SPHERICAL CHARACTERS ON

CERTAIN P-ADIC SYMMETRIC SPACES

Jiandong Guo

Abstract. Let E be a quadratic extension field of a p-adic local field F. We prove that spherical characters on the symmetric space GL(2n, F)/GL(n, E) are locally integrable functions which are locally constant on the set of regular elements.

1. Introduction

Suppose F is a nonarchimedean local field of characteristic zero and G is a connected reductive algebraic group defined over F. Let G =G(F). Harish-Chandra proved in [HC1] that the character of an irreducible admissible representation π of G is a locally integrable distribution on G in the sense that there is a locally integrable function χ_{π} on G such that

traceπ(f) = ∫_{G} f(g)χ_{π}(g)dg, for anyf ∈ C_{c}^{∞}(G).

Furthermore the function χ_{π} is locally constant on the set of regular elements of G, and has asymptotic expansions around singular elements in terms of Fourier transforms of nilpotent distributions on the tangent spaces of the centralizers of these elements.

Now let θ be an involution of G and let H ⊂ G be the subgroup of fixed elements of θ. Then G/H is a symmetric space over F. An irreducible admissible representation π of G is called class one if there exists a nonzero H-invariant linear form l_{π} on π as well as a nonzero H-invariant linear form lπ̃ on the representation π̃ contragredient to π. Then the bi-H-invariant distribution Θ_{π} on G defined by

Θ_{π}(f) =< l_{π}, π(f)lπ̂ >, f ∈ C_{c}^{∞}(G)

is called a spherical character for the class one representation π. These distributions come up in the spectral decomposition of global trace formulas on the symmetric spaces. It is one of the main object in the study of the theory of harmonic analysis on

Typeset by AMS-TEX

the p-adic symmetric space G/H. Due to the work of Jacquet and his coworkers on trace formulas on symmetric spaces, this theory is currently of considerable interest. Here a natural question is whether or not a spherical character has the properties of the characters in the group case described above. This question was largely answered in a recent work of Rader and Rallis ([RR]). By extending certain techniques of Harish-Chandra ([HC1]) as well as some results of Howe ([Ho1], [Ho2]) from the group case to the symmetric space case, they proved that a spherical character is a locally constant function on the set of θ-regular elements and has an asymptotic expansion around the origin similar to Harish-Chandra’s description of the characters in the group case. But they also showed by examples that, unlike the characters in the group case,spherical characters are generally not locally integrable functions.

However in certain cases we can still expect the local integrability of the spherical characters, a result which usually has important applications in the theory of automorphic forms. An example for this is provided by the symmetric space G(E)/G(F) where E is a quadratic extension field of F, as proved by Hakim in [Ha1]. (In fact this is the only example of a symmetric space of general rank, other than the group case, we know so far where the spherical characters are locally integrable.)

In this paper we will investigate the case where G = GL(2n, F) and H ⊂ G is a subgroup which is isomorphic to GL(n, E) over F. We also assume that the residual characteristic of F is odd. In this case (G, H) is a Gelfand pair ([G3]) in the sense that the dimensions of both the space of the H-invariant linear forms on π and the space of the H-invariant linear forms on π̃ are at most one. So the spherical character for a class one representation is unique up to a scalar. We will prove the following theorem.

Theorem 1.1. Let G and H be the groups as above. Then the spherical character Θ_{π} for a class one representation π of G is locally integrable on G which is locally constant on the set of θ-regular elements.

Since the set of θ-regular elements in the symmetric space G/H is open and dense with complement in G/H of measure zero, the next corollary follows immediately from the above theorem.

Corollary 1.2. The spherical character Θ_{π} for a class one representation π of G is nonzero when restricted to the set of θ-regular elements.

To explain the motivation of the work, we go to a global setting. Let F be a number field and let E be a quadratic extension field of F. Let G = GL(2n, F) and H ⊂ G be a subgroup which is isomorphic to GL(n, E). We denote by Z the center of G. For an automorphic cuspidal representation π of G, it is conjectured by Jacquet that if there is a cusp form ϕ in the space of π such that the period integral

P (ϕ) = ∫_{H(F_{A})/H(F)Z(F_{A})} ϕ(h)dh

is nonzero, then the exterior square L-function of π has a pole at 1 and |P (ϕ)|^{2} is essentially the product L(1/2, π)L(1/2, π ⊗ η) (where F_{A} is the adele ring of F and η is the quadratic idele class character of F attached to E, see [G2] for details). When n = 1, this is a fundamental result of Waldspurger as proved by him by using the theory of Weil representations ([W]). Jacquet reproved this result of Waldspurger later by comparing the trace formula for the symmetric space GL(2, F)/E^{∗} and the trace formula with weight factor η for the symmetric space GL(2, F)/F^{∗} × F^{∗} ([J1], [J2], see also [G1] for further applications). In extending the method of Jacquet to the GL(2n) case, it is expected that the result in this paper will play the same role as the one played by Harish-Chandra’s result in the problems concerning the comparison of Arthur-Selberg trace formula.

I would like to thank professor Jacquet for his help and encouragement. I would also like to thank professor Bump for his encouragement and support, and professor Rallis for his helpful comments. I am grateful to Max-Planck-Institut für Mathematik in Bonn for its hospitality and support.

2. The spherical characters

Suppose F is a p-adic local field of characteristic zero and odd residual characteristic and suppose G is a connected reductive algebraic group defined over F. Let G=G(F) and let θ be an involution in G. We denote by H the subgroup of fixed points of θ in G. Let S be the subset of G of elements of the form gθ(g^{−1}) for g ∈ G. The group G acts on S by twisted conjugation

(g, s) ↦ gsθ(g^{−1}), g ∈ G, s ∈ S.

Because θ(h) = h for any h ∈ H, the subgroup H acts on S by conjugation

(h, s) ↦ hsh^{−1}.

Then the surjective map ρ : G → S defined by ρ(g) = gθ(g)^{−1} for g ∈ G induces an isomorphism between the symmetric space G/H and the space S as G-spaces.

Let π be an irreducible admissible representation of G on the complex vector space V. We say π is H-distinguished if there is a nonzero H-invariant linear form l_{π} on V. Let Ṽ be the space of smooth linear forms on V and let π̃ be the representation of G on Ṽ contragredient to π. Then the representation π is called class one if both π and π̃ are H-distinguished. Now let π be a class one representation of G and let l_{π} and lπ̃ be nonzero H-invariant linear forms on π and π̃ respectively. For any f ∈ C_{c}^{∞}(G) then π(f)lπ̃ can be regarded as a vector in V. A spherical character of π is a distribution on G defined by

Θ_{π}(f) =< l_{π}, π(f)lπ̃ >, f ∈ C_{c}^{∞}(G).

Clearly Θ_{π} is bi-H-invariant.

An element s ∈ S is called θ-regular or simply regular if it is semisimple in G and has maximal dimensional H-orbit([R],[RR]). We denote by S′ the set of the regular elements in S. It is a dense open subset of S with the complement of measure 0. It is proved in [RR] that the spherical character Θ_{π} is represented by a locally constant function on S′.

We now consider the behavior of Θ_{π} around the singular elements in S. Let 𝔤 be the Lie algebra of G. Then G acts on 𝔤 by adjoint representation:

Adg(X) = gXg^{−1}, g ∈ G, X ∈ 𝔤.

We fix a nondegenerate G-invariant bilinear form B on 𝔤 as well as a nontrivial additive character ψ on F. Let 𝔥 and 𝔰 be the 1 and −1 eigenspaces of θ in 𝔤. Then 𝔥 is the Lie algebra of H and 𝔰 is the tangent space for S at the coset ¯1. We have 𝔤 = 𝔥 + 𝔰. Given a semisimple element s in S, we denote by M_{G} the identity component of the centralizer of s in G and we let

M_{G} = M_{G}(F), M_{H} = M_{G} ∩ H.

