April 2, 1999

LOCAL CHARACTER EXPANSIONS

Dan Barbasch and Allen Moy

Abstract. In this paper the authors develop a method to compute the local character expansion of a depth zero representation of a p–adic group. The main idea is to use the generalized Gelfand–Graev characters for finite groups as test functions to plug into the character formula. This is possible due to results of Waldspurger on the validity of the local character expansion in a large enough neighborhood of the identity. The method leads to a classification of the unipotent orbits in terms of parahoric subalgebras.

Dans cet article, les auteurs développent une méthode pour calculer l’expansion locale d’un caractère d’une representation de profondeur zero. L’idée principale est d’ utiliser les caractères de Gelfand–Graev généralisés des groupes finis. C’est faisable grace aux resultats the J.– L. Waldspurger qui montre que le developpement du caractère est valide sur un voisinage suffisament grand. La méthode donne une classification des orbites unipotentes en termes des sous-algebres parahoriques.

1. Introduction

Suppose k is a nonarchimedean local field of characteristic zero and G is a reductive group defined over k with Lie algebra Lie(G). Let G(k) be the k–rational points of G and Θ_{π} the character of an irreducible admissible representation π of G(k). Harish–Chandra proved that Θ_{π} can be represented as a locally constant integrable function on the regular set. Generalizing a result of Howe for GL(n) [Ho], Harish–Chandra also showed that Θ_{π} is a linear combination of Fourier transforms of nilpotent orbits in a sufficiently small neighborhood of zero in 𝔤 = Lie(G)(k), i.e. there exist constants c_{O}(π), one for each nilpotent orbit O in 𝔤 such that

Θ_{π}( exp X ) = ∑_{O} c_{O}(π)μ̂_{O}(X)

for X ∈ 𝔤, a regular element sufficiently close to 0. Much qualitative and quantitative information on the representation π can be gleaned from the Harish-Chandra–Howe local character expansion. For instance, the asymptotic growth of the space of vectors of π fixed by parahoric filtration subgroups P_{x,r_{k}} as k → ∞ is controlled by the orbits O of maximal dimension with c_{O} ≠ 0. These same orbits also relate to the existence of various generalized Whittaker models for π ([MW]). Another example of the quantitative information in the local character expansion occurs when π is a discrete series representation. Here, the formal degree of π equals (−1)^{r}c_{0}(π) where r is the split rank of G over k and c_{0}(π) is the coefficient of the zero orbit [Ro].

Two natural questions to ask in regard to the Harish-Chandra–Howe local character expansion are

(1) what can be said about the size of the neighborhood in 𝔤 on which the local character expansion is valid and

(2) what techniques are there to compute the coefficients c_{O}?

In regard to the first question, Waldspurger [Wa] has proved a conjectured sharpening (due to Hales [Ha]) of the results of Howe and Harish–Chandra. A representation π is defined to have depth zero, if there exists a parahoric P so that π possesses non zero vectors fixed by P_{+}, the maximal normal pro-p-subgroup of P. Under mild conditions, Waldspurger has shown that if G is essentially a group of classical type and π is a depth zero representation, then the local character expansion is valid on each neighborhood log(Q_{+}) (Q a parahoric subgroup). In particular, Waldspurger’s theorem gives a very complete and satisfactory answer to the first question for representations of depth zero.

In regards to the second question, one type of answer can be found in [MW] in terms of generalized Whittaker models.

It is the purpose of this paper to show how Waldspurger’s results in [Wa] also leads to a solution of the second question along very different lines from [MW]. We show that the validity of the local character expansion for depth zero representations, can be used as a trace formula from which one can completely determine the coefficients c_{O}(π) in terms of the depth zero K–types of π. Our method is to inflate the characters of generalized Gelfand–Graev representations for reductive groups over finite fields associated to parahoric subgroups of G(k) and show that the inflations can be used as test functions in the local character expansion. In brief, if p is sufficiently large, then the Gelfand–Graev characters can be pulled back via the exponential map to the Lie algebra. The support of the Fourier tranforms of the Gelfand–Graev characters on the topologically nilpotent set in the Lie algebra is compatible with the partial order of nilpotent orbits in a finite field Lie algebra. In section 3 (Propositions 3.5 and 3.8) we give a classification of the nilpotent orbits of a Lie algebra 𝔤 in terms of associate classes of parahoric subalgebras 𝔭_{x} and nilpotent orbits of 𝔭_{x}/𝔭_{x}^{+}. This classification highlights properties of the cosets N + 𝔭_{x}^{+} and their closure relations (Propositions 3.12 and 3.16) which had been noticed independently by Howe [Ho] for GL(n) at an earlier time. These results generalize the results in [Wb] (rather [Wc] which only seems to be available in preprint form). While [Wc] proceeds case by case and deals only with the classical cases, our methods are quite general and treat all cases at the same time.

The results in section 3 lead to a recursive method for calculating the coefficients c_{O}(π) via the partial ordering on the orbits O’s.

The first author would like to thank the University of Paris VII for their hospitality during part of the time these results were proved. We would like to thank Magdy Assem, Mark Gross, Gerard Laumon and Fiona Murnaghan for discussions during the period this paper was written. We are also thankful to Gopal Prasad for aid in the proofs of section 3, and the referee for valuable suggestions.

The final writing of this paper occurred while both authors were members of the Institute for Advanced Study. We thank the Institute for its hospitality and support. Both authors were supported in part by the National Science Foundation.

