = ẇ_{λ},  p′_{λ}^{−1} F′_{λ}(p′_{λ}) = ẇ′_{λ}.

These allow us to write (19) in the form

(21)  s · p′_{λ}^{−1}u′_{λ} F(p′_{λ}) = p_{λ}^{−1}u_{λ} F(p_{λ}) · F(s).

Equation (21) shows that the element h := p_{λ}sp′_{λ}^{−1} satisfies h ∗ u′_{λ} = u_{λ}. We have h ∈ G_{λ} since p_{λ}, s, p′_{λ} are all in G_{λ}. It is clear that Ad(h)(T′_{λ}, χ′_{λ}) = (T_{λ}, χ_{λ}), which then implies that Ad(h)_{∗}κ′_{λ} = κ_{λ}. □

4.5. Definition of the L-packets. Given a TRSELP φ, an element λ ∈ X_{w} and a set of choices (C_{λ}, u_{λ}, p_{λ}), define

π_{λ} := Ind_{Z^{F}G{F \atop λ} ; λ}^{G{F \atop λ}} κ_{λ},

where Ind denotes smooth induction. (The functions in π_{λ} automatically have compact support modulo Z^{F}.) In this notation we have suppressed the choices (C_{λ}, u_{λ}, p_{λ}), but by Lemma 4.4.2 the G-orbit (in fact the G_{λ}-orbit) of (u_{λ}, π_{λ}) is independent of these choices.

Lemma 4.5.1. The representation π_{λ} of GF_{λ} is irreducible supercuspidal.

Proof. By [44, 6.6] it suffices to show that κ_{λ} induces irreducibly to the group

(G_{λ}^{⋆})F_{λ} = {g ∈ GF_{λ} : g · J_{λ} = J_{λ}},

which is the normalizer of GF_{λ} ; λ in GF_{λ}. For this, it is enough to show the stabilizer of κ_{λ} in (G_{λ}^{⋆})F_{λ} is just ZF_{λ}GF_{λ} ; λ.

Suppose h ∈ (G_{λ}^{⋆})F_{λ} and Ad(h)_{∗}κ_{λ} = κ_{λ}. By [20, Thm. 6.8], there is g ∈ GF_{λ} ; λ such that

Ad(gh)_{∗}(T_{λ}, χ_{λ}) = (T_{λ}, χ_{λ}).

Then by [14] there is ℓ ∈ (G_{λ}^{+})F_{λ} such that

Ad(ℓgh)_{∗}(T_{λ}, χ_{λ}) = (T_{λ}, χ_{λ}).

That is, ℓgh ∈ N(G, T_{λ})F_{λ} and fixes χ_{λ}. Hence p_{λ}^{−1}ℓghp_{λ} ∈ N F_{w} and fixes χ. Let z be the projection of p_{λ}^{−1}ℓghp_{λ} to W_{o}. By Lemma 4.3.1 we have ẑ ∘ s = s, but CĜ(s) = T̂, so z = 1. It follows that ℓgh ∈ T F_{λ} ; λ ∩ (G_{λ}^{⋆})F_{λ} ⊂ ZF_{λ}GF_{λ} ; λ. Since ℓ and g are in GF_{λ} ; λ, this implies that

h ∈ ZF_{λ}GF_{λ} ; λ. □

At this point we have a supercuspidal representation π_{λ} ∈ Irr(GF_{λ}) for every λ ∈ X_{w}. We now show that the G-orbit [u_{λ}, π_{λ}] := Ad(G) · (u_{λ}, π_{λ}) depends only on the character ρ_{λ} ∈ Irr(C_{φ}) corresponding to the image of λ in [X/(1 − wϑ)X]_{tor} = Irr(C_{φ}) (see Section 4.1).

Lemma 4.5.2. Given φ, along with λ, μ ∈ X_{w}, make choices (C_{λ}, u_{λ}, p_{λ}), (C_{μ}, u_{μ}, p_{μ}) as above. Then ρ_{λ} = ρ_{μ} if and only if there exists g ∈ G such that

(1) g ∗ u_{λ} = u_{μ};

(2) g · J_{λ} = J_{μ};

(3) Ad(g)_{∗}κ_{λ} ≃ κ_{μ}.

Proof. Suppose ρ_{λ} = ρ_{μ}. This is equivalent to having μ = λ + (1 − wϑ)ν for some ν ∈ X, which amounts to the following equation in W ⋊ ⟨ϑ⟩:

t_{ν}σ_{λ}t_{ν}^{−1} = t_{ν}t_{λ}wϑt_{ν}^{−1} = t_{μ}wϑ = σ_{μ}.

Lifting to N, we have

(22)  t_{ν}ẇ_{λ}u_{λ} F(t_{ν})^{−1} = tẇ_{μ}u_{μ},

for some t ∈ ^{0}T. Arguing as in the proof of Lemma 4.4.2, there is s ∈ ^{0}T_{μ} such that

p_{μ}tp_{μ}^{−1} = s^{−1} F_{μ}(s).

Using Equations (20) we then find that

g ∗ u_{λ} = u_{μ},

where g = sp_{μ}t_{ν}p_{λ}^{−1}.

Since σ_{λ} and σ_{μ} have unique fixed-points x_{λ} and x_{μ} in A_{ad}, we must have t_{ν} · x_{λ} = x_{μ}, hence t_{ν} · J_{λ} = J_{μ}, from which 2 is immediate.

Finally, we have

Ad(g)_{∗}(T_{λ}, χ_{λ}) = Ad(sp_{μ}t_{ν})_{∗}(T, χ) = Ad(s)_{∗}(T_{μ}, χ_{μ}),

so Ad(g)_{∗}κ_{λ} ≃ κ_{μ}.

Turning to the converse, suppose we have g ∈ G satisfying items 1-3 above. By 2 and 3 and [20, Thm. 6.8], the pairs

(Ad(g)T_{λ}, Ad(g)_{∗}χ_{λ}), (T_{μ}, χ_{μ})

are conjugate in GF_{μ} ; μ, so without loss of generality, we may assume these two pairs are equal. Then, the element n := p_{μ}^{−1}gp_{λ} belongs to N. By 1, Ad(n) preserves T F_{w}, and it preserves the F_{w}-regular character χ, since Ad(g)_{∗}χ_{λ} = χ_{μ}. It follows that n ∈ T. Let t_{ν} be the image of n in W. As in the first paragraph of the proof, it suffices to prove that t_{ν}σ_{λ}t_{ν}^{−1} = σ_{μ}. But this follows from the equation

Ad(n) ∘ Ad(ẇ_{λ}u_{λ}) ∘ F ∘ Ad(n)^{−1} = Ad(ẇ_{μ}u_{μ}) ∘ F,

which is proved using Equations (20) as before. □

Now we have our first main result.

Theorem 4.5.3. Given a TRSELP φ with associated w ∈ W_{o}, let r : X_{w} → H^{1}(F, G) be as in section 2.8. For each ω ∈ H^{1}(F, G) define

Π(φ, ω) := {[u_{λ}, π_{λ}] : λ ∈ r^{−1}(ω)}.

Then we have a well-defined bijection Irr(C_{φ}, ω) →^{∼} Π(φ, ω), as follows. Given ρ ∈ Irr(C_{φ}, ω), choose any λ ∈ r^{−1}(ω) such that ρ_{λ} = ρ, and associate to ρ the G-orbit [u_{λ}, π_{λ}] ∈ Π(φ, ω).

Proof. Recall that r(λ) = ω if and only if ρ_{λ} ∈ Irr(φ, ω). Suppose we have λ, μ ∈ X_{w} such that ρ_{λ}, ρ_{μ} ∈ Irr(φ, ω). From [44, 6.2] it follows that conditions 1-3 of Lemma 4.5.2 are equivalent to having g ∈ G such that

Ad(g) · (u_{λ}, π_{λ}) = (u_{μ}, π_{μ}).

So we have proved that

[u_{λ}, π_{λ}] = [u_{μ}, π_{μ}] ⇔ ρ_{λ} = ρ_{μ},

as desired. □ Remark 4.5.4. Recall that Irr(C_{φ}, ω) is equal to the fiber over ω under the composition

Irr(C_{φ}) →^{∼} [X/(1 − wϑ)X]_{tor} → [X̄/(1 − ϑ)X̄]_{tor} →^{∼} H^{1}(F, G),

whereby ρ = ρ_{λ} ↦ r(λ). By Lemma 2.7.3 we have u_{λ} ∈ ω_{λ} = r(λ) = ω. Hence our representation π_{λ} lives on an inner twist of G belonging to the class ω ∈ H^{1}(F, G), in accordance with the conjectures in Section 3. 4.6. Choosing representatives in an L-packet. We now use Section 2.8 to choose representatives, living on a single group, of each G-orbit in an L-packet Π(φ, ω). We fix u ∈ ω ∩ N, and for each λ ∈ r^{−1}(ω) we choose m_{λ} as in Section 2.8. For each ρ ∈ Irr(C_{φ}, ω), define

π_{u}(φ, ρ) := Ad(m_{λ})_{∗}π_{λ} ∈ Irr(GF_{u}),

for any λ ∈ r^{−1}(ω) such that ρ_{λ} = ρ. We have seen that the isomorphism class of π_{λ} is independent of the choice of λ. Two choices of m_{λ} differ by an element of GF_{u}, so the isomorphism class of π_{u}(φ, ρ_{λ}) likewise does not depend on the choice of m_{λ}. The normalized L-packet is then defined as

Π_{u}(φ) := {π_{u}(φ, ρ) : ρ ∈ Irr(C_{φ}, ω)}.

More explicitly, the representation π_{u}(φ, ρ) is given as follows. Recall that m_{λ} ·C_{λ} is our fixed F_{u}-stable alcove C. The facet I_{λ} := m_{λ} · J_{λ} is contained in C̄, and is likewise F_{u}-stable. The F_{u}- minisotropic torus S_{λ} = Ad(m_{λ})T_{λ} = Ad(q_{λ})T (see Section 2.8) has the property that S_{λ} ∩ G_{I_{λ}} projects to an F_{u}-minisotropic torus S_{λ} in G_{I_{λ}}. The character θ_{λ} := Ad(m_{λ})_{∗}χ_{λ} = Ad(q_{λ})_{∗}χ is F_{u}-regular, and gives a(n inflated) Deligne-Lusztig representation

κ_{λ}^{0} := ε(G_{I_{λ}}, S_{λ}) · R_{S_{λ},θ_{λ}}^{G_{I_{λ}}} ∈ Irr(GF_{u} ; I_{λ}),

and an extension of κ_{λ}^{0} to a representation κ_{λ} of Z^{F} GF_{u} ; I_{λ}. Finally, we have

π_{u}(φ, ρ) = Ind_{Z^{F} G{F \atop u} ; I_{λ}}^{G{F \atop u}} κ_{λ}.

5. Normalizations of measures and formal degrees

We now move toward Harmonic Analysis. The first step is a uniform normalization of Haar measures on groups of the form G^{F}, where G = G(K) and G is a connected reductive k-group, split over K. We then verify the equality of formal degrees in an L-packet, according to the conjectures in Section 3. (Note that the group C_{φ} is abelian for these L-packets.) Except where noted, our Frobenius on G is now unspecified, and is denoted by F, according to our conventions.

5.1. Haar measure. We denote the Lie algebra of G by 𝔤, and again let F denote the induced Frobenius action on 𝔤.

Suppose x ∈ B(G) or B(G_{ad}). Just as we could attach a parahoric G_{x} and its pro-unipotent radical G_{x}^{+} to x, so we can define lattices 𝔤_{x} and 𝔤_{x}^{+} in 𝔤 (see [43, §3.2], [3, §2.2], where the corresponding objects are called 𝔤_{x,0} and 𝔤_{x,0^{+}}). As before, the lattices 𝔤_{x} and 𝔤_{x}^{+} are independent of the facet to which x belongs. If J is any subset of a facet and x ∈ J, then we set 𝔤_{J} = 𝔤_{x} and 𝔤_{J}^{+} = 𝔤_{x}^{+}. If J is an F -stable subset of a facet, then

L_{J} := 𝔤_{J}/𝔤_{J}^{+}

is the Lie algebra of G_{J}, and we have

L_{J}^{F} = 𝔤_{J}^{F} /(𝔤_{J}^{+F}).

Let dg denote the Haar measure on G^{F}, normalized so that

meas_{dg}(G_{J}^{F}) = \frac{|G_{J}^{F} |}{|L_{J}^{F} |^{1/2}}

for one, in fact every, F -stable facet J in B(G).

Let dX denote the Haar measure on 𝔤^{F}, normalized so that

meas_{dX}(𝔤_{J}^{F}) = |L_{J}^{F} |^{1/2}

for one (in fact, every) F -stable facet J in B(G).

To show that these normalizations are independent of the choice of J as claimed, it is enough to show that if J and J′ are F -stable facets in B(G) with J′ ⊂ J̄, then

meas_{dg}(G_{J}^{F}) = \frac{|G_{J}^{F} |}{|L_{J}^{F} |^{1/2}}

implies

meas_{dg}(G_{J′}^{F}) = \frac{|G_{J′}^{F}|}{|L_{J′}^{F}|^{1/2}}

(and similarly for the measure dX on 𝔤). Since J′ ⊂ J̄, we have

G_{J′}^{+} ⊂ G_{J}^{+} ⊂ G_{J} ⊂ G_{J′}.

Moreover, the image of G_{J} in G_{J′} is a parabolic 𝔣-subgroup with unipotent radical G_{J}^{+}/G_{J′}^{+} and Levi component isomorphic to G_{J}. A short calculation gives the desired result.

Remark 5.1.1. The above expression for meas_{dg}(G_{J}^{F}) can be simplified a bit. Let G be a connected reductive group over 𝔣 with Frobenius F. Let T ⊂ B be an F -stable maximal torus and an F -stable Borel subgroup in G. Then

|G^{F} | = [G^{F} : B^{F}] · |B^{F} | = q^{ν}[G^{F} : B^{F}] · |T^{F} |,

where ν is the number of (absolute) roots of T in B. The latter two factors are prime to p, so

|G^{F} |_{p′} = [G^{F} : B^{F}] · |T^{F} |,

where | · |_{p′} is the largest factor of | · | which is prime to p. We have dim G = dim T + 2ν. It follows that

meas_{dg}(G_{J}^{F}) = q^{− rk(G)/2}|G_{J}^{F} |_{p′}.

where rk(G) is the absolute rank of G.

This normalization applies as well to the largest k-split torus Z of the center of G, and gives

meas_{dz}(^{0}Z^{F}) = q^{− rk(Z)/2}|Z^{F} | = (q^{1/2} − q^{−1/2})^{rk(Z)},

where Z = ^{0}Z/^{0}Z^{+}.

For any irreducible admissible representation π of G^{F} which is square-integrable modulo Z^{F}, let Deg(π) denote the formal degree of π with respect to the quotient measure dg/dz on G^{F} /Z^{F} (c.f. [26]).

5.2. Formal degree of the Steinberg representations. The formal degree conjectures in Section 3 require Haar measures for which the formal degree of the Steinberg representation of G^{F} is unchanged by inner twists of F, for F = F_{u}. In this section we show that the measures dg defined above have this property. First we consider some constants arising in this formal degree.

Recall that the quasi-split Frobenius F acts on X = X_{∗}(T) by the automorphism ϑ, and that Z denotes the largest k-split torus in the center of G. Note that G/Z = (G/Z)^{I}.

Let X_{1} = X_{∗}(T/Z) and let C_{1} be the projection to the apartment of T/Z in B(G/Z) of the ϑ-stable alcove C in A. Let Ω_{1} be the stabilizer of C_{1} in the affine Weyl group of T/Z in G/Z. The inclusion X_{∗}(Z) ↪ X projects to an embedding

X_{∗}(Z) ↪ (X/X^{∘})^{ϑ} ≃ Ω_{C}^{ϑ},

where X^{∘} is the co-root lattice of T. Identifying, as we may, X^{∘} with the co-root lattice of T/Z, we have

Ω_{1} ≃ X_{1}/X^{∘} ≃ Ω_{C}/X_{∗}(Z),

and a finite subgroup Ω_{2} := Ω_{C}^{ϑ}/X_{∗}(Z) ↪ Ω_{1} fitting into the exact sequence

1 ⟶ Ω_{2} ⟶ Ω_{1} ⟶^{1−ϑ} Ω_{1} ⟶ Ω_{1}/(1 − ϑ)Ω_{1} ⟶ 1,

showing that

(23)  |Ω_{2}| = |Ω_{1}/(1 − ϑ)Ω_{1}| = |H^{1}(F, G/Z)|.

Now take a cocycle u ∈ Z^{1}(F, N_{C}), with corresponding twist F_{u} = Ad(u)∘F as before. Since u ∈ N_{C} and Ω_{C} is abelian, we have Ω_{C}^{uϑ} = Ω_{C}^{ϑ}. It follows that Ω_{2} is unchanged if we replace ϑ by an inner twist uϑ. Of course this also follows from (23).

Next, let V_{1} = X_{1} ⊗ℂ, let R be the graded ℂ-algebra of W_{o}-invariant polynomial functions on the ℂ-vector space V_{1}, and let 𝔪 be the maximal ideal in R of functions vanishing at 0 ∈ V_{1}. Then V := 𝔪/𝔪^{2} is a vector space of dimension ℓ := dim V_{1}. The space V inherits a grading from R, written V = ⊕V (d). Moreover, ϑ acts naturally on R and V, preserving the grading. Choose a basis of eigenvectors for ϑ in each V (d) and let f_{1}, …, f_{ℓ} be the collection of eigenvectors obtained. Let d_{j} = deg(f_{j}) and let ϵ_{j} be the eigenvalue of ϑ on f_{j}.

Define the constant

c(G/Z) := |Ω_{2}| · ∏_{i=1}^{ℓ} \frac{q^{d_{i}} − ϵ_{i}}{q^{d_{i}−1} − ϵ_{i}} .

The denominators in c(G/Z) are nonzero because each ϵ_{i} is a root of unity and V (1)^{ϑ} = {0}. Since u acts trivially on R and |Ω_{2}| is invariant under inner-twists, it follows that c(G/Z) is invariant under inner-twists.

Let G_{C_{1}} be the Iwahori subgroup of G/Z at the alcove C_{1}. From [5, 5.3] and [55, 3.10], (see also [22, 5.5]) it follows that the formal degree of the Steinberg representation St_{u} of GF_{u} is given, using our normalizations in section 5.1, by

Deg(St_{u}) = \frac{|TF_{u}/ZF_{u}|}{c(G/Z)} · \frac{1}{meas_{dg/dz}(GF_{u} ; C_{1})}

= \frac{|GF_{u} ; C_{1}|_{p′}}{c(G/Z)} · \frac{q^{rk(G/Z)/2}}{|GF_{u} ; C_{1}|_{p′}}

= \frac{q^{rk(G/Z)/2}}{c(G/Z)} .

This last expression is independent of u, as claimed.

5.3. Formal degrees in our L-packets. Now suppose π is an irreducible cuspidal representation of G^{F} of the sort considered in 4.5, namely π = Ind_{G_{J}^{F} Z^{F}}^{G^{F}} κ, for some minimal F -stable facet J ⊂ B(G) and κ ∈ Irr(G_{J}^{F} Z^{F}). The formal degree of π is given by

Deg(π) = dim κ · \frac{meas_{dz}(^{0}Z^{F})}{meas_{dg}(G_{J}^{F})}.

Recall also that κ is of the following form. We have an F -minisotropic torus S < G such that A(S)^{F} = J^{F}, a regular character θ ∈ Irr(S^{F}) whose restriction to S ∩ G_{J}^{F} factors through S^{F} = S ∩ G^{F} /S ∩ G_{J}^{+F}, and on G_{J}^{F} we have

κ = ε(G_{J}, S) · R_{S,θ}^{G_{J}}.

By [20, Thm. 7.1] we have

dim κ = \frac{|G_{J}^{F} |_{p′}}{|S^{F} |} .

Using also Remark 5.1.1, we find that

Deg(π) = \frac{|Z^{F} |}{|S^{F} |}q^{rk(G/Z)/2}.

Now if F = F_{u} and π = π_{u}(φ, ρ) as in 4.6, then the torus S is k-isomorphic to the platonic torus T with twisted Frobenius F_{w} (see 2.8). Therefore, we have

Deg(π_{u}(φ, ρ)) = \frac{q^{rk(G/Z)/2}}{|TF_{w}/Z^{F}|} .

The right side of this equation is independent of u and ρ, so all representations in an L-packet Π(φ) (see Section 4.5) have the same formal degree.

6. Generic Representations

In this section we determine the generic representations in our L-packets Π(φ). Only quasisplit groups have generic representations, so these can only occur in packets Π(φ, ω) for ω belonging to the kernel of the map j_{G} : H^{1}(F, G) → H^{1}(F, G_{ad}) induced by the adjoint map j : G → G_{ad}.

Let B be a Borel subgroup of G defined over k, and let U be the unipotent radical of B. We may and shall assume that B contains our fixed maximal torus T, which is the centralizer of a maximal k-split torus S.

A character ψ : U^{F} ⟶ ℂ^{×} is generic if ψ is nontrivial on each simple root group of S^{F} in U^{F}. A representation π ∈ Irr(G^{F}) is generic if Hom_{U^{F}}(π, ψ) ≠ 0, for some generic character ψ of U^{F}. We say that π is ψ-generic if we want to specify ψ.

If ω ∈ ker j_{G} and ρ ∈ Irr(C_{φ}, ω), we say the class π(φ, ρ) ∈ Π(φ, ω) is generic if some (equivalently, every) representation in π(φ, ρ) is generic.

Generic characters and representations for finite reductive groups are defined similarly.

6.1. Depth-zero generic characters and representations. This first section of this chapter concerns all generic depth-zero supercuspidal representations, not just those arising in our Lpackets.

Given a hyperspecial vertex x ∈ A_{ad}^{ϑ}, set U_{x} := U ∩ G_{x}, U_{x}^{+} := U ∩ G_{x}^{+}. The quotient U_{x} := U_{x}/U_{x}^{+} is the unipotent radical of an F-stable Borel subgroup of G_{x}. We say that a character ψ : U^{F} ⟶ ℂ^{×} has depth-zero at x if the restriction of ψ to U_{x}^{F} factors through a generic character ψ_{x} of U_{x}^{F}. Note that a depth-zero character at x is automatically generic for U^{F}, since x is hyperspecial. Moreover, any generic character ψ_{x} of U_{x}^{F} arises from some ψ having depth-zero at x (using, for example, [27, 24.12]).

Let κ^{∘} ∈ Irr(G_{x}^{F}) be the inflation of an irreducible cuspidal representation of G_{x}^{F}, and let κ be an extension of κ^{∘} to Z^{F}G_{x}^{F}. In this chapter, it is convenient to use the notation

(24)  π(x, κ) := ind_{Z^{F}G_{x}^{F}}^{G^{F}} κ

for the compactly induced representation of G^{F}. Since x is hyperspecial, the normalizer of G_{x}^{F} in G^{F} is Z^{F}G_{x}^{F}, so [44, 6.6] implies that π(x, κ) is an irreducible depth-zero supercuspidal representation of G^{F}.

Lemma 6.1.1. Let x ∈ A_{ad}^{ϑ} be a hyperspecial vertex, let ψ be a character of U^{F} having depthzero at x, and let ψ_{x} be the corresponding generic character of U_{x}^{F} as above. Assume that κ^{∘} is ψ_{x}-generic. Then π(x, κ) is ψ-generic.

Proof. This follows from Frobenius reciprocity: Let V ⊂ π(x, κ) be the space of functions supported on Z^{F}G_{x}^{F}U^{F}. Then V ≃ ind_{U_{x}^{F}}^{U^{F}} κ as representations of U^{F}, and V is a U^{F}-stable direct summand of π(x, κ). We have

0 ≠ Hom_{U_{x}^{F}}(κ, ψ_{x}) = Hom_{U^{F}}(ind_{U_{x}^{F}}^{U^{F}} κ, ψ)
= Hom_{U^{F}}(V, ψ)
⊂ Hom_{U^{F}}(π(x, κ), ψ).
□

The next result shows that all generic depth-zero supercuspidals are of the form π(x, κ) as constructed in (24) above.

Lemma 6.1.2. Let ψ be a generic character of U^{F}, and let π be an irreducible supercuspidal depth-zero ψ-generic representation of G^{F}. Then there is a hyperspecial vertex x ∈ A_{ad}^{ϑ} and a cuspidal representation κ^{∘} of G_{x}^{F} (which we inflate to a representation of G_{x}^{F}), such that the following hold.

(1) ψ has depth-zero at x, and κ^{∘} is a ψ_{x}-generic representation of G_{x}^{F}.

(2) There is an an extension of κ^{∘} to a representation κ of Z^{F}G_{x}^{F}, such that π ≃ π(x, κ).

Proof. From [44, 6.8] there is a vertex z ∈ A_{ad}^{ϑ}, a cuspidal representation κ_{z} of G_{z}^{F}, and a representation κ̇_{z} of the normalizer Ġ_{z}^{F} of G_{z}^{F} in G^{F} such that κ_{z} appears in κ̇_{z}|_{G_{z}^{F}} and π ≃

indG^{F} ; Ġ_{z}^{F} κ̇_{z}.

We may assume that z is contained in the closure of our fixed alcove jC^{ϑ} ⊂ A_{ad}^{ϑ}. Let Φ̃ be the set of affine roots of S in G. For any point y in the closure of jC^{ϑ}, we set

\begin{aligned}Φ̃_{y} & := \{α̃ ∈ Φ̃ : α̃(y) = 0\}, \\ Φ̃_{y}^{+} & := \{α̃ ∈ Φ̃_{y} : α̃|_{C} > 0\}. \end{aligned}

Then Φ̃_{y} is a spherical root system, and Φ̃_{y}^{+} is a set of positive roots in Φ̃_{y}. We let Π̃_{y} be the unique base of Φ̃_{y} contained in Φ̃_{y}^{+}.

Let Φ_{y}, Φ_{y}^{+}, Π_{y} be the respective sets of gradients of the affine roots in Φ̃_{y}, Φ̃_{y}^{+}, Π̃_{y}. Each of these sets lies in Φ_{o}, a set upon which W_{o}^{ϑ} acts. The roots in Π_{y} are non-divisible in Φ_{y}, and form a base of the reduced root system consisting of non-divisible roots in Φ_{y}.

Let

zW_{o}^{ϑ} := {w ∈ W_{o}^{ϑ} : w^{−1}Π_{z} ⊂ Φ_{o}^{+}}.