Then M_{G} is invariant under θ and M_{H} = {m ∈ M_{G}|θ(m) = m}. Let 𝔐_{𝔤} be the centralizer of s in 𝔤 and let

𝔐_{𝔰} = 𝔰 ∩ 𝔐_{𝔤}, 𝔐_{𝔥} = 𝔥 ∩ 𝔐_{𝔤}.

Then 𝔐_{𝔤} is the Lie algebra of M_{G}, 𝔐_{𝔥} is the Lie algebra of M_{H} and 𝔐_{𝔰} is the tangent space at ¯1 of the induced symmetric space

M_{G}/M_{H} ⋍ M_{S} = {mθ(m^{−1})|m ∈ M_{G}}

Let x be a fixed element in G such that ρ(x) = s. We consider the map

H × M_{G} × H → G, (h_{1}, g, h_{2}) ↦ h_{1}xgh_{2}.

Let U_{x} be the set of g ∈ M_{G} such that the above map is submersive at (1, g, 1) (hence at (h_{1}, g, h_{2}) for any h_{1}, h_{2} ∈ H) and let Ω_{x} be the subset HxU_{x}H of G. Then U_{x} is an open and bi-M_{H}-invariant neighborhood of 1 in M_{G}, and Ω_{x} is an open and bi-H-invariant neighborhood of x in G (§3, [RR]). By the standard theory of Harish-Chandra ([HC2]), there exists a surjective map α ↦ f_{α} from C_{c}^{∞}(H × U_{x} × H) to C_{c}^{∞}(Ω_{x}) such that

∫_{Ω_{x}} f_{α}(t)F (t)dt = ∫_{H×U_{x}×H} α(h_{1}, g, h_{2})F (h_{1}xgh_{2})dh_{1}dgdh_{2}

for all F ∈ C_{c}^{∞}(Ω_{x}). Then there is a unique bi-M_{H}-invariant distribution Θ_{M} on U_{x} (see, for example, p 103 of [JR]) such that

Θ_{π}(f_{α}) = Θ_{M}(β_{α})

for all α ∈ C^{∞}(H × U_{x} × H) where

β_{α}(g) = ∫_{H} ∫_{H} α(h_{1}, g, h_{2})dh_{1}dh_{2}, g ∈ U_{x}.

To state the germ expansion theorem, we need to fix some further notation. We first note that the map ρ induces a surjective map f ↦ f^{ρ} from C_{c}^{∞}(M_{G}) to C_{c}^{∞}(M_{S}) by the formula

f^{ρ}(mθ(m^{−1})) = ∫_{M_{H}} f(mh)dh, f ∈ C_{c}^{∞}(M_{G}).

We denote by Θ_{M}^{ρ} the M_{H}-invariant distribution on M_{S} defined by Θ_{M}^{ρ}(f^{ρ}) = Θ_{M}(f) for f ∈ C_{c}^{∞}(U_{x}). Next we recall that the exponential map exp induces a diffeomorphism from a sufficiently small neighborhood of 0 in 𝔐_{𝔰} onto a neighborhood of ¯1 in M_{S} (Lemma 7.1, [RR]). Let 𝔑_{𝔰} be the set of nilpotent elements in 𝔰 and let 𝔑_{𝔐_{𝔰}} = 𝔑_{𝔰} ∩ 𝔐_{𝔰}. Then the Fourier transform of a function f ∈ C_{c}^{∞}(𝔐_{𝔰}) is a function f̂ in C_{c}^{∞}(𝔐_{𝔰}) defined by

f̂(Y) = ∫_{𝔐_{𝔰}} f(X)ψ(B(Y, X))dX, X, Y ∈ 𝔐_{𝔰}.

The Fourier transform of a distribution D on 𝔐_{𝔰} is defined by

D̂(f) = D(f̂), f ∈ C_{c}^{∞}(𝔐_{𝔰}).

We have:

Theorem 2.1. Let Λ_{1}, ..., Λ_{l} be a basis for the space of the M_{H}-invariant distributions on 𝔐_{𝔰} supported in 𝔑_{𝔐_{𝔰}}. Then for a class one representation π of G, there exist unique complex numbers c_{1},..., c_{l} such that

Θ_{π}(f_{α}) = Θ_{M}(β_{α}) = Θ_{M}^{ρ}(β_{α}^{ρ}) = ∑_{i=1}^{l} c_{i}Λ̂_{i}(β_{α}^{ρ} ∘ exp)

for all α ∈ C_{c}^{∞}(H ×U_{x} ×H) with β_{α} supported in a sufficiently small bi-M_{H}-invariant neighborhood of 1 in U_{x}.

This result was proved by Rader-Rallis ([RR]) for x = 1 for a general symmetric space, and by Hakim ([Ha1]) for any semisimple elements for the symmetric space G(E)/G(F) where E is a quadratic extension field of F. For completeness, we will formulate a proof of this theorem in this general setting based on their work in the next section. Then it is easily seen that Θ_{π} is locally integrable for any class one representation π of G if the distributions Λ̂_{i} are locally integrable for each pair (M_{H}, 𝔐_{𝔰}) (§2, [Ha1]). But unfortunately this is not the case generally (see [RR] for examples).

The classical case where these distributions Λ̂_{i} are known to be locally integrable is for the pair (G, 𝔤) with a reductive group G acting on its Lie algebra 𝔤 by the adjoint map. This result is stated in the following theorem of Harish-Chandra ([HC1]).

Theorem 2.2. Consider the pair (G, 𝔤) with the adjoint action of a reductive group G on its Lie algebra 𝔤. Then each nilpotent orbit in 𝔤 carries an invariant measure with locally integrable Fourier transform. Furthermore these measures form a basis for the space of G invariant distributions on 𝔤 supported in the nilpotent subset of 𝔤.

To put this theorem in the setting of this paper, we consider the symmetric space G × G/G where the involution θ on G × G is defined by θ(g_{1}, g_{2}) = (g_{2}, g_{1}) and G is embedded in G × G diagonally. Then it is easily seen that G × G/G is isomorphic to G as G-spaces. So 𝔤 is the tangent space for G×G/G at (1, 1)^{−}. In this case any class one representation of G × G/G is of the form π × π̃ where π is a representation of G. Then the spherical character for π × π̃ coincides with the usual character for π which is locally integrable by theorem 2.1 and theorem 2.2 as already proved by Harish-Chandra ([HC1]). On the other hand it is not hard to see that the tangent space for each of the induced symmetric spaces for the symmetric space G(E)/G(F) is also of the type as in theorem 2.2. So it follows again from this theorem that the spherical characters for G(E)/G(F) are locally integrable as proved by Hakim ([Ha1]).

The case of interest in this paper has the following setting. Let E be a quadratic extension field of F. Let G = GL(2n, F) and let H ⊂ G be a subgroup isomorphic to GL(n, E) over F (see section 4 for a precise description of the situation). Then we can show that the pair (H, 𝔰) is isomorphic to the pair

(GL(n, E), M(n, E))

with GL(n, E) acts on M(n, E) by twisted conjugation

(g, α) ↦ gαḡ^{−1}, g ∈ GL(n, E), α ∈ M(n, E).

So theorem 2.2 does not apply in this case. In fact one of the main tasks of this paper is to establish a result similar to theorem 2.2 for this new pair. More precisely we will prove

Theorem 2.3. Consider the symmetric space G/H with the groups G, H as above. Then each nilpotent H-orbit of 𝔰 carries an H-invariant measure with locally integrable Fourier transform.

It follows then easily (see, for example, proposition 1.2 in [RR]) that these measures corresponding to the nilpotent orbits form a basis for the space of H-invariant distributions on 𝔰 supported in the nilpotent set. In section 4 we will prove that for each semisimple element s ∈ S the pair (M_{H}, 𝔐_{𝔰}) is isomorphic to a product of pairs of the two types dealt with in theorem 2.2 and 2.3. It follows then from theorem 2.1 that in this case the spherical characters are locally integrable. This is the assertion of theorem 1.1 in the introduction.