2. Generalized Gelfand–Graev representations

2.1. We summarize the construction of the generalized Gelfand–Graev representations. Let 𝔽_{q} be a finite field of characteristic p. Fix an algebraic closure \overline{𝔽_{q}} of 𝔽_{q}. Let G be a simple adjoint group defined over 𝔽_{q} and let 𝔤 be its Lie algebra. We identify G (resp. 𝔤) with its \overline{𝔽_{q}} points. Denote the Frobenius actions on G and 𝔤 by F. In particular, the F -fixed points G^{F} and 𝔤^{F} correspond to the 𝔽_{q}–rational points of G and 𝔤 respectively. We recall some basic results from chapter 5 of [Ca] on the nilpotent elements in 𝔤. Let T ⊂ G be a maximal torus in G and let Δ(G, T) (resp. Φ(G, T) = {α_{1}, …, α_{r}}) be the roots (resp. simple roots). The height ht(α) of a root α = ∑_{i}n_{i}α_{i} is ht(α) = ∑_{i}n_{i}. Set

m = max_{α} ht(α). The relation m = h − 1, where h is the Coxeter number, holds. Consider

the inclusion 𝔤↪^{ad}𝔤𝔩(𝔤).

(1) If N ∈ 𝔤 is nilpotent, then ad(N)^{2h−1} = 0, [Ca:5.5.2]. In particular, if p > 2h − 1, then the exponential map from 𝔤𝔩(𝔤) into GL(𝔤) is defined on the nilpotent elements of 𝔤.

(2) Suppose N ∈ 𝔤 is nilpotent, non zero and ad(N)^{n} = 0. If p − 2 > n, then the Jacobson–Morozov theorem is valid for N, [Ca:5.3.2], i.e. N determines an embedding

ϕ_{N} : 𝔰𝔩(2) ⟶ 𝔤

so that ϕ_{N}(e) = N, where e, h, f are the usual elements in 𝔰𝔩(2). If N lies in 𝔤 = 𝔤^{F}, then this 𝔰𝔩(2) can be taken to be defined over 𝔽_{q}. Furthermore, two embeddings ϕ_{N} and ϕ_{N′} are equivalent under the adjoint action of G^{F} if and only if N and N′ belong to the same G^{F} conjugacy class.

(3) In case (2), if p > 3(h − 1), then 𝔤 is a completely reducible 𝔰𝔩(2) module under the adjoint action. These modules are “the same” as over ℂ ([Ca:5.5.5]).

Hence, following [Ca], we assume that

(2.1.1)  p > 3(h − 1).

Then both the log and exp maps are defined on the unipotent and nilpotent elements of G and 𝔤 respectively and properties (2) and (3) are valid. Furthermore, by the theorem of Dynkin–Kostant, there are finitely many nilpotent conjugacy classes in 𝔤.

2.2. Let N be a nilpotent element in 𝔤^{F} (with associated 𝔰𝔩(2)–triple N = e, h, f). Let

\begin{aligned}𝔤(i) & = \{ X ∈ 𝔤 | [h, X] = iX \} i ∈ ℤ \\ 𝔪 & = 𝔤(0) \\ 𝔲_{+} & = ⊕_{i>0}𝔤(i) \\ 𝔲_{−} & = ⊕_{i<0}𝔤(i) \\ 𝔭 & = 𝔪 + 𝔲_{+} \end{aligned}

and let M (resp. U, P, etc. ) be the algebraic group corresponding to 𝔪 (resp. 𝔲_{+}, 𝔭, etc. ). Via the Killing form, identify 𝔲_{−} with the dual 𝔲_{+}^{∗}. Let O_{U} = O_{U}(−f) be the U^{F} coadjoint orbit of the element −f. Note that e and −f are conjugate via SL(2)^{F} but that e and f need not be conjugate.

2.3. By Kirillov theory for nilpotent groups, the coadjoint orbit O_{U} determines a |O_{U}|^{\frac{1}{2}} – dimensional representation η_{N} of U^{F}. The induced representation

Γ_{N} = Ind_{U^{F}}^{G^{F}} η_{N}

is called the generalized Gelfand–Graev representation of G^{F} associated to N. When N is regular, the Gelfand–Graev representation is the finite field analogue of a space of Whittaker vectors. Let γ_{N} denote the character of Γ_{N}. Various properties of γ_{N} are collected together in the following proposition.

2.4. Proposition. (Kawanaka [Ka,Kb])

(1) γ_{N} depends only on the Ad(G^{F})–orbit of N.

(2) |O_{U}|^{\frac{1}{2}} = q^{\frac{(dim(U)−dim(Ad(P)N)}{2}}

(3) dim(γ_{N}) = |G^{F} |q^{− \frac{dim(Ad(G)N)}{2}} , dim(Ad(G)N) = dim(U)+ dim(Ad(P)N)

(4) The support of γ_{N} is contained in the closure of Ad(G)(exp N).

Let N denote the nilpotent cone in 𝔤. We see that the support of γ_{N} is contained in the set of unipotent elements and therefore via the exponential map, we can transfer γ_{N} to an Ad(G^{F})–invariant function on the Lie algebra 𝔤^{F}. The function γ_{N} ∘exp is supported inside the closure of O(N). For ease of notation, when the context is clear, we use γ_{N} to denote the function on 𝔤^{F} which is equal to γ_{N} ∘ exp on the nilpotent elements and zero elsewhere. This will be done in the next section.

2.5. Fix ψ be a nonzero additive character of 𝔽_{q}. Define the Fourier transform (with respect to the Cartan–Killing form ⟨ , ⟩) of a function f on 𝔤^{F} by

FT_{𝔤^{F}} (f)(x) = f̂(x) = ∑_{x′∈𝔤^{F}} ψ(⟨x, x′⟩)f(x′).

The Fourier transform of a function supported on N^{F}, e.g. γ_{N}, will in general not have support contained in N^{F}. The following two results of Lusztig give a general expression for the values of γ̂_{N} at an arbitrary element y ∈ 𝔤^{F} as well as more detailed information at a nilpotent element. Set

(2.5.1)  Σ = −f + C_{𝔤}(e)

where C_{𝔤}(e) is the centralizer of N = e in 𝔤 and let

(2.5.2)

r(N) = \frac{1}{2}dim𝔤(1) + dim𝔤(i ≥ 2)

= \frac{1}{2}(dim𝔤 − dim(C_{𝔤}(N)))

2.6. Proposition. (Lusztig [L] Prop.2.5.) For any y ∈ 𝔤^{F}

γ̂_{N}(y) = q^{r(N)} #{ g ∈ G^{F} | Ad(g)(y) ∈ Σ }

In particular, γ̂_{N} is integer valued.