Since π is ψ-generic and is a quotient of ind_{G_{z}^{F}}^{G^{F}} κ_{z}, we have

Hom_{G^{F}}(ind_{G_{z}^{F}}^{G^{F}} κ_{z}, Ind_{U^{F}}^{G^{F}} ψ) ≠ 0.

As in the proof of [47, Lemma 4], this implies that there exists n ∈ N^{F} whose image v ∈ W_{o}^{ϑ} belongs to_{z}W_{o}^{ϑ} and such that n_{∗}ψ|_{G_{z}^{+}∩^{n}U^{F}} is trivial, while n_{∗}ψ|_{G_{z}∩^{n}U^{F}} appears in κ_{z}|_{G_{z}∩^{n}U^{F}}.

By [47, Lemma 2], the image V_{z}^{F} of G_{z} ∩ ^{n}U^{F} in G_{z}^{F} is the maximal unipotent subgroup of G_{z}^{F} generated by root groups U_{β}^{F} for β ∈ Π_{z}. Let θ be the character of V_{z}^{F} obtained from the restriction of^{n}ψ to G_{z} ∩^{n}U^{F}. We have seen that θ appears in κ_{z}|_{V_{z}^{F}}.

We claim that v^{−1}Π_{z} ⊂ Π_{o}. Suppose not, and choose β ∈ Π_{z} such that v^{−1}β ∈ Φ_{o}^{+} ∖ Π_{o}. Then the root group U_{v^{−1}β}^{F} is contained in the kernel of ψ, so θ is trivial on the simple root group U_{β}^{F} in V_{z}^{F}. This contradicts the cuspidality of κ_{z}. So v^{−1}Π_{z} ⊂ Π_{o}, hence in fact v^{−1}Π_{z} = Π_{o}, since |Π_{z}| = |Π_{o}| = dim A_{ad}^{ϑ}.

We have shown, moreover, that for each α ∈ Π_{o}, the character^{n}ψ is trivial on G_{z}^{+} ∩ U_{vα}^{F}, and nontrivial on G_{z} ∩ U_{vα}^{F}. Hence ψ is trivial on G_{n^{−1}·z}^{+} ∩ U_{α}^{F} and nontrivial on G_{n^{−1}·z} ∩ U_{α}^{F}.

It now suffices to prove that the vertex z is hyperspecial. For then the previous paragraph shows that ψ has depth-zero at x := n^{−1} · z, and taking κ^{∘} := Ad(n^{−1})_{∗}κ_{z}, κ = Ad(n^{−1})_{∗}κ̇_{z} will satisfy the conclusions of the lemma.

Since vΠ_{o} = Π_{z}, it is clear that z is special, but not immediately clear that it is hyperspecial. Let α ∈ Π_{o}. Since vα ∈ Π_{z}, there is k_{α} ∈ ℤ such that vα − k_{α} ∈ Π̃_{z}. It follows that

z = ∏_{α∈Π_{o}} t_{k_{α}vλ_{α}} · o,

where {λ_{α} : α ∈ Π_{o}} ⊂ X_{ad} is the dual basis of Π_{o}. Hence z = t · o, for an element t ∈ T_{ad}^{F}. Since Ad(t) is a k-rational automorphism of G^{F}, it follows that z is hyperspecial. □

6.2. Generic representations in our L-packets. Fix a TRSELP φ with corresponding w ∈ W_{o}. We identify

Irr(C_{φ}) = H^{1}(F_{w}, T) = [X/(1 − wϑ)X]_{tor}.

We likewise identify

H^{1}(F_{w}, T_{ad}) = X_{ad}/(1 − wϑ)X_{ad}.

(Note that the latter group is finite.) For λ ∈ X_{w}, let ρ_{λ} denote the image of λ in H^{1}(F_{w}, T), and ρ_{jλ} the image of jλ in H^{1}(F_{w}, T_{ad}). Then ρ_{jλ} = j_{w}(ρ_{λ}), where

j_{w} : H^{1}(F_{w}, T) ⟶ H^{1}(F_{w}, T_{ad})

is the map induced by the map j : G ⟶ G_{ad}. Recall that x_{λ} is the unique fixed-point of t_{λ}wϑ in A_{ad}.

Lemma 6.2.1. For λ ∈ X_{w}, the following are equivalent.

(1) ρ_{jλ} = 1;

(2) the vertex x_{λ} is hyperspecial;

(3) the representation π_{λ} of Section 4.5 is generic.

Proof. The representations κ_{λ} are generic, by [21, 3.10]. The equivalence of 2 and 3 now follows from Lemmas 6.1.1 and 6.1.2.

To prove the equivalence of 1 and 2, recall that x_{λ} is defined by the relation

(1 − wϑ)x_{λ} = t_{jλ} · o.

Now x_{λ} is hyperspecial iff x_{λ} ∈ X_{ad} · o, iff jλ ∈ (1 − wϑ)X_{ad}, iff ρ_{jλ} = 1. □

For ω ∈ ker j_{G}, we set

Irr(C_{φ}, ω)_{gen} := {ρ ∈ Irr(C_{φ}, ω) : π(φ, ρ) is generic}.

Lemma 6.2.2. For ω ∈ ker j_{G}, we have

| Irr(C_{φ}, ω)_{gen}| = [X_{ad}^{ϑ} : j(X^{ϑ})].

In particular, the number of generic representations in Π(φ, ω) is independent of the TRSELP φ and the class ω ∈ ker j_{G}.

Proof. We give the proof assuming that p ∤ [X_{ad} : jX]. The argument for general p is more complicated (see [19]). In this proof only, we change notation and let Z denote the full center of G, and set Z = G ∩ Z. We have a diagram of group homomorphisms

H^{1}(k, Z) ⟶^{ι} H^{1}(F, G) ⟶^{j_{G}} H^{1}(F, G_{ad})
∥  r ↑
H^{1}(k, Z) ⟶^{ι_{w}} H^{1}(F_{w}, T) ⟶^{j_{w}} H^{1}(F_{w}, T_{ad})
∥
Irr(C_{φ})

induced by the inclusions Z ↪ T ↪ G, the adjoint map j : G ⟶ G_{ad}, and Ad(p_{0}) : T ⟶ G, where p_{0}^{−1} F(p_{0}) = ẇ (see Section 2.7). The rows are exact at the middle term [51, Prop.38], and ι = r ∘ ι_{w}. Recall that r^{−1}(ω) = Irr(C_{φ}, ω). We prove the result by computing | ker ι| in two ways.

We have ker ι_{w} ⊆ ker ι, so the ι-fibers are unions of ι_{w}-fibers. From Lemma 6.2.1 it follows that

ι_{w}(ι^{−1}(ω)) = Irr(C_{φ}, ω)_{gen}.

This implies that

| ker ι| = |ι^{−1}(ω)| = | Irr(C_{φ}, ω)_{gen}| · | ker ι_{w}|.

Now

ker ι_{w} ≃ T F_{w} ; ad/j(T F_{w}),

and we have

T F_{w} ; ad = X_{ad}^{wϑ} × (^{0}T_{ad})F_{w}, j(T F_{w}) = j(X^{wϑ}) × j(^{0}T F_{w}).

Since X_{ad}^{wϑ} = {0}, it follows that

(25)  | ker ι| = | Irr(C_{φ}, ω)_{gen}| · |(^{0}T_{ad})F_{w}/j(^{0}T F_{w})|.

On the other hand, we have

| ker ι| = |G_{ad}^{F}/j(G^{F})|.

Since G is quasi-split, [8, 5.6] implies that the inclusion T_{ad} ↪ G_{ad} induces a bijection

T_{ad}^{F}/j(T^{F}) ⟶^{∼} G_{ad}^{F}/j(G^{F}).

Since T_{ad}^{F} = X_{ad}^{ϑ} × (^{0}T_{ad})^{F}, we have

T_{ad}^{F}/j(T^{F}) = [X_{ad}^{ϑ}/j(X^{ϑ})] × (^{0}T_{ad})^{F}/j(^{0}T^{F}),

so

(26)  | ker ι| = |X_{ad}^{ϑ}/j(X^{ϑ})| · |(^{0}T_{ad})^{F}/j(^{0}T^{F})|.

Comparing Equations (25) and (26), the proof boils down to showing that

(27)  |(^{0}T_{ad})F_{w}/j(^{0}T F_{w})| = |(^{0}T_{ad})^{F}/j(^{0}T^{F})|.

If p ∤ [X_{ad} : jX], then jX ⊗ R_{K}^{×} = X_{ad} ⊗ R_{K}^{×}, so we have an exact sequence

1 ⟶ ^{0}T ∩ Z ⟶ ^{0}T ⟶^{j} ^{0}T_{ad} ⟶ 1.

Since H^{1}(F_{w},^{0}T) = H^{1}(F,^{0}T) = 1 and w acts trivially on Z, it follows that both sides of Equation (27) are equal to |H^{1}(F,^{0}T ∩ Z)|. □

It follows from [60, 2.5] that [X_{ad}^{ϑ} : j(X^{ϑ})] is the number of GF_{u}-orbits of hyperspecial vertices in B(GF_{u}). Lemma 6.2.2 leads one to expect that each of these orbits supports a unique generic representation in Π_{u}(φ). We will prove this in a few steps, as follows.

Lemma 6.2.3. Let F_{u} be a quasi-split Frobenius, and let S be an F_{u}-minisotropic torus in G. Assume that the unique fixed-point x of SF_{u} in B(G_{ad})F_{u} is hyperspecial. Then

N(GF_{u}, S)/SF_{u} = N(G, SF_{u})/S.

Proof. Let n ∈ N(G, SF_{u}) ⊂ N(G, S). Since x is hyperspecial and is contained in the apartment of S in B(G_{ad}), we have N(G, S) = N(G_{x}, S)S, so we may assume n ∈ N(G_{x}, SF_{u}). Then F_{u}(n) = nt for some t ∈ S ∩ G_{x} = ^{0}S. Choose d ≥ 1 such that F_{u}^{d}(n) = n. If d = 1 there is nothing to prove, so assume d > 1. This implies that t F_{u}(t) ⋯ F_{u}^{d−1}(t) = 1. By Lemma 2.3.1 there is s ∈ ^{0}T such that t = s F_{u}(s)^{−1}, so that F_{u}(ns) = ns. □

Returning to the notation of Section 4.6, let u ∈ ω ∈ ker j_{G}, and suppose λ, μ ∈ r^{−1}(ω) are such that ρ_{λ}, ρ_{μ} ∈ ker j_{w}. It follows from Lemma 6.2.1 that v_{λ} := m_{λ} · x_{λ} and v_{μ} := m_{μ} · x_{μ} are hyperspecial vertices in AF_{u} ; ad. The representations π_{u}(φ, ρ_{λ}) and π_{u}(φ, ρ_{μ}) are induced from the stabilizers in GF_{u} of v_{λ} and v_{μ}, respectively.

Lemma 6.2.4. Assume that v_{λ} and v_{μ} are GF_{u}-conjugate hyperspecial vertices. Then ρ_{λ} = ρ_{μ}.

Proof. We first prove that S_{λ} and S_{μ} are GF_{u}-conjugate. Since GF_{u} = GF_{u} ; v_{λ}N F_{u}GF_{u} ; v_{μ}, there is n ∈ N F_{u} such that n · v_{μ} = v_{λ}. The F_{u}-minisotropic tori

S_{1} := S_{λ}, S_{2} := ^{n}S_{μ}

both have v_{λ} as their unique fixed-point in B(G_{ad})F_{u}. Let T and S_{i} be the images of T ∩ G_{v_{λ}} and S_{i} ∩ G_{v_{λ}}, respectively, in G_{v_{λ}}.

Set

k_{1} := q_{λ}m_{λ}^{−1}, k_{2} := nq_{μ}m_{μ}^{−1}n^{−1}.

Then k_{i} ∈ G_{v_{λ}} and S_{i} = Ad(k_{i})T for i = 1, 2. Let k̄_{i} be the image of k_{i} in G_{v_{λ}}, so that S_{i} = Ad(k̄_{i})T.

Using Equation (10) we find that

k_{1}^{−1} F_{u}(k_{1}) ≡ m_{λ} · wu^{−1} · F_{u}(m_{λ})^{−1} mod T,
k_{2}^{−1} F_{u}(k_{2}) ≡ nm_{μ} · wu^{−1} · F_{u}(nm_{μ})^{−1} mod T.

Since v_{λ} is hyperspecial, every class in N/T has a representative in N ∩ G_{v_{λ}}. Applying this to m_{λ}T, nm_{μ}T and wu^{−1}T, it follows that k̄_{1}^{−1} F_{u}(k̄_{1}) and k̄_{2}^{−1} F_{u}(k̄_{2}) are F_{u}-conjugate in the Weyl group of T in G_{v_{λ}}. This means (c.f. [12, 3.3.3]) that S_{1} and S_{2} are GF_{u} ; v_{λ}-conjugate. The uniqueness part of Lemma 8.0.10 then implies that S_{1} and S_{2} are GF_{u}-conjugate. Hence S_{λ} and S_{μ} are GF_{u}-conjugate, as claimed.

By Lemma 2.11.1 there is z_{o} ∈ W_{o}^{wϑ} such that

λ ≡ z_{o}μ mod (1 − wϑ)X.

But Lemmas 6.2.3 and 2.11.2 imply that

W_{o}^{wϑ} = W_{o,μ}^{wϑ}.

Hence λ ≡ μ mod (1 − wϑ)X, so ρ_{λ} = ρ_{μ}. □

Remark 6.2.5. Lemma 6.2.3 and the last step in the above proof can be seen in another way, as follows. Since v_{λ} is hyperspecial, Lemma 6.2.2 implies that ρ_{λ} ∈ ker j_{w} = im i_{w} : [H^{1}(k, Z) → H^{1}(F_{w}, T)]. Since W_{o}^{wϑ} acts trivially on H^{1}(k, Z), it follows that ρ_{λ} is a W_{o}^{wϑ}-fixed point in H^{1}(F_{w}, T).

Combining Lemmas 6.2.2 and 6.2.4 yields the promised result:

Corollary 6.2.6. There is a bijection between the set of generic representations in Π_{u}(φ) and the set of GF_{u}-orbits of hyperspecial vertices in B(G_{ad})F_{u}, such that a generic representation is induced from the stabilizer of any hyperspecial vertex in the corresponding orbit.

Remark 6.2.7. If G has connected center, then G_{x} has connected center for any hyperspecial vertex x ∈ B(G). Assume x is F_{u}-stable. It follows from Proposition 5.26 and Theorems 6.8 and 10.7 of [20] that every cuspidal generic representation of G_{x} is of the form ±R_{S,θ}^{G_{x}} for some F_{u}- minisotropic maximal torus S ⊂ G_{x} and θ ∈ Irr(SF_{u}) in general position. Using Lemma 6.1.2, this implies that every depth-zero generic supercuspidal representation of GF_{u} appears in Π_{u}(φ) for some TRSELP φ.

7. Topological Jordan decomposition

We define the set of compact elements in G by

G_{0} := ⋃_{x∈B(G)} G_{x},

and the set of topologically unipotent elements in G by

G_{0^{+}} = ⋃_{x∈B(G)} G_{x}^{+}.

We define 𝔤_{0} and 𝔤_{0^{+}} similarly. These G ⋊ ⟨F ⟩-stable subsets of G will play an important role in this paper.

Remark 7.0.8. From [16] we have that if x ∈ B(G), then

G_{0} ∩ stab_{G}(x) = G_{x}.

Let p denote the characteristic of 𝔣. Choose m such that for all F -stable facets J in B(G) and all elements g ∈ G_{J}^{F} we have g^{(p^{m})} = s where s denotes the semisimple component in the Jordan decomposition of g.

Suppose γ ∈ G_{0}^{F}. Let J ⊂ B(G) be any F -stable facet such that γ ∈ G_{J}. Since γ ∈ G_{J}^{F}, it follows that we can define

γ_{s} := lim_{n→∞} γ^{(p^{mn})}.

This limit does not depend on m, and the element γ_{s} has finite order prime to p. We set

γ_{u} = γ · γ_{s}^{−1}.

The topological Jordan decomposition is the commuting factorization

γ = γ_{s}γ_{u} = γ_{u}γ_{s}.

We have γ_{s}, γ_{u} ∈ G_{J}^{F}. Moreover, γ_{s} is semisimple and has semisimple image in G_{J} while γ_{u} has unipotent image in G_{J}. In particular, γ_{u} is topologically unipotent. We say that γ is topologically semisimple if γ = γ_{s}, that is, if γ = γ^{p^{m}}.

The topological Jordan decomposition γ = γ_{s}γ_{u} is the unique commuting factorization of γ as a product of a topologically semisimple element and a topologically unipotent element. This implies that if g ∈ G is chosen so that^{g}γ ∈ G^{F}, then^{g}(γ_{s}) = (^{g}γ)_{s} and^{g}(γ_{u}) = (^{g}γ)_{u}.

Lemma 7.0.9. Suppose γ ∈ G_{0}^{F} has topological Jordan decomposition γ = γ_{s}γ_{u}. Then γ, γ_{s}, and γ_{u} all belong to G_{γ_{s}}. Moreover, if γ ∈ G^{rss}, then γ_{u} ∈ G_{γ_{s}}^{rss}.

Proof. Choose a Borel subgroup B < G containing γ. Since B ∩ G is a closed subgroup of G, both γ_{s} and γ_{u} belong to B ∩ G. Since γ_{s} is semisimple, it follows from [6, Theorem 10.6 (5ii)] that the centralizer in B of γ_{s} is connected. Thus, γ, γ_{s}, and γ_{u} belong to B_{γ_{s}} ∩ G ⊂ G_{γ_{s}}.

The centralizer of γ_{u} in G_{γ_{s}} has finite index in the centralizer of γ in G. This implies the last assertion. □

Since γ_{s} is compact and has finite order prime to p, the results of [46] combined with Remark 7.0.8 allow us to identify

(28)  B(G_{γ_{s}}) = B(G)^{γ_{s}}.

More precisely, there is an unramified maximal torus S of G containing γ_{s}, and a bijection from the apartment of S in B(G_{γ_{s}}) to the apartment of S in B(G) which extends to a G_{γ_{s}}-equivariant bijection B(G_{γ_{s}}) ⟶^{∼} B(G)^{γ_{s}}. In particular, G_{γ_{s}} and G have the same K-rank.

For an exhaustive treatment of the topological Jordan decomposition, see [52].

8. Unramified and minisotropic maximal tori

Recall that we are assuming G is K-split, and that we say a subgroup S < G is a maximal unramified torus in G if S = S(K), where S is a K-split maximal torus in G such that S is defined over k.

All maximal unramified tori in G can be found as follows.

Lemma 8.0.10. Suppose we are given a nonempty F -stable subset J of a facet in B(G) or B(G_{ad}) and an F -stable maximal torus S < G_{J}. Then there exists a maximal unramified torus S in G such that

(1) J ⊂ A(S);

(2) the image of S ∩ G_{J} in G_{J} is exactly S. Moreover, S is unique up to conjugacy by G_{J}^{+F}.

Proof. The existence of S is shown in the proof of [10, 5.1.10]. The uniqueness is proved in [14, Lemma 2.2.2]. □

Such an S is called a lift of (J, S).

A maximal unramified torus S in G is called F -minisotropic in G if X_{∗}(S)^{F} = X_{∗}(Z), where Z is the identity component of the maximal k-split torus in the center of G.

Likewise, a maximal 𝔣-torus S in a reductive 𝔣-group G with Frobenius F is called F -minisotropic in G if X_{∗}(S)^{F} = X_{∗}(Z), where Z is the maximal 𝔣-split torus the center of G.

Let 𝔗(G) be the set of F -minisotropic maximal tori in G. If S ∈ 𝔗(G), then there exists a unique F -stable facet J ⊂ B(G) such that

A(S)^{F} = J^{F} .

The unique parahoric subgroup^{0}S of S is given by

0S = S ∩ G_{J}.

Note that N(G, S)^{F} preserves A(S)^{F}, hence normalizes G_{J}^{F} and G_{J}^{+F}. In particular, G_{J}^{+F} N(G, S)^{F} is a subgroup of G^{F}.

Let S be the image of S ∩ G_{J} in G_{J}. Then S is an F -minisotropic torus in G_{J}, and S is a lift of (J, S).

Fix now S ∈ 𝔗(G) and a topologically semisimple element γ ∈ G_{0}^{F}. For our later integral calculations we must consider the two sets

\begin{aligned}E(γ, S) & := \{g ∈ G^{F} : ^{g}γ ∈ G_{J}, \overline{^{g}γ} ∈ S\}, \\ D̃(γ, S) & := \{d ∈ G^{F} : ^{d}γ ∈ S\}. \end{aligned}

In other terms, D̃(γ, S) is the set of elements of G^{F} which conjugate S into G_{γ}, and E(γ, S) is the set of elements of G^{F} which send some G_{J}^{+F} -conjugate of S into G_{γ} and whose inverse sends J into B(G_{γ}). Since γ ∈ G_{0}^{F}, we have D̃(γ, S) ⊂ E(γ, S).

We have obvious actions, by multiplication, of G_{J}^{+F} N(G, S)^{F} × G_{γ}^{F} on E(γ, S), and of N(G, S)^{F} × G_{γ}^{F} on D̃(γ, S).

Lemma 8.0.11. The inclusion D̃(γ, S) ↪ E(γ, S) induces a bijection

N(G, S)^{F} \D̃(γ, S)/G_{γ}^{F} ⟶^{∼} G_{J}^{+F} N(G, S)^{F} \E(γ, S)/G_{γ}^{F} .

Both sets of double cosets are finite.

Proof. The set N(G, S)^{F} \D̃(γ, S)/G_{γ}^{F} parametrizes G_{γ}^{F} -conjugacy classes of F -minisotropic tori in G_{γ} which lie in the G^{F} -conjugacy class of S. Since G_{γ}^{F} has only finitely many conjugacy classes of unramified maximal tori, the set N(G, S)^{F} \D̃(γ, S)/G_{γ}^{F} is finite.

We now prove injectivity. Suppose we have d, d′ ∈ D̃(γ, S), and h ∈ G_{J}^{+F} , n ∈ N(G, S)^{F} , g ∈ G_{γ}^{F}, such that d′ = nhdg. Replacing d′ by n^{−1}d′g^{−1}, we may assume without loss of generality that d′ = hd. This means that^{d}γ and^{hd}γ both belong to S, and being compact,^{d}γ and^{hd}γ in fact belong to S∩G_{J}. Since h ∈ G_{J}^{+F}, both^{d}γ and^{hd}γ have the same image in S. Hence we can write hdγ = ^{d}γγ_{1}, where γ_{1} ∈ G_{J}^{+F} ∩ S is topologically unipotent. But then ^{d}γγ_{1} = γ_{1}^{d}γ, and since hdγ is topologically semisimple, we must have γ_{1} = 1, by uniqueness of the topological Jordan decomposition. It follows that S and h^{−1}Sh are two lifts of (S, J) in ^{d}G_{γ_{s}}. By Lemma 8.0.10, there is k ∈ (^{d}G_{γ_{s}})_{J}^{+F} such that kSk^{−1} = h^{−1}Sh. This implies that h ∈ N(G, S)^{F} · ^{d}(G_{γ_{s}}^{F}), proving injectivity.

For surjectivity, suppose g ∈ E(γ, S), and let H = ^{g}G_{γ}. Then ^{g}γ fixes J pointwise, so by Equation (28), J is contained in a facet in the building B(H) of H. We let H_{J} denote the corresponding parahoric subgroup of H. Then^{g}γ ∈ H_{J}.

Considering root data, we find a 𝔣-isomorphism ι : (G_{J})\overline{^{g}γ} ⟶^{∼} H_{J} making the following diagram commutative.

\begin{matrix}H ∩ G_{J} & = & H_{J} \\ ↓ & & ↓ \\ (G_{J})\overline{^{g}γ} & ⟶^{ι} & H_{J} \end{matrix}.

We have S < (G_{J})\overline{^{g}γ} by hypothesis, hence ιS is an F -stable maximal torus in H_{J}. Choose a lift S′ in H of (J, ιS). Then S′ ∩ H_{J} = S′ ∩ G_{J} so S′ is a lift of (J, S) in G. But S is also a lift of (J, S) in G, so by 8.0.10 there is k ∈ G_{J}^{+F} such that ^{k}S′ = S. Since ^{g}γ ∈ S′, we have ^{kg}γ ∈ S. This means kg ∈ D̃(γ, S), proving surjectivity. □

9. Some character computations

In this chapter we give an integral formula for the characters of the representations constructed in Section 4.4. In fact, we define a set of integrals on G^{F} which include these characters as a subset. Our eventual goal is to express these integrals as combinations of similar integrals on the set of topologically unipotent elements, in the same way that a Deligne-Lusztig character is expressed as a combination of Green functions.

9.1. Harish-Chandra’s character formula. Recall that Z denotes the group of K-rational points of the maximal k-split torus in the center of G.

Suppose that Q is an open subgroup of G^{F} containing Z^{F} such that Q is compact modulo Z^{F}. Suppose also that κ is a representation of Q for which the compactly-induced representation π := ind_{Q}^{G^{F}} κ of G^{F} is irreducible. Let χ̇_{κ} denote the extension by zero of the character of κ to a function on G^{F}. In [26] Harish-Chandra showed that the value of the character of π at γ ∈ (G^{rss})^{F} is given by the formula

\frac{Deg(π)}{χ_{κ}(1)} ∫_{G^{F} /Z^{F}} dg^{∗} ∫_{L} χ̇_{κ}(^{gl}γ) dl.

Here dg^{∗} denotes the quotient measure on G^{F} /Z^{F} with respect to Haar measures dg and dz on G^{F} and Z^{F}, respectively, Deg(π) denotes the formal degree of π with respect to dg^{∗} = dg/dz (see Section 5), and L is an arbitrary compact open subgroup of G^{F} with Haar measure dl normalized so that meas_{dl}(L) = 1.

9.2. The character integral. Let S be an F -minisotropic maximal torus in G, and let J be the unique minimal F -stable facet in B(G) such that A(S)^{F} = J^{F}. Recall that^{0}S^{F} = S^{F} ∩ G_{J}, and S ∩ G_{J} projects onto an F -minisotropic torus S in G_{J}.