3. The germ expansion

We keep to the notation described above theorem 2.1 in the last section. As we noted before when s = 1 the assertion of theorem 2.1 is proved by Rader and Rallis in [RR]. Their proof can be extended to the case of a general θ-semisimple element s when combined with certain results in [Ha1]. For completeness, we summarize here the main steps of a proof of theorem 2.1 and refer the reader to the papers cited above for details.

We first recall Howe’s Kirillov-orbit method as adapted by Rader-Rallis and Hakim in the case of symmetric spaces. Let L be a θ-stable lattice in 𝔐_{𝔤} with L = L_{+} + L_{−} where L_{+} = L ∩ 𝔐_{𝔥} and L_{−} = L ∩ 𝔐_{𝔰}. Suppose L is ee (see p.158 of [RR] for the definition). Then exp maps L homeomorphically to a compact open subgroup K of M_{G}. We denote by ε(K) the set of equivalence classes of irreducible unitary representations of K. Let K_{+} = expL_{+}. Let ε(K/K_{+}) be the subset of ε(K) consisting of representations which possess nonzero K_{+} invariant vectors. For μ ∈ ε(K/K_{+}), we denote by d(μ) the degree of μ. The spherical function ϕ_{μ} of μ is a function ϕ_{μ} on K defined by ϕ_{μ}(k) =< μ(k)e_{1}, e_{1} > for k ∈ K where e_{1} is the K_{+}-invariant unit vector in the space of μ. This unit vector is unique since (K, K_{+}) is a Gelfand pair ([Ha2]). Let L_{−}^{⊥} be the annihilator of L_{−} in 𝔐_{𝔰} with respect to ψ ∘ B. It is invariant under K_{+}. So K_{+} acts on 𝔐_{𝔰}/L_{−}^{⊥} by conjugation (Here we remind the reader who is familiar with Howe’s work that we assume the residual characteristic of F is odd, so we have \frac{1}{2}L = L and exp\frac{1}{2}L = K^{1/2} = K).

Lemma 3.1. There is a bijection μ ↦ O_{μ} from ε(K/K_{+}) to the set of K_{+}-orbits in 𝔐_{𝔰}/L_{−}^{⊥} such that

d(μ)ϕ_{μ}(exp X) = ∑_{Z∈O_{μ}/L_{−}^{⊥}} ψ(B(Z, X)), X ∈ L_{−}.

Here we regard O_{μ} as a L_{−}^{⊥}- invariant subset of 𝔐_{𝔰}. This lemma is the symmetric space version of Howe’s parameterization of representations of small compact subgroups ([Ho2]). See proposition 7.8 of [RR] and theorem 3 of [Ha2] for the proof of this lemma. Here we note that if μ corresponds to the K-orbit O′_{μ} in 𝔐_{𝔤}/L^{⊥} by Howe’s parameterization of ε(K) as in [Ho2], then O_{μ} = O′_{μ} ∩ 𝔐_{𝔰}.

We now fix a norm || || on 𝔐_{𝔤} as in [RR]. Let

S_{1} = {X ∈ 𝔐_{𝔤} | ||X|| = 1}

and for ϵ > 0, let

V (ϵ) = {X ∈ S_{1} | ||X − n|| < ϵ for some n ∈ 𝔑_{𝔐_{𝔤}} ∩ S_{1}}.

So V (ϵ) is a neighborhood of 𝔑_{𝔐_{𝔤}} ∩ S_{1} in S_{1}. Then we have

Lemma 3.2. We choose ϵ > 0 sufficiently small and R > 0 sufficiently large. Let μ be a representation in ε(K/K_{+}) such that Θ_{M}^{ρ}(ϕ̄_{μ}) ≠ 0 and ||Z|| > R for some Z ∈ O_{μ}. Then we have O_{μ} ⊂ F V (ϵ).

Here ϕ̄_{μ} is the complex conjugate of ϕ_{μ}. This is theorem 4 in section 7 in [Ha1]. We remark that although the main purpose of [Ha1] is to prove the local integrability of the spherical characters for a certain symmetric space, but the results in §2 –§7 in [Ha1] are worked out for a general symmetric space and are generalizations of the corresponding results of Harish-Chandra in the group case ([HC1], [C]).

Next we recall a formula for the spherical functions as given by Rader-Rallis.

Lemma 3.3. Assume μ ∈ ε(K/K_{+}). Then we have

d(μ)ϕ_{μ}(exp X) = κ̂_{O_{μ}}(X)/vol(L_{−}^{⊥}), X ∈ L_{−},

where κ_{O_{μ}} is the characteristic function of O_{μ} in 𝔐_{𝔰} and vol(L_{−}^{⊥}) is the volumn of L_{−}^{⊥}.

This is proposition 7.10 (2) of [RR].

We now choose an M_{H}-invariant open and closed subset 𝔐_{0} of 𝔐_{𝔰} as in page 168 in [RR] such that exp defines an analytic isomorphism from 𝔐_{0} to an M_{H}-invariant open and closed subset U_{0} of ρ(U_{x}) ⊂ M_{S}. We restrict Θ_{M}^{ρ} to U_{0}, then lift it to 𝔐_{0} by the exponential map, and extend it to 𝔐_{𝔰} by 0. In this way we obtain a distribution Θ_{0} on 𝔐_{𝔰}. For Z ∈ 𝔐_{𝔰}, let f_{Z} be the characteristic function of Z + L_{−}^{⊥} in 𝔐_{𝔰}.

Lemma 3.4. If ϵ is small enough and R is large enough, then for Z ∈ 𝔐_{𝔰} such that ||Z|| > R and Θ̂_{0}(f_{Z}) ≠ 0, we have ||Z − n|| ≤ ϵ||n|| for some n ∈ 𝔑_{𝔐_{𝔰}}.

Proof. Let O_{μ} be the K_{+}-orbit of −Z in 𝔐_{𝔰}/L_{−}^{⊥} which is associated to a class one representation μ ∈ ε(K/K_{+}) by lemma 3.1. We have κ_{O_{μ}} = ∑_{Z′∈O_{μ}/L_{−}^{⊥}} f_{Z′}. Then by Lemma 3.3 we obtain

d(μ)ϕ̄_{μ}(exp X) = ∑_{Z′∈O_{μ}/L_{−}^{⊥}} f̂_{−Z′}(X)/vol(L_{−}^{⊥}), X ∈ L_{−}.

Thus we have

Θ_{M}^{ρ}(ϕ̄_{μ}) = card(O_{μ}/L_{−}^{⊥})Θ̂_{0}(f_{Z})/vol(L_{−}^{⊥}).

Here we use the fact that Θ̂_{0} is M_{H}-invariant. It follows that if Θ̂_{0}(f_{Z}) ≠ 0, then Θ_{M}^{ρ}(ϕ̄_{μ}) ≠ 0. Since || − Z|| = ||Z|| > R, we obtain −Z ∈ O_{μ} ⊂ F V (ϵ) by Lemma 3.2. Thus we have Z ∈ F V (ϵ). In other words there exists a ∈ F^{∗} and n_{1} ∈ 𝔑_{𝔐_{𝔤}} such that ||a^{−1}Z − n_{1}|| ≤ ϵ. Let n = an_{1} ∈ 𝔑_{𝔐_{𝔤}}. Then ||Z − n|| ≤ ϵ||n||. Following the same argument as in the second paragraph of the proof of theorem 7.11 of [RR], we can even choose n ∈ 𝔑_{𝔐_{𝔰}}. This proves the lemma.