2.7. Proposition. (Lusztig [L] Prop.6.13.) For N′ ∈ N^{F}, let O(N′) be the Ad(G)–orbit of N′ in 𝔤.

(1) If γ̂_{N}(N′) ≠ 0, then N must lie in the closure of O(N′),

(2) if N ∈ O(N′) and γ̂_{N}(N′) ≠ 0, then N′ is in the G^{F} –orbit of N,

(3) γ̂_{N}(N) = q^{r(N)} #C_{G}(N)^{F}.

Roughly speaking, Proposition 2.7 says that the Fourier transform inverts the support of a generalized Gelfand–Graev representation γ_{N}, i.e. γ_{N}(N′) = 0 if N is not in the closure of Ad(G)(N′) in 𝔤; furthermore, if O(N) = O(N′), then γ_{N}(N′) ≠ 0 precisely if N′ is G^{F} conjugate to N.

3. Transition to the p–adic group

3.1. Let k be a p-adic field, with ring of integers 𝔬_{k} and prime ideal 𝔭_{k}. Let 𝔽_{q} = 𝔬_{k}/𝔭_{k} be the residue field. We wish to lift generalized Gelfand–Graev characters to functions on the p-adic group. These functions will be supported on the topologically unipotent/nilpotent set. Let K be a fixed maximal unramfied extension of k. Let 𝔤 = Lie(G)(K) (resp. 𝔤 = Lie(G)(k)) denote the K–rational (resp. k–rational) points of the Lie algebra of G. Let ℬ_{K} and ℬ_{k} be the Bruhat–Tits building of the group G over the fields K and k respectively. The Galois group Gal(K/k) acts on ℬ_{K} so that the building ℬ_{k} can be naturally identified as the fixed points of Gal(K/k). Given a point x ∈ ℬ_{K}, let P_{x}, 𝔤_{x,0}, 𝔤_{x,0^{+}}, 𝔤_{x,r_{i}}, etc. be the parahoric subgroup and various lattices associated to the point x as in [MPa]. The quotient 𝔪_{x} = 𝔤_{x,0}/𝔤_{x,0^{+}} is the Lie algebra of M_{x} = P_{x,0}/P_{x,0^{+}}. If the point x lies in ℬ_{k}, then both 𝔪_{x} and M_{x} are defined over 𝔽_{q}. We assume now that x ∈ ℬ_{k}. Let P_{x} = P_{x} ∩ G(k), and let 𝔤_{x,r_{i}} = 𝔤_{x,r_{i}} ∩ 𝔤.

Let τ : G ⟶GL(V) be a representation of G such that the corresponding representation T_{τ} : 𝔤 ⟶ 𝔤𝔩(V (k)) of the k–rational points 𝔤 of Lie(G) is faithful. The following lemma is elementary, e.g. see [Wa:III.3] for a proof.

3.2. Lemma. If T_{τ} : 𝔤 ⟶ 𝔤𝔩(V (k)) is faithful, and the characteristic p of k satisfies

(3.2.1)  \frac{p − 1}{dim(τ)} > val_{k}(p),

then the exponential map from 𝔤𝔩(V (k)) to GL(V (k)) is defined for any element Y ∈ 𝔤_{x,0^{+}} (x ∈ ℬ_{k}).

We will mainly use this in the case when τ is the adjoint representation, in which case exp(𝔤_{x,0^{+}}) = P_{x,0^{+}}.

3.3. Let ψ be an additive character of the p–adic field which is trivial on ℘ but not on 𝔬. The character ψ induces naturally a nontrivial additive character of the residue field 𝔽_{q} = 𝔬/℘. We shall denote both characters by ψ. The Fourier transform

FT_{𝔤}(f)(x) = f̂(x) = ∫_{𝔤} ψ(⟨x, x′⟩)f(x′)dx′

is related to the finite field Fourier transform as follows. Given a function f on 𝔤_{x,0}/𝔤_{x,0^{+}}, we denote by f̃ its lift to 𝔤_{x,0}. We have

(3.3.1)  FT_{𝔤}(f̃) = vol(𝔤_{x,0^{+}})\begin{matrix} \\ FT & 𝔪_{x}^{F} \end{matrix}(f).

We shall need to evaluate the nilpotent orbital integrals of the inflated functions f̃. Suppose N′ is a nilpotent element in 𝔤 and f̃ is a function on 𝔤 which is supported on 𝔤_{x,0} and constant on cosets of 𝔤_{x,0^{+}}. The orbital integral of f̃ with respect to the orbit O′ = O(N′) can be written as

(3.3.2)  \begin{matrix}∫_{O′} f̃(y)dy = & ∑_{Y} ∫_{O′∩Y} f(Y)dy \\ = & ∑_{Y}′ f(Y) μ_{O′}(O′ ∩ Y) \end{matrix}

where ∑ is the sum over the cosets Y = y + 𝔤_{x,0^{+}} ⊂ 𝔤_{x,0} and ∑′ is the sum over those cosets whose image in the Lie algebra 𝔪_{x} = 𝔪_{x}^{F} = 𝔤_{x,0}/𝔤_{x,0^{+}} is nilpotent. Here and in the later sections we let μ_{O} denote a nonzero invariant measure supported on the orbit of O. The normalization will not be so important for us, except perhaps in the example at the end of the paper.