Let Irr_{0}(S^{F}) denote the set of depth-zero characters of S^{F}. For θ ∈ Irr_{0}(S^{F}), the restriction of θ to ^{0}S^{F} factors through S^{F}, and thus defines a Deligne-Lusztig virtual character R_{S,θ}^{G_{J}}. Let Ṙ_{S,θ}^{G_{J}} denote the natural inflation of R_{S,θ}^{G_{J}} to a function on G_{J}^{F}, extended by zero to the rest of G^{F}.

Define a function R(G, S, θ) on (G^{rss})^{F} by the integral

R(G, S, θ)(γ) := \frac{meas_{dz}(Z_{J}^{F})}{meas_{dg}(G_{J}^{F})} · ∫_{G^{F} /Z^{F}} dg^{∗} ∫_{L} Ṙ_{S,θ}^{G_{J}}(^{gl}γ) dl.

Here L and the measures dg^{∗}, dl are as in Section 9.1. (The integral converges; see, for example, Lemma 10.0.7.)

Remark 9.2.1. For h ∈ G^{F}, a change of variables shows that

R(G,^{h}S, h_{∗}θ) = R(G, S, θ),

where h_{∗}θ = θ ∘ Ad(h)^{−1}. If T is a G^{F} -orbit of pairs (S, θ) with S ∈ 𝔗(G) and θ ∈ Irr_{0}(S^{F}), we sometimes write

R(G, T ) := R(G, S, θ),

for any (S, θ) ∈ T .

9.3. Relation to characters. Suppose θ ∈ Irr_{0}(S^{F}) is regular, in the sense that θ has trivial stabilizer in N(G, S^{F})/S. There is a unique representation κ of Z^{F} G_{J}^{F} such that

(1) the restriction to G_{J}^{F} of κ has the character ε(G_{J}, S) · Ṙ_{S,θ}^{G_{J}}, and

(2) the restriction of κ to Z^{F} is given by the scalar character θ|_{Z^{F}}, times the identity.

We have seen that the induced representation π := Ind_{Z^{F} G_{J}^{F}}^{G^{F}} κ is irreducible and supercuspidal.

Lemma 9.3.1. Let Θ_{π} be the character of the representation π just defined. Then Θ_{π} vanishes off the set Z^{F} G_{0}^{F}. For z ∈ Z^{F} and regular semisimple γ ∈ G_{0}^{F} we have

Θ_{π}(zγ) = ε(G_{J}, S) · θ(z) · R(G, S, θ)(γ).

Proof. Harish-Chandra’s integral formula (see Section 9.1) makes the vanishing assertion obvious and gives, for z ∈ Z^{F} and regular semisimple γ ∈ G_{0}^{F}, the formula

Θ_{π}(zγ) = θ(z) · \frac{meas_{dg}(G_{J}^{F})}{meas_{dz}(Z_{J}^{F})} · \frac{Deg(π)}{Ṙ_{S,θ}^{G_{J}}(1)} · R(G, S, θ)(γ).

Consequently, we need to show that

\frac{meas_{dg}(G_{J}^{F})}{meas_{dz}(Z_{J}^{F})} · \frac{Deg(π)}{Ṙ_{S,θ}^{G_{J}}(1)} = ε(G_{J}, S).

But, from Remark 5.3 we have

Deg(π) · meas_{dg^{∗}}(Z^{F} G_{J}^{F} /Z^{F}) = dim(κ),

and the claim follows. □

9.4. Stable conjugacy of tori and their characters. We want to produce a sum of character integrals that will be be stable. In the situation of Section 9.3, these sums will specialize to the sum of characters over an L-packet, as defined in Section 4.6. Our integral sums are based on the notion of stable conjugacy of unramified tori and their characters.

Recall that 𝔗(G) denotes the set of F -minisotropic maximal tori in G. We say that two tori S_{1}, S_{2} ∈ 𝔗(G) are G-stably conjugate if there is g ∈ G such that ^{g}(S_{1}^{F}) = S_{2}^{F}. This defines an equivalence relation on 𝔗(G), whose equivalence classes are called G-stable classes . The set of G-stable classes injects into H^{1}(F, N/T) as follows. Any two maximal unramified tori in G are conjugate by an element of G. For S ∈ 𝔗(G), write S = ^{g}T, for g ∈ G. Since F (S) = S, we have an element n := g^{−1}F (g) ∈ Z^{1}(F, N). Projecting to N/T gives an element n̄ := g^{−1}F (g)T ∈ Z^{1}(F, N/T). One checks that the class [n̄] of n̄ in H^{1}(F, N/T) is independent of g. Note that S^{F} = ^{g}(T^{F_{n}}), where, as usual, F_{n} = Ad(n) ∘ F.

Lemma 9.4.1. Suppose h ∈ G, and n, m ∈ N. Then

h^{−1}mF (h) ∈ nT ⇔ ^{h}(T^{F_{n}}) = (^{h}T)^{F_{m}}.

Proof. Implication ⇒ is straightforward. For the converse, choose a strongly regular element t ∈ T^{F_{n}}. From the equation F_{m}(^{h}t) = ^{h}t, we find that the element h^{−1}mF (h)n^{−1} centralizes t, hence lies in T. □

For v = nT ∈ N/T, set F_{v} = F_{n}, and define

T_{v} := {S ∈ 𝔗(G) : S^{F} = ^{g}(T^{F_{v}}) for some g ∈ G}.

Lemma 9.4.2. The sets T_{v} have the following properties.

(1) If T_{v} is nonempty, then T_{v} is a G-stable class in 𝔗(G).

(2) Every G-stable class in 𝔗(G) is of the form T_{v} for some v ∈ N/T.

(3) For v, v′ ∈ N/T, we have T_{v} = T_{v′} if and only if [v] = [v′] in H^{1}(F, N/T).

(4) If G is k-quasi-split, then T_{v} is nonempty.

Proof. See [14]. □

For each S ∈ T_{v}, Lemma 9.4.1 implies that there is g ∈ G such that S = ^{g}T and g^{−1}F (g) ∈ v. Note that the choice of g is not uniquely determined by S; two choices of g differ by an element of N(G, S^{F}). The map Ad(g) : T ⟶ S intertwines (T, F_{v}) and (S, F). For each depth-zero character χ ∈ Irr_{0}(T^{F_{v}}), we have a corresponding character g_{∗}χ ∈ Irr_{0}(S^{F}), which depends on the choice of g.

This dependence on g is eliminated by passing to a “covering” of T_{v}, as follows. Consider the set of pairs

𝔗̂(G) := {(S, θ) : S ∈ 𝔗(G) and θ ∈ Irr_{0}(S^{F})}.

We say that two pairs (S_{1}, θ_{1}), (S_{2}, θ_{2}) ∈ 𝔗̂(G) are G-stably conjugate if there is g ∈ G such that

(1) ^{g}(S_{1}^{F}) = S_{2}^{F}, and

(2) g_{∗}θ_{1} = θ_{2}. The G-stable classes of pairs (S, θ) ∈ 𝔗̂(G) are parametrized as follows. Fix v ∈ N/T, χ ∈ Irr_{0}(T^{vF}), and define

T̂_{v,χ} := {(S, θ) ∈ 𝔗̂(G) : there exists g ∈ G such that S^{F} = ^{g}(T^{F_{v}}), and θ = g_{∗}χ}.

Lemma 9.4.3. (1) If T_{v} is nonempty, then T̂_{v,χ} is a nonempty G-stable class in 𝔗̂(G). (2) Every G-stable class in 𝔗̂(G) is of the form T̂_{v,χ} for some v ∈ N/T, χ ∈ Irr_{0}(T^{F}).

(3) For v ∈ N/T, χ, χ′ ∈ Irr_{0}(T^{F_{v}}), we have T̂_{v,χ} = T̂_{v,χ′} if and only if there is n ∈ N(G, T^{F_{v}}) such that n_{∗}χ = χ′.

Proof. This follows easily from Lemma 9.4.2. □

Thus, we have a partition

𝔗̂(G) = ∐_{\begin{matrix}v∈N/T \\ T_{v}≠∅ \end{matrix}} ∐_{χ∈Irr_{0}(T^{Fv})/N(G,T^{Fv})} T̂_{v,χ}

of 𝔗̂(G) into nonempty G-stable classes.

Projection onto the first factor is a surjection p_{1} : T̂_{v,χ} ⟶ T_{v}. Given S ∈ T_{v} we can project the fiber p_{1}^{−1}(S) onto the second factor. This gives a map

p_{2} : p_{1}^{−1}(S) ⟶ Irr_{0}(S^{F}).

We define

θ_{S}^{χ} = ∑_{θ ∈ p_{2}p_{1}^{−1}(S)} θ.

To see the dependence on χ, choose g as in the definition of T̂_{v,χ} above. Then

θ_{S}^{χ} = ∑_{n̄∈N(G,S^{F})/S}(ng)_{∗}χ,

and the sum is independent of the choice of g.

The character sums θ_{S}^{χ} have the following stability property.

Lemma 9.4.4. Suppose S_{1}, S_{2} ∈ T_{v}, γ ∈ S_{1}^{F}, and χ ∈ Irr_{0}(T^{F_{v}}). Then for any h ∈ G such that h(S_{1}^{F}) = S_{2}^{F}, we have

θ_{S_{1}}^{χ}(γ) = θ_{S_{2}}^{χ}(^{h}γ).

Proof. This is immediate from the observation that h_{∗}[p_{2}p_{1}^{−1}(S_{1})] = p_{2}p_{1}^{−1}(S_{2}). □

9.5. The stable character integral. Fix a G-stable class T̂_{st} ⊂ 𝔗̂(G). The group G^{F} acts on 𝔗̂(G) via g · (S, θ) = (^{g}S, g_{∗}θ), and T̂_{st} is the union of finitely many G^{F} -orbits in 𝔗̂(G). By Remark 9.2.1 the function R(G, S, θ) depends only on the G^{F} -orbit of (S, θ). We can therefore define a function R(G, T̂_{st}) on (G^{rss})^{F} by

R(G, T̂_{st}) := ∑_{(S,θ)∈T̂_{st}/G^{F}} R(G, S, θ),

where R(G, S, θ) was defined in Section 9.2. Our eventual goal is to show that the function R(G, T̂_{st}) is stable. But first, we relate R(G, T̂_{st}) to the sum of characters in an L-packet.

9.6. Relation to L-packets. In this section we show that the sum of characters in an L-packet, as defined in Section 4.6, can be expressed, up to a sign, as one of the functions R(G, T̂_{st}) as defined in Section 9.5. We return to the notation used in Section 4.6 and previously, so that F = F_{u}. Set v = ẇu^{−1}T ∈ N/T, and let χ ∈ Irr_{0}(T F_{w}) be regular. Note that F_{v} = F_{w}, and by the proof of Lemma 2.11.2, we may identify

N(G, T F_{w})/T = W_{o}^{wϑ}.

For each λ ∈ r^{−1}(ω) we have the pair (S_{λ}, θ_{λ}) = q_{λ} · (T, χ) ∈ T̂_{v,χ}. Recall from Lemma 2.6.1 the commutative diagram

[X/(1 − wϑ)X]_{tor} ⟶ [X̄/(1 − ϑ)X̄]_{tor}
≃↓  ↓≃
H^{1}(F_{w}, T)  ⟶^{Ad(p_{0})}  H^{1}(F, G)

where the vertical maps are bijections. Recall that [r^{−1}(ω)] denotes the fiber of the map in the top row, and this fiber carries a natural action of W_{o}^{wϑ}.

Lemma 9.6.1. Recall that v = ẇu^{−1}T, and χ ∈ Irr_{0}(T F_{w}) is regular. The mappings λ ↦ (S_{λ}, θ_{λ}), λ ↦ S_{λ}, respectively, induce bijections

α : [r^{−1}(ω)] ⟶^{∼} T̂_{v,χ}/GF_{u}, β : [r^{−1}(ω)]/W_{o}^{wϑ} ⟶^{∼} T_{v}/GF_{u},

which make the following diagram commute.

[r^{−1}(ω)]  ⟶^{α} T̂_{v,χ}/GF_{u}
p ↓  ↓ p̄_{1}
[r^{−1}(ω)]/W_{o}^{wϑ} ⟶^{β} T_{v}/GF_{u}

Here p is the quotient map and p̄_{1} is induced by the projection p_{1} onto the first factor.

Proof. The map β is well-defined and bijective, by Lemma 2.11.1.

If λ, μ ∈ r^{−1}(ω) are congruent modulo (1 − wϑ)X, then from the proof of Lemma 2.10.1 there exists s ∈ S_{λ} such that q_{μ}q_{λ}^{−1}s ∈ GF_{u}. Since

q_{μ}q_{λ}^{−1}s · (S_{λ}, θ_{λ}) = (S_{μ}, θ_{μ}),

this shows that the map α is well-defined.

The fiber of p̄_{1} over the GF_{u}-orbit of S_{λ} in T_{v} is in evident bijection with N(G, SF_{u} ; λ)/N(GF_{u}, S_{λ}). By Lemma 2.11.2, the latter is in bijection with the fiber of p over the class of λ in [r^{−1}(ω)]/W_{o}^{wϑ}.

It therefore suffices to prove that α is injective. Suppose g ∈ GF_{u}, λ, μ ∈ r^{−1}(ω) and g · (S_{μ}, θ_{μ}) = (S_{λ}, θ_{λ}). As in the proof of Lemma 2.11.1, the element q_{λ}^{−1}gq_{μ} belongs to N(T F_{w}), and projects to an element z_{o} ∈ W_{o}^{wϑ} such that z_{o}μ ≡ λ mod (1 − wϑ)X. But also g_{∗}θ_{μ} = θ_{λ}, which means that z_{0} fixes χ. Since χ is regular, we have z_{o} = 1, hence μ ≡ λ mod (1 − wϑ)X. □

Recall that for λ ∈ r^{−1}(ω), u ∈ ω, and a TRSELP φ we defined in Section 4.6 the representation

π_{u}(φ, ρ_{λ}) = Ad(m_{λ})_{∗}π_{λ} ∈ Irr(GF_{u}),

where m_{λ} is as in Lemma 2.8.1. This construction involved the character χ = χ_{φ} ∈ Irr_{0}(T F_{w}) corresponding to φ as in Section 4.3.

Lemma 9.6.2. Let G_{u} be the inner twist of G given by the cocycle u ∈ ω, and let T_{w} be the twist of T determined by w. Then for λ ∈ r^{−1}(ω) we have

ε(G_{λ}, T_{λ}) = ε(G_{u}, T_{w}).

Hence, this sign is independent of λ ∈ r^{−1}(ω).

Proof. The 𝔣-rank of G_{λ} equals the k-rank of G_{u_{λ}}, and G_{u_{λ}} ≃ G_{u} over k. Likewise, we have seen that T_{λ} ≃ T_{w} over k. □

For λ ∈ r^{−1}(ω), let Θ_{ρ_{λ}} be the character of π_{u}(φ, ρ_{λ}). By construction, the function Θ_{ρ_{λ}} depends only on the class of λ in [r^{−1}(ω)]. We can now prove the desired result of this section.

Lemma 9.6.3. Let v = wu^{−1} and let χ = χ_{φ} be as in Section 4.3. Then

∑_{λ∈[r^{−1}(ω)]} Θ_{ρ_{λ}} = ε(G_{u}, T_{w}) · R(G, T̂_{v,χ}).

Proof. By Lemma 9.3.1, we have

Θ_{ρ_{λ}} = ε(G_{λ}, T_{λ}) · R(G, S_{λ}, θ_{λ}),

so the claim follows from Lemmas 9.6.1 and 9.6.2. □

10. Reduction formulae for character integrals

If G is a connected reductive 𝔣-group with Frobenius F, S is a maximal 𝔣-torus in G, and θ ∈ Irr(S^{F}), then from [20, Thm 4.2] we have the reduction formula

(29)  R_{S,θ}^{G}(x) = ∑_{\begin{matrix}g∈G^{F} \\ ^{g}S⊂G_{s} \end{matrix}} g_{∗}θ(s) · Q_{gSg^{−1}}^{G_{s}}(u),

where x = su ∈ G^{F} is the Jordan decomposition, and for any maximal 𝔣-torus S_{1} ⊂ G_{s}, the normalized Green function Q_{S_{1}}^{G_{s}} is defined on all of G^{F} by

(30)  Q_{S_{1}}^{G_{s}}(h) := \begin{cases}\frac{1}{|G_{s}^{F} |}R_{S_{1},θ_{1}}^{G_{s}}(h) & if h ∈ G_{s}^{F} and h is unipotent \\ 0 & otherwise , \end{cases}

the right side being independent of θ_{1} ∈ Irr(S_{1}^{F}).

In this section we prove an analogue of Equation (29) for our functions R(G, S, θ), using now the topological Jordan decomposition.

Fix a pair (S, θ) ∈ 𝔗̂(G), and let T̂ denote the G^{F} -orbit of (S, θ). For γ ∈ G_{0}^{F} ∩ G^{rss} with topological Jordan decomposition γ = γ_{s}γ_{u}, we define

T̂(γ_{s}) := {(S′, θ′) ∈ T̂ : γ_{s} ∈ S′}.

Then G_{γ_{s}}^{F} preserves T̂(γ_{s}), and acts on T̂(γ_{s}) with finitely many orbits.

Our reduction formula for R(G, S, θ) is as follows.

Lemma 10.0.4. For γ = γ_{s}γ_{u} as above, we have

R(G, S, θ)(γ) = ∑_{(S′,θ′)∈T̂(γ_{s})/G_{γs}^{F}} θ′(γ_{s}) · R(G_{γ_{s}}, S′, 1)(γ_{u}).

The proof of Lemma 10.0.4 will require some preliminary steps. Let J be the facet in A(S) such that J^{F} = A(S)^{F}, and let S be the image of S ∩ G_{J} in G_{J}. Any compact element δ ∈ S^{F} belongs to S ∩ G_{J}, and we let \overline{δ} ∈ S denote the image of δ.

Applying Equation (30) with G = G_{J}, s = \overline{γ}_{s}, and S_{1} = S, we have the normalized Green function Q_{S}^{(G_{J})_{\overline{γ}_{s}}} defined on all of G_{J}^{F}. We let Q̇_{S}^{(G_{J})_{\overline{γ}_{s}}} denote the natural inflation of Q_{S}^{(G_{J})_{\overline{γ}_{s}}} to a function on G_{J}^{F}, extended by zero to the rest of G^{F}.

Lemma 10.0.5. Let γ ∈ G_{0}^{F} be regular semisimple with γ_{s} ∈ S, and let L_{γ_{s}} be a compact open subgroup of G_{γ_{s}} with Haar measure di. Then the support of the function on G_{γ_{s}}^{F} given by

h ↦ ∫_{L_{γs}} Q̇_{S}^{(G_{J})_{\overline{γ}_{s}}}(^{hi}γ_{u}) di,

is compact modulo the center of G_{γ_{s}}^{F}.

Proof. The function Q_{S}^{(G_{J})_{\overline{γ}_{s}}} on the unipotent set in (G_{J})_{\overline{γ}_{s}} is the restriction of R_{S,θ}^{(G_{J})_{\overline{γ}_{s}}}, for any

θ ∈ Irr_{0}(S^{F}). Take θ to be regular. Since S is F -minisotropic in (G_{J})_{\overline{γ}_{s}}, the function Ṙ_{S,θ}^{(G_{J})_{\overline{γ}_{s}}} is a matrix coefficient of a supercuspidal representation of G_{γ_{s}}^{F}, constructed as in Section 4.4 with G^{F} there replaced by G_{γ_{s}}^{F}. Hence the function Q̇_{S}^{(G_{J})_{\overline{γ}_{s}}} is the restriction of a supercuspidal matrix coefficient to the compact topological unipotent set in G_{γ_{s}}^{F}. The result now follows from [26, Lemma 23, p. 59]. □

The restriction of θ to S^{F} ∩ G_{J} is the inflation of a character θ_{0} ∈ Irr(S^{F}). Let θ̇ denote the function on G^{F} defined by

θ̇(δ) = \begin{cases}θ_{0}(δ̄) & if δ ∈ G_{J}^{F} and δ̄ ∈ S, \\ 0 & otherwise. \end{cases}

For each regular semisimple element γ ∈ G_{0}^{F}, define a locally constant function f_{γ} on G^{F} by

f_{γ}(g) := θ̇(^{g}γ_{s}) · Q̇_{S}^{(G_{J})_{\overline{^{g}γ}_{s}}}(^{g}γ_{u}).

Note that f_{γ} is supported on the set E(γ_{s}, S) defined in Section 8, and is left-invariant under G_{J}^{+F}.

Lemma 10.0.6. Let γ ∈ G_{0}^{F} be regular semisimple, and let L_{γ_{s}} be a compact open subgroup of G_{γ_{s}} with Haar measure di. Then the function τ_{γ}: G^{F} → ℂ defined by

τ_{γ}(g) := ∫_{L_{γs}} f_{γ}(gi) di

is locally constant and compactly supported modulo Z^{F}.

Proof. Since τ_{γ}(jg) = τ_{γ}(g) for all j ∈ (G_{J}^{+})^{F} and g ∈ G^{F}, it is clear that τ_{γ} is locally constant.

Without loss of generality, we assume that γ ∈ G_{J}^{F} and \overline{γ}_{s} ∈ S. By Lemma 8.0.10, there is a lift of (J, S) in G_{γ_{s}}. Any such lift is F -minisotropic in G. It follows that the center of G_{γ_{s}}^{F} is compact modulo Z^{F}.

Choose a set D(γ_{s}, S) of representatives for the double cosets in

N(G, S)^{F} \D̃(γ_{s}, S)/G_{γ_{s}}^{F}.

By Lemma 8.0.11 the set D(γ_{s}, S) is finite, and the support of τ_{γ} is contained in

E(γ_{s}, S) = ∐_{d∈D(γ_{s},S)} G_{J}^{+F} N(G, S)^{F} dG_{γ_{s}}^{F}.

Since S is F -minisotropic, the group N(G, S)^{F} is compact modulo Z^{F}. It suffices therefore to show, for fixed d ∈ D(γ_{s}, S), that the function h ↦ τ_{γ}(dh) on G_{γ_{s}}^{F} has compact support modulo the center of G_{γ_{s}}^{F}. This is Lemma 10.0.5 with γ there replaced by^{d}γ. □

The key to the reduction formula is the following “localization” result.

Lemma 10.0.7. Suppose γ ∈ G_{0}^{F} is regular semisimple and L is a compact open subgroup of G^{F}, and let L_{γ_{s}} = L ∩ G_{γ_{s}}. Normalize Haar measures so that meas_{dℓ}(L) = meas_{di}(L_{γ_{s}}) = 1. Then the integrals

∫_{G^{F} /Z^{F}} dg^{∗} ∫_{L_{γs}} f_{γ}(gi) di

and

∫_{G^{F} /Z^{F}} dg^{∗} ∫_{L} f_{γ}(gl) dl

both converge and are equal. Moreover, these integrals are independent of L.

Proof. The first integral is

∫_{G^{F} /Z^{F}} τ_{γ}(g) dg^{∗}.

Lemma 10.0.6 shows that this integral converges and allows us to rewrite it as

∫_{G^{F} /Z^{F}} τ_{γ}(g) dg^{∗} = ∫_{G^{F} /Z^{F}} dg^{∗} ∫_{L} τ_{γ}(gl) dl
= ∫_{G^{F} /Z^{F}} dg^{∗} ∫_{L} dl ∫_{L_{γs}} f_{γ}(gli) di
= ∫_{G^{F} /Z^{F}} dg^{∗} ∫_{L} f_{γ}(gl) dl,

absorbing i into the integral over L.

To see that the integrals are independent of L, it suffices to show they are unchanged if we replace L by a compact open subgroup L′ < L. We have

∫_{G^{F} /Z^{F}} dg^{∗} ∫_{L_{γs}} f_{γ}(gl) dl = ∫_{G^{F} /Z^{F}} dg^{∗} ∫_{L_{γs}} dl ∫_{(L′)_{γs}} f_{γ}(gll′) dl′.

By Lemma 10.0.6 again, the integral over (L′)_{γ_{s}} has compact support as a function on G^{F} /Z^{F} × L_{γ_{s}}. Hence we may switch the integrals over G^{F} /Z^{F} and L_{γ_{s}}. The claim follows. □

Now we can prove Lemma 10.0.4. From Equation (29), we have

\frac{meas_{dg}(G_{J}^{F})}{meas_{dz}(^{0}Z^{F})}R(G, S, θ)(γ) = ∫_{G^{F} /Z^{F}} dg^{∗} ∫_{L} Ṙ_{S,θ}^{G_{J}}(^{gl}γ) dl

= ∑_{x∈G_{J}^{F} /G_{J}^{+F}} ∫_{G^{F} /Z^{F}} dg^{∗} ∫_{L} f_{γ}(xgl) dl.

DEPTH-ZERO SUPERCUSPIDAL L-PACKETS AND THEIR STABILITY 57

Absorbing x into the integral over G^{F} /Z^{F} and using Lemma 10.0.7 , we get

\frac{meas_{dg}(G_{J}^{F})}{meas_{dz}(^{0}Z^{F})}R(G, S, θ)(γ) = |G_{J}^{F} | ∫_{G^{F} /Z^{F}} dg^{∗} ∫_{L_{γs}} f_{γ}(gi) di.

During the rest of this calculation only, we use the abbreviations

N := N(G, S)^{F} , U := G_{J}^{+F} , H = G_{γ_{s}}.

Let D(γ_{s}, S) be as in the proof of 10.0.6. The integral over L_{γ_{s}} is supported on

E(γ_{s}, S) = ∐_{d∈D(γ_{s},S)} ∐_{n̄∈N/N_{d}} UndH^{F} /Z^{F} ,

where N_{d} = ^{d}H ∩ N. Consequently, we have

\frac{meas_{dg}(G_{J}^{F})}{meas_{dz}(^{0}Z^{F})}R(G, S, θ)(γ) = |G_{J}^{F} | ∑_{d∈D(γ_{s},S)} ∑_{n̄∈N/N_{d}} ∫_{UndH^{F} /Z^{F}} dg^{∗} ∫_{L_{γs}} f_{γ}(gi) di. (31)

Note that the map (d, n̄) ↦ (nd)^{−1} · (S, θ) induces a bijection

D(γ_{s}, S) × N/N_{d} ⟶^{∼} T̂(γ_{s})/H.

Hence the sum in Equation (31) matches the sum in Lemma 10.0.4.