It follows then from a fundamental result of Howe (see theorem 6.8 of [RR]) that

Θ̂_{0} ∗ κ_{L_{−}^{⊥}}/volL_{−}^{⊥} ∈ J (ω) ∗ κ_{L_{−}^{⊥}}/volL_{−}^{⊥}.

Here ω is a compact set in 𝔐_{𝔰} and J (ω) is the set of M_{H} invariant distributions on 𝔐_{𝔰} supported in AdM_{H}(ω). By another result of Howe (see theorem 6.9 of [RR]), if L is sufficiently large, then the space J (ω) ∗ κ_{L_{−}^{⊥}}/volL_{−}^{⊥} is spanned by the projections Λ_{i} ∗ κ_{L_{−}^{⊥}}/volL_{−}^{⊥} for i = 1, ..., l. Thus theorem 2.1 follows by taking the inverse Fourier transform. For details see the last paragraph in the proof of theorem 7.11 in [RR].

4. Induced infinitesimal symmetric spaces

Let F be a p-adic local field with odd residual characteristic, and let E(\sqrt{τ}) be a quadratic extension field of F. We denote by σ : z ↦ z̄ the Galois conjugation in E. The norm map N from E to F is defined by N(z) = zz̄ for z ∈ E. For any positive integer m, we use G_{m} to denote the linear group GL(m) of m × m invertible matrices. Suppose G is the subgroup of G_{2n}(E) consisting of the elements of the form

( \begin{matrix}α & β \\ β̄ & ᾱ \end{matrix} ) ∈ G_{2n}(E).

Regarded as an algebraic group over F, it is isomorphic to GL_{2n}(F). We consider an involution θ of G defined by θ(g) = ϵgϵ^{−1} where

ϵ = ( \begin{matrix}\sqrt{τ} & 0 \\ 0 & −\sqrt{τ} \end{matrix} ) .

Then the subgroup H = {g ∈ G|θ(g) = g} of G consists of the elements of the form

h_{g} = ( \begin{matrix}g & 0 \\ 0 & ḡ \end{matrix} ) , g ∈ G_{n}(E).

It is clear that H is isomorphic to G_{n}(E). Now we consider the variety

S = {gθ(g)^{−1}|g ∈ G} ⊂ G.

Then for g ∈ G we have that g ∈ S if and only if (gϵ)(gϵ) = τI_{2n} (for a proof of this fact see page 281 in [G3]). This condition becomes

(4.1)  α^{2} = I_{n} + ββ̄, αβ = βᾱ

if we write

g = ( \begin{matrix}α & β \\ β̄ & ᾱ \end{matrix} ) .

The Lie algebra of G is

𝔤 = \{( \begin{matrix}α & β \\ β̄ & ᾱ \end{matrix} ) |α, β ∈ M_{n}(E)\}.

The Lie subalgebra of H in 𝔤 is

𝔥 = \{( \begin{matrix}α & 0 \\ 0 & ᾱ \end{matrix} ) |α ∈ M_{n}(E)\}

and the −1 eigenspace of Ad(ϵ) in 𝔤 is the subspace 𝔰 of 𝔤 of the element of the form

s_{α} = ( \begin{matrix}0 & α \\ ᾱ & 0 \end{matrix} ) , α ∈ M(n, E).

As in the last section, for a semisimple element s ∈ 𝔰, let 𝔐_{𝔤} be the centralizer of s in 𝔤. Let M_{H} = M_{G} ∩ H and 𝔐_{𝔰} = 𝔐_{g} ∩ 𝔰. The task in this section is to determine the pair (M_{H}, 𝔐_{𝔰}) for each semisimple element s of S. If necessary we sometimes write H and 𝔰 as H_{n} and 𝔰_{n} to indicate they are related to the group G_{2n}.

Up to a conjugation by H, we may assume that s is the matrix

(4.2)  ( \begin{matrix}α & 0 & β & 0 \\ 0 & μ_{n_{1},n_{2}} & 0 & 0 \\ β̄ & 0 & α & 0 \\ 0 & 0 & 0 & μ_{n_{1},n_{2}} \end{matrix} )

where

μ_{n_{1},n_{2}} = ( \begin{matrix}I_{n_{1}} & 0 \\ 0 & −I_{n_{2}} \end{matrix} )

and α is a matrix in M_{n−n_{1}−n_{2}}(F) without eigenvalues {1, −1} (see [G2]). To continue, we first consider two special cases.

Case I Suppose we have n_{1} = n_{2} = 0. Then

s = ( \begin{matrix}α & β \\ β̄ & α \end{matrix} )

where α is in M_{n}(F) such that α^{2} − I_{n} = ββ̄ is nonsingular and αβ = βα. Let γ = β(I_{n} + α)^{−1}. Then it is an element in G_{n}(E). We have

I_{n} − γγ̄ = I_{n} − ββ̄(I_{n} + α)^{−2}
= I_{n} + (I_{n} − α)(I_{n} + α)^{−1} = 2(I_{n} + α)^{−1}

which is also nonsingular. Then by a simple computation we get

(4.3)  α = (I_{n} + γγ̄)(I_{n} − γγ̄)^{−1}, β = 2γ(I_{n} − γγ̄)^{−1}.

It follows immediately from these relations that if g ∈ G_{n}(E), then gγḡ^{−1} = γ if and only if gα = αg, gβ = βḡ, which in turn are equivalent to the condition

h_{g}s = ( \begin{matrix}g & 0 \\ 0 & ḡ \end{matrix} ) s = s ( \begin{matrix}g & 0 \\ 0 & ḡ \end{matrix} ) = sh_{g}.

Let M_{n}(E)_{γ}^{σ} be the set of the element ν in M_{n}(E) where νγ = γν̄ and let G_{n}(E)_{γ}^{σ} = M_{n}(E)_{γ}^{σ} ∩ G_{n}(E). Then we may regard G_{n}(E)_{γ}^{σ} as the set of F -points of a reductive algebraic group over F, with M_{n}(E)_{γ}^{σ} as its Lie algebra ([AC]). It is now clear that the centralizer of s in H is the subgroup

M_{H} = {h_{g} ∈ H|g ∈ G_{n}(E)_{γ}^{σ}}.

Next we determine the centralizer 𝔐_{𝔰} of s in 𝔰. Let

x = s_{λ} = ( \begin{matrix}0 & λ \\ λ̄ & 0 \end{matrix} )

be an element in 𝔰. Then xs = sx if and only if λα = αλ, λβ̄ = βλ̄. Let λβ^{−1} = ν. We will show that the above conditions for λ is equivalent to the condition νγ = γν̄.

First of all if λα = αλ and λβ̄ = βλ̄, then we have

ναβ = νβα = λα = αλ = ανβ.

Thus να = αν. We also have that

νββ̄ = λβ̄ = βλ̄ = βν̄β̄.

So we have νβ = βν̄. This implies that

νγ = νβ(I_{n} + α)^{−1} = βν̄(I_{n} + α)^{−1} = β(I_{n} + α)^{−1}ν̄ = γν̄.

On the other hand, if νγ = γν̄, then

νγγ̄ = γν̄γ̄ = γγ̄ν.

It follows from (4.3) that

να = αν, νβ = βν̄.

This implies

λα = νβα = ναβ = ανβ = αλ, λβ̄ = νββ̄ = βν̄β̄ = βλ̄.

So we obtain

𝔐_{𝔰} = \{s_{νβ} = ( \begin{matrix}0 & νβ \\ νβ̄ & 0 \end{matrix} ) ∈ 𝔰|ν ∈ M_{n}(E)_{γ}^{σ}\}.

Note that for h_{g} ∈ M and s_{νβ} ∈ 𝔐_{𝔰} where g ∈ G_{n}(E)_{γ}^{σ} and ν ∈ M_{n}(E)_{γ}^{σ}, we have

h_{g}s_{νβ}h_{g}^{−1} = ( \begin{matrix}g & 0 \\ 0 & ḡ \end{matrix} ) ( \begin{matrix}0 & νβ \\ ν̄β̄ & 0 \end{matrix} ) ( \begin{matrix}g^{−1} & 0 \\ 0 & ḡ^{−1} \end{matrix} ) = s_{gνβḡ^{−1}} = s_{gνg^{−1}β}.