3.4. We find it very useful to make a connection between nilpotent elements in 𝔤 and those in the Lie algebras 𝔪_{x} attached to a point x ∈ ℬ_{k}. As mentioned in the introduction, these results are related to results of Waldspurger in [Wb]. Recall the Jacobson–Morozov theorem. If N is a nonzero nilpotent element in 𝔤 =Lie(G)(k), then there exists a nontrivial homomorphism ϕ :SL(2) ⟶ G (with associated differential T_{ϕ} : 𝔰𝔩(2) ⟶ 𝔤) such that ϕ(e) = N.

3.5. Proposition. Let ϕ :SL(2) ⟶ G be a nontrivial homomorphism defined over k, and let J = ϕ(SL(2, 𝔬_{k})). Then

(1) There exists a minimal parahoric subgroup P_{x} (x ∈ ℬ_{k}) containing J.

(2) If P_{x} and P_{y} (x, y ∈ ℬ_{k}) are two minimal parahoric subgroups containing J, then the natural maps from P_{x} ∩ P_{y} to M_{x} and M_{y} are both surjective. In particular, P_{x} and P_{y} are associate parahoric subgroups (in the sense of [MPa:5.1]).

Proof. Consider the action of the compact group J_{K} = J×Gal(K/k) on the building ℬ_{K}. By Bruhat–Tits theory, the fixed point set F = ℬ_{K}^{J_{K}} (a subset of ℬ_{k}) is nonempty. The set F is convex, a union of facets and J ⊂ P_{x} for any x ∈ F. Furthermore, P_{x} is a minimal parahoric subgroup containing J precisely if the facet determined by x, i.e. the facet of smallest dimension containing x, is maximal, i.e. not contained in a facet of strictly larger dimension. To proceed with the proof of part (2), we introduce some notation and prove a lemma.

Define a linear simplex of dimension r in an Euclidean space E to be the convex closure of r + 1 points in general position, i.e. the r + 1 points do not lie in an (r − 1)–dimensional affine subspace. If S is an r–dimensional linear simplex, let A(S) be the smallest affine subspace containing S. The dimension of A(S) is r.

3.6. Lemma. Suppose F is a convex subset of an Euclidean space E and suppose F has a (locally finite) cellular decomposition by linear simplicies. If S is a simplex in the cellular decomposition with the property that S is not properly contained in any larger simplex of F, then F ⊂ A(S).

Proof of Lemma. Suppose F ⊈ A(S). Choose x ∈ F − A(S). Since F is convex, it contains the convex closure of S and x. This implies S is contained in a (strictly) larger simplex of F, a contradiction. Hence F ⊂ A(S). □

Suppose S_{1} and S_{2} are two maximal facets of the convex set F = ℬ_{K}^{J_{K}}. Let A be an apartment in ℬ_{K} which contains both S_{1} and S_{2} and consider the convex set F_{A} which is the intersection of F with A. We note that A, see section 3.1 of [MPb], corresponds to a maximal K-split torus T of G which is defined over k. Let A(S_{1}) and A(S_{2}) be the smallest affine subspaces of A containing S_{1} and S_{2} respectively. By Lemma 3.6, we conclude that i) A(S_{1}) = A(S_{2}) (call this common affine subspace A′), ii) that F_{A} ⊂ A′ and iii) S_{1} and S_{2} are open subsets of A′. As in section 6.1 of [MPb], if z ∈ A, then M_{z} is generated by the image of the maximal compact subgroup T_{b} of the K–split torus associated to A and the affine root groups U_{α} such that α(z) = 0. Since x and y are in the interiors of maximal facets of A′, it it follows that α(x) = 0 if and only if α(y) = 0 and therefore the natural maps from P_{x} ∩ P_{y} to M_{x} and M_{y} are onto. This proves part (2) of the proposition. □

As an important corollary of Proposition 3.5, we have

3.7. Corollary. Given a nonzero nilpotent element N ∈ 𝔤, let N = e, h, f be a 𝔰𝔩(2) triple associated to N. There exists a point x ∈ ℬ_{k} so that

(1) e, h, f ∈ 𝔤_{x,0}, and the images of e, h, f in 𝔪_{x} = 𝔤_{x,0}/𝔤_{x,0^{+}} generate a 𝔰𝔩(2) triple in 𝔪_{x}

(2) Among those parahoric subgroups P_{x′} such that 𝔤_{x′,0} satisfies (1), P_{x} is minimal.

(3) If P_{y} is another parahoric subgroup minimal among those parahoric subgroup for which 𝔤_{y,0} satisfies (1), then P_{x} and P_{y} are associates.

Corollary 3.7 allows us to associate to any nilpotent orbit in 𝔤 a unique associate class of parahoric subgroups. In the converse direction to Corollary 3.7, we have

3.8. Proposition. Given an 𝔰𝔩(2) triple (defined over 𝔽_{q}) \overline{e}, \overline{h}, \overline{f} in 𝔪_{x}, there is a lift to a 𝔰𝔩(2) triple e, h, f in 𝔤.

Proof. We mimic the proof of the Jacobson–Morozov theorem in [Ko]. We will use the following variant of Nakayama’s Lemma.

3.9. Lemma. Suppose V and W are 𝔬–lattices, i.e. free finite rank 𝔬–modules and

A : V ⟶ W is an 𝔬–module map

such that the quotient map

\overline{A} : V/℘V ⟶ W/℘W is surjective.

Then, A is surjective.