Fix d ∈ D(γ_{s}, S) and n̄ ∈ N/N_{d}, and set

J′ = (nd)^{−1}J, U′ = G_{J′}^{+F} = Ad(nd)^{−1}U, (S′, θ′) = (nd)^{−1} · (S, θ), γ′_{s} = ^{nd}γ_{s}. We then have

∫_{UndH^{F} /Z^{F}} dg^{∗} ∫_{L_{γs}} f_{γ}(gi) di = \frac{meas_{dg}(U)}{meas_{dh}(H ∩ U′)} ∫_{H^{F} /Z^{F}} dh^{∗} ∫_{L_{γs}} f_{γ}(ndhi) di.

From the definitions, we have

f_{γ}(ndhi) = θ̇(γ′_{s}) · Q̇_{S}^{(G_{J})_{\overline{γ′}_{s}}}(^{ndhi}γ_{u}) = θ̇′(γ_{s}) · Q̇_{S′}^{(G_{J′})_{\overline{γ}_{s}}}(^{hi}γ_{u}).

As in the proof of Lemma 8.0.11, the projection H ∩ G_{J′} ⟶ G_{J′} allows us to identify

(G_{J′})_{\overline{γ}_{s}} = H_{J′},

so that

f_{γ}(ndhi) = θ̇′(γ_{s}) · Q̇_{S′}^{H_{J′}}(^{hi}γ_{u}).

Since U = G_{J}^{+F} and H_{J′}^{+F} = H ∩ U′, we have

∫_{UndH^{F} /Z^{F}} dg^{∗} ∫_{L_{γs}} f_{γ}(gi) di = \frac{meas_{dg}(G_{J}^{+F})}{meas_{dh}(H_{J′}^{+F})} ∫_{H^{F} /Z^{F}} dh^{∗} ∫_{L_{γs}} θ̇′(γ_{s}) · Q̇_{S′}^{H_{J′}}(^{hi}γ_{u}) di.

Since the center of H is contained in the F -minisotropic torus S′, we conclude that Z is the group of K-rational points of the maximal k-split torus in the center of G_{γ_{s}}. Hence, from the definition of R(H, S′, 1), we have

|G_{J}^{F} | ∫_{UndH^{F} /Z^{F}} dg^{∗} ∫_{L_{γs}} f_{γ}(gi) di = \frac{meas_{dg}(G_{J}^{F})}{meas_{dz}(^{0}Z^{F})}θ′(γ_{s}) · R(H, S′, 1)(γ_{u}).

Inserting this into Equation (31) completes the proof of Lemma 10.0.4.

10.1. Characters in a simple case. We illustrate Lemma 10.0.4 in the simple case where γ ∈ G_{0}^{F} is strongly regular and topologically semisimple. We have γ_{s} = γ and γ_{u} = 1.

Let T̂ ⊂ 𝔗̂(G) be a G^{F} -orbit. We write

R(G, T̂) := R(G, S′, θ′)

for any (S′, θ′) ∈ T̂. Then T̂(γ) is nonempty if and only if (S, θ) ∈ T̂, where S = G_{γ} and θ ∈ Irr_{0}(S^{F}), in which case we have

T̂(γ) = {(S, n_{∗}θ) : n ∈ N(G, S)^{F} /S^{F} }.

Since R(G_{γ}, S, 1)(1) = 1, Lemma 10.0.4 gives the formula

R(G, T̂)(γ) = ∑_{n∈N(G,S)^{F} /S^{F}} n_{∗}θ(γ)

if T̂(γ) is nonempty, and R(G, T̂)(γ) = 0 otherwise.

Return now to the situation of Section 9.6, with F = F_{u} etc. By Lemma 9.6.1, T̂ contains

(S_{λ}, θ_{λ}) = Ad(q_{λ}) · (T, χ),

for some λ ∈ r^{−1}(ω). If S is not GF_{u}-conjugate to S_{λ}, then R(G, T̂)(γ) = 0. Suppose S = ^{h}S_{λ} for some h ∈ GF_{u}. Let θ = h_{∗}θ_{λ}, so that

T̂(γ) = {(S, n_{∗}θ) : n ∈ N(G, S)F_{u}/S}.

From Lemmas 2.11.2 and 10.0.4 it follows that

R(G, T̂)(γ) = ∑_{y∈W_{o,λ}^{wϑ}}(hq_{λ}y)_{∗}χ(γ).

From Lemma 9.3.1 we get the following character values.

Proposition 10.1.1. Suppose γ ∈ GF_{u} ; 0 is strongly regular and topologically semisimple. Then Θ_{ρ_{λ}}(γ) = 0 unless γ lies in a GF_{u}-conjugate of S_{λ}, and if γ ∈ ^{h}S_{λ} for h ∈ GF_{u}, we have

Θ_{ρ_{λ}}(γ) = ε(G_{λ}, T_{λ}) ∑_{y∈W_{o,λ}^{wϑ}}(hq_{λ}y)_{∗}χ(γ).

11. Reduction formula for stable character integrals

In this section we prove the analogue of Lemma 10.0.4 for stable character integrals. Fix a G-stable class T̂_{st} ⊂ 𝔗̂(G). Recall from Lemma 9.4.3 that there is v ∈ N/T and χ ∈ Irr_{0}(T^{F_{v}}) such that every pair (S, θ) ∈ T̂_{st} is of the form

(S, θ) = (^{g}T , g_{∗}χ)

for some g ∈ G with g^{−1}F (g) ∈ v.

Given γ ∈ G_{0}^{F} regular semisimple, with topological Jordan decomposition γ = γ_{s}γ_{u}, we define

T̂_{st}(γ_{s}) := {(S, θ) ∈ T̂_{st} : γ_{s} ∈ S}.

This set is a finite disjoint union

T̂_{st}(γ_{s}) = ∐_{î∈Î(γ_{s})} T̂_{st}(γ_{s}, î),

where each T̂_{st}(γ_{s}, î) is a G_{γ_{s}}-stable class in 𝔗̂(G_{γ_{s}}), and Î(γ_{s}) is an index set for these G_{γ_{s}}-stable classes.

Applying p_{1}, we have

T_{st}(γ_{s}) := p_{1}[T̂_{st}(γ_{s})] = ∐_{i∈I(γ_{s})} T_{st}(γ_{s}, i),

where each T_{st}(γ_{s}, i) is a G_{γ_{s}}-stable class in 𝔗(G_{γ_{s}}), and I(γ_{s}) is an index set for these G_{γ_{s}}-stable classes. There is a surjective map î ↦ i from Î(γ_{s}) to I(γ_{s}), such that

p_{1}[T̂_{st}(γ_{s}, î)] = T_{st}(γ_{s}, i).

The fiber of this map over i ∈ I(γ_{s}) has cardinality

N(i) := |N(G_{γ_{s}}, S^{F})/S|,

where S is any element of T_{st}(γ_{s}, i).

For any G_{γ_{s}}-stable class T_{st}^{1} ⊂ 𝔗(G_{γ_{s}}), we set

Q(G_{γ_{s}}, T_{st}^{1}) := ∑_{S∈T_{st}^{1}/G_{γs}^{F}} |N(G_{γ_{s}}, S^{F})/N(G_{γ_{s}}^{F}, S)| · R(G_{γ_{s}}, S, 1).

This will turn out to be a stable p-adic analogue of a Green function. We will consider the sums Q(G_{γ_{s}}, T_{st}(γ_{s}, i)), for i ∈ I(γ_{s}). But first we need more notation.

For each î ∈ Î(γ_{s}) and S ∈ T_{st}(γ_{s}, î), we have a character sum

θî ; S = ∑_{θ∈pî ; 2(pî ; 1)^{−1}(S)} θ,

where pî ; 1 (resp. pî ; 2) is the restriction of p_{1} (resp. p_{2}) to T̂_{st}(γ_{s}, î).

60 STEPHEN DEBACKER AND MARK REEDER

In fact, this sum is independent of S: Given two tori S, S′ ∈ T_{st}(γ_{s}, i), we have

θî ; S(γ_{s}) = θî ; S′(γ_{s}),

as a special case of Lemma 9.4.4. We therefore define

θχ ; î(γ_{s}) := θî ; S(γ_{s}),

for any S ∈ T_{st}(γ_{s}, i). Note that the sum

θ_{i}^{χ}(γ_{s}) := ∑_{î↦i} θχ ; î(γ_{s})

is none other than the character sum θ_{S}^{χ}(γ_{s}), for any S ∈ T_{st}(γ_{s}, i), as defined in Section 9.4.

Finally, recall (Section 10) that for each G^{F} -orbit T̂ ⊂ T̂_{st}, we have defined

T̂(γ_{s}) = {(S, θ) ∈ T̂ : γ_{s} ∈ S}.

Now we are ready to state the reduction formula for stable character integrals. Lemma 11.0.2. For γ ∈ G_{0}^{F} regular semisimple, with topological Jordan decomposition γ = γ_{s}γ_{u}, we have

R(G, T̂_{st})(γ) = ∑_{i∈I(γ_{s})} \frac{θ_{i}^{χ}(γ_{s})}{N(i)} · Q(G_{γ_{s}}, T_{st}(γ_{s}, i))(γ_{u}).

Proof. Using Lemma 10.0.4, we compute

R(G, T̂_{st})(γ) = ∑_{T̂∈T̂_{st}/G^{F}} R(G, T̂)(γ)

= ∑_{T̂∈T̂_{st}/G^{F}} ∑_{(S,θ)∈T̂(γ_{s})/G_{γs}^{F}} θ(γ_{s}) · R(G_{γ_{s}}, S, 1)(γ_{u})

= ∑_{î∈Î(γ_{s})} ∑_{(S,θ)∈T̂_{st}(γ_{s},î)/G_{γs}^{F}} θ(γ_{s}) · R(G_{γ_{s}}, S, 1)(γ_{u})

= ∑_{î∈Î(γ_{s})} ∑_{S∈T_{st}(γ_{s},i)/G_{γs}^{F}} \frac{θî ; S(γ_{s})}{|N(G_{γ_{s}}^{F}, S)/S^{F} |} · R(G_{γ_{s}}, S, 1)(γ_{u})

= ∑_{i∈I(γ_{s})} θ_{i}^{χ}(γ_{s}) ∑_{S∈T_{st}(γ_{s},i)/G_{γs}^{F}} \frac{1}{|N(G_{γ_{s}}^{F}, S)/S^{F} |} · R(G_{γ_{s}}, S, 1)(γ_{u})

= ∑_{i∈I(γ_{s})} \frac{θ_{i}^{χ}(γ_{s})}{N(i)} · Q(G_{γ_{s}}, T_{st}(γ_{s}, i))(γ_{u}).

□

11.1. A bijection between stable classes of unramified tori. Lemma 11.0.2 reduces the proof of stability to the topologically unipotent set, as follows. Let T_{st}(γ_{s})/G_{γ_{s}} denote the set of G_{γ_{s}}- stable classes in T_{st}(γ_{s}). So T_{st}(γ_{s})/G_{γ_{s}} is indexed by I(γ_{s}).

We now assume that γ ∈ G_{0}^{F} is in fact strongly regular semisimple, that is, the centralizer of γ in G is a torus. Then if g ∈ G and^{g}γ is again in G^{F}, we can construct a bijection

ι_{g} : T_{st}(γ_{s})/G_{γ_{s}} ⟶^{∼} T_{st}(^{g}γ_{s})/G_{^{g}γ_{s}}

as follows. Let S ∈ T_{st}(γ_{s}). Since γ ∈ G^{F} and has connected centralizer, we have g^{−1}F (g) ∈ Z^{1}(F, G_{γ_{s}}).

Let Z_{γ_{s}} be the maximal k-split torus in the center of G_{γ_{s}}. Since S is F -minisotropic in G_{γ_{s}}, the group of co-invariants of F in X_{∗}(S) has the same rank as X_{∗}(Z_{γ_{s}}). It then follows from [35, Thm.1.2] (see also Lemma 2.6.1) that the map H^{1}(F, S) ⟶ H^{1}(F, G_{γ_{s}}) is surjective.

This means there is h ∈ G_{γ_{s}} such that (gh)^{−1}F (gh) ∈ S. Hence Ad(gh) : S ⟶ ^{gh}S commutes with F, so^{gh}S ∈ T_{st}(^{g}γ_{s}).

Suppose also S′ ∈ T_{st}(γ_{s}), and (S′)^{F} = ^{k}(S^{F}) for some k ∈ G_{γ_{s}}. This implies that k^{−1}F (k) ∈ S. As above, there exists h′ ∈ G_{γ_{s}} such that (gh′)^{−1}F (gh′) ∈ S′. Then the element j := gh′kh^{−1}g^{−1} ∈ G_{^{g}γ_{s}} satisfies j^{−1}F (j) ∈ ^{gh}S,^{jgh}S = ^{gh′}S′, which means that^{gh}S is G_{^{g}γ_{s}}-stably conjugate to ^{gh′}S′. Therefore, sending the G_{γ_{s}}-stable class of S to the G_{^{g}γ_{s}}-stable class of ^{gh}S gives a well-defined injection ι_{g}, as above. It is straightforward to check that ι_{(g^{−1})} is the inverse of ι_{g}, so ι_{g} is actually a bijection.

We may view ι_{g} as a bijection on index sets:

ι_{g} : I(γ_{s}) ⟶ I(^{g}γ_{s}).

This map has the property that

N(i) = N(ι_{g}(i)),

for each i ∈ I(γ_{s}).

Lemma 11.1.1. Let γ ∈ G_{0}^{F} be strongly regular semisimple, with topological Jordan decomposition γ = γ_{s}γ_{u}, and let g ∈ G be such that ^{g}γ ∈ G^{F}. Let T̂_{st} be a G-stable class in 𝔗̂(G), and assume that for all i ∈ I(γ_{s}) we have

Q(G_{γ_{s}}, T_{st}(γ_{s}, i))(γ_{u}) = Q(G_{^{g}γ_{s}}, T_{st}(^{g}γ_{s}, ι_{g}(i)))(^{g}γ_{u}).

Then we have

R(G, T̂_{st})(γ) = R(G, T̂_{st})(^{g}γ).

Proof. From Lemma 11.0.2 we have

\begin{aligned}& R(G, T̂_{st})(γ) & = ∑_{i∈I(γ_{s})} \frac{θ_{i}^{χ}(γ_{s})}{N(i)} · Q(G_{γ_{s}}, T_{st}(γ_{s}, i))(γ_{u}). \\ (32) \end{aligned}

On the other hand, by Lemma 9.4.4 again (this time in full force) we have

θ_{i}^{χ}(γ_{s}) = θ_{ι_{g}(i)}^{χ}(^{g}γ_{s}).

It follows that

(33)  R(G, T̂_{st})(^{g}γ) = ∑_{i∈I(γ_{s})} \frac{θ_{i}^{χ}(γ_{s})}{N(i)} · Q(G_{^{g}γ_{s}}, T_{st}(^{g}γ_{s}, ι_{g}(i)))(^{g}γ_{u}),

whence the result. □

11.2. Stable characters in a simple case. We illustrate Section 11.1 by considering the stable version of Section 10.1. As in the latter section, we suppose γ ∈ G^{F} is strongly regular and topologically semisimple, and let S = G_{γ}. Let T̂_{st} ⊂ 𝔗̂(G) be a G-stable class.

Let us describe the objects in 11.0.2 in this case. If T̂_{st}(γ) is empty, then R(G, T̂_{st})(γ) = 0. Assume T̂_{st}(γ) is nonempty. Then there is θ ∈ Irr_{0}(S^{F}) such that

T̂_{st}(γ) = {(S, n_{∗}θ) : n ∈ N(G, S^{F})/S}.

Thus, we may identify Î(γ) = N(G, S^{F})/S, and for each n ∈ Î(γ), we have

T̂_{st}(γ, n) = {(S, n_{∗}θ)}.

The index set I(γ) consists of a single element, i, and

Q(G_{γ}, T_{st}(γ, i))(γ_{u}) = Q(S, {S})(1) = 1.

In terms of tori, the map ι_{g} simply sends S to ^{g}S. Hence the conditions of Lemma 11.1.1 hold trivially, so that R(G, T̂_{st}) is constant on the G-stable class of γ.

Lemma 11.0.2 gives the formula

R(G, T̂_{st})(γ) = ∑_{n∈N(G,S^{F})/S} n_{∗}θ(γ).

From Lemma 9.6.3 it follows that the sum of characters in the L-packet Π(φ, ω) is constant on the G-stable class of γ.

12. Transfer to the Lie algebra

Lemma 11.1.1 reduces the proof of stability to the following.

Lemma 12.0.1. Assume as above that γ ∈ G_{0}^{F} is strongly regular semisimple, and g ∈ G is such that^{g}γ ∈ G^{F}. Let T_{st} be a G_{γ_{s}}-stable class in 𝔗(G_{γ_{s}}). Then

Q(G_{γ_{s}}, T_{st})(γ_{u}) = Q(G_{^{g}γ_{s}}, ι_{g}T_{st})(^{g}γ_{u}).

We will prove Lemma 12.0.1 under some restrictions on k, to be installed as they are needed. The first step in the proof of Lemma 12.0.1 is to transfer the calculation to the Lie algebras 𝔤_{γ_{s}} and 𝔤_{^{g}γ_{s}} of G_{γ_{s}} and G_{^{g}γ_{s}} respectively. We then invoke a deep result of Waldspurger [63], which states that, for groups which are inner forms of each other, the fundamental lemma for the Lie algebra is true.

12.1. Orbital Integrals. Fix γ and g as in the statement of Lemma 12.0.1. Since the calculation takes place mostly in the groups G_{γ_{s}} and G_{^{g}γ_{s}}, we adjust the notation slightly for clarity. Let H = G_{γ_{s}}, and let 𝔥 = Lie(H) be the Lie algebra of H. We fix an additive character Λ : k ⟶ ℂ^{×} which is trivial on the prime ideal of R but non-trivial on R. Suppose B is a nondegenerate, symmetric, ⟨F ⟩⋉H-invariant bilinear form on 𝔥. For f ∈ C_{c}^{∞}(𝔥^{F}), the space of locally constant, compactly supported functions on 𝔥^{F}, we define the Fourier transform (with respect to B) of f by

f̂(X) = ∫_{𝔥^{F}} f(Y) · Λ(B(X, Y)) dY,

where dY is Haar measure on 𝔥, normalized as in Section 5.

Suppose X is a regular semisimple element of 𝔥^{F}. For f ∈ C_{c}^{∞}(𝔥^{F}) we define μ_{X}^{H^{F}} (f), the orbital integral of f with respect to X, by

μ_{X}^{H^{F}} (f) := ∫_{H^{F} /(C′_{H}(X))^{F}} f(^{h}X) \frac{dh}{dt}

where C′_{H}(X) is the maximal unramified torus in the torus C_{H}(X) and dh, dt are Haar measures on H^{F}, C′_{H}(X)^{F}, respectively, normalized as in Section 5.

Remark 12.1.1. If X′ ∈ 𝔥^{F} is H-conjugate to X, then the tori C′_{H}(X) and C′_{H}(X′) are Hconjugate. Consequently, if dt′ denotes the Haar measure on C′_{H}(X′), it follows that the measures \frac{dh}{dt} and \frac{dh}{dt′} determine the same multiple of the top degree form on the orbit^{H}X = ^{H}X′.

We define μ̂_{X}^{H^{F}} (f) := μ_{X}^{H^{F}} (f̂) for f ∈ C_{c}^{∞}(𝔥^{F}). In this way, we have a distribution μ̂_{X}^{H^{F}} on C_{c}^{∞}(𝔥^{F}). Thanks to Harish-Chandra [25, Theorem 4.4], we know that μ̂_{X}^{H^{F}} is represented on 𝔥^{F} by a function, which we also denote by μ̂_{X}^{H^{F}}. (The same result is true for the Fourier transform of any orbital integral.)

12.2. A result of Waldspurger. In this section, H is any connected reductive k-group splitting over K. As usual, F is the Frobenius action on both H := H(K), and 𝔥 := Lie(H). For X ∈ 𝔥^{F} regular semisimple, write

[Ad(H)X]^{F} = ∐_{i} Ad(H^{F})X_{i}

where the X_{i} run over a (finite) set of representatives for the Ad(H^{F})-orbits in [Ad(H)X]^{F} (see Section 2.9.1). We set

Ŝ_{X}^{𝔥} = ∑_{i} μ̂_{X_{i}}^{H^{F}} .

The measures used for each orbital integral are compatible, in the sense of Remark 12.1.1.

Let H^{∗} denote a k-quasi-split inner form of H, and let H_{ad}^{∗} be the adjoint group of H^{∗}. Let H^{∗} and H_{ad}^{∗} denote the groups of K-rational points of H^{∗} and H_{ad}^{∗}, respectively, and let F^{∗} denote the action of Frobenius on H^{∗}, H_{ad}^{∗} and 𝔥^{∗} = Lie(H^{∗}). Choose an inner twist

ϕ: H → H^{∗}.

That is, ϕ is a K-isomorphism, and there is h_{ϕ}^{∗} ∈ H_{ad}^{∗} such that

Ad(h_{ϕ}^{∗}) = F^{∗} ∘ ϕ ∘ F^{−1} ∘ ϕ^{−1} ∈ Aut_{K}(H^{∗}).

Here we implicitly use the isomorphism H^{1}(F, H_{ad}) = H^{1}(k, H_{ad}), see Section 2.2. The choice of ϕ defines an injective map S_{ϕ} from the set of stable regular semisimple orbits in 𝔥^{F} to the set of stable regular semisimple orbits in (𝔥^{∗})^{F^{∗}}, as follows. If X ∈ 𝔥^{F}, then F^{∗}(dϕ(X)) = Ad(h_{ϕ}^{∗})dϕ(X), so the Ad(H^{∗})-orbit of dϕ(X) is F^{∗}-stable. If X is regular semisimple, then so too is dϕ(X). The existence of an F^{∗}-stable Kostant section shows that the Ad(H^{∗})-orbit of dϕ(X) contains an F^{∗}-fixed point X^{∗} (see, for example, [56, 9.5] or [36]). Finally, S_{ϕ} sends [Ad(H)X]^{F} to [Ad(H^{∗})X^{∗}]^{F^{∗}}.

Suppose now that H′ is any inner form of H. Let H′ denote the group of K-rational points of H′ and let F′ denote the action of Frobenius on H′ and 𝔥′ = Lie(H′). Suppose X ∈ 𝔥^{F} and X′ ∈ (𝔥′)^{F′} are regular semisimple elements, and ϕ: H → H^{∗} and ϕ′: H′ → H^{∗} are inner twists. We say that X and X′ are (ϕ, ϕ′)-comparable provided that

S_{ϕ}([Ad(H)X]^{F} ) = S_{ϕ′}([Ad(H′)X′]^{F′})

as stable regular semisimple orbits in (𝔥^{∗})^{F^{∗}}.

Example 12.2.1. Take H = G_{γ_{s}} as in the situation of Section 12.1. Let log: G_{0^{+}} → 𝔤 be any injective ⟨F ⟩ ⋉ G-equivariant map which takes regular semisimple elements to regular semisimple elements. (The existence of such a map with just these properties follows from [9, p. 333, §7.6, Proposition 10].) Then C_{G}(γ) is a torus in H, and F (g) = gs for some s ∈ C_{H}(γ). Moreover, Ad(g) : H ⟶ H′ := ^{g}H is an inner twist, with F′ = F. Let X := log(γ_{u}). From [29, Theorem 13.4(a)] it follows that X ∈ 𝔥 and ^{g}X ∈ ^{g}𝔥 are regular semisimple elements. Since F and Ad(s) fix γ_{u}, it follows that F (X) = X, and F (^{g}X) = ^{g}X.

Suppose ϕ: H → H^{∗} is an inner twist, and let X^{∗} ∈ [Ad(H^{∗})dϕ(X)]^{F^{∗}}. One checks that the map ϕ′ := ϕ ∘ Ad(g^{−1}): ^{g}H → H^{∗} is also an inner twist, and that X^{∗} ∈ [Ad(H^{∗})dϕ′(^{g}X)]^{F^{∗}}. It follows that X and^{g}X are (ϕ, ϕ′)-comparable.

Example 12.2.2. Continue with the notation of Example 12.2.1 and also Section 11.1. Let T_{st} be an H-stable class in 𝔗(H), and let T′_{st} = ι_{g}T_{st}, an H′-stable class in 𝔗(H′), be as in Section 11.1. Suppose S_{0} ∈ T_{st} and X_{0} is a 𝔤-regular element of Lie(S_{0})^{F}. Let X ∈ [Ad(H)X_{0}]^{F}. Note that X is regular in 𝔤. As in the definition of ι_{g}, there is h ∈ H such that (gh)^{−1}F (gh) ∈ C_{G}(X), and the elements X and X′ := ^{gh}X ∈ [Ad(H′)(^{g}X_{0})]^{F} are (ϕ, ϕ ∘ Ad(g)^{−1})-comparable.

Lemma 12.2.3. Let ϕ: H → H^{∗} and ϕ′: H′ → H^{∗} denote inner twists. Suppose X, Y (resp. X′, Y′) are regular semisimple elements in 𝔥^{F} (resp. (𝔥′)^{F′}). If X and X′ are (ϕ, ϕ′)-comparable elliptic elements and Y and Y′ are (ϕ, ϕ′)-comparable elements, then we have

Ŝ_{X}^{𝔥}(Y) = ε(H, H′) · Ŝ_{X′}^{𝔥′}(Y′).

Remark 12.2.4. The above lemma may be viewed as more evidence for Kottwitz’ sign conjecture [33].

Proof. Without loss of generality, H′ is k-quasi-split, and we may replace H′ by H^{∗}. Waldspurger has already shown [63, Théorème 1.5] that for X, Y, X′, Y′ as in the statement of the lemma, we have

Ŝ_{X}^{𝔥}(Y) = c · Ŝ_{X′}^{𝔥^{∗}}(Y′)

where c is an eighth root of unity. (In the notation of [63], this is actually the special case s = 1, ξ = I of [63, Théorème 1.5], and c = γ_{Λ}(𝔥^{∗})/γ_{Λ}(𝔥).) We will give two proofs that c = ε(H, H^{∗}).

The first proof uses Shalika germs. For all n ∈ ℤ we have

Ŝ_{X}^{𝔥}(ϖ^{n}Y) = c · Ŝ_{X′}^{𝔥^{∗}}(ϖ^{n}Y′).