So if we identify 𝔐_{𝔰} with M_{n}(E)_{γ}^{σ} by sending s_{νβ} to ν and identify M_{H} with G_{n}(E)_{γ}^{σ} in the obvious way, then the pair (M_{H}, 𝔐_{𝔰}) is the pair (G_{n}(E)_{γ}^{σ}, M_{n}(E)_{γ}^{σ}) with G_{n}(E)_{γ}^{σ} acting on its Lie algebra M_{n}(E)_{γ}^{σ} by the adjoint representation.

Case II Suppose n_{1} or n_{2} is n. Then s is either I_{2n} or −I_{2n}. In both cases we have M_{H} = H and 𝔐_{𝔰} = s.

We now consider a general semisimple element s of S in the form of (4.2). Then for an element h_{g} ∈ H, we have that h_{g}s = sh_{g} if and only if

gdiag(α, I_{n_{1}}, −I_{n_{2}})g^{−1} = diag(α, I_{n_{1}}, −I_{n_{2}})

and

gdiag(β, 0, 0)ḡ^{−1} = diag(β, 0, 0).

Set n_{0} = n − n_{1} − n_{2}. Since α has no eigenvalues 1 and −1, the matrix g must be of the form

diag(g_{0}, g_{1}, g_{2}), g_{0} ∈ G_{n_{0}}(E), g_{1} ∈ G_{n_{1}}(E), g_{2} ∈ G_{n_{2}}(E)

such that g_{0}αg_{0}^{−1} = α and g_{0}βḡ_{0}^{−1} = β. Thus if we set γ = β(I_{n_{0}} + α)^{−1} as before, these conditions for g_{0} mean g_{0} ∈ G_{n_{0}}(E)_{γ}^{σ}.

So the centralizer of s in H is

(4.4)  M_{H} = G_{n_{0}}(E)_{γ}^{σ} × H_{n_{1}} × H_{n_{2}}.

Similarly we can show that the centralizer 𝔐_{𝔰} of s in 𝔰 consists of the elements of the form s_{λ} where

λ = diag(λ_{0}β, λ_{1}, λ_{2}), λ_{0} ∈ M_{n_{0}}(E)_{γ}^{σ}, λ_{1} ∈ M_{n_{1}}(E), λ_{2} ∈ M_{n_{2}}(E).

So we obtain

(4.5)  𝔐_{𝔰} = M_{n_{0}}(E)_{γ}^{σ} × 𝔰_{n_{1}} × 𝔰_{n_{2}}.

Thus we have proved the following proposition.

Proposition 4. Suppose s is a semisimple element in 𝔰. Let M_{H} and 𝔐_{𝔰} be the centralizers of s in H and 𝔰 respectively. Then there is a partition n = n_{0} + n_{1} + n_{2} of n such that (M_{H}, 𝔐_{𝔰}) is isomorphic to

(G_{n_{0}}(E)_{γ}^{σ}, M_{n_{0}}(E)_{γ}^{σ}) × (H_{n_{1}}, 𝔰_{n_{1}}) × (H_{n_{2}}, 𝔰_{n_{2}})

where γ is an element in G_{n_{0}}(E) related to s.

5. The nilpotent orbits (I)

We continue with the notation of the last section. By proposition 4 and theorem 2.2, we are reduced to study the pair (H, 𝔰). From now on in this paper, we will concentrate on the proof of theorem 2.3. In this section we will show that each nilpotent orbit of 𝔰 carries an H-invariant measure by deriving an explicit formula for this measure . We will also give an explicit formula for its Fourier transform.

Consider the G-invariant F -bilinear form B on 𝔤 defined by B(X, Y) = tr(XY) for X, Y ∈ 𝔤. Then the restriction of B on 𝔰 is

B(s_{α}, s_{β}) = tr(s_{α}s_{β}) = tr(αβ̄ + ᾱβ) α, β ∈ M_{n}(E)

which is nondegenerate. We fix a nontrivial additive character ψ of F of conductor O_{F}. The Fourier transform of a function f ∈ C_{c}^{∞}(𝔰) is a function f̂ ∈ C_{c}^{∞}(𝔰) defined by

(5.1)  f̂(s_{α}) = ∫_{𝔰} f(s)ψ(B(s, s_{α}))ds, s_{α} ∈ 𝔰

and the Fourier transform D̂ of a distribution D on 𝔰 is defined by D̂(f) = D(f̂).

We now consider the nilpotent orbits in 𝔰. In [G3] we proved that each nilpotent orbit of 𝔰 contains an element s_{α_{0}} where α_{0} is a nilpotent matrix in M_{n}(F). Let V be the n-dimensional vector space E^{n} of column vectors over E. The matrix α_{0} acts on V as well as on the F -vector space F^{n} which is a subset of V. Let V_{i} = kerα_{0}^{i}. Then we have a flag

{0} = V_{0} ⊆ V_{1} ⊆ ... ⊆ V_{l} = E^{n}

of E^{n} where l is the smallest positive integer such that α_{0}^{l} = 0. Let

𝔓 = {p ∈ M_{n}(E)|p(V_{i}) ⊂ V_{i}, i = 0, 1, ..., l}

and

𝔑 = {x ∈ M_{n}(E)|x(V_{i}) = V_{i−1}, i = 1, ..., l}.

Then 𝔓 is a parabolic subalgebra of M_{n}(E) and 𝔑 is the nilpotent radical of 𝔓. Since α_{0} is an element in M_{n}(F), we have that V̄_{i} = V_{i} for any i. This implies that ᾱ ∈ 𝔓 (resp. 𝔑) if and only if α ∈ 𝔓 (resp. 𝔑). Let P = 𝔓 ∩ G_{n}(E) be the corresponding parabolic subgroup in G_{n}(E) and let N be the unipotent radical of P. We set

P_{H} = {h_{g} ∈ H|g ∈ P }, N_{H} = {h_{g} ∈ H|g ∈ N}.

Then we let

𝔓_{𝔰} = {s_{α} ∈ 𝔰|α ∈ 𝔓}, 𝔑_{𝔰} = {s_{α} ∈ 𝔰|α ∈ 𝔑}.

Finally let K_{E} = G_{n}(O_{E}) and let K_{H} = {h_{k}|k ∈ K_{E}}.

Suppose Λ is the distribution on 𝔰 determined by the orbital integral on the H-orbit of s_{α_{0}}. Then for f ∈ c_{c}^{∞}(𝔰), we have

Λ(f) = ∫_{H/H_{sα_{0}}} f(hs_{α_{0}}h^{−1})dh

where H_{s_{α_{0}}} is the centralizer of s_{α_{0}} in H (for some normalization of the quotient measure dh).

Proposition 5.1. For f ∈ C_{c}^{∞}(𝔰), the above integral for Λ(f) is convergent. In fact we have

(5.2)  Λ(f) = ∫_{𝔑_{𝔰}}(∫_{K_{H}} f(kuk^{−1})dk)du

and

(5.3)  Λ̂(f) = ∫_{𝔓_{𝔰}}(∫_{K_{H}} f(kpk^{−1})dk)dp

with suitably normalized Haar measures on 𝔑 and 𝔓.