We first show that \overline{e} lifts to a nilpotent element e′ ∈ 𝔤_{x,0}. Let G_{1} be the 1-parameter (multiplicative) subgroup in M_{x} so that \overline{h} lies in the Lie algebra Lie(G_{1}). Lift G_{1} to a 1parameter k-subgroup G_{1} of G so that G_{1}(𝔬_{k}) = G_{1}(k) ∩ P_{x}, and identify the (algebraic) characters of G_{1} and G_{1}. Let 𝔤(i) be the i-th weight space (i ∈ ℤ) for G_{1} under Ad. Let 𝔤_{x,0}(i) = 𝔤(i) ∩ 𝔤_{x,0} and 𝔤_{x,0^{+}}(i) = 𝔤(i) ∩ 𝔤_{x,0^{+}}. Note that

𝔤_{x,0} = ⨁_{i} 𝔤_{x,0}(i)

and similarly for 𝔤_{x,0^{+}}. Choose an e′ ∈ 𝔤_{x,0}(2) whose image is \overline{e}. Note that e is nilpotent. It is the desired nilpotent lift of \overline{e}.

Let m be the smallest nonnegative integer so that (ade′)^{m+1} = 0. An application of Nakayama’s lemma shows that ad(e′)^{2} maps 𝔤_{x,0}(−2) onto 𝔤_{x,0}(2). In particular there is an f′ ∈ 𝔤_{x,−2}(0) such that ad(e′)^{2}(f′) = e′. Let h′′ = [f′, e′]. By the proof of Lemma 3.4 in [Ko], the ad eigenvalues of h′′ on Cent_{𝔤_{x,0}}(e′) are i/2 with 0 ≤ i ≤ m. If the characteristic p is large enough, in particular if p satisfies (2.1.1) for all the simple factors of 𝔪_{x}, then we conclude that ad(\overline{h′′}) + 1 is nonsingular. By Nakayama’s lemma, we conclude that ad(h′′) + 1 is onto when restricted to Cent_{𝔤_{x,0}}(e′). This is Corollary 3.4 in [Ko]. Then the proof of Theorem 3.4 in [Ko] applies and we find the desired triple e, h, f. □

Remark. Propositions 3.5 and 3.8 give a classification of the unipotent orbits of G in terms of associate classes of parahoric subgroups P_{x} and unipotent orbits in M_{x}.

3.10. For an arbitrary element Y ∈ 𝔤_{x,0} (= 𝔤_{x,0} ∩ 𝔤), denote by \overline{Y} the image in 𝔪_{x}^{F}. Consider e, h, f, an 𝔰𝔩(2) triple as in the first part of Corollary 3.7. Let 𝔤(i) be the i–th weight space of the element h, and 𝔤(≤ i) be the sum of the eigenspaces with eigenvalues ≤ i. Note that any 𝔤_{x,r_{j}} is a direct sum of the spaces 𝔤_{x,r_{j}}(i) = 𝔤(i)∩𝔤_{x,r_{j}}. Since we are assuming that p satisfies (2.1.1) for all the simple factors of 𝔪_{x}, it follows that the representations of 𝔰𝔩(2)(𝔽_{q}) occuring in 𝔤_{x,0}/ϖ𝔤_{x,0} behave as in the characteristic zero theory – an irreducible representation is multiplicity free on restriction to the Cartan subalgebra determined by \overline{h}. Also, \overline{e} (resp. \overline{f}) acts as a raising (resp. lowering) operator.

3.11. Lemma. Suppose that X = e + z + w (z ∈ 𝔤_{x,0}(≤ 0), w ∈ 𝔤_{x,r_{1}}). Then there exists g ∈ P_{x} such that Ad(g)X = e + z′ where z′ ∈ 𝔤_{x,0}(≤ 0) and z′ = z mod 𝔤_{x,r_{1}}.

Proof. For any index r_{j}, the quotient L = 𝔤_{x,r_{j}}/𝔤_{x,r_{j+1}} is a 𝔤_{x}–module. Let L(i) = {x ∈ L|[\overline{h}, x] = ix\} be the i-th weight space of \overline{h} in L. Define L(≤ 0) and L(≥ 0) in the obvious way. A simple calculation shows that given Y ∈ L(≥ 0), there exists a U ∈ L such that

(3.11.1)  ad(\overline{e} + \overline{z})(U) = Y modulo L(≤ 0).

Claim. If r_{j} > 0, then any element X = e + z′ + w′ with z′ ∈ 𝔤_{x,0}(≤ 0) and w′ ∈ 𝔤_{x,r_{j}}(≥ 0) can be conjugated by an element g ∈ P_{x,r_{j}} so that Ad(g)(X) = e+z′′ +w′′ with z′′ ∈ 𝔤_{x,0}(≤ 0) and w′′ ∈ 𝔤_{x,r_{j+1}}.

Indeed, choose U ∈ 𝔤_{x,r_{j}} so that ad(\overline{e}+\overline{z′})(\overline{U}) = −\overline{w′} modulo 𝔤_{x,r_{j}}(≤ 0)+𝔤_{x,r_{j+1}}. It follows that Ad(exp(U))(X) has the required property stated in the claim. The proof of the lemma now follows from the claim and a standard Hensel’s lemma argument. □

3.12. Proposition. If X = e + z′ as in Lemma 3.11 is nilpotent, then e ∈ \overline{O(X)}.

Proof. Let λ_{X} : G_{m} ⟶ G be a one parameter subgroup so that Ad(λ_{X}(t))X = t^{2}X. The element h also determines a one parameter subgroup λ : G_{m} ⟶ G whose weight spaces are the 𝔤(i)’s. Write X = e + ∑_{i≤0} z_{i} where z_{i} ∈ 𝔤_{x,0,i}. We have

\begin{aligned}Ad(λ(t^{−1})Ad(λ_{X}(t))X) & = Ad(λ(t^{−1})t^{2}X) \\ & = e + ∑_{i≤0} t^{2−i}z_{i}. \end{aligned}

As t → 0, we conclude e lies in the closure of the adjoint orbit of X. □

3.13. Corollary. Suppose X ∈ e + 𝔤_{x,0^{+}} is nilpotent. Then e ∈ \overline{O(X)}.

3.14. In this section we work over the field K. We identify the residue field of K with \overline{𝔽_{q}}. Suppose e, h, f is an 𝔰𝔩(2) triple as in the first part of Proposition 3.6. Then for an arbitrary element in z ∈ 𝔤_{x,0}, denote by \overline{z} its image in 𝔪_{x}. We denote by Σ_{x} the analogue of (2.5.1), namely,

Σ_{x} = \overline{e} + C_{𝔪_{x}}(\overline{f}).