From Harish-Chandra [25, Theorem 5.1.1], for all n ∈ ℤ sufficiently large we have

\begin{aligned}Ŝ_{X}^{𝔥}(ϖ^{2n}Y) & = ∑_{O∈O_{𝔥}(0)} c_{O}^{𝔥}(X) · μ̂_{O}(ϖ^{2n}Y) \\ & = c_{0}^{𝔥}(X) + ∑_{O∈O_{𝔥}(0)\\{0\}} c_{O}^{𝔥}(X) · q^{−n·dim O} · μ̂_{O}(Y) \end{aligned}

where O_{𝔥}(0) denotes the set of nilpotent H^{F} -orbits in 𝔥^{F}, the c_{O}^{𝔥}(X) are complex constants, and 0 denotes the zero orbit {0}. A similar statement is true for Ŝ_{X′}^{𝔥^{∗}}. Thus,

c_{0}^{𝔥}(X) = lim_{n→∞} Ŝ_{X}^{𝔥}(ϖ^{2n}Y)

= lim_{n→∞} c · Ŝ_{X′}^{𝔥^{∗}}(ϖ^{2n}Y′)

= c · c_{0}^{𝔥^{∗}}(X′).

Let X_{1}, X_{2}, …, X_{m} be representatives for the H^{F} -orbits in [Ad(H)X]^{F}. From [25, Theorem 8.1] we have

c_{0}^{𝔥}(X) = ∑_{j=1}^{m} Γ_{0}^{C_{𝔥}(X_{j})}(X_{j}),

where Γ_{0}^{C_{𝔥}(X_{j})}(X_{j}) denotes the evaluation of the (unnormalized) Shalika germ corresponding to the zero orbit at X_{j}.

If the center of H^{F} is compact, then so too is the center of (H^{∗})^{F^{∗}}. Thanks to Rogawski [49] we have

Γ_{0}^{C_{𝔥}(X_{j})}(X_{j}) = \frac{ε(H, Z)}{Deg(St^{H})}.

Thus, if the center of H^{F} is compact, we conclude from the above, the fact that Deg(St^{H}) > 0, and the fact that c is an eighth root of unity that c = ε(H, H^{∗}).

Suppose that the center Z^{F} of H^{F} is not compact. Let H_{d} denote the derived group of H and let 𝔥_{d} denote the Lie algebra of H_{d} = H_{d}(K). The center of H_{d} is finite and H^{F} /(H_{d}^{F})Z^{F} is a finite group. Without loss of generality, we assume X ∈ 𝔥_{d}^{F}. From Lemma 2.9.1, we have that two regular semisimple elements of 𝔥_{d}^{F} are H-stably conjugate if and only if they are H_{d}-stably conjugate. However, since two regular semisimple elements of 𝔥_{d}^{F} may be H^{F} -conjugate without being H_{d}^{F} -conjugate, for 1 ≤ i ≤ m we introduce the group

H_{i}^{F} := {h ∈ H^{F} : there is an h′ ∈ H_{d}^{F} such that ^{h′h}X_{i} = X_{i}}.

We have H_{d}^{F} Z^{F} ⊴ H_{i}^{F} ⊴ H^{F}. Thus, we can write

Ŝ_{X}^{𝔥_{d}} = ∑_{i=1}^{m} ∑_{h̄∈H^{F} /H_{i}^{F}} μ̂_{^{h^{−1}}X_{i}}^{H_{d}^{F}}.

Suppose we can show that the restriction of Ŝ_{X}^{𝔥} to 𝔥_{d}^{F} equals e · Ŝ_{X}^{𝔥_{d}} for some constant e > 0. We would then have c_{0}^{𝔥}(X) = e · c_{0}^{𝔥_{d}}(X). Arguing as in the previous paragraph, we would again conclude that c = ε(H, H^{∗}).

To complete the proof, we now show that such a constant e exists. We will use Harish-Chandra’s integral formula for the Fourier transform of a regular semisimple orbital integral [25, Lemma 7.9]. Since we only wish to establish the positivity of e, in what follows we are not careful about specifying our invariant measures nor about accounting for the (positive) constants that occur. Let L be a compact open subgroup of H^{F} which lies in H_{d}^{F} Z^{F}. There is a positive constant const so that for regular semisimple Y ∈ 𝔥_{d}^{F}

Ŝ_{X}^{𝔥}(Y) = const · ∑_{i} ∫_{H^{F} /Z^{F}} dg^{∗} ∫_{L} Λ(B(^{gℓ}Y, X_{i})) dℓ

= const · ∑_{i} ∑_{h̄∈H^{F} /H_{i}^{F}} ∑_{ḡ_{1}∈H_{i}^{F} /H_{d}^{F} Z^{F}} ∫_{H_{d}^{F} Z^{F} /Z^{F}} dg_{2}^{∗} ∫_{L} Λ(B(^{g_{1}g_{2}ℓ}Y,^{h^{−1}}X_{i})) dℓ

which, from the definition of H_{i}^{F}, becomes

= const · ∑_{i} ∑_{h̄∈H^{F} /H_{i}^{F}} |H_{i}^{F} /H_{d}^{F} Z^{F} | · ∫_{H_{d}^{F} Z^{F} /Z^{F}} dg_{2}^{∗} ∫_{L} Λ(B(^{g_{2}ℓ}Y,^{h^{−1}}X_{i})) dℓ.

We claim that for 1 ≤ i, j ≤ m we have

|H_{i}^{F} /H_{d}^{F} Z^{F} | = |H_{j}^{F} /H_{d}^{F} Z^{F} | .

In fact, we will show that the group H_{i}^{F} is independent of i. Note that H_{i}^{F} /H_{d}^{F} Z^{F} can be characterized as the set of cosets in H^{F} /H_{d}^{F} Z^{F} which intersect (C_{H}(X_{i}))^{F} nontrivially. Thus, it is enough to show that for h ∈ H^{F} we have

h(H_{d}^{F} Z^{F}) ∩ (C_{H}(X_{i}))^{F} ≠ ∅ ⟺ h(H_{d}^{F} Z^{F}) ∩ (C_{H}(X_{j}))^{F} ≠ ∅.

Suppose h ∈ H^{F} and g ∈ H_{d}^{F} Z^{F} so that hg ∈ (C_{H}(X_{i}))^{F}. It is enough to produce a g′ ∈ H_{d}^{F} Z^{F} such that hg′ ∈ (C_{H}(X_{j}))^{F}. Since X_{i} and X_{j} are H_{d}-stably conjugate, there is an h′ ∈ H_{d} so that ^{h′}X_{i} = X_{j}. Since C_{H}(X_{i}) is abelian, this implies that ^{h′}((C_{H}(X_{i}))^{F}) = (C_{H}(X_{j}))^{F}. Consequently, ^{h′}(hg) ∈ (C_{H}(X_{j}))^{F} and ^{h′}(hg) = h(h^{−1}h′hg(h′)^{−1}) ∈ hH_{d}Z^{F}. Set g′ := (h^{−1}h′hg(h′)^{−1}). Note that g ∈ H_{d}^{F} Z^{F} implies g′ ∈ H_{d}(Z^{F}). But also g′ ∈ h^{−1}(C_{H}(X_{j}))^{F} ≤ H^{F}, so in fact g′ ∈ H_{d}^{F} Z^{F} and hg′ ∈ (C_{H}(X_{j}))^{F}, as desired.

Therefore,

Ŝ_{X}^{𝔥}(Y) = const′ · ∑_{i} ∑_{h̄∈H^{F} /H_{i}^{F}} ∫_{H_{d}^{F} Z^{F} /Z^{F}} dg_{2}^{∗} ∫_{L} Λ(B(^{g_{2}ℓ}Y,^{h^{−1}}X_{i})) dℓ

= const′ · ∑_{i} ∑_{h̄∈H^{F} /H_{i}^{F}} ∫_{H_{d}^{F} /(H_{d}^{F} ∩Z^{F})} dg_{2}^{∗} ∫_{L} Λ(B(^{g_{2}ℓ}Y,^{h^{−1}}X_{i})) dℓ.

□

12.3. Another calculation of Waldspurger’s sign. In this section we give a second proof of Lemma 12.2.3 in terms of our pure inner forms G_{λ}. This proof continues in the vein of [63].

For λ ∈ X_{w}, we have a pure inner form G_{λ} of G with Frobenius F_{λ} = Ad(u_{λ}) ∘ F. In particular G = G_{0} is k-quasi-split. Note that G_{λ} = G as groups; the subscript indicates the variation in k-structure. To simplify the notation, we write σ := wϑ, and set Σ := ⟨σ⟩.

We define an inner twisting ϕ_{λ} : G_{λ} ⟶ G_{0} by ϕ_{λ} = Ad(h_{λ}), where h_{λ} = p_{0}p_{λ}^{−1} and p_{λ}, p_{0} ∈ G satisfy the equations

(34)  p_{0}^{−1} F(p_{0}) = ẇ,  p_{λ}^{−1}u_{λ} F(p_{λ}) = t_{λ}ẇ

of Chapter 2.7. Let Φ(T) denote the set of roots of T in G. Likewise, let T_{λ} = Ad(p_{λ})T, and let Φ(T_{λ}) denote the set of roots of T_{λ} in G_{λ}.

The map Ad(p_{λ}) : T ⟶ T_{λ} intertwines F_{w} on T with F_{λ} on T_{λ}. It induces a map Φ(T) ⟶ Φ(T_{λ}) given by

α ↦ α_{λ} := α ∘ Ad(p_{λ})^{−1},

satisfying

F_{λ} ·α_{λ} = (σ · α)_{λ}.

(Recall that F_{w} acts on Φ(T) via σ.)

Fix λ ∈ X_{w}. By Hilbert’s Theorem 90, there exists a set {E_{α_{λ}} : α ∈ Φ(T)} of T_{λ}-root vectors in 𝔤 having the property that

F_{λ} ·E_{α_{λ}} = E_{(σ·α)_{λ}}.

The transformed root vectors

E_{α}^{∗} := ϕ_{λ}(E_{α_{λ}})

are only preserved by F up to scalar multiples. That is, for each α ∈ Φ(T) there is c_{α_{λ}} ∈ k̄ such that

F(c_{α_{λ}}E_{α}^{∗}) = c_{(σ·α)_{λ}}E_{σ·α}^{∗}.

A straightforward computation shows, for each α ∈ Φ(T), that

(35)  Frob(c_{α_{λ}}) = ϖ^{⟨λ,σ·α⟩} · c_{(σ·α)_{λ}}.

Following [63], a Σ-orbit in Φ(T) is called symmetric if it is closed under α ↦ −α and anti-symmetric otherwise. Let Ṡym(T) be a set of representatives for the symmetric Σ-orbits in Φ(T).

For any α ∈ Φ(T), define

Σ_{α} = {τ ∈ Σ : τ · α = α},

and let k_{α} ⊂ k̄ be the fixed field of the pre-image of Σ_{α} in Gal(k̄/k).

For each α ∈ Ṡym(T), define

Σ_{±α} = {τ ∈ Σ : τ · α = ±α},

and let k_{±α} ⊂ k̄ be the fixed field of the pre-image of Σ_{±α} in Gal(k̄/k). There is an integer m = m(α) such that

Σ_{±α} = ⟨σ^{m}⟩, Σ_{α} = ⟨σ^{2m}⟩.

We have σ^{m}α = −α. The extension k_{α}/k is unramified of degree 2m.

Moreover, k_{α}/k_{±α} is an unramified quadratic extension, hence corresponds via class-field theory to the character χ_{α} : k_{±α}^{×} ⟶ {±1} given by

χ_{α}(x) = (−1)^{v(x)},

where v is the valuation on K. Using [63], Lemma 12.2.3 is equivalent to the following formula.

Lemma 12.3.1.

∏_{α∈Ṡym(T)} χ_{α}(c_{α_{λ}} · c_{−α_{λ}}) = ε(G_{λ}, G_{0}).

The proof requires a few steps.

Lemma 12.3.2. We have

c_{α_{λ}} · c_{−α_{λ}} = n · ϖ^{−⟨λ,α+σ·α+···+σ^{(m−1)}·α⟩},

where m = m(α) and n ∈ k_{±α}^{×} is a norm from k_{α}.

.

Proof. Applying Equation (35) repeatedly, we have

c_{(σ^{k}·α)_{λ}} = Frob^{k}(c_{α_{λ}}) · ϖ^{−⟨λ,σ·α+···+σ^{k}·α⟩},

for k ≥ 1. Since σ^{m} · α = −α, we have

c_{−α_{λ}} = Frob^{m}(c_{α_{λ}}) · ϖ^{−⟨λ,σ·α+···+σ^{(m−1)}·α−α⟩}.

Hence

c_{α_{λ}} · c_{−α_{λ}} = c_{α_{λ}} · Frob^{m}(c_{α_{λ}}) · ϖ^{2⟨λ,α⟩} · ϖ^{−⟨λ,α+σ·α+···+σ^{m−1}·α⟩}.

Since c_{α} ∈ k_{α}, this proves the claim, with

n = c_{α_{λ}} · Frob^{m}(c_{α_{λ}}) · ϖ^{2⟨λ,α⟩}.
□

Choose a set of positive roots Φ^{+}(T) ⊂ Φ(T), and set

2ρ = ∑_{β∈Φ^{+}(T)} β.

Lemma 12.3.3.

∑_{α∈Ṡym(T)}⟨λ, α + σ · α + ⋯ + σ^{m(α)−1} · α⟩ ≡ ⟨λ, 2ρ⟩ mod 2.

Proof. Let O′_{1}, …, O′_{p} be a choice of one from each pair {O′_{i}, −O′_{i}} of anti-symmetric Σ-orbits in Φ(T), and let O_{1}, …, O_{q} be the symmetric Σ-orbits. For α ∈ Φ(T), define |α| = α if α ∈ Φ^{+}(T), and |α| = −α if −α ∈ Φ^{+}(T). Set

∥O′_{i}∥ = {|α| : α ∈ O′_{i}}, O_{j}^{+} = O_{j} ∩ Φ^{+}(T).

Then we have a disjoint union

Φ^{+}(T) = ∐_{i=1}^{p} ∥O′_{i}∥ ⊔ ∐_{j=1}^{q} O_{j}^{+}.

For any 1 ≤ i ≤ p, we have

∑_{β∈∥O′_{i}∥} β ≡ ∑_{α∈O′_{i}} α mod 2ℤΦ(T).

The latter sum is Σ-invariant, hence it vanishes, since σ is elliptic. It follows that

⟨λ, 2ρ⟩ ≡ ∑_{j=1}^{q} ∑_{β∈O_{j}^{+}}⟨λ, β⟩ mod 2.

Working modulo two, we can replace each sum over O_{j}^{+} by

∑_{k=0}^{m(α)−1}⟨λ, σ^{k} · α⟩,

for any α ∈ O_{j}. This proves the lemma. □

Combining Lemmas 12.3.2 and 12.3.3, we get

Corollary 12.3.4.

∏_{α∈Ṡym(T)} χ_{α}(c_{α_{λ}} · c_{−α_{λ}}) = (−1)^{⟨λ,2ρ⟩}.

We next give another expression for ε(G_{λ}, G_{0}). Let z_{λ} ∈ W_{o} be the projection of u_{λ}. Then z_{λ} and ϑ act linearly on the ℚ-vector space V := X ⊗ ℚ (recall that X = X_{∗}(T)), and the k-rank of G_{λ} is given by

rk(G_{λ}) = dim V^{z_{λ}ϑ}.

Let det(A) denote the determinant of an operator A ∈ GL(V).

Lemma 12.3.5.

ε(G_{λ}, G_{0}) = det(z_{λ}).

Proof. Since z_{λ}ϑ has finite order and preserves the lattice X ⊂ V, we have

det(z_{λ}ϑ) = (−1)^{dim V −dim V^{z_{λ}ϑ}}.

Likewise,

det(ϑ) = (−1)^{dim V −dim V^{ϑ}}.

Together, these give

det(z_{λ}) = (−1)^{dim V^{z_{λ}ϑ}−dim V^{ϑ}} = ε(G_{λ}, G_{0}).
□

To prove Lemma 12.3.1 it remains to prove

Lemma 12.3.6.

det(z_{λ}) = (−1)^{⟨λ,2ρ⟩}.

Proof. From the definitions we see that z_{λ} = z_{λ+ν} for any ν ∈ X^{∘}+X^{W}, where X^{∘} is the co-root lattice of T. Likewise, the parity of ⟨λ, 2ρ⟩ depends only on the class of λ in X/(X^{∘} + X^{W}). We have det(z_{λ}) = (−1)^{⟨λ,2ρ⟩} = +1 if λ ∈ X^{∘} + X^{W}.

Assume now that λ ∉ X^{∘} + X^{W}. Recall that T_{ad} is the image of T in the adjoint group G_{ad} of G, and that X_{ad} = X_{∗}(T_{ad}). We may view X^{∘} as a subgroup of X_{ad}. The natural map X → X_{ad} induces an injection

X/(X^{∘} + X^{W}) ↪ X_{ad}/X^{∘}.

The nontrivial elements in the group X_{ad}/X^{∘} are represented by the minuscule co-weights of T_{ad} [9, p. 240]. Hence the class of λ in X/(X^{∘} + X^{W}) determines a simple root α ∈ Φ^{+}(T) such that ⟨λ, β⟩ = 0 for all simple roots β ≠ α, and ⟨λ, α⟩ = 1. Moreover, we have a disjoint union

Φ(T) = Φ_{−1} ⊔ Φ_{0} ⊔ Φ_{1},

where

Φ_{i} = {β ∈ Φ(T) : ⟨λ, β⟩ = i}

(see [9, p. 239]).

Iwahori-Matsumoto [30, 1.18] show that z_{λ} is W_{o}-conjugate to the unique element of W_{o} whose set of positive roots made negative is exactly Φ_{1}. This implies that

det(z_{λ}) = (−1)^{|Φ_{1}|}.

On the other hand, since

Φ^{+}(T) = [Φ_{0} ∩ Φ^{+}(T)] ⊔ Φ_{1},

it follows that

⟨λ, 2ρ⟩ = ∑_{β∈Φ_{1}}⟨λ, β⟩ = |Φ_{1}|.

This proves the present lemma, as well as Lemma 12.3.1.

□

12.4. Murnaghan-Kirillov theory. In this section, H is any connected reductive k-group, split over K, with Frobenius F on H := H(K). Let H_{0^{+}}, 𝔥_{0^{+}} denote respectively the sets of topologically unipotent elements in H, and topologically nilpotent elements in 𝔥 = Lie(H).

We make the following restrictions on k and H. Recall that q, a power of a prime p, is the cardinality of the residue field 𝔣. Let e denote the ramification degree of k over ℚ_{p}, and let ν(H) be the number of positive roots in H.

Restrictions 12.4.1. (1) q ≥ ν(H).

(2) There is a faithful k-embedding φ : H ↪ GL_{n} such that p ≥ (2 + e)n.

Note that if G is as in the previous part of the paper, H is the identity component of the centralizer of a topological semisimple element in G_{0}, and Restrictions 12.4.1 hold for G and some n, then they hold for H, with the same n.

In Appendices A and B we will prove:

Lemma 12.4.2. Assume Restrictions 12.4.1 hold. Then we have:

(1) For every F -stable facet J ⊆ B(H), and maximal F -stable torus S ⊂ H_{J} with Lie algebra L_{S}, there is an element X̄_{S} ∈ L_{S}^{F} whose centralizer in H_{J} is exactly S.

(2) There is an ⟨F ⟩ ⋉ H-equivariant bijection log : H_{0^{+}} ⟶ 𝔥_{0^{+}}, which induces, for every minimal F -stable facet J ⊆ B(H), an ⟨F ⟩ ⋉ H_{J}-equivariant bijection from the set of unipotent elements of H_{J} to the set of nilpotent elements of the Lie algebra of H_{J}.

Recall that for S ∈ 𝔗(H) there is a unique F -stable facet J ⊂ B(H) such that J^{F} = B(S)^{F}, and that S denotes the image of S in H_{J}. Let Z denote the maximal k-split torus in the center of H. The following lemma is a special case of a result in [18].

Lemma 12.4.3. Assume Restrictions 12.4.1 hold. For each S ∈ 𝔗(H), with (S, J) as above, and any X_{S} ∈ Lie(S) ∩ 𝔥_{J}^{F} whose projection to L_{S}^{F} is an element X̄_{S} as in 12.4.2, we have the equality

R(H, S, 1)(γ) = ε(H, Z) · μ̂_{X_{S}}^{H^{F}} (log(γ)),

for every regular semisimple γ ∈ H_{0^{+}}^{F}, where log is as in Lemma 12.4.2

Proof. Fix a regular semisimple γ ∈ H_{0^{+}}^{F} . Let dℓ denote the Haar measure on H_{J}^{F} with meas_{dℓ}(H_{J}^{F}) = 1. We have

R(H, S, 1)(γ) = \frac{meas_{dz}(Z_{J}^{F})}{meas_{dh}(H_{J}^{F})} · ∫_{H^{F} /Z^{F}} dh^{∗} ∫_{H_{J}^{F}} Ṙ_{S,1}^{H_{J}}(^{hℓ}γ) dℓ.

On the other hand, from [2, Proposition 3.3.1], we can write

(36)  μ̂_{X_{S}}^{H^{F}} (X) = \frac{meas_{dz}(Z_{J}^{F})}{meas_{ds}((C_{H}(X_{S}))_{J}^{F})} · ∫_{H^{F} /Z^{F}} dh^{∗} ∫_{H_{J}^{F}} dℓ ∫_{H_{J}^{F}} Λ(B(^{ℓ′hℓ}X, X_{S})) dℓ′

where X = log γ and ds is the Haar measure on (C_{H}(X_{S}))^{F}, normalized as in Section 5. (Note that in [2] the quotient measure is normalized slightly differently.)

From [2, Lemma 6.1.1] we see that the inner integral in Equation (36) is zero unless^{hℓ}X ∈ 𝔥_{J}^{F}. Consequently, it is enough to show that if^{hℓ}X ∈ 𝔥_{J}^{F}, then

(37)  Ṙ_{S,1}^{H_{J}}(^{hℓ}γ) = \frac{ε(H, Z) · meas_{dh}(H_{J}^{F})}{meas_{ds}((C_{H}(X_{S}))_{J}^{F})} · ∫_{H_{J}^{F}} Λ(B(^{ℓ′hℓ}X, X_{S})) dℓ′.

But since

meas_{ds}((C_{H}(X_{S}))_{J}^{F}) = \frac{|S^{F} |}{|L_{S}^{F} |^{1/2}} ,

and

ε(H, Z) = ε(H_{J}, S)

because S is minisotropic, Equation (37) follows immediately from [31, Theorem 3] and the properties of the map log in Lemma 12.4.2. □

12.5. Completion of the proof of stability. In this section, we prove 12.0.1, assuming that Restrictions 12.4.1 are in place.

Let T_{st} be an H-stable class in 𝔗(H). We fix S_{0} ∈ T_{st} and X_{0} := X_{S_{0}} ∈ 𝔥^{F} as in 12.4.3.

Lemma 12.5.1. The map X ↦ C_{G}(X) induces a surjective map

c : [Ad(H) · X_{0}]^{F} /H^{F} ⟶ T_{st}/H^{F} ,

whose fiber over the H^{F} -orbit of S ∈ T_{st} is in bijection with N(H, S^{F})/N(H^{F} , S).

Proof. Note that

[Ad(H) · X_{0}]^{F} = {^{h}X_{0} : h ∈ H, and h^{−1}F (h) ∈ S_{0}},

and recall that

T_{st} = {^{h}S_{0} : h ∈ H, and ^{h}(S_{0}^{F}) = (^{h}S_{0})^{F} }.

Since h^{−1}F (h) ∈ S_{0} if and only if^{h}(S_{0}^{F}) = (^{h}S_{0})^{F}, it follows that

{C_{G}(X) : X ∈ [Ad(H) · X_{0}]^{F} } = T_{st}.

One checks that for k, h ∈ H with ^{k}X,^{h}X ∈ 𝔥^{F}, we have that C_{G}(^{k}X) is H^{F} -conjugate to C_{G}(^{h}X) if and only if there is ℓ ∈ H^{F} such that k^{−1}ℓh ∈ N(H, S_{0}^{F}). It follows that the fiber of c over the H^{F} -orbit of^{k}S_{0} consists of the distinct H^{F} -orbits Ad(H^{F})(^{gn}X), as n ranges over N(H, (^{k}S_{0})^{F})/N(H^{F} ,^{k}S_{0}). □

Lemma 12.5.2. If Restrictions 12.4.1 hold, then Lemma 12.0.1 holds.

Proof. Using Lemmas 12.4.3 and 12.5.1, we have

Q(H, T_{st})(γ_{u}) = ∑_{S∈T_{st}/H^{F}} |N(H, S^{F})/N(H^{F} , S)| · R(H, S, 1)(γ_{u})

= ε(H, Z) · ∑_{S∈T_{st}/H^{F}} |N(H, S^{F})/N(H^{F} , S)| · μ̂_{X_{S}}^{H^{F}} (log(γ_{u}))

= ε(H, Z) · ∑_{X∈[Ad(H)·X_{0}]^{F} /H^{F}} μ̂_{X}^{H^{F}} (log(γ_{u}))

= ε(H, Z) · Ŝ_{X_{0}}^{𝔥}(log(γ_{u})).

A similar result holds for Q(^{g}H, ι_{g}T_{st})(^{g}γ_{u}). The result now follows from Examples 12.2.1, 12.2.2 and Lemma 12.2.3. □

13. L-packets arising from the opposition involution

We illustrate our L-packets with a canonical example. For simplicity, take G to be absolutely quasi-simple and simply-connected, and let w_{o} be the unique element of W_{o} such that w_{o} · C = −C. Up to isomorphism, there is a unique K-split k-structure on G for which the Frobenius F acts on X by ϑ = −w_{o}. This k-structure is quasi-split, and we have H^{1}(F, G) = 1.

We tabulate the groups G below, using their names from the tables of [60], and give the number r := [X_{ad}^{ϑ} : j(X^{ϑ})] of generic representations in an L-packet Π(φ) (see Lemma 6.2.2).

\begin{matrix}G & ^{2}A′_{2m} & ^{2}A′_{2m−1} & B_{n} & C_{n} & D_{2m} & ^{2}D_{2m+1} & G_{2} & F_{4} & ^{2}E_{6} & E_{7} & E_{8} \\ r & 1 & 2 & 2 & 2 & 4 & 2 & 1 & 1 & 1 & 2 & 1 \end{matrix}

Now let φ be a TRSELP whose associated w is w_{o}. Since w_{o}ϑ = − Id, the L-packet Π(φ) is parametrized by

Irr(C_{φ}) = X/2X,

where X = X^{∘} is the co-root lattice of T in G. In particular, |Π(φ)| = 2^{n}, where n is the absolute rank of G. With Haar measure normalized as in Section 5.3, each representation π ∈ Π(φ) has formal degree

Deg(π) = (q^{1/2} + q^{−1/2})^{−n}.