For convenience in the proof, we identify H with G_{n}(E) and 𝔰 with M_{n}(E). Then the action of H on 𝔰 is equivalent to the action of G_{n}(E) on M_{n}(E) by twisted conjugation. If we denote this action by Tc, then we have Tcg(α) = gαḡ^{−1} for g ∈ G_{n}(E) and α ∈ M_{n}(E). From now on, unless otherwise specified, the action of G_{n}(E) on M_{n}(E) always means the twisted conjugation. We may regard B as a F - bilinear form on M_{n}(E). Then for α, β ∈ M_{n}(E), we have B(α, β) = tr(αβ̄+ ᾱβ). We use the same notation Λ to denote the G_{n}(E) invariant distribution on M_{n}(E) defined by the orbital integral on the twisted conjugacy class of α_{0} under G_{n}(E). Then in this setting the formulas (5.2) and (5.3) become

(5.4)  Λ(f) = ∫_{𝔑}(∫_{K} f(kuk̄^{−1})dk)du

and

(5.5)  Λ̂(f) = ∫_{𝔓}(∫_{K} f(kuk̄^{−1})dk)dp

for f ∈ C_{c}^{∞}(M_{n}(E)). The rest of this section is devoted to the proof of these two formulas. We will follow closely the arguments of Howe in [Ho1].

We start the proof with two lemmas. Given p_{1}, p_{2} ∈ P, and α ∈ 𝔑, we have

p_{1}αp_{2}V_{i} ⊆ p_{1}αV_{i} ⊆ p_{1}V_{i−1} ⊆ V_{i−1}

for any i. So p_{1}𝔑p_{2} ⊆ 𝔑. In particular we have p_{1}α_{0}p_{2} ∈ 𝔑. It is proved in lemma 2 of [Ho1] that the intersection of 𝔑 with the set

AdG_{n}(E)(α_{0}) = {gα_{0}g^{−1}|g ∈ G_{n}(E)}

is precisely AdP (α_{0}) which is open and dense in 𝔑. We first prove a twisted version of this result.

Lemma 5.2. If gα_{0}ḡ^{−1} ∈ 𝔑 for g ∈ G_{n}(E), then g ∈ P. Furthermore the set TcP (α_{0}) is open and dense in 𝔑.

Proof. Let g ∈ G_{n}(E) such that α = gα_{0}ḡ^{−1} ∈ 𝔑. We want to prove that g(V_{i}) ⊂ V_{i} for i = 0, 1, ..., l. We distinguish two cases.

If i = 2j, we have (αᾱ)^{j} = g(α_{0})^{i}g^{−1}. So ker(αᾱ)^{j} = g(kerα_{0}^{i}) = g(V_{i}). On the other hand since α ∈ 𝔑, so is ᾱ ∈ N. Thus αᾱV_{m} ⊂ αV_{m−1} ⊂ V_{m−2} for any m. We have

(αᾱ)^{j}V_{i} ⊂ (αᾱ)^{j−1}V_{i−2} ⊂ ... ⊂ V_{0} = {0}.

Thus V_{i} ⊂ ker(αᾱ)^{j} = g(V_{i}). But dimV_{i} = dim (g(V_{i})). So g(V_{i}) = V_{i} for i = 2j. If i = 2j + 1, we can prove similarly that

ḡ(V_{i}) = ker[(αᾱ)^{j}α] = V_{i}.

Since V̄_{i} = V_{i}, So we have

\overline{g(V_{i})} = ḡ(V̄_{i}) = ḡ(V_{i}) = V_{i} = V̄_{i}.

Thus gV_{i} = (V_{i}). Therefore g ∈ P. This proves the first assertion of the lemma. To continue we may assume that

(5.6)  α_{0} = ( \begin{matrix}0 & I_{n_{1}n_{2}} & 0 & … & 0 \\ & 0 & I_{n_{2}n_{3}} & … & 0 \\ & & ... & ... & ⋮ \\ & & & ... & I_{n_{l−1}n_{l}} \\ & & & & 0 \end{matrix} ) , n_{1} ≤ n_{2} ≤ ⋯ ≤ n_{l}

where we use I_{n_{i}n_{i+1}} to denote the n_{i} × n_{i+1} matrix ( I_{n_{i}} 0 ) .

Then 𝔑 consists of the matrices of the form

(5.7)  α = ( \begin{matrix}0 & A_{1} & A_{13} & … & A_{1l} \\ & 0 & A_{2} & … & A_{2l} \\ & & ... & ... & ⋮ \\ & & & ... & A_{l−1} \\ & & & & 0 \end{matrix} ) .

Let M be the subgroup of G_{n}(E) of the matrices of the form diag(B_{1}, B_{2}, …, B_{l}) with B_{i} ∈ G_{n_{i}}(E). Then it is a Levi subgroup of P. We claim that TcP (α_{0}) is precisely the subset 𝔑_{1} = {α ∈ 𝔑|rank(A_{i}) = n_{i}, i = 1, 2, ..., l − 1} of 𝔑.

Clearly TcP (α_{0}) is contained in 𝔑_{1}. Suppose α is an element in 𝔑_{1} in the form of (5.7). We need to show that α is twisted conjugate to α_{0}. Because the rank of A_{1} is n_{1}, we can find matrices B_{i}, i = 1, 2, ..., l such that A_{1}B_{i} = A_{1i}. Then twisted-conjugating α by the matrix

( \begin{matrix}I \\ & I & B̄_{3} & … & B̄_{l} \\ & & I \\ . & . & . \\ & & & & I \end{matrix} ) ∈ P,

we make A_{13}, …, A_{1l} vanish. So, continuing in this way, we have A_{ij} = 0 for all i, j in the expression (5.7) of α. Then for such α it is easily seen that we can find matrices D_{i} ∈ G_{n_{i}}(E), i = 2, ..., l inductively such that A_{1}D_{2} = I_{n_{1}n_{2}} and D̄_{i}^{−1}A_{i+1}D_{i+1} = I_{n_{i}n_{i+1}} for i ≥ 2. Let D = diag(I, D̄_{2}^{−1}, ..., D̄_{l}^{−1}) Then we have DαD̄^{−1} = α_{0}. Hence TcP (α_{0}) = 𝔑_{1} which is open and dense in 𝔑.

Remark: It is easily seen that the set AdP (α_{0}) is also 𝔑_{1} by the similar argument. So we have AdP (α_{0}) = TcP (α_{0}).

Next let M_{n_{i}n_{j}}(E) be the set of all the n_{i} × n_{j} matrices over E. We may regard it as a vector space over E. Then for A ∈ G_{n_{i}}(E) and B ∈ G_{n_{j}}(E), the Jacobian of the linear map from M_{n_{i}n_{j}} to itself defined by X ↦ AXB^{−1} is

\frac{|detA|^{n_{j}}}{|detB|^{n_{i}}} = \frac{|detA|^{n_{j}}}{|det B̄|^{n_{i}}}

which is also the Jacobian of the map X ↦ AXB̄^{−1}. Here we use | | to denote the absolute value on E. It follows immediately from this simple fact that |det(Tcm|_{𝔑})| = |det(Adm|_{𝔑})|. So for p = mn ∈ P where m ∈ M and n ∈ N, we have

|det(Tcp|_{𝔑})| = |det(Tcm ∘ Tcn|_{𝔑})| = |det(Tcm|_{𝔑})| = |det(Adm|_{𝔑})|.

Recall that if we denote by δ the modular function on P, then δ(p) = |det(Adm|_{𝔑})|. Thus we have proved:

Lemma 5.3. For p ∈ P, we have |det(Tcp|_{𝔑})| = δ(p).

We now prove the formula (5.4). Here the the argument of Howe in the proof of proposition 5 in [Ho1] goes through largely unchanged. Let ρ be the representation of G_{n}(E) on the vector space W = C_{c}^{∞}(M_{n}(E)) defined by ρ(g)f(α) = f(gαḡ^{−1}) for g ∈ G_{n}(E), f ∈ W and α ∈ M_{n}(E). Let ρ′ be the contragredient representation of G on the linear dual of W. Suppose w′ is a linear functional on W defined by

w′(f) = ∫_{𝔑} f(α)dα.