Because adh has only integer eigenvalues, the inverse image of Σ_{x} is contained in

∑_{i≤0}𝔤_{x,0}(i) + 𝔤_{x,0^{+}}.

3.15. Lemma. Consider the map

ψ : M_{x} × Σ_{x} ⟶ 𝔪_{x}, ψ(g, z) = Ad(g)(z).

Then there exists a Zariski open set U containing \overline{e}, which is in the image of ψ.

Proof. The map ψ is smooth with differential T\overline{e}ψ which is onto, and all the varieties in question are nonsingular. □

3.16. Proposition. Suppose Y = y + 𝔤_{x,0^{+}} is a nilpotent coset and that the closure of the adjoint orbit of \overline{Y} in 𝔪_{x} contains \overline{e}. If X ∈ y + 𝔤_{x,0^{+}} is nilpotent, then e ∈ \overline{O(X)(K)} (the closure is taken in the p–adic topology).

Proof. This is an application of Proposition 3.5. By the assumption, the orbit O(\overline{Y}) must meet every Zariski open set containing \overline{e}. By Lemma 3.15, there is a \overline{g} ∈ M_{x}^{F} and \overline{z} ∈ \overline{Σ} such that \overline{Y} = \overline{g}(\overline{e} + \overline{z}). Thus we may replace Y by a conjugate (under P_{x}) such that \overline{Y} = \overline{e} + \overline{z}. By Lemma 3.11, we can find a representative y ∈ Y of the form y = e + z, where z ∈ 𝔤_{x,0}(≤ 0). Thus Proposition 3.12 applies and we get the desired result. □

4. The Main Theorem

4.1 An element u ∈ G(k) (resp. n ∈ 𝔤) is called topologically unipotent (resp. topologically nilpotent) if u is contained in some P_{x,0^{+}}, (resp. n is contained in some 𝔤_{x,0^{+}}). In particular, lim_{m→∞}u^{p^{m}} = 1 (resp. lim_{m→∞}ad(n)^{p^{m}} = 0). Observe also that if τ : G(k) ⟶ GL(V (k)) is a faithful rational representation of G(k) and (3.2.1) is satisfied, then the exponential map from 𝔤𝔩(V (k)) to GL(V (k)) is defined on the topologically nilpotent elements and it (the exponential) maps the topologically nilpotent elements to the topologically unipotent elements of G(k).

4.2. We recall now the main result in [Wa]. Let G be a group of the type Γ as in [Wa:II.1] (essentially a quotient of a product of groups of classical type associated to self– dual lattices). Attached to such a group are positive integers e(G) and d(G) [Wa:II.1], as well as a set of primes P(G).

4.3. Theorem. ([Wa:I.3]) Suppose that p is a prime number satisfying

p ∉ P(G), p ≥ 2d(G)e(G)(2d(G) + 1) and val_{k}(p) ≤ \frac{p−1}{4e(G)d(G)}.

Let (π, V_{π}) be an irreducible admissible representation such that V_{π}^{P_{x,0^{+}}} ≠ {0} for some parahoric subgroup P_{x}. Then, the local character expansion

Θ_{π}( exp X ) = ∑_{O} c_{O}(π)μ̂_{O}(X)

is valid for all regular topologically nilpotent elements X ∈ 𝔤.

4.4. We shall now assume the following conditions on the characteristic p of the field k.

(1) p satisfies the conditions of Theorem 4.3,

(2) p satisfies Lemma 3.3.2 with ρ the adjoint representation,

(3) p satisfies the condition of §2.1 for all M_{x}.

4.5. Theorem. Under the hypothesis that the characteristic p of k satisfies 4.4, suppose that e, h, f ∈ 𝔤_{x,0^{+}} is an 𝔰𝔩(2) triple. Let f_{x,O} be the character of the generalized Gelfand– Graev representation of M_{x}^{F} = P_{x,0}/P_{x,0^{+}} attached to \overline{e} ∈ 𝔪_{x}^{F}. Then

(1) f_{x,O} is supported on the topologically unipotent set,

(2) μ̂_{O′}(f_{x,O}) = 0 unless O lies in the closure \overline{O′}.

(3) Suppose O_{1} and O_{2} are two nilpotent orbits in 𝔤 which belong to the same nilpotent

orbit in 𝔤.

i) If O_{1} and O_{2} are distinct in 𝔤, then μ̂_{O_{1}}(f_{x,O_{2}}) = 0.

ii) If O_{1} = O_{2} = O in 𝔤, then μ̂_{O}(f_{x,O}) ≠ 0.

(4) For any irreducible smooth admissible representation π,

Θ_{π}(f_{x,O}) = ∑_{σ∈M̂_{x}^{F}}m_{π}(σ)⟨f_{x,O}, σ⟩

where m_{π}(σ) (resp. ⟨f_{x,O}, σ⟩) is the multiplicity of σ in π (resp. f_{x,O}).

In statements (2) and (3), for ease of notation, we have denoted by f_{x,O}, the function on the Lie algebra 𝔤 obtained from f_{x,O} via the exponential map.