Since W_{o}^{w_{o}ϑ} = W_{o}, the full Weyl group W_{o} acts on Irr(C_{φ}). This action has several interpretations.

First, by Lemma 9.6.1, the W_{o}-orbits on X/2X are in bijection with the G^{F}-orbits in the G-stable class T_{w_{o}}. The tori in this stable class are k-isomorphic to U_{1}^{n}.

Second, the W_{o}-orbits in X/2X are in bijection, via evaluation at −1, with conjugacy-classes of 2-torsion elements in G (or G, since G is simply-connected, and Lemma 2.9.1 applies). For each λ ∈ X, we have

x_{λ} = \frac{1}{2}t_{λ} · o,

and the root datum, with F_{λ}-action, of G_{x_{λ}} is that of the centralizer in G of λ(−1). The generic representations in Π(φ) correspond to the 2-torsion elements in the center of G.

For exceptional groups, the 2-torsion picture places a strong limitation on the type of inducing parahorics that appear in Π(φ). For example, in E_{8} there are three W_{o}-orbits in X/2X. The L-packet Π(φ) has 256 = 1 + 120 + 135 representations, induced from parahoric subgroups of type E_{8}, A_{1}E_{7}, D_{8}, respectively.

Third and finally, the generic representations in Π(φ) are parametrized by the W_{o}-invariants:

Irr(C_{φ})_{gen} = Irr(C_{φ})^{W_{o}}.

Containment “⊆” is shown in Remark 6.2.5. For the other containment, note that a W_{o}-invariant element in X_{ad}/2X_{ad} corresponds to a central 2-torsion element in G_{ad}, hence must be trivial. Containment “⊇” now follows from Lemma 6.2.1.

Appendix A. Good bilinear forms and regular elements

In the appendices, we prove various results used in the proof of stability. Here G is any connected reductive k-group, not necessarily split over K, and F is the corresponding Frobenius automorphism of G.

A.1. Good bilinear forms. We say that a symmetric bilinear form B on 𝔤 is “good” if B is ⟨F ⟩ ⋉ G-equivariant, nondegenerate, and restricts to the Killing form, B′, on the derived algebra 𝔤′ = [𝔤, 𝔤] of 𝔤.

Let 𝔤_{x,t}, 𝔤_{x,t^{+}} be the Moy-Prasad filtration subalgebras of 𝔤 attached to x ∈ B(G) and t ∈ ℝ. (See Section B.5 below for a brief introduction to Moy-Prasad filtrations.)

Lemma A.1.1. If p > n + 1, where n ≥ 2 is the dimension of a faithful k-representation of G, then there exists a good bilinear form B on 𝔤 which induces, for all x ∈ B(G) and for all t ∈ ℝ, a nondegenerate pairing

𝔤_{x,t}/𝔤_{x,t^{+}} × 𝔤_{x,(−t)}/𝔤_{x,(−t)^{+}} → 𝔉.

Remark A.1.2. If B satisfies Lemma A.1.1 and x is F -fixed, then the induced pairing,

𝔤_{x,t}/𝔤_{x,t^{+}} × 𝔤_{x,(−t)}/𝔤_{x,(−t)^{+}} → 𝔉,

is ⟨F ⟩ ⋉ G_{x}-equivariant.

Proof. The existence of such a form B follows from the proof of [4, Proposition 4.1] under the condition that p ∤ B′(H_{α}, H_{α}) for any root α of a maximal torus T ⊂ G, where H_{α} is the corresponding Chevalley basis vector in the Lie algebra of T.

Let 𝔤_{1}, …, 𝔤_{r} be the simple factors of 𝔤′. Let m_{i}^{∗} be the sum of the coefficients in the expression of the highest co-root of 𝔤_{i} in terms of simple co-roots. From [54, I.4.8], any prime dividing B′(H_{α}, H_{α}) must divide 6(m_{i}^{∗} + 1), where 𝔤_{i} is the factor containing α.

Let m^{∗} = max{m_{i} : 1 ≤ i ≤ r}. We have n ≥ m^{∗}. To prove this, one may assume 𝔤 simple, and check the result case-by-case (recall that k has characteristic zero). The result follows. □

A.2. Regular elements. Suppose J is an F -stable facet in B(G) and S is a maximal 𝔣-torus in G_{J}. We wish to establish conditions on p and q which will guarantee that the Lie algebra L_{S}^{F} contains a regular semisimple element of L_{J}.

Let F_{0} be the q-power Frobenius of 𝔉/𝔣. Let Φ_{J} be the set of 𝔉-roots of G_{J} with respect to S, and let ℓ = dim L_{S}. There is a permutation τ of Φ_{J} such that

α ∘ F = F_{0} ∘ τ(α)

for all α ∈ Φ_{J}. Let d be the order of τ. Let Φ̄_{J} be the set of orbits in Φ_{J} under the group generated by τ and α ↦ −α.

Lemma A.2.1. If p ≠ 2 and q > |Φ̄_{J} |, then L_{S}^{F} contains a regular element of L_{J}.

Proof. Set 𝔣_{d} := 𝔣^{F_{0}^{d}}, L_{S}^{d} := L_{S}^{F^{d}}. The 𝔣-linear map

tr : L_{S}^{d} ⟶ L_{S}^{F} ,

given by

tr X := ∑_{j=0}^{d−1} F^{j}(X)

has the property that for all α ∈ Φ_{J}, the composition α ∘ tr is not identically zero on L_{S}^{d}. Indeed, suppose there exists α ∈ Φ_{J} for which α ∘ tr is zero. Since S is 𝔣_{d}-split, we can assume that the Chevalley basis vector H_{α} belongs to L_{S}^{d}. For all t ∈ 𝔣_{d}, we have

0 = α(tr(tH_{α}))
= α(tH_{α} + t^{q}H_{τα} + ⋯ + t^{q^{d−1}}H_{τ^{d−1}α})
= t⟨α, α̌⟩ + t^{q}⟨α, τ̌α⟩ + ⋯ + t^{q^{d−1}}⟨α, τ̌^{d−1}α⟩.

Since p ≠ 2, we have ⟨α, α̌⟩ ≠ 0. Hence we have a nonzero polynomial of degree at most q^{d−1} but with q^{d} zeros in 𝔉, a contradiction.

Thus, for each α ∈ Φ_{J} we have a nonzero 𝔣-linear map

α ∘ tr : L_{S}^{d} ⟶ 𝔣_{d}.

Let Z_{α} be the kernel of this linear map. Since

F_{0}(τ(α)(tr X)) = α(tr X),

we have Z_{τ(α)} = Z_{α}. Also, we have Z_{α} = Z_{−α}. Hence the subspace Z_{α} depends only on the image of α in Φ̄_{J}.

It suffices to show that the set

L_{S}^{∘} := L_{S}^{d} \ ⋃_{α∈Φ̄_{J}} Z_{α}

is nonempty. We have

|L_{S}^{∘}| = |L_{S}^{d}| − | ⋃_{ᾱ∈Φ̄_{J}} Z_{α}| ≥ q^{ℓd} − |Φ̄_{J} | |Z_{β}| ,

where β is chosen so that |Z_{β}| = max{|Z_{α}| : α ∈ Φ̄_{J}}. Since dim_{𝔣} Z_{β} ≤ ℓd − 1, we have |Z_{β}| ≤ q^{ℓd−1}. Consequently,

|L_{S}^{∘}| ≥ q^{ℓd} − |Φ̄_{J} | q^{ℓd−1}.

Therefore q > |Φ̄_{J} | ensures that L_{S}^{∘} is nonempty. □

Note that

|Φ̄_{J}| ≤ ν(𝔤),

where ν(𝔤) is the number of positive (absolute) roots in 𝔤.

If p is not a torsion prime for G_{J}, then the centralizer in G_{J} of any semisimple element in L_{J} is connected [58, Theorem 3.14]. The torsion primes of G_{J} are also torsion primes of G. Consequently, if p is not a torsion prime for G, then any regular element of L_{S} has centralizer equal to S. The torsion primes of G are less than the number m^{∗} defined in the proof of Lemma A.1.1. Putting all this together with Lemma A.2.1 gives the following result. Let n be as in Lemma A.1.1.

Lemma A.2.2. If p > n + 1 and q > ν(𝔤), then for every F -stable facet J in B(G), and every maximal F -stable maximal torus S ⊂ G_{J}, the Lie algebra L_{S}^{F} contains an element whose centralizer in G_{J} is exactly S.

Appendix B. A logarithm mapping for G

Let e denote the ramification degree of k over ℚ_{p}, and let φ : G ⟶ GL_{n} be a faithful k-representation. We suppose that ν(K^{×}) = ℤ where ν is the valuation on K. For notational convenience we sometimes write (G^{F})_{0^{+}} instead of (G_{0^{+}})^{F}.

The purpose of this appendix is to prove the following Lemma.

Lemma B.0.3. If p ≥ (2 + e)n, then there exists a ⟨F ⟩ ⋉ G-equivariant bijective map

log: G_{0^{+}} → 𝔤_{0^{+}}

which, for each F -stable facet J in B(G), induces a ⟨F ⟩ ⋉ G_{J}-equivariant bijective map from the set of unipotent elements in G_{J} to the set of nilpotent elements in L_{J}.

B.1. The exponential map for the general linear group. Recall that q is the order of the residue field of k. For each X ∈ 𝔤𝔩_{n}(k) we have X ∈ 𝔤𝔩_{n}(k)_{0^{+}} if and only if |μ| ≤ q^{−1/n} for each eigenvalue μ of X. For each g ∈ GL_{n}(k), we have g ∈ GL_{n}(k)_{0^{+}} if and only if |μ − 1| ≤ q^{−1/n} for each eigenvalue μ of g.

We begin with a technical result.

Lemma B.1.1. If p > en + 1, then

(1) \frac{q^{−j/n}}{|j!|} ≤ q^{−1/n} for j ≥ 2 and

(2) \frac{q^{−j/n}}{|j!|} → 0.

Proof. Set

A(j) := ⌊\frac{j}{p}⌋ + ⌊\frac{j}{p^{2}} ⌋ + ⌊\frac{j}{p^{3}} ⌋ + · · · .

Note that

\frac{q^{−j/n}}{|j!|} = \frac{q^{−j/n}}{q^{−eA(j)}} = q^{(neA(j)−j)/n}.

To establish item (1) it is enough to show

(38)  neA(j) − j ≤ −1,

and to establish item (2) it is enough to show

(39)  neA(j) − j ≤ \frac{−j}{(p − 1)}.

Write

j = ∑_{i=0}^{ℓ} b_{i}p^{i}

with b_{i} ∈ {0, 1, 2, …, (p − 1)} and b_{ℓ} ≠ 0. We have

⌊\frac{j}{p^{t}} ⌋ = ∑_{t}^{ℓ} b_{i}p^{i−t}

for 1 ≤ t ≤ ℓ. Consequently, (p − 1)A(j) = ∑_{i=0}^{ℓ} b_{i}(p^{i} − 1) = j − ∑ b_{i}. Thus,

enA(j) < (p − 1)A(j) ≤ (j − 1),

establishing (38), and

(p − 1)(neA(j) − j)
= nej − ne ∑ b_{i} − (p − 1)j
< (ne − p + 1)j
≤ −j

establishing (39). □

Our assumption p ≥ (2 + e)n ensures that p > en + 1. Thus, thanks to Lemma B.1.1 and [25, §10.1], the map exp defined by

X ↦ ∑_{ℓ=0}^{∞} \frac{X^{ℓ}}{ℓ!}

converges to a GL_{n}(k)-equivariant bijective analytic map from 𝔤𝔩_{n}(k)_{0^{+}} to GL_{n}(k)_{0^{+}}. We extend exp to a ⟨F ⟩ ⋉ GL_{n}(K)-equivariant bijective analytic map from 𝔤𝔩_{n}(K)_{0^{+}} to GL_{n}(K)_{0^{+}} as follows. For each m ∈ ℤ_{≥1}, by replacing k by K^{F^{m}} in the discussion above, we obtain an analytic map exp_{m}: (𝔤𝔩_{n}(K)_{0^{+}})^{F^{m}} → (GL_{n}(K)_{0^{+}})^{F^{m}}. Thus, if X ∈ 𝔤𝔩_{n}(K)_{0^{+}}, we may choose m ∈ ℤ_{≥1} so that X ∈ (𝔤𝔩_{n}(K)_{0^{+}})^{F^{m}} and define exp(X) := exp_{m}(X) ∈ (GL_{n}(K)_{0^{+}})^{F^{m}} ⊂ GL_{n}(K)_{0^{+}}. This gives a well-defined ⟨F ⟩ ⋉ GL_{n}(K)-equivariant bijective analytic map from 𝔤𝔩_{n}(K)_{0^{+}} to GL_{n}(K)_{0^{+}}

For each facet J in B(GL_{n}(K)), the map exp takes 𝔤𝔩_{n}(K)_{J} ∩ 𝔤𝔩_{n}(K)_{0^{+}} to GL_{n}(K)_{J} ∩ GL_{n}(K)_{0^{+}}. Finally, the map exp also takes the Haar measure on 𝔤𝔩_{n}(k) into the Haar measure on GL_{n}(k).

B.2. The logarithmic mapping ψ. From [9, III, §7.3, 2, Proposition 3], there is a neighborhood V of 0 in 𝔤^{F} and a map ϕ: V → G^{F} such that ϕ(V) is an open subgroup of G^{F} and ϕ: V → ϕ(V) is a k-analytic isomorphism of analytic manifolds with the property that ϕ(mX) = ϕ(X)^{m} for all m ∈ ℤ and for all X ∈ V. From [9, III, §7.6, 6, Proposition 10], there is a neighborhood U of the identity in (G^{F})_{0^{+}} and a unique k-analytic map ψ: (G^{F})_{0^{+}} → 𝔤^{F} such that ψ(U) = V, ϕ ∘ ψ = 1_{U}, and ψ(g^{m}) = mψ(g) for all g ∈ (G^{F})_{0^{+}} and all m ∈ ℤ. Note that ψ is locally injective, hence injective.

Recall that the exponential map, exp, for the general linear group was defined in section B.1. The unique map from GL_{n}(k)_{0^{+}} to 𝔤𝔩_{n}(k) determined (in the sense of the previous paragraph) by exp is called log. It has the usual power series expansion. Since p ≥ (2 + e)n > en + 1, the map log: 𝔤𝔩_{n}(k)_{0+} → GL_{n}(k)_{0^{+}} is the inverse of exp: GL_{n}(k)_{0^{+}} → 𝔤𝔩_{n}(k)_{0^{+}} (see, for example, [25, Lemma 10.1]).

From [9, III, §4.4, Corollary 2] there is a neighborhood V′ ⊂ V in 𝔤^{F} such that

(40)  φ(ϕ(X)) = exp(dφ(X))

for all X ∈ V′ and

dφ(ψ(g)) = log(φ(g))

for all g ∈ ϕ(V′). Suppose g ∈ (G)_{0^{+}}^{F}. Choose m ∈ ℤ_{≥1} so that g^{p^{m}} ∈ ϕ(V′). We have

dφ(ψ(g)) = p^{−m} · dφ(ψ(g^{p^{m}})) = p^{−m} · log(φ(g^{p^{m}})) = log(φ(g)).

Thus

(41)  dφ(ψ(g)) = log(φ(g))

for all g ∈ (G_{0^{+}})^{F}.

B.3. An extension of ψ. The map ψ has a unique extension, which we shall also call ψ, to a ⟨F ⟩ ⋉ G-equivariant map from G_{0^{+}} to 𝔤. Indeed, for each m ∈ ℤ_{≥1}, by replacing k by K^{F^{m}} in the discussion above, we obtain a (unique) K^{F^{m}}-analytic map ψ_{m}: (G_{0^{+}})^{F^{m}} → 𝔤^{F^{m}} for which

dφ(ψ_{m}(g)) = log(φ(g))

for all g ∈ (G_{0^{+}})^{F^{m}}. Thus, since dφ is injective, for m′ ≥ m ≥ 1 we have ψ_{m′}(g) = ψ_{m}(g) whenever g ∈ (G_{0^{+}})^{F^{m}}. In particular, ψ_{m}(g) = ψ_{1}(g) whenever g ∈ (G_{0^{+}})^{F}. Thus, we may define ψ: G_{0^{+}} → 𝔤 by setting ψ(g) = ψ_{m}(g) whenever g ∈ (G_{0^{+}})^{F^{m}}. To see that ψ is ⟨F ⟩ ⋉ G - equivariant, it is enough to check that it is F -equivariant. Since dφ is injective, it is enough to check that dφ(ψ(F g)) = dφ(F (ψ(g))) for all g ∈ G_{0^{+}}. However,

\begin{aligned}dφ(ψ(F g)) & = log(φ(F g)) = F log(φ(g)) \\ & = F dφ(ψ(g)) = dφ(F ψ(g)). \end{aligned}

B.4. The adjoint representation and ψ. Suppose Y ∈ 𝔤𝔩_{n}(K)_{0^{+}}. Since the valuations of the eigenvalues of ad(Y) are bounded (below) by those of Y, and p ≥ (2 + e)n, the power series for exp(ad(Y)) converges in GL(𝔤𝔩_{n})(K), and we have

(42)  exp(ad(Y)) = Ad(exp(Y)).

Similarly, for all g ∈ GL_{n}(K)_{0^{+}} we have

(43)  log(Ad(g)) = ad(log(g)).

For h ∈ G_{0^{+}}, we define log(Ad(h)) ∈ 𝔤𝔩(𝔤)(K) by

log(Ad(h)) := − ∑_{m≥1} \frac{(1 − Ad(h))^{m}}{m} .

Thus, for all h ∈ G_{0^{+}} and X ∈ 𝔤,

dφ[ad(ψ(h))X] = [ad(dφ(ψ(h)))]dφ(X)

(from Equation (41)) = [ad(log(φ(h)))]dφ(X) (from Equation (43)) = [log(Ad(φ(h)))]dφ(X)

= dφ(log(Ad(h))(X)).

Since dφ is injective, we conclude that

(44)  log(Ad(h))X = ad(ψ(h))X

for all h ∈ G_{0^{+}} and X ∈ 𝔤.

B.5. A brief introduction to the filtrations of Moy and Prasad. We recall here what we need from the theory of Moy-Prasad filtration lattices ([44, 43]).

Let T denote the group of K-rational points of a maximally K-split torus in G. Let A denote the apartment in B(G) corresponding to T, let Φ denote the set of roots of G with respect to T, and let Δ denote the set of affine roots of G with respect to T and our valuation on K. The elements of Δ are affine functions on A. For δ ∈ Δ, we let δ̇ ∈ Φ denote the gradient of δ.

For α ∈ Φ, let 𝔤_{α} denote the corresponding root space in 𝔤. For δ ∈ Δ, define the lattices 𝔤_{δ}^{+} and 𝔤_{δ} in 𝔤δ̇ as follows: Choose a facet J in A on which δ is zero. Set

𝔤_{δ} := 𝔤_{J} ∩ 𝔤δ̇ and 𝔤_{δ}^{+} := 𝔤_{J}^{+} ∩ 𝔤δ̇.

These definitions are independent of the choice of J.

Since G is K-quasi-split, the centralizer M := C_{G}(T) is the group of K-rational points of a maximal K-torus M of G. Let 𝔪 denote the Lie algebra of M. For s ∈ ℝ, we define

𝔪_{s} := {X ∈ 𝔪 | ν(dχ(X)) ≥ s for all χ ∈ X^{∗}(M)}.

For x ∈ A and s ∈ ℝ, we define the lattice

𝔤_{x,s} := 𝔪_{s} ⊕ ∑_{δ∈Δ; δ(x)≥s} 𝔤_{δ}.

For t ≥ s we have 𝔤_{x,t} ⊂ 𝔤_{x,s}; in fact,

⋃ 𝔤_{x,s} = 𝔤, and ⋂_{s} 𝔤_{x,s} = {0}.

We set

𝔤_{x,s^{+}} := ⋃_{t>s} 𝔤_{x,t}.

If y is in B(G), then there is a g in G so that gy ∈ A. For s ∈ ℝ, we define

𝔤_{y,s} := ^{g}𝔤_{x,s} and 𝔤_{y,s^{+}} := ^{g}𝔤_{x,s^{+}}.

This is independent of the choice of g.

Recall [3, §3] that, for s ∈ ℝ, we have the closed, open, G-invariant subsets

𝔤_{s} := ⋃_{x∈B(G)} 𝔤_{x,s} and 𝔤_{s^{+}} := ⋃_{t>s} 𝔤_{t}.

For each s ∈ ℝ_{≥0} and each x ∈ B(G), we also define, in a completely analogous manner, Moy-Prasad filtration subgroups G_{x,s} ≤ G_{x,0} = G_{x} (see [43]).

The Moy-Prasad filtration lattices and subgroups have the following properties (which we shall use without further comment). The first two properties are proved in [45, §2], the third is a formal consequence of the definitions, and the final is [1, Proposition 1.4.3].

(1) For s, t ∈ ℝ and x ∈ B(G), we have [𝔤_{x,t}, 𝔤_{x,s}] ⊂ 𝔤_{x,(t+s)}.

(2) For s, t ∈ ℝ_{≥0} and x ∈ B(G), we have (G_{x,s}, G_{x,t}) ⊂ G_{x,(t+s)}.

(3) For s ∈ ℝ and x ∈ B(G) we have

ϖ · 𝔤_{x,s} = 𝔤_{x,(s+1)}.

(4) For t ∈ ℝ_{≥0}, s ∈ ℝ, and x ∈ B(G), we have (Ad(g) − 1)𝔤_{x,s} ⊂ 𝔤_{x,s+t} for all g ∈ G_{x,t}.

B.5.1. A technical result. The purpose of this section is to establish a (weak) connection between the Moy-Prasad filtrations for 𝔤 and those for 𝔤𝔩_{n}(K). We do this so as to avoid introducing another constant (r_{G} below) into our hypotheses.

Fix a facet J ⊂ B(G). Define a continuous, piecewise-linear function r: J → ℝ_{>0} by sending x ∈ J to the unique real number r(x) for which

𝔤_{J}^{+} = 𝔤_{x,r(x)} ≠ 𝔤_{x,r(x)^{+}}

After extending by zero, the function r becomes a continuous function on the closure of J. Hence, we may choose x_{J} ∈ J so that

r(x_{J}) ≥ r(x)

for all x in the closure of J. Define r_{J} := r(x_{J}). The (rational) number r_{J} depends only on the G-conjugacy class of J. We set

r_{G} := min_{J} r_{J}.

Note that, if J is F -stable, then, from the concavity of r and the Bruhat-Tits fixed-point theorem (see, for example, [60, §2.3.1]), we may assume that x_{J} is F -fixed.

Lemma B.5.1. If C is an alcove in B(G), then r_{G} = r_{C}.

Proof. Without loss of generality, G is semisimple. We can write

B(G) = ∏_{i=1}^{m} B(G_{i})

where the G_{i} are the simple factors of G. This decomposition respects the polysimplicial structure of B(G). If, with respect to this decomposition, x ∈ J is written as

(x_{1}, …, x_{m}),

then, from the way in which the Moy-Prasad filtration lattices are defined,

r(x) = min{r_{1}(x_{1}), …, r_{m}(x_{m})}.

Here r_{1}, …, r_{m} are the analogues of r: J → ℝ. Hence, we may in fact assume that G is simple.

Let J be a facet in B(G) and let C be an alcove in B(G). We shall show that r_{J} ≥ r_{C}. After conjugating, we may assume that J is contained in the boundary of C and that C ⊂ A. Let Δ_{C} denote the set of simple affine roots in Δ determined by C. Let Δ_{J} be the set

{δ ∈ Δ_{C} | res_{J} δ ≠ 0} = {δ ∈ Δ_{C} | res_{J} δ > 0}.

We set

r′_{J} := max_{x∈J} min_{δ∈Δ_{J}} δ(x),

and we let s denote the smallest positive number for which 𝔪_{s} ≠ 𝔪_{s^{+}}. From the way in which the Moy-Prasad filtration lattices are defined, we have

r_{J} = min{s, r′_{J}} and r_{C} = min{s, r′_{C}}.

Thus, it is enough to show that r′_{C} ≤ r′_{J}.

One can show that

r′_{J} = [ ∏_{δ∈Δ_{J}} r_{δ}]/[ ∑_{δ′∈Δ_{J}}( ∏_{δ∈Δ_{J} ∖{δ′}}r_{δ})]

where r_{δ} denotes the maximum value that δ obtains on the closure of J (and hence, on the closure of C).

Suppose J′ is a facet in the closure of C such that J is contained in the closure of J′ and dim(J′) = dim(J) + 1. Let δ̃ ∈ Δ_{C} denote the affine root for which Δ_{J′} = Δ_{J} ∪ {δ̃}. Algebraic manipulation yields

r′_{J′} = \frac{rδ̃}{rδ̃ + r′_{J}} · r′_{J} < r′_{J}.

By iterating the above process, we conclude that r′_{C} ≤ r′_{J}.

□

Remark B.5.2. If G is simple modulo its center and K-split, then

r_{G} = (1 + ∑ m_{α})^{−1},

where m_{α} runs over the coefficients of the simple roots in the expression for the highest root. In particular, for G = GL_{n}(K), we have r_{G} = n^{−1}.

Lemma B.5.3.

𝔤_{0^{+}} = 𝔤_{r_{G}} ≠ 𝔤_{r_{G}^{+}}.

Proof. Let C be an alcove in B(G). For all J in the closure of C we have 𝔤_{J}^{+} ⊂ 𝔤_{C}^{+}. Consequently

𝔤_{0^{+}} = ⋃_{g∈G}^{g}𝔤_{C}^{+}.

Thus, the equality follows from the fact that 𝔤_{C}^{+} = 𝔤_{x_{C},r_{G}}.

As in the proof of Lemma B.5.1, we may assume that G is simple; we use the notation of that proof.