Then for any p ∈ P, we have

ρ′(p)w′(f) = ∫_{𝔑} f(pαp̄^{−1})dα = |det(Tcp^{−1}|_{𝔑})|w′(f) = δ(p)^{−1}w′(f).

It follows from proposition 5 in [Ho1] that ρ′(χ_{K})w′ is G_{n}(E) invariant, where χ_{K} is the characteristic function of K. If we denote by D the distribution defined by ρ′(χ_{K})w′, then We have

D(f) = ∫_{𝔑} ∫_{K} f(kαk̄^{−1})dkdα.

Clearly D is supported on the closure of TcK(𝔑). Recall that we have G_{n}(E) = P K and TcP (α_{0}) is dense in 𝔑. Hence D is a G_{n}(E) invariant distribution which is supported on the closure of TcG_{n}(E)α_{0}. So we must have Λ = D.

It remains to compute the Fourier transform Λ̂ of Λ. Let ω be an uniformizer of E. For any positive integer m, let L_{m} be the lattice of M_{n}(E) consisting of matrices with entries in ω^{m}O_{E}, and let L_{m}^{⊥} = {α ∈ M_{n}(E)|tr(αβ̄ + ᾱβ) ∈ O_{E}}. Then by an easy computation we find L_{m}^{⊥} = L_{−m} . We normalize the Haar measure on E such that measure(O_{E}) = 1. Given f ∈ C_{c}^{∞}(M_{n}(F)), there is an m such that f(α_{1}+α_{2}) = f(α_{1}) for α_{1} ∈ M_{n}(E) and α_{2} ∈ L_{m}. Then we have

f̂(β) = ∫_{M_{n}(E)} f(α)ψ(tr(αβ̄ + ᾱβ))dα

= ∫_{M_{n}(E)/L_{m}} f(α_{1})[∫_{L_{m}} ψ(tr(α_{1}β̄ + ᾱ_{1}β))ψ(tr(α_{2}β̄ + ᾱ_{2}β))dα_{2}]dα_{1}.

It follows immediately from this expression that f̂ is supported in L_{m}^{⊥}. So we have

Λ̂(f) = Λ(f̂) = ∫_{𝔑} ∫_{K} f̂(kαk̄^{−1})dkdα = ∫_{𝔑∩L_{m}^{⊥}} ∫_{K} f̂(kαk̄^{−1})dkdα

= ∫_{𝔑∩L_{m}^{⊥}} ∫_{K} ∫_{M_{n}(E)} f(β)ψ(tr(β̄kαk̄^{−1} + βk̄ᾱk^{−1}))dβdkdα

= ∫_{𝔑∩L_{m}^{⊥}} ∫_{K} ∫_{M_{n}(E)} f(β)ψ(tr[k̄(k̄^{−1}β̄k)αk̄^{−1} + k(k^{−1}βk̄)ᾱk^{−1}])dβdkdα

= ∫_{𝔑∩L_{m}^{⊥}} ∫_{K} ∫_{M_{n}(E)} f(kβk̄^{−1})ψ(tr(β̄α + βᾱ))dβdkdα.

Now let 𝔑^{−} be the nilpotent radical of the parabolic subalgebra 𝔓^{−}of M_{n}(E) which is opposite to 𝔓. Then we have

M_{n}(E) = 𝔓 + 𝔑^{−} = 𝔑 + 𝔓^{−}.

It is easily seen that tr(βᾱ + β̄α) = 0 for β ∈ 𝔑, α ∈ 𝔓, and that

𝔑^{−} ∩ L_{m} = {α ∈ 𝔑^{−}|tr(βᾱ + β̄α) ∈ O_{E} for any β ∈ 𝔑 ∩ L_{m}^{⊥}}.

By our normalization of the Haar measure, we have

measure(𝔑^{−} ∩ L_{m})measure(𝔑 ∩ L_{m}^{⊥}) = 1.

So the above integral becomes

∫_{𝔓} ∫_{K} ∫_{𝔑^{−}} f(kβ_{1}k̄^{−1})[∫_{𝔑∩L_{m}^{⊥}} ψ(tr(β̄_{2}α + β_{2}ᾱ))dα]dβ_{2}dkdβ_{1}

= ∫_{𝔓} ∫_{K} f(kβk̄^{−1})dkdp.

Thus we obtain the formula (5.5). This ends the proof of proposition 5.1.

So we obtain the first assertion of theorem 2.3. To finish the proof of this theorem, we still need to show that the distribution Λ̂ on 𝔰 as expressed by formula (5.5) is represented by a locally integrable function. The basic tools for doing this will be certain type of Weyl integration formula as well as a descent formula for the regular orbital integrals on 𝔰. These are the subjects of the next section.

6. Cartan subspaces and orbital integrals

A Cartan subspace of 𝔰 is a maximal abelian subspace consisting of semisimple elements. An element s ∈ 𝔰 is called θ-regular or simply regular if it is semisimple and its centralizer in 𝔰 is a Cartan subspace. We denote by 𝔰′ the set of θ-regular elements in 𝔰. We first give a description of the θ-regular elements and the Cartan subspaces.

Lemma 6.1. Let s = s_{α} be an element in 𝔰 with α ∈ M_{n}(E). Then the following statements are equivalent.

(1) s_{α} is θ-regular.

(2) αᾱ is a regular element in G_{n}(E).

(3) s_{α} is a regular element in 𝔤.

Proof. We first note that if two elements s_{α} and s_{β} of 𝔰 are H-conjugate, then αᾱ and ββ̄ are G_{n}(E) conjugate. According to a result in [G3], we may assume, up to a H conjugation, that

α = ( \begin{matrix}α_{1} & 0 \\ 0 & α_{2} \end{matrix} )

where α_{1} ∈ G_{m_{1}}(E) with α_{1}ᾱ_{1} = ᾱ_{1}α_{1} ∈ G_{n}(F) and α_{2} is a nilpotent element in M_{m_{2}}(F). Here m_{1} + m_{2} = n. We note that in this case

s_{α} = ( \begin{matrix}0 & 0 & α_{1} & 0 \\ 0 & 0 & 0 & 0 \\ ᾱ_{1} & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \end{matrix} ) + ( \begin{matrix}0 & 0 & 0 & 0 \\ 0 & 0 & 0 & α_{2} \\ 0 & 0 & 0 & 0 \\ 0 & ᾱ_{2} & 0 & 0 \end{matrix} )

is the Jordan Decomposition for s_{α} in 𝔰. Now given an element s_{β} ∈ 𝔰, then it commutes with s_{α} if and only βᾱ = αβ̄. This is equivalent to

β = ( \begin{matrix}β_{1} & 0 \\ 0 & β_{2} \end{matrix} ) , α_{1}β̄_{1} = β_{1}ᾱ_{1}, β_{2}α_{2} = α_{2}β̄_{2}.

We first consider the subspace 𝔱_{𝔰_{m_{1}}} of 𝔰_{m_{1}} consists of elements s_{β_{1}} ∈ 𝔰_{m_{1}} with α_{1}β̄_{1} = β_{1}ᾱ_{1}. Let β_{1}α_{1}^{−1} = γ. Then it is easily seen that the above condition for β_{1} is equivalent to γα_{1} = α_{1}γ̄. By an easy computation we find that 𝔱_{𝔰_{m_{1}}} is an abelian subspace if and only if the subspace

M_{m_{1}}(E)_{α_{1}}^{σ} = {γ ∈ M_{n_{1}}(E)|γα_{1} = α_{1}γ̄}

of M_{n_{1}}(E) is abelian. By [AC], this in turn is equivalent to the fact that α_{1}ᾱ_{1} is regular in G_{m_{1}}(F), and if this is the case then M_{m_{1}}(E)_{α_{1}}^{σ} = M_{m_{1}}(F)_{α_{1}ᾱ_{1}}- the centralizer of α_{1}ᾱ_{1} in M_{m_{1}}(F). Then we have

𝔱_{𝔰_{m_{1}}} = {s_{γα_{1}}|γ ∈ M_{m_{1}}(F)_{α_{1}ᾱ_{1}}}

which is a Cartan subspace of 𝔰_{m_{1}}.