Proof. Let Γ\overline{e} denote the generalized Gelfand–Graev character of P_{x,0}/P_{x,0^{+}} attached to \overline{e} ∈ 𝔪_{x}^{F}, so that f_{x,O} = Γ̃\overline{e}. Statement (1) is part (4) of Proposition 2.4. To prove part (2), we need to show μ_{O′}(FT_{𝔤}(f_{x,O})) = 0 unless O ⊂ \overline{O′}. By the calculations in section 3.3,

(4.5.1)  μ_{O′}(FT_{𝔤}(f_{x,O})) = vol(𝔤_{x,0^{+}})∑_{Y}′ (FT_{𝔪_{x}^{F}} Γ\overline{e})(Y) μ_{O′}(O′ ∩ Y)

Note that μ_{O′}(O′ ∩ Y) ≠ 0 precisely when the intersection O′ ∩ Y ≠ ∅. Suppose y ∈ O′ ∩ Y. By Proposition 2.7 (1), (FT_{𝔪_{x}^{F}} Γ\overline{e})(Y) ≠ 0 only if O_{M_{x}}(\overline{e}) is contained in the closure of O_{M_{x}}(Y). Hence, by Proposition 3.16, the product

(FT_{𝔤^{F}} Γ\overline{e})(Y)μ_{O′}(O′ ∩ Y)

is non zero only when O is contained in the closure of O′. This proves (2). In regards to (3), part i) follows from Proposition 2.7 (2). To show (3ii), observe that the sum in (4.5.1) is over the M_{x}^{F} conjugates of \overline{e}. There are [M_{x}^{F} : C_{M}F_{x} (\overline{e})] such conjugates. The value of (FT_{𝔤^{F}} Γ\overline{e}) at these elements is, by Proposition 2.7 (3), q^{r(\overline{e})}|C_{M}F_{x} (\overline{e})|. Whence,

μ̂_{O}(f_{x,O}) = μ_{O′}(FT_{𝔤}(f_{x,O}))
= vol(𝔤_{x,0^{+}}) [M_{x}^{F} : C_{M}F_{x} (\overline{e})] q^{r(\overline{e})} |C_{M}F_{x} (\overline{e})| μ_{O}(O ∩ e + 𝔤_{x,0^{+}})
= q^{r(\overline{e})} vol(P_{x,0}) μ_{O}(O ∩ e + 𝔤_{x,0^{+}})

Statement (4) is immediate. □

4.6. Let O_{1}, O_{2}, …, O_{m} be the k–rational nilpotent conjugacy classes in 𝔤, and let ⪯ be the partial ordering O ⪯ O′ if O is contained in the closure of O′. Choose e_{i} ∈ O_{i} and an 𝔰𝔩(2) triple e_{i}, h_{i}, f_{i}. By Proposition 3.5, we can select a point x_{i} ∈ ℬ_{k} so that e_{i}, h_{i}, f_{i} ∈ 𝔤_{x_{i},0}.

Let f_{x_{i},O_{i}} be as in Theorem 4.5. Let

(4.6.1)  Θ_{π} = ∑_{O} c_{O}(π)μ̂_{O}

be the local character expansion of Θ_{π}. Each of the functions f_{x_{i},O_{i}} is supported in the topologically unipotent set. They can therefore be used as test functions in the local character expansion (4.6.1). This leads to an invertible triangular linear system with the c_{O_{i}}(π)’s as the unknowns. In particular, c_{O_{j}}(π) can be obtained by using the test functions f_{x_{i},O_{i}} with O_{i} ⪰ O_{j}.

4.7. Corollary. The complete local character expansion of a depth 0 representation can be computed via the f_{x_{i},O_{i}}’s.

5. Wave Front Set

5.1 Assume the characteristic p of k satisfies the conditions of section 4.4. Recall that the Fourier transform μ̂_{O} of a nilpotent orbit O satisfies the homogeneity condition

(5.1.1)  μ̂_{O}(t^{2}X) = |t|^{−dimO}μ̂_{O}(X)

If x is a point in ℬ_{k}, r ∈ ℝ and ℓ is defined as in [MPa] (ℓ = 1 if G is split), then 𝔤_{x,r+ℓ} = ϖ𝔤_{x,r}. For r > 0, we have P_{x,r} is the image of 𝔤_{0,r} under the exponential map. It will be useful to only consider r equal to some r_{i}. Let 1_{x,r_{i}} be the characteristic function of P_{x,r_{i}}. When the context is clear we also let 1_{x,r_{i}} denote the characteristic function of 𝔤_{0,r_{i}}. If (π, V_{π}) is an arbitrary admissible representation of G(k), the local character expansion Θ_{π} = ∑_{O} c_{O}(π)μ̂_{O} will be valid in a small neighborhood of the identity and thus for r_{i}

sufficiently large

\begin{aligned}vol(P_{x,r_{i}}) dim V^{P_{x,r_{i}}} & = Θ_{π}(1_{x,r_{i}}) \\ & = ∑_{O} c_{O}(π) μ̂_{O}(1_{x,r_{i}}) \end{aligned}

Suppose m is a natural integer. By the homogeneity condition (5.1.1), we have

μ̂_{O}(1_{x,2mℓ}) = vol(𝔤_{x,2mℓ})μ̂_{O}(1_{x,0}).

Hence,

dim V^{P_{x,2mℓ}} = ∑_{O} q^{m dimO}c_{O}(π)μ̂_{O}(1_{x,0})

In particular, the growth (as a function in m) of the dimension of the spaces V^{P_{x,2mℓ}} is governed by the orbits O of maximal dimension whose coefficients c_{O} are non zero. The set

W F (π) = ⋃_{c_{π}(O)≠0}\overline{O}

is called the wave front set of π. Thus, the dimension of the wave front set governs the growth of the spaces V^{P_{x,2mℓ}}.

Define a nilpotent orbit O to be a wave front set nilpotent for π if O is an open subset of W F (π). In such a situation, choose e ∈ O and f_{x,O} as in 4.5. If O′ is a strictly larger nilpotent orbit than O, by the definition of W F (π), we have c_{π}(O′) = 0. Hence,

Θ_{π}(f_{x,O}) = c_{O}(π) q^{r(\overline{e})} vol(P_{x,0}) μ_{O}(O ∩ e + 𝔤_{x,0^{+}})

This means a wave front nilpotent O forces π to contain a representation σ of P_{x,0}/P_{x,0^{+}} which appears in the generalized Gelfand–Graev representation Γ\overline{e}.