Suppose 𝔤_{r_{G}} = 𝔤_{r_{G}^{+}}. Under this assumption, from [3, Corollary 3.2.2] we have 𝔤_{x_{C},r_{G}} ⊂ 𝔤_{x_{C},r_{G}^{+}} + N. Thus, from, for example, [17, §4.1.2] or [43, Proposition 4.3], every coset Ξ in 𝔤_{x_{C},r_{G}}/𝔤_{x_{C},r_{G}^{+}} is killed by a one-parameter subgroup of M := M_{0}/M_{0}^{+}; that is, for each Ξ there exists a one parameter subgroup μ = μ_{Ξ} of the 𝔣-group M so that lim_{t→0} ^{μ(t)}Ξ = 0. Consequently, in order to show that 𝔤_{r_{G}} ≠ 𝔤_{r_{G}^{+}}, it is enough to find an X ∈ 𝔤_{x_{C},r_{G}} for which the coset Ξ_{X} := X + 𝔤_{x_{C},r_{G}^{+}} is not killed by any one-parameter subgroup of M.

If s < r′_{C}, then choose X ∈ 𝔪_{s} ∖ 𝔪_{s^{+}}. Since M is abelian, no one parameter subgroup of M can kill Ξ_{X}. If s ≥ r′_{C}, then for each δ ∈ Δ_{C}, we may choose X_{δ} in the root space corresponding to the gradient of δ so that X_{δ} ∈ 𝔤_{x_{C},r_{G}} yet X_{δ} ∉ 𝔤_{x_{C},r_{G}^{+}}. From, for example, [13, Proposition 1.2], the coset Ξ_{X} for

X := ∑_{δ} X_{δ}

cannot be killed by a one-parameter subgroup of M.

□

Lemma B.5.4. We have r_{G} ≥ n^{−1}. In particular, G_{C}^{+} = G_{x_{C},1/n} and 𝔤_{C}^{+} = 𝔤_{x_{C},1/n}.

Proof. Since we are assuming that p ≥ (2+e)n, it follows that every K-torus in G or GL_{n} splits over a tame extension of K. Hence, from the discussion in [3, §3.6] we have

𝔤 ∩ 𝔤𝔩_{n}(K)_{s^{+}} = 𝔤_{s^{+}}

and

𝔤 ∩ 𝔤𝔩_{n}(K)_{s} = 𝔤_{s}

for all s. From Remark B.5.2 and Lemma B.5.3, we have 𝔤𝔩_{n}(K)_{0^{+}} = 𝔤𝔩_{n}(K)_{1/n} ≠ 𝔤𝔩_{n}(K)_{1/n^{+}}. We conclude that 1/n ≤ r_{G}. For the last assertion, note that G_{x_{C},r} = G_{x_{C},0^{+}} for 0 < r ≤ r_{G}, and likewise for 𝔤_{x_{C},r}. □

B.6. A logarithmic map for semisimple groups. Suppose that G is semisimple.

Moy-Prasad filtrations and the adjoint representation.

Lemma B.6.1. Suppose x ∈ B(G), t ∈ ℝ, and X ∈ 𝔤_{x,t}. We have

X ∉ 𝔤_{x,t^{+}}

if and only if there exist q ∈ ℝ and Q ∈ 𝔤_{x,q} ∖ 𝔤_{x,q^{+}} such that

ad(X)Q ∈ 𝔤_{x,(t+q)} ∖ 𝔤_{x,(t+q)^{+}}.

Proof. “⇐”: Suppose X ∈ 𝔤_{x,t^{+}}. Then for all q ∈ ℝ and for all Q ∈ 𝔤_{x,q} we have

ad(X)Q ∈ 𝔤_{x,(t+q)^{+}},

a contradiction.

“⇒”: From Lemma A.1.1, there exists Y ∈ 𝔤_{x,−t} ∖ 𝔤_{x,(−t)^{+}} such that

(45)  B(X, Y) ∈ R^{×}.

For all s ∈ ℝ we have

(ad(X) ad(Y))𝔤_{x,s} ⊂ 𝔤_{x,s}.

Since G is semisimple, we have that B is the Killing form. We conclude from Equation (45) that there exist a q ∈ ℝ and a Z ∈ 𝔤_{x,(t+q)} ∖ 𝔤_{x,(t+q)^{+}} such that

ad(X)(ad(Y)Z) ∈ 𝔤_{x,(t+q)} ∖ 𝔤_{x,(t+q)^{+}}.

Let Q := ad(Y)Z ∈ 𝔤_{x,q}. Since ad(X)Q ∈ 𝔤_{x,(t+q)} ∖ 𝔤_{x,(t+q)^{+}}, we conclude that Q ∈ 𝔤_{x,q} ∖ 𝔤_{x,q^{+}}. □

Corollary B.6.2. Suppose x ∈ B(G), s ∈ ℝ, and X ∈ 𝔤. We have

X ∈ 𝔤_{x,s}

if and only if for all q ∈ ℝ and for all Q ∈ 𝔤_{x,q} we have

ad(X)Q ∈ 𝔤_{x,(s+q)}.

Proof. “⇒”: There is nothing to prove.

“⇐”: If X ∉ 𝔤_{x,s}, then there exists t < s such that X ∈ 𝔤_{x,t} ∖ 𝔤_{x,t^{+}}. From Lemma B.6.1, as X ∉ 𝔤_{x,t^{+}}, there exist q ∈ ℝ and Q ∈ 𝔤_{x,q} such that ad(X)Q ∉ 𝔤_{x,(t+q)^{+}}. But 𝔤_{x,(s+q)} ⊂ 𝔤_{x,(t+q)^{+}}, so ad(X)Q ∉ 𝔤_{x,(s+q)}. □

Moy-Prasad filtrations and ψ, I.

Remark B.6.3. Since p ≥ (2 + e) · n, we have m ≥ n · ν(m) + 2 for m ≥ 2. If we assume that m ≥ (2n − 1), then we have m ≥ n · (2 + ν(m)) − 1.

Lemma B.6.4. Suppose x ∈ B(G) and t ∈ ℝ_{≥1/n}. If g ∈ G_{x,t}, then for all q ∈ ℝ and for all Q ∈ 𝔤_{x,q} we have

log(Ad(g))Q ≡ (Ad(g) − 1)Q modulo 𝔤_{x,(2t+q)}.

Proof. Fix q ∈ ℝ and Q ∈ 𝔤_{x,q}. For m > 1 we have

\frac{(1 − Ad(g))^{m}}{m} Q

belongs to

\begin{aligned}\frac{1}{m} · 𝔤_{x,(q+tm)} & ⊂ 𝔤_{x,(q+tm−ν(m))} \\ & ⊂ 𝔤_{x,(q+2t+t(m−2)−ν(m))}. \end{aligned}

From Remark B.6.3 we have m ≥ n · ν(m) + 2. Thus

𝔤_{x,(q+2t+t(m−2)−ν(m))} ⊂ 𝔤_{x,(q+2t)}.

Consequently,

log(Ad(g))Q ≡ (Ad(g) − 1)Q modulo 𝔤_{x,(q+2t)}.
□

Corollary B.6.5. For all x ∈ B(G) and for all s ≥ 1/n, we have

ψ(G_{x,s}) ⊂ 𝔤_{x,s}.

Proof. From Corollary B.6.2, it is enough to show that for all q ∈ ℝ, for all Q ∈ 𝔤_{x,q}, and for all g ∈ G_{x,s}, we have

ad(ψ(g))Q ∈ 𝔤_{x,(s+q)}.

However, from Equation (44) we have

ad(ψ(g))Q = log(Ad(g))Q,

and log(Ad(g))Q ∈ 𝔤_{x,(s+q)} from Lemma B.6.4. □

Logarithmic behavior of ψ.

Lemma B.6.6. Suppose x ∈ B(G) and s, t ∈ ℝ_{>0} with s ≤ t. If g ∈ G_{x,s} and h ∈ G_{x,t}, then for all q ∈ ℝ and for all Q ∈ 𝔤_{x,q} we have

(1 − Ad(gh))^{m}Q ≡ (1 − Ad(g))^{m}Q modulo 𝔤_{x,(t+q+(m−1)s)}

for all m ∈ ℤ_{≥1}.

Proof. We will argue by induction on m. Suppose x ∈ B(G), s, t ∈ ℝ_{>0} with s ≤ t, g ∈ G_{x,s}, and h ∈ G_{x,t}. For q ∈ ℝ and Q ∈ 𝔤_{x,q}, we define T = T (Q, h) ∈ 𝔤_{x,(q+t)} by T := ^{h}Q − Q.

When m = 1, we have that for all q ∈ ℝ and for all Q ∈ 𝔤_{x,q}

(1 − Ad(gh))Q = Q −^{gh}Q = Q −^{g}Q −^{g}T
≡ Q −^{g}Q modulo 𝔤_{x,(q+t)}
= (1 − Ad(g))Q.

If

(1 − Ad(gh))^{m}Q′ ≡ (1 − Ad(g))^{m}Q′ modulo 𝔤_{x,(q′+t+(m−1)s)}

for all q′ ∈ ℝ and for all Q′ ∈ 𝔤_{x,q′}, then for all q ∈ ℝ and for all Q ∈ 𝔤_{x,q}

(1 − Ad(gh))^{(m+1)}Q = (1 − Ad(gh))^{m}[(1 − Ad(gh))Q]
(since s ≤ t, we have (1 − Ad(gh))Q ∈ 𝔤_{x,(q+s)})
≡ (1 − Ad(g))^{m}[(1 − Ad(gh))Q] modulo 𝔤_{x,(q+s+t+(m−1)s)}
(≡ (1 − Ad(g))^{m}[(1 − Ad(gh))Q] modulo 𝔤_{x,(q+t+ms)})
= [(1 − Ad(g))^{(m+1)}Q] − [(1 − Ad(g))^{m}(^{g}T)]
(since^{g}T ∈ 𝔤_{x,(t+q)} and s ≤ t)
≡ (1 − Ad(g))^{(m+1)}Q modulo 𝔤_{x,(t+q+ms)}.
□

Lemma B.6.7. Suppose x ∈ B(G) and s, t ∈ ℝ with t ≥ s ≥ 1/n. For all g ∈ G_{x,s} and for all h ∈ G_{x,t}, we have

ψ(gh) ≡ ψ(g) + ψ(h) modulo 𝔤_{x,(s+t)}.

Proof. Suppose x, s, t, g, and h are as in the statement of the lemma. From Corollary B.6.2, it will be enough to show that if q ∈ ℝ and Q ∈ 𝔤_{x,q}, then

ad[ψ(gh) − ψ(g) − ψ(h)]Q ∈ 𝔤_{x,(q+s+t)}.

Thus, from Equation (44), it will be enough to show that

[log(Ad(gh)) − log(Ad(g)) − log(Ad(h))]Q ∈ 𝔤_{x,(q+s+t)}.

Since

\frac{(1 − Ad(gh))^{m}}{m} , \frac{(1 − Ad(g))^{m}}{m} , and \frac{(1 − Ad(h))^{m}}{m}

all tend to zero in 𝔤𝔩(𝔤)(K), there exists N ∈ ℤ_{>1}, independent of q and Q, so that

[log(Ad(gh)) − log(Ad(g)) − log(Ad(h))]Q

is equivalent to

− ∑_{1}^{N} \frac{1}{m} · [(1 − Ad(gh))^{m} − (1 − Ad(g))^{m} − (1 − Ad(h))^{m}]Q

modulo 𝔤_{x,(s+t+q)}. Fix 2 ≤ m ≤ N. From Lemma B.6.6 we have

[(1 − Ad(gh))^{m}−(1 − Ad(g))^{m} − (1 − Ad(h))^{m}]Q
≡ −(1 − Ad(h))^{m}Q modulo 𝔤_{x,(t+q+s(m−1))}
(since t ≥ s)
≡ 0 modulo 𝔤_{x,(t+q+s(m−1))}.

Thanks to Remark B.6.3, for m ≥ 2 we have

s(m − 2) − ν(m) ≥ \frac{1}{n}(m − 2) − ν(m) ≥ 0,

We conclude that for m ≥ 2

\frac{1}{m}[(1 − Ad(gh))^{m} − (1 − Ad(g))^{m} − (1 − Ad(h))^{m}]Q

belongs to 𝔤_{x,(q+t+s)}. Consequently,

[log(Ad(gh)) − log(Ad(g)) − log(Ad(h))]Q

is equivalent to

−[(1 − Ad(gh)) − (1 − Ad(g)) − (1 − Ad(h))]Q

modulo 𝔤_{x,(q+t+s)}. But the latter is (Ad(g) − 1)(Ad(h) − 1)Q, which belongs to 𝔤_{x,(q+t+s)}. □

Remark B.6.8. The condition t ≥ s in Lemma B.6.7 is not required. Suppose x, g, h are as in the statement of the lemma and 1/n ≤ t < s. Choose u ∈ G_{x,(s+t)} so that^{g}h = hu. Then

ψ(gh) = ψ((^{g}h)g) ≡ ψ(^{g}h) + ψ(g) modulo 𝔤_{x,(s+t)}
= ψ(hu) + ψ(g) ≡ ψ(h) + ψ(u) + ψ(g) modulo 𝔤_{x,(2t+s)}

(from Corollary B.6.5) ≡ ψ(h) + ψ(g) modulo 𝔤_{x,(s+t)}

= ψ(g) + ψ(h).

We can now reformulate Lemma B.6.7 as follows.

Corollary B.6.9. Suppose x ∈ B(G) and s, t ∈ ℝ_{≥1/n}. For all g ∈ G_{x,s} and for all h ∈ G_{x,t}, we have

ψ(gh) ≡ ψ(g) + ψ(h) modulo 𝔤_{x,(s+t)}.

Filtration quotients and ψ.

Remark B.6.10. Since every torus of G splits over a tamely ramified extension of K, for all t ∈ ℝ_{>0} and for all x ∈ B(G) we have an isomorphism of abelian groups

G_{x,t}/G_{x,t^{+}} ≅ 𝔤_{x,t}/𝔤_{x,t^{+}}.

This isomorphism has the property that for each coset Ξ_{G} in G_{x,t}/G_{x,t^{+}} the isomorphism identifies a coset Ξ_{𝔤} in 𝔤_{x,t}/𝔤_{x,t^{+}} so that for all q ∈ ℝ and for each Q ∈ 𝔤_{x,q} we have

ad(X)Q ≡ (Ad(g) − 1)Q modulo 𝔤_{x,(t+q)^{+}}

for all X ∈ Ξ_{𝔤} and for all g ∈ Ξ_{G}. See [64, Corollary 2.4] or [65] for details.

Lemma B.6.11. Suppose x ∈ B(G) and t ∈ ℝ_{>0}. If g ∈ G_{x,t} ∖ G_{x,t^{+}}, then there exist q ∈ ℝ and Q ∈ 𝔤_{x,q} ∖ 𝔤_{x,q^{+}} such that

(Ad(g) − 1)Q ∈ 𝔤_{x,(t+q)} ∖ 𝔤_{x,(t+q)^{+}}.

Proof. Choose g as in the statement of the lemma. If X ∈ Ξ_{𝔤}, where Ξ_{𝔤} ∈ 𝔤_{x,t}/𝔤_{x,t^{+}} corresponds to the coset gG_{x,t^{+}} in G_{x,t}/G_{x,t^{+}}, then X ∈ 𝔤_{x,t}∖𝔤_{x,t^{+}}. From Lemma B.6.1 we can choose q ∈ ℝ and Q ∈ 𝔤_{x,q} ∖ 𝔤_{x,q^{+}} so that

ad(X)Q ∈ 𝔤_{x,(t+q)} ∖ 𝔤_{x,(t+q)^{+}}.

Since, from Remark B.6.10,

(Ad(g) − 1)Q ≡ ad(X)Q modulo 𝔤_{x,(t+q)^{+}},

the lemma follows. □

Lemma B.6.12. Suppose t ≥ 1/n and x ∈ B(G). The restriction of ψ to G_{x,t} induces an isomorphism

G_{x,t}/G_{x,t^{+}} ≅ 𝔤_{x,t}/𝔤_{x,t^{+}}

of abelian groups.

Proof. Fix t ≥ 1/n and x ∈ B(G). Since ψ: G_{0^{+}} → 𝔤 is injective and from Corollary B.6.5

ψ(G_{x,t^{+}}) ⊂ 𝔤_{x,t^{+}}

while

ψ(G_{x,t}) ⊂ 𝔤_{x,t},

from Lemma B.6.7 we have that ψ induces a group homomorphism

G_{x,t}/G_{x,t^{+}} → 𝔤_{x,t}/𝔤_{x,t^{+}}.

We will show that this map is surjective. Since G_{x,t}/G_{x,t^{+}} and 𝔤_{x,t}/𝔤_{x,t^{+}} are finite-dimensional 𝔉-vector spaces of the same dimension, injectivity will follow.

To show that the induced map is surjective, we must show that for each X ∈ 𝔤_{x,t} there is a g ∈ G_{x,t} for which

X − ψ(g) ∈ 𝔤_{x,t^{+}}.

Equivalently, from Corollary B.6.2, we need that for all q ∈ ℝ and for all Q ∈ 𝔤_{x,q}

[ad(X) − ad(ψ(g))]Q ∈ 𝔤_{x,(q+t)^{+}}.

Suppose X ∈ 𝔤_{x,t}. From Remark B.6.10, there is a g ∈ G_{x,t} so that for all q ∈ ℝ and each Q ∈ 𝔤_{x,q} we have

(46)  (Ad(g) − 1)Q ≡ ad(X)Q modulo 𝔤_{x,(q+t)^{+}}.

Now, for all q ∈ ℝ and for all Q ∈ 𝔤_{x,q}, we have

[ad(X) − ad(ψ(g))]Q = [ad(X) − log(Ad(g))]Q

(from Corollary B.6.4) ≡ [ad(X) − (Ad(g) − 1)]Q modulo 𝔤_{x,(q+2t)}

(from Equation (46)) ≡ 0 modulo 𝔤_{x,(q+t)^{+}}.

Thus [ad(X) − ad(ψ(g))]Q ∈ 𝔤_{x,(q+t)^{+}}. □

Moy-Prasad filtrations and ψ, II. We begin with an abstract result about maps between complete topological groups.

Lemma B.6.13. Suppose H and L are complete topological groups. Let f: H → L be a map for which there exist neighborhood bases at the identity

{H_{i} ≤ H | H_{1} := H ≥ H_{2} ≥ H_{3} ≥ ⋯ ≥ {1}}

and

{L_{i} ≤ L | L_{1} := L ≥ L_{2} ≥ L_{3} ≥ ⋯ ≥ {1}}

for H and L so that (1) f(H_{i}) ⊂ L_{i} for all i and

(2) if h ∈ H_{i} and h′ ∈ H_{j}, then f(hh′) ≡ f(h)f(h′) modulo L_{i+j}. If the induced map

H_{i}/H_{(i+1)} → L_{i}/L_{(i+1)},

is surjective for all i, then f is surjective.

Remark B.6.14. Note that the first condition on f implies that it is continuous at the identity, while the second implies that f is continuous everywhere.

Proof. Suppose ℓ ∈ L. Fix j_{0} ∈ ℤ_{≥1} so that ℓ ∈ L_{j_{0}} ∖ L_{(j_{0}+1)}. By hypothesis, there is an h_{0} ∈ H_{j_{0}} such that f(h_{0}) ≡ ℓ modulo L_{(j_{0}+1)}. Fix j_{1} > j_{0} so that f(h_{0})^{−1}ℓ ∈ L_{j_{1}} ∖ L_{(j_{1}+1)}. Since the induced map

H_{j_{1}}/H_{(j_{1}+1)} → L_{j_{1}}/L_{(j_{1}+1)}

is surjective, there is an h′_{1} ∈ H_{j_{1}} so that f(h′_{1}) ≡ f(h_{0})^{−1}ℓ modulo L_{(j_{1}+1)}. Set h_{1} := h_{0}h′_{1}. We have

f(h_{1}) ≡ f(h_{0})f(h′_{1}) modulo L_{j_{1}+j_{0}}.

Thus, f(h_{1}) ≡ ℓ modulo L_{j_{1}+1}.

Choose j_{2} > j_{1} so that f(h_{1})^{−1}ℓ ∈ L_{j_{2}} ∖ L_{(j_{2}+1)}. Since the induced map

H_{j_{2}}/H_{(j_{2}+1)} → L_{j_{2}}/L_{(j_{2}+1)}

is surjective, there is an h′_{2} ∈ H_{j_{2}} so that f(h′_{2}) ≡ f(h_{1})^{−1}ℓ modulo L_{(j_{2}+1)}. Set h_{2} := h_{1}h′_{2}. We have

f(h_{2}) ≡ f(h_{1})f(h′_{2}) modulo L_{j_{2}+j_{1}}.

Thus, f(h_{2}) ≡ ℓ modulo L_{j_{2}+1}.

Continuing in this fashion, we produce a convergent sequence (h_{i}) in H. If h = lim h_{i}, then f(h) = ℓ. □

Lemma B.6.15. For all facets J ⊂ B(G) and for all s > 0 we have ψ(G_{x_{J} ,s}) = 𝔤_{x_{J} ,s}.

Proof. From Lemma B.5.4 we may assume that s ≥ 1/n. Thanks to Corollary B.6.5 it suffices to prove surjectivity.

Choose m′ ∈ ℤ_{≥1} so that J is F^{m′}-stable. We let x = x_{J}. It will be enough to show that for all m ∈ ℤ_{≥m′} we have

ψ(G_{x,s}^{F^{m}}) = 𝔤_{x,s}^{F^{m}}.

Note that G_{x,s}^{F^{m}} and 𝔤_{x,s}^{F^{m}} are complete topological groups. Thanks to Corollary B.6.5, Lemma B.6.7, and Lemma B.6.12, the result follows from Lemma B.6.13. □

Corollary B.6.16.

ψ(G_{0^{+}}) = 𝔤_{0^{+}}.

Proof. From Lemma B.6.15, for all facets J in B(G) we have ψ(G_{J}^{+}) = 𝔤_{J}^{+}. Since

G_{0^{+}} = ⋃_{J} G_{J}^{+}

and

𝔤_{0^{+}} = ⋃_{J} 𝔤_{J}^{+},

the result follows. □

The map over the residue field induced by ψ.

Lemma B.6.17. Suppose x ∈ B(G). If t ≥ 1/n and g ∈ G_{x,t}, then for all q ∈ ℝ and for each Q ∈ 𝔤_{x,q}, we have

log(Ad(g))Q ≡ − ∑_{m=1}^{2(n−1)} \frac{(1 − Ad(g))^{m}}{m} Q

modulo 𝔤_{x,(q+2−1/n)}.

Proof. Fix t ≥ 1/n and g ∈ G_{x,t}. Suppose q ∈ ℝ and Q ∈ 𝔤_{x,q}. For all m ∈ ℤ_{≥1} we have

\frac{(1 − Ad(g))^{m}}{m} Q ∈ 𝔤_{x,(q+mt−ν(m))}.

Since p ≥ (2 + e)n, we conclude that for 1 ≤ m ≤ (3n − 2),

\frac{(1 − Ad(g))^{m}}{m} Q ∈ 𝔤_{x,(q+mt)}

(since m is a unit). In particular, as t ≥ 1/n, we conclude that

∑_{m=1}^{2(n−1)} \frac{(1 − Ad(g))^{m}}{m} Q ≡ ∑_{m=1}^{(3n−2)} \frac{(1 − Ad(g))^{m}}{m} Q modulo 𝔤_{x,(q+2−1/n)}.

To finish the proof, it is enough to show that if m ≥ (3n − 1), then mt − ν(m) ≥ 2 − 1/n. This follows from Remark B.6.3.

□

Lemma B.6.18. Suppose J ⊂ B(G) is a facet and C ⊂ B(G) is an alcove which contains J in its closure. If g ∈ G_{C}^{+} and h ∈ G_{J}^{+}, then

ψ(gh) ∈ ψ(g) + 𝔤_{J}^{+}.

Proof. Since G_{J}^{+} ≤ G_{C}^{+}, both g and gh belong to G_{C}^{+} = G_{x_{C},1/n} (see Lemma B.5.4). Consequently, from Lemma B.6.15, both ψ(g) and ψ(gh) belong to 𝔤_{C}^{+} ≤ 𝔤_{J}. Hence, the images of ψ(g) and ψ(gh) in L_{J} belong to the nilradical of the Borel subgroup of G_{J} corresponding to C. Hence, they both belong to the derived Lie algebra of L_{J}. Since the restriction of B to 𝔤_{J} induces the Killing form on the derived Lie algebra of L_{J}, it will be enough to show that for all Q ∈ 𝔤_{J} we have

[ad(ψ(gh)) − ad(ψ(g))]Q ∈ 𝔤_{J}^{+}.

Fix Q ∈ 𝔤_{J}. Since ϖQ ∈ 𝔤_{J}^{+} , we have

Q ∈ ϖ^{−1}𝔤_{J}^{+} ≤ ϖ^{−1}𝔤_{C}^{+} = 𝔤_{x_{C},1/n−1}.

From Lemma B.6.17 and Equation (44) we have

ad(ψ(gh))Q = log(Ad(gh))Q ≡ − ∑_{m=1}^{2(n−1)} \frac{(1 − Ad(gh))^{m}}{m} Q

modulo 𝔤_{x_{C},1} = ϖ𝔤_{C} ≤ ϖ𝔤_{J} ≤ 𝔤_{J}^{+}. (Note: m is a unit for 1 ≤ m ≤ 2(n − 1).) Similarly,

ad(ψ(g))Q ≡ − ∑_{m=1}^{2(n−1)} \frac{(1 − Ad(g))^{m}}{m} Q

modulo 𝔤_{J}^{+}. Consequently,

[ad(ψ(gh) − ψ(g))]Q ≡ ∑_{m=1}^{2(n−1)} [\frac{(1 − Ad(g))^{m} − (1 − Ad(gh))^{m}}{m} ] Q

modulo 𝔤_{J}^{+}. Since h acts trivially on 𝔤_{J}/𝔤_{J}^{+}, we have

(1 − Ad(gh))^{m}Q ≡ (1 − Ad(g))^{m}Q

modulo 𝔤_{J}^{+}. The result follows. □

Corollary B.6.19. Suppose J is an F -stable facet in B(G). The restriction of ψ to G_{0^{+}} ∩ G_{J} induces a ⟨F ⟩⋉G_{J}-equivariant bijective map from U_{J}, the 𝔣-variety of unipotent elements in G_{J}, to N_{J}, the 𝔣-variety of nilpotent elements in L_{J}.

Proof. If ḡ ∈ U_{J}, then there exist an alcove C and a g ∈ G_{C}^{+} such that J ⊂ C̄ and g is a lift of ḡ. From Lemma B.6.15 we have ψ(g) ∈ 𝔤_{C}^{+} ⊂ 𝔤_{J}. Thus, the image of ψ(g) in L_{J} belongs to N_{J}. From Lemma B.6.18, the image of ψ(g) in L_{J} is independent of the choice of g. Hence ψ induces a map ψ̄: U_{J} → N_{J}. As ψ is ⟨F ⟩ ⋉ G_{J}-equivariant, it follows that ψ̄ is ⟨F ⟩ ⋉ G_{J}-equivariant.