5.2. Proposition. The wave front set of a smooth admissible representation π of depth zero can be determined from the wave front sets of the finite field representations.

6. An Example

We give an example of the technique. Assume p ≠ 2 and let E be the unramified quadratic extension of k. Let G be the unitary group associated to the hermitian form in three variables given by 2x_{−1}\overline{x_{1}} + x_{0}\overline{x_{0}}. Let i) O(N_{r}) be the regular nilpotent class, ii) O(N_{u}) and O(N_{ϖ}) be the two subregular classes (we choose O(N_{u}) so that there exists

N ∈ O(N_{u}) whose image in Lie(G)(𝔽_{q}) is a subregular nilpotent), iii) N_{0} the trivial class.

Normalize Haar measure on G(k) so that the measure of an Iwahori subgroup is equal to \frac{q^{4}−1}{(q+1)(q^{3}+1)}. Normalize the invariant measure on a nilpotent orbit O so that if N ∈ O and Γ_{N} is the associated generalized Gelfand–Graev representation, then μ_{O}(Γ_{N}) = 1. With these normalizations, the coefficient, c_{O}, of a wave front set orbit O is an integer equal to the dimension of generalized Whittaker functionals associated to O [MW]. If π is a square integrable representation, the coefficient of the trivial orbit in the local character expansion of Θ_{π} is equal to the negative of the formal degree of π, e.g. with these normalizations, the formal degree of the Steinberg representation is 1 and its local character expansion is μ̂_{r} − μ̂_{0}.

The following table lists the nilpotent orbital integrals for the four associated generalized Gelfand–Graev representations.

Γ_{N_{r}} Γ_{N_{u}} Γ_{N_{ϖ}}  Γ_{N_{0}}
N_{r}  1  q  1  q^{3} + 1
N_{u}  0  1  0 \frac{(q^{3}+1)q^{2}}{(q+1)(q^{2}+1)}
N_{ϖ}  0  0  1 \frac{(q^{3}+1)(q−1)}{(q^{4}−1)}
N_{0}  0  0  0  1

Given this table, it is easy to compute the local character expansion of various depth zero representations. As two particularly interesting representation, we have

(1) Let Θ_{10} be the character of the supercuspidal representation induced from the irreducible cuspidal unipotent θ_{10} representation of G(𝔽_{q}) inflated to K = G(𝔬_{k}). In particular, the formal degree of Θ_{10} is equal to the degree of θ_{10} divided by the measure of K. The representation θ_{10} does not have a finite field Whittaker model; therefore, by Proposition 5.5.2, neither does Θ_{10} and so the coefficient of μ̂_{r} in the local character expansion is 0. We see that

Θ_{10} = μ̂_{u} − \frac{q(q^{2} − 1)}{q^{4} − 1}μ̂_{0}.

(2) Let Θ_{DS} be the character of a non Steinberg discrete series representation π which possesses a nonzero Iwahori fixed vector. The representation π has a unique Whittaker model, i.e. c_{O(N_{r})} = 1. Let K′ be a maximal compact subgroup of G not conjugate to K and let K′_{0+} be as in [MPb]. The space of K′_{0+}-fixed vectors in π is q dimensional. We have

Θ_{DS} = μ̂_{r} − μ̂_{ϖ} − \frac{q(q^{2} − 1)}{q^{4} − 1}μ̂_{0}.

It is known that Θ_{10} and Θ_{DS} form an L–packet, and we have been informed by Kottwitz that μ̂_{r}, μ̂_{u} − μ̂_{ϖ} and μ̂_{0} are all stable distributions.

Note. After this manuscript was submitted, Waldspurger announced that he had determined the stable combinations of nilpotent orbital integrals for classical groups.

References

[Ca] R. Carter, Finite groups of Lie type (1985), Wiley Interscience.

[Ha] T. Hales, A simple definition of Transfer factors for unramified groups, Contemporary Math. 145 (1993), 109–134.

[Ho] R. Howe, The Fourier Transform and Germs of Characters, Math. Ann. 208 (1974), 305–322.

[Ka] N. Kawanaka, Fourier Transforms of Nilpotently Supported Invariant Functions on a Simple Lie Algebra over a Finite Field, Invent. Math. 69 (1982), 411-435.

[Kb] , Shintani lifting and Gelfand–Graev representations, Proc. Symposia in Pure Math. 47 (1987), 147-163.

[Ko] B. Kostant, The principal three–dimensional subgroup and the Betti numbers of a complex Lie group, Amer. Jour. of Math 81 (1959), 973–1032.

[L] G. Lusztig, A Unipotent Support for Irreducible Representations, Advances in Math. 94 (1992), 139–179.

[MW] C. Moeglin and J.–L. Waldspurger, Modèles de Whittaker dégeénérés pour les groupes p–adiques, Math. Z. 3 (1987), 427–452.

[MPa] A. Moy and G. Prasad, Unrefined minimal K-types for p-adic groups, Invent. Math. 116 (1994), 393-4089.

[MPb] , Jacquet functors and unrefined minimal K-types, Comment. Math. Helv. 71 (1996), 98-121.

[Ro] J. Rogawski, An application of the building to orbital integrals, Compositio Math. 42 (1980/81), 417–423.

[Wa] J. Waldspurger, Homogeneite de certaines distributions sur les groupes p–adiques, Publ. Math. IHES 81 (1995), 25–72.

[Wb] , Quelques questions sur les integrales orbitales unipotentes et les algebres de Hecke, Bull. Soc. Math. France 124 no. 1 (1996), 1–34.

[Wc] , Quelques resultats de finitude concernant les distributions invariantes sur les algebres de Lie p–adiques, preprint.