To see that ψ̄: U_{J} → N_{J} is bijective, we note that p ≥ (2 + e)n implies (see for example [12, §1.15]) that there is a (non-unique) bijective, G_{J}-equivariant 𝔣-morphism identifying U_{J} with N_{J}. Thus, for all m ∈ ℤ_{≥1} the sets U_{J}^{F^{m}} and N_{J}^{F^{m}} have the same cardinality. Consequently, it is enough to show that the restriction to U_{J}^{F^{m}} of ψ̄ surjects onto N_{J}^{F^{m}}. If X̄ ∈ N_{J}^{F^{m}}, then there exist an F^{m}-stable alcove C and X ∈ (𝔤_{C}^{+})^{F^{m}} such that J ⊂ C̄ and X is a lift of X̄. From the proof of Lemma B.6.15 there exists a g ∈ (G_{C}^{+})^{F^{m}} such that ψ(g) = X.

Since U_{J} is the image of G_{0^{+}} ∩ G_{J} in G_{J}, the corollary follows. □

B.7. An extension to reductive groups. Drop the assumption that G is semisimple. In this section, we prove that the ⟨F ⟩ ⋉ G- equivariant map ψ: G_{0^{+}} → 𝔤 has the properties described in Lemma B.0.3.

Let G′ denote the group of K-rational points of the derived group of G. Let Z denote the group of K-rational points of Z, the identity component of the center of G. We recall that Z ∩ G′ is finite. We let 𝔤′ (resp. 𝔷, resp. 𝔷) denote the Lie algebra of G′ (resp. Z, resp. Z).

In Section B.6 we proved that the map

res_{G′} ψ: G′_{0^{+}} → 𝔤′

has the properties required by Lemma B.0.3. From [9, III, §7, Proposition 11] we have

(47)  ψ(zh) = ψ(z) + ψ(h)

for all z ∈ Z_{0^{+}} and all h ∈ G′_{0^{+}}.

Suppose S is any torus in G. Let S denote the group of K-rational points of S. Let 𝔰 (resp. 𝔰) denote the Lie algebra of S (resp. S).

Lemma B.7.1. With our assumptions on p, we have

ψ(S_{0^{+}}) = 𝔰_{0^{+}}.

Proof. Let E denote the splitting field of S over K. Since φ: G → GL_{n} is faithful and φ(S) is a torus in GL_{n}, the field E is a tame Galois extension of K and ν_{E}(p) ≤ nν(p).

Since E is a tame Galois extension of K, from [2, Lemma 2.2.3], we have

S_{0^{+}} = S(E)_{0^{+}} ∩ S.

By an argument similar to that given in Section B.3, there is a unique Gal(E/K) ⋉ G(E)- equivariant extension of ψ to a map ψ: G(E)_{0^{+}} → 𝔤(E). From Equation (44) the image of the restriction to S(E)_{0^{+}} of this map lies in 𝔰(E). It will be enough to show that ψ(S(E)_{0^{+}}) = 𝔰(E)_{0^{+}}.

Since S is E-split, there is an E-isomorphism φ_{S} from S to (GL_{1})^{j} for some j. Since

p ≥ (2 + ν(p))n ≥ 2n + ν_{E}(p),

we have p ≥ 2 + ν_{E}(p). We conclude (see the discussion concerning GL_{n} in §B.1) that

log((GL_{1}(E))_{0^{+}}^{j}) = (M_{1}(E))_{0^{+}}^{j}.

Since φ_{S} and dφ_{S} are E-isomorphisms, the result follows from the fact that dφ_{S}(ψ(s)) = log(φ_{S}(s)) for s ∈ S(E)_{0^{+}} (see Equation (41)). □

Lemma B.7.2. Under our assumptions on p, the map (z, h) ↦ zh from Z_{0^{+}} × G′_{0^{+}} to G_{0^{+}} is bijective.

The proof below is due to Loren Spice; it is shorter than our original proof.

Proof (Spice). Since for each x ∈ B(G) we have Z_{0^{+}} ⊆ G_{x,0^{+}} and G′_{x,0^{+}} ⊆ G_{x,0^{+}}, it suffices to check that the map i_{x}: Z_{0^{+}} × G′_{x,0^{+}} → G_{x,0^{+}} which sends (z, h) to zh is bijective for all x ∈ B(G).

Fix x ∈ B(G). To show i_{x} is bijective, it is enough to check that the induced map on successive quotients of Moy-Prasad filtration subgroups is bijective. Fix r ∈ ℝ_{>0}. From [64, Corollary 2.4], it is enough to check that the induced map

𝔷_{r}/𝔷_{r^{+}} × 𝔤′_{x,r}/𝔤′_{x,r^{+}} → 𝔤_{x,r}/𝔤_{x,r^{+}}

is bijective. From [4, Proposition 3.2], it is surjective. If (Z̄, X̄) is in its kernel, then there exist Z ∈ 𝔷_{r} (resp., X ∈ 𝔤′_{x,r}) lifting Z̄ (resp., X̄) so that Z + X ∈ 𝔤_{x,r^{+}}. From [4, Proposition 3.2], we conclude that Z ∈ 𝔷_{r^{+}} and X ∈ 𝔤′_{x,r^{+}}. Thus, the map is injective as well. □

Thanks to Equation (47), from Lemma B.7.1 and Lemma B.7.2, the map ψ is a bijective ⟨F ⟩ ⋉ G-equivariant map from G_{0^{+}} to 𝔤_{0^{+}} = 𝔷_{0^{+}} + 𝔤′_{0^{+}}. Moreover, since, for all x ∈ B(G), the image of 𝔷_{0^{+}} in L_{x} is trivial, it follows from Lemma B.7.1 (with S = Z) that ψ has the properties required by Lemma B.0.3.

Index of selected notation and terms

elliptic Langlands parameter  25
F -regular  8
F -minisotropic  8
generic representation  40
G-stable conjugacy-class  21
G-stably conjugate  50
G-stable classes  50
lift of (J, S)  47
(ϕ, ϕ′)-comparable  64
rational classes  21
regular semisimple  8
strongly regular semisimple  8
tame regular semisimple  27
topological Jordan decomposition  46
topologically semisimple  46
TRSELP  28
unramified torus  8

∗  g ∗ u := guF (g)^{−1}  11
Ad  adjoint action of G  7
A(S)  apartment of unramified torus S  9
A  apartment corresponding to T  9

[Ad(H)X]^{F} ∐_{i=1}^{n} Ad(H^{F})X_{i}  63
B  nondegenerate, symmetric, ⟨F ⟩ ⋉ H-invariant bilinear form on 𝔥  63
B(G)  Bruhat-Tits building of G  9
C_{λ}  an alcove in A which contains J_{λ} in its closure  17
C_{φ}  component group of CĜ(φ)  25
deg(π)  formal degree of π  3
D̃(γ, S)  {d ∈ G^{F} : ^{d}γ ∈ S}  47
ε(·, ·)  sign depending on relative ranks  8
E(γ, S)  \{g ∈ G^{F} : ^{g}γ ∈ G_{J}, \overline{^{g}γ} ∈ S\}  47
f  element of N̂  28
f̂  Fourier transform of f with respect to B  63
𝔣  residue field of k  7
𝔣_{d}  the degree d extension of 𝔣  12
𝔉  residue field of K  7
Frob  topological generator for Γ  8
F  automorphism of G arising from k-structure on G  9
F  the automorphism F when G is k-quasisplit  10
F_{u}  Ad(u) ∘ F  10
F_{λ}  Ad(u_{λ}) ∘ F  19
G_{0}  compact elements of G  45
G_{0^{+}}  topologically unipotent elements in G  45
Ĝ  dual group of G  16
LG  ⟨ϑ̂⟩ ⋉ Ĝ  25
G^{rss}  set of regular semisimple elements of G  8
G^{srss}  set of strongly regular semisimple elements of G  8
G_{ad}  group of K-rational points of the adjoint group of G  9
G_{J}  parahoric subgroup of G corresponding to J ⊂ B(G)  9
G_{λ}  G_{J_{λ}}  19
G_{J}^{+}  pro-unipotent radical of G_{J}  9
G_{J}  connected reductive 𝔣-group associated to J ⊂ B(G)  9
G_{λ}  G_{λ}/G_{λ}^{+}  32
𝔤_{J}  lattice in 𝔤 attached to J ⊂ B(G)  37
𝔤_{J}^{+}  sublattice in 𝔤_{J}  37
𝔤_{0}  compact elements of 𝔤  45
𝔤_{0^{+}}  topologically nilpotent elements in 𝔤  45
Γ  Gal(k̄/k)/I  7
G_{γ}  identity component of the centralizer of γ in G  8
γ_{s}  topologically semisimple part of γ  46
γ_{u}  topologically unipotent part of γ  46
ind  compact induction functor  3
Ind  smooth induction functor  26

Irr  set of irreducible representations  8
Irr^{2}  set of irreducible square-integrable representations  8
Irr_{0}  set of irreducible depth-zero representations  49
Irr(C_{φ}, ω)  representations ρ ∈ Irr(C_{φ}) with ω_{ρ} = ω  25
I  inertia subgroup of Gal(k̄/k)  7
I^{+}  wild inertia subgroup  25
I_{t}  tame inertia group  25
I(γ_{s})  index set for certain G_{γ_{s}}-stable classes  59
Î(γ_{s})  index set for certain G_{γ_{s}}-stable classes  59
ι_{g}  map from I(γ_{s}) to I(^{g}γ_{s})  61
J_{λ}  facet in A preserved by σ_{λ}  17
k  finite extension of ℚ_{p}  7
K  maximal unramified extension of k  7
κ_{λ}^{0}  ϵ(G_{λ}, T_{λ}) · R_{T_{λ},χ_{λ}^{0}}^{G_{λ}} ∈ Irr(GF_{λ} ; λ)  33
κ_{λ}  representation of Z^{F}GF_{λ} ; λ  33
L_{J}  Lie algebra of G_{J}; identified with 𝔤_{J}/𝔤_{J}^{+}  37
m_{λ}  element of N for which m_{λ} ∗ u_{λ} = u and m_{λ} · C_{λ} = C  19
μ_{X}^{H^{F}} (f)  H^{F} -orbital integral of f with respect to X  63
μ̂_{X}^{H^{F}}  function representing Fourier transform of μ_{X}^{H^{F}}  63
N(G, S)  normalizer of a subgroup S ⊂ G  7
N  N(G, T)  9
N_{o}  N ∩ G_{o}  10
N(i)  |N(G_{γ_{s}}, S^{F})/S|, where S ∈ T_{st}(γ_{s}, i)  59
Ω_{C}  {ω ∈ W : ω · C = C} for some alcove C in A  9
ω_{λ}  unique element of t_{λ}W^{∘} ∩ Ω_{C}  15
ω̇_{λ}  element of Z^{1}(F, N_{C}) with image ω_{λ} in W  15
ω  fixed element of H^{1}(F, G)  19
o  F-fixed hyperspecial vertex in A_{ad}  10
p  characteristic of the residue field 𝔣  7
p_{λ}  element of G_{λ} for which p_{λ}^{−1} F_{λ}(p_{λ}) = ẇ_{λ}  19
π_{λ}  Ind_{Z^{F}G{F \atop λ} ; λ}^{G{F \atop λ}} κ_{λ}  34
π_{u}(φ, ρ)  Ad(m_{λ})_{∗}π_{λ} ∈ Irr(GF_{u})  36
Π_{u}(φ)  normalized L-packet  36
p_{1}  surjective projection onto first factor: T̂_{v,χ} ⟶ T_{v}  51
p_{2}  projection on second factor: p_{1}^{−1}(S) ⟶ Irr_{0}(S^{F})  51
q  cardinality of the residue field 𝔣  7
q_{λ}  m_{λ}p_{λ} ∈ G  20
Q̇_{S}^{(G_{J})_{\overline{γ}_{s}}}  natural inflation of Q_{S}^{(G_{J})_{\overline{γ}_{s}}}, extended by zero to G^{F}  54
Q(G_{γ_{s}}, T_{st}^{1}) stable p-adic analogue of a Green function  59
r  map X_{w} → H^{1}(F, G)  19

[r^{−1}(ω)]  image of r^{−1}(ω) in [X/(1 − wϑ)X]_{tor}  21
R(G, S, θ)  function on (G^{rss})^{F}  49
R(G, T )  R(G, S, θ), where T is the G^{F} -orbit of (S, θ)  49
R(G, T̂_{st})  ∑_{(S,θ)∈T̂_{st}/G^{F}} R(G, S, θ)  52
s  homomorphism s : I_{t} ⟶ T̂ with CĜ(s) = T̂  28
0S  maximal bounded subgroup of an unramified torus S  8
σ_{λ}  t_{λ}wϑ ∈ W ⋊ ⟨ϑ⟩  17
S_{λ}  Ad(q_{λ})T  20
St  Steinberg representation  3
Ŝ_{X}^{𝔥}  Fourier transform of the stable orbital integral associated to X  63
T  fixed maximally k-split K-split torus in G  9
0T  maximal bounded subgroup of T  9
T_{λ}  Ad(p_{λ})T  32
T̂  Y ⊗ ℂ^{×}  16
t_{λ}  element of T or W corresponding to λ ∈ X  9
𝔗(G)  set of F -minisotropic maximal tori in G  47
𝔗̂(G)  {(S, θ) : S ∈ 𝔗(G) and θ ∈ Irr_{0}(S^{F})}  51
T_{v}  {S ∈ 𝔗(G) : S^{F} = ^{g}(T^{F_{v}}) for some g ∈ G}  50
T̂_{v,χ}  (S, θ) for which there is g ∈ G so that S^{F} = ^{g}(T^{F_{v}}) and θ = g_{∗}χ  51
T̂_{st}  fixed G-stable class in 𝔗̂(G)  52
T̂  G^{F} -orbit in 𝔗(G)  54
T̂(γ_{s})  {(S′, θ′) ∈ T̂ : γ_{s} ∈ S′}  54
T̂_{st}(γ_{s})  {(S, θ) ∈ T̂_{st} : γ_{s} ∈ S}  59
θ_{S}^{χ}  ∑_{θ ∈ p_{2}p_{1}^{−1}(S)} θ  52
θ_{i}^{χ}(γ_{s})  θ_{S}^{χ}(γ_{s}), for any S ∈ T_{st}(γ_{s}, i)  60
Θ_{ρ_{λ}}  character of π_{u}(φ, ρ_{λ})  53
u  fixed representative of ω  19
u_{λ}  element of Z^{1}(F, N) which lifts y_{λ}  18
ϖ  fixed uniformizer of k  7
ϑ  automorphism of X, X_{ad}, A, A_{ad}, W, or W_{ad}  10
W  Weil group of k  25
W_{t}  tame Weil group  25
W  N/^{0}T  9
W^{∘}  generated by reflections in the walls of an alcove C  9
W_{o}  image of N_{o} in W  10
Ẇ_{o}  Tits extension of W_{o}  16
W_{λ}  generated by reflections in hyperplanes containing J_{λ}  17
W_{o}^{wϑ}  {z_{o} ∈ W_{o} : wϑ(z_{o})w^{−1} = z}  22
W_{o,λ}^{wϑ}  stabilizer in W_{o}^{wϑ} of the class of λ in [r^{−1}(ω)]  23
w  element of W_{o}  16

ẇ  fixed lift in Ẇ_{o} of w  18
w_{λ}  unique element in W_{λ} for which σ_{λ} · C_{λ} = w_{λ} · C_{λ}  17
ẇ_{λ}  unique lift of w_{λ} in N satisfying t_{λ}ẇ = ẇ_{λ}u_{λ}  18
x_{λ}  unique fixed-point in A for t_{λ}wϑ  2
X_{∗}(H)  group of algebraic one-parameter subgroups of H  8
X  X_{∗}(T)  9
X^{∘}  co-root sublattice in X  9
X̄  X/X^{∘}  15
X_{w}  preimage in X of [X/(1 − wϑ)X]_{tor}  17
χ_{φ}  depth zero character corresponding to φ  30
χ_{λ}  Ad(p_{λ})_{∗}χ ∈ Irr(T F_{λ} ; λ)  33
Y  algebraic character group of T  16
y_{λ}  w_{λ}^{−1}t_{λ}w  17
Z^{1}(F, U)  continuous cocycles Γ ⟶ U  11
Ẑ  center of Ĝ  16

References

[1] J. D. Adler, Refined anisotropic K-types and supercuspidal representations, Pacific J. Math., 185 (1998), pp. 1–32.

[2] J. Adler and S. DeBacker, Murnaghan-Kirillov theory for supercuspidal representations of general linear groups, J. Reine Angew. Math., 575 (2004), pp. 1–35.

[3] , Some applications of Bruhat-Tits theory to harmonic analysis on the Lie algebra of a reductive p-adic group, Michigan Math. J., 50 (2002), no. 2, pp. 263–286.

[4] J. Adler and A. Roche, An intertwining result for p-adic groups, Canad. J. Math., 52 (2000), no. 3, pp. 449–467.

[5] A. Borel, Admissible representations of a semisimple group over a local field with vectors fixed under an Iwahori subgroup, Invent. Math., vol. 35, 1976.

[6] , Linear algebraic groups, second enlarged edition, Graduate Texts in Mathematics, vol. 126, Springer-Verlag New York, 1991.

[7] , Automorphic L-functions, Automorphic forms, representations, and L-functions (A. Borel and W. Casselman, eds.), Proc. Symp. Pure Math., vol. 33, part 2, American Mathematical Society, 1979, pp. 27– 61.

[8] A. Borel and J. Tits, Groupes réductifs, Inst. Hautes Études Sci. Publ. Math., No. 27, (1965), pp. 55–151.

[9] N. Bourbaki, Lie groups and Lie algebras. Chapters 1–3, translated from the French, reprint of the 1989 English translation. Elements of Mathematics (Berlin). Springer-Verlag, Berlin, 1998.

[10] F. Bruhat and J. Tits, Groupes réductifs sur un corps local. II. Schémas en groupes. Existence d’une donnée radicielle valuée, Inst. Hautes Études Sci. Publ. Math., No. 60, (1984), pp. 197–376.

[11] , Groupes algébriques sur un corps local Chapitre III. Compléments et applications à la cohomologie galoisienne, J. Fac. Sci. Univ. Tokyo, 34 (1987), pp. 671-698.

[12] R. Carter, Finite groups of Lie type. Conjugacy classes and complex characters, Reprint of the 1985 original, Wiley Classics Library, John Wiley & Sons, Ltd., Chichester, 1993.

[13] J. Dadok and V. Kac, Polar Representations, J. of Algebra, 92 (1985), pp. 504-524.

[14] S. DeBacker, Parameterizing conjugacy classes of maximal unramified tori via Bruhat-Tits theory, Michigan Math. J., 54 (2006), no. 1, pp. 157–178.

[15] , Homogeneity results for invariant distributions of a reductive p-adic group, Ann. Sci. École Norm. Sup., 35 no. 3, (2002), pp. 391–422.

[16] , Some applications of Bruhat-Tits theory to harmonic analysis on a reductive p-adic group, Michigan Math. J., 50 (2002), no. 2, pp. 241–261.

[17] , Parametrizing nilpotent orbits via Bruhat-Tits theory, Ann. of Math. (2) , 156 (2002), no. 1, pp. 295– 332.

[18] S. DeBacker and D. Kazhdan, Murnaghan-Kirillov theory for depth zero supercuspidal representations, work in progress.

[19] S. DeBacker and M. Reeder, On generic very cuspidal representations, work in progress.

[20] P. Deligne and G. Lusztig, Representations of reductive groups over finite fields, Ann. of Math., 103 (1976) pp. 103-161

[21] F. Digne, G.I. Lehrer, and J. Michel, The characters of the group of rational points of a reductive group with non-connected centre, J. Reine Angew. Math., 425 (1992), pp. 155-192.

[22] B. Gross, On the motive of a reductive group, Invent. Math., 130 (1997), pp. 287-313.

[23] B. Gross and D. Prasad, On the decomposition of a representation of SO_{n} when restricted to SO_{n−1}, Canad. J. Math., 44 (1992), pp. 974-1002.

[24] B. Gross and M. Reeder, From Laplace to Langlands via representations of orthogonal groups, Bull. A.M.S., 43 (2006), pp. 163-205.

[25] Harish-Chandra, Admissible invariant distributions on reductive p-adic groups, Preface and notes by Stephen DeBacker and Paul J. Sally, Jr. University Lecture Series, 16, American Mathematical Society, Providence, RI, 1999.

[26] , Harmonic Analysis on Reductive p-adic Groups, Springer Lecture Notes, 162 (1970).

[27] E. Hewitt and K. Ross, Abstract Harmonic Analysis I, Springer Verlag, New York-Heidelberg, 1979.

[28] J. Humphreys, Conjugacy classes in semisimple algebraic groups, Mathematical Surveys and Monographs, 43, American Mathematical Society, Providence, RI, 1995.

[29] , Linear algebraic groups, Graduate Texts in Mathematics, No. 21, Springer-Verlag, New York-Heidelberg, 1975.

[30] N. Iwahori and H. Matsumoto, On some Bruhat decompositions and the structure of the Hecke rings of 𝔭-adic Chevalley groups, Inst. Hautes Études Sci. Publ. Math., No. 25, (1965), pp. 5–48.

[31] D. Kazhdan, Proof of Springer’s hypothesis, Israel J. Math., 28 (1977), pp. 272–286.

[32] D. Kazhdan and Y. Varshavsky, Endoscopic decomposition of characters of certain cuspidal representations, Electron. Res. Announc. Amer. Math. Soc., 10 (2004), pp. 11-20.

[33] R. Kottwitz, Sign changes in harmonic analysis on reductive groups, Trans. Amer. Math. Soc., 278 (1983), no. 1, pp. 289–297.

[34] , Stable trace formula: cuspidal tempered terms, Duke Math. J., 51, (1984) pp. 611-650.

[35] , Stable trace formula: elliptic singular terms, Math. Ann., 275 (1986), pp. 365-399.

[36] , Transfer factors for Lie algebras, Represent. Theory , 3 (1999), pp. 127–138 (electronic).

[37] R. Langlands, Representations of abelian algebraic groups, Pacific J. Math., (1997), pp. 231–250.

[38] G. Lusztig, Green functions and character sheaves, Ann. Math., 131 (1990), pp. 355–408.

[39] , Classification of unipotent representations of simple p-adic groups, Int. Math. Res. Not., (1995) pp. 517-589.

[40] , Classification of unipotent representations of simple p-adic groups II, Rep. Thy, (2001), (electronic).

[41] C. Moeglin and J.-L. Waldspurger, Paquets stables de reprsentations tempres et de rduction unipotente pour SO(2n+1), Invent. Math., 152 (2003), no. 3, pp. 461–623.

[42] L. Morris, Level zero G-types, Compositio Math., 118 (1999), pp. 135–157.

[43] A. Moy and G. Prasad, Unrefined minimal K-types for p-adic groups, Inv. Math. , 116 (1994), pp. 393–408.

[44] , Jacquet functors and unrefined minimal K-types, Comment. Math. Helvetici, 71 (1996), pp. 98–121.

[45] G. Prasad and M. S. Raghunathan, Topological central extensions of semi-simple groups over local fields, I, Ann. of Math., 119 (1984), pp. 143–201.

[46] G. Prasad and J.-K. Yu, On finite group actions on reductive groups and buildings, Invent. Math., 147 (2002), pp. 545–560.

[47] M. Reeder, Whittaker models and unipotent representations of p-adic groups, Math. Ann., 308 (1997), pp.587-592.

[48] , Formal degrees and L-packets of unipotent discrete series of exceptional p-adic groups, J.Reine Angew. Math., 520 (2000), pp.37-93.

[49] J. D. Rogawski, An application of the building to orbital integrals, Compositio Math., 42 (1980/81), no. 3, pp. 417–423.

[50] J.P. Serre, Local fields, Springer-Verlag, 1979.

[51] , Galois cohomology, Springer-Verlag, 2002.

[52] L. Spice, Topological Jordan Decompositions, (2006) preprint.

[53] T. A. Springer, Trigonometric sums, Green functions of finite groups and representations of Weyl groups, Invent. Math., 36 (1976), pp. 173–207.

[54] T. A. Springer and R. Steinberg, Conjugacy Classes, Seminar in algebraic groups and related finite groups, Lecture Notes in Math., 131 (1970), pp. 167–266.

[55] R. Steinberg, Endomorphisms of linear algebraic groups, Mem. Amer. Math. Soc., 80 (1968).

[56] , Regular elements of semisimple algebraic groups, Inst. Hautes Études Sci. Publ. Math., No. 25 (1965), pp. 281-312

[57] , Lectures on Chevalley groups, Yale University, 1967.

[58] , Torsion in reductive groups, Advances in Math., 15 (1975), pp. 63–92.

[59] J. Tits, Sur les constantes de structure et le théorèm d’existence des algèbres de Lie semi-simples, Inst. Hautes Études Sci. Publ. Math., No. 31 (1966), pp. 21-58.

[60] , Reductive groups over p-adic fields, Automorphic forms, representations, and L-functions (A. Borel and W. Casselman, eds.), Proc. Symp. Pure Math., vol. 33, part 1, Amer. Math. Soc., Providence, RI, 1979, pp. 29–69.

[61] E.J. Todd, Les représentations modérées de GL_{N} sur un corps local, C.R. Acad. Sci., Sér. I Math. 317, no. 1 (1993) pp. 33–36.

[62] D. Vogan, The local Langlands conjecture, Representation theory of groups and algebras, Contemp. Math., 145, Amer. Math. Soc., Providence, RI, 1993, pp. 305–379.

[63] J.-L. Waldspurger, Le lemme fondamental implique le transfert, Compositio Math., 105 (1997), no. 2, pp. 153– 236.

[64] J.-K. Yu, Construction of tame supercuspidal representations, J. of the Amer. Math. Soc., 14 (2001), no. 3, pp. 579–622.

[65] , Smooth models associated to concave functions in Bruhat-Tits theory, (2002) preprint.

  E-mail address: smdbackr@umich.edu

The University of Michigan, Ann Arbor, MI 48109

  E-mail address: reederma@bc.edu

Boston College, Chestnut Hill, MA 02467