As for the subspace 𝔱_{2} = {s_{β_{2}}|β_{2}α_{2} = α_{2}β̄_{2}} of 𝔰_{m_{2}}, it is easily seen that it is not abelian unless m_{2} = 0. In fact if we assume that α_{2} is in the form of (5.6), then both s_{I_{m_{2}}} and s_{β_{2}} with β_{2} = diag(\sqrt{τ}I_{n_{1}}, −\sqrt{τ}I_{n_{2}}, \sqrt{τ}I_{n_{3}}, ...) are in 𝔱_{2}. Clearly s_{I_{m_{2}}} and s_{β_{2}} are not commutative.

It follows that (1) is equivalent to (2).

Now let α be an element in M_{n}(E). Then we have

(λI_{2n} − s_{α})(λI_{2n} + s_{α}) = ( \begin{matrix}λI_{n} & −α \\ −ᾱ & λI_{n} \end{matrix} ) ( \begin{matrix}λI_{n} & α \\ ᾱ & λI_{n} \end{matrix} ) = ( \begin{matrix}λ^{2}I_{n} − αᾱ & 0 \\ 0 & λ^{2}I_{n} − ᾱα \end{matrix} ) .

Since we have

ϵ(λI_{2n} + s_{α})ϵ^{−1} = λI_{2n} + θ(s_{α}) = λI_{2n} − s_{α},

then

det(λI_{2n} − s_{α}) = det(λ^{2}I_{n} − αᾱ) = (λ^{2} − λ_{1})...(λ^{2} − λ_{n})

where λ_{1},..., λ_{n} are the eigenvalues of αᾱ in an algebraic closed closure of F. Thus s_{α} has 2n different eigenvalues if and only λ_{i} ≠ λ_{j}, if i ≠ j and λ_{i} ≠ 0 for all i. It follows that (2) is equivalent to (3). This ends the proof of the proposition.

We also proved the following corollary in the proof of the lemma.

Corollary 6.2. Each Cartan subspace of 𝔰 is conjugate to a Cartan subspace of the form

𝔱_{α} = \{s_{γα} = ( \begin{matrix}0 & γα \\ γᾱ & 0 \end{matrix} ) |γ ∈ M_{n}(F)_{αᾱ}\}

where αᾱ is a regular element in G_{n}(F) and M_{n}(F)_{αᾱ} is the centralizer of αᾱ in M_{n}(F).

Define a function D_{𝔤}(x) on 𝔤 by

det(t − adx) = D_{𝔤}(x)t^{2n} + terms with higher degrees, x ∈ 𝔤.

Then D_{𝔤}(x) is a nonzero polynomial on 𝔤. It is well known that x ∈ 𝔤 is regular if and only if D_{𝔤}(x) ≠ 0. So for s ∈ 𝔰, we have that s ∈ s′ if and only if D_{𝔤}(s) ≠ 0.

We now consider the H-invariant orbital integrals on 𝔰. Recall that we denote by 𝔥 the Lie algebra of H in 𝔤. Let 𝔱_{𝔰} be a Cartan subspace of 𝔰 and let 𝔱′_{𝔰} = 𝔱_{𝔰} ∩ 𝔰′ be the set of regular elements in 𝔱_{𝔰}. Suppose 𝔱_{𝔥} is the centralizer of 𝔱_{𝔰} in 𝔥 and T_{H} is the centralizer of 𝔱_{𝔰} in H. Then, by corollary 6.2, T_{H} is a torus in H which is conjugate to a Cartan subgroup in G_{n}(F), and 𝔱_{𝔥} is the Lie algebra of T_{H} in 𝔥. The orbital integral of f ∈ C_{c}^{∞}(𝔰) at a regular element s in 𝔱_{𝔰} is defined by the formula

Φ(s; f; 𝔰/𝔱_{𝔰}) = ∫_{H/T_{H}} f(hsh^{−1})dh.

Since the H orbit of a semisimple element in 𝔰 is closed, the above integral is convergent. Recall that 𝔥, 𝔰 are the 1, −1 eigenspaces of θ in 𝔤. It is easily seen that 𝔱_{𝔤} = 𝔱_{𝔥} + 𝔱_{𝔰} is the centralizer of 𝔱_{𝔰} in 𝔤. Since θ-regular elements in 𝔰 are in 𝔤′, we find that 𝔱_{𝔤} is actually a Cartan algebra in 𝔤. Now we have 𝔱_{𝔤}∩𝔰 = 𝔱_{𝔰} and 𝔱_{𝔤}∩𝔥 = 𝔱_{𝔥}. So we may regard 𝔰/𝔱_{𝔰} and 𝔥/𝔱_{𝔥} as subspaces of 𝔤/𝔱_{𝔤}. We identify 𝔤/𝔱_{𝔤} with the orthogonal complement 𝔱_{g}^{+} of 𝔱_{𝔤} in 𝔤 under the bilinear form B. Note that ad(s) for s ∈ 𝔰 maps 𝔰 into 𝔥 and 𝔥 into 𝔰. So for s ∈ 𝔱′_{𝔰}, we have

D_{𝔤}(s) = det(−ads|_{𝔤/𝔱_{𝔤}}) = det(−ads|_{𝔥/𝔱_{𝔥}+𝔰/𝔱_{𝔰}}) = det(−ads|_{𝔱_{𝔥}^{+}+𝔱_{𝔰}^{+}}),

where 𝔱_{𝔥}^{+} is the orthogonal complement of 𝔱_{𝔥} in 𝔥 and 𝔱_{𝔰}^{+} is the orthogonal complement of 𝔱_{𝔰} in 𝔰. Let T_{H}^{∗} be the normalizer of 𝔱_{𝔰} in H and let W_{H}(𝔱_{𝔰}) = T_{H}^{∗}/T_{H} be the Weyl group. Then W_{H}(𝔱_{𝔰}) is a finite group. We denote by |W_{H}(𝔱_{𝔰}))| the number of the elements in W_{H}(𝔱_{𝔰}). Then we have (p 106, [RR]):

Weyl Integration Formula. For f ∈ C_{c}^{∞}(𝔰), then

∫_{𝔰} f(s)ds = ∑_{{𝔱_{𝔰}}} |W_{H}(𝔱_{𝔰})|^{−1} ∫_{𝔱′_{𝔰}} |D_{𝔤}(s)|^{1/2}Φ(s; f; 𝔰/𝔱_{𝔰})ds

where the sum is over the H-conjugacy classes of the Cartan subspaces of 𝔰 and the Haar measure involved are suitably normalized.

Next we derive a descent formula for the orbital integrals in the spirit of Harish-Chandra. Let 𝔓 be a parabolic subalgebra of M_{n}(E). Let 𝔐 be a Levi subalgebra and 𝔑 be the nilpotent radical of 𝔓. We denote by 𝔓_{𝔤} the parabolic subalgebra of 𝔤 of elements of the form

( \begin{matrix}α & β \\ β̄ & ᾱ \end{matrix} ) , α, β ∈ 𝔓.

Suppose 𝔐_{𝔤} and 𝔑_{𝔤} are the Levi subalgebra and the nilpotent radical of 𝔓_{𝔤} corresponding to 𝔐 and 𝔑 respectively. Let 𝔓_{𝔰} = 𝔓_{𝔤} ∩ 𝔰, 𝔐_{𝔰} = 𝔐_{𝔤} ∩ 𝔰 and 𝔑_{𝔰} = 𝔑_{𝔤} ∩ 𝔰. Then 𝔓_{𝔰} = {s_{α}|α ∈ 𝔓}. The same thing is true for 𝔐_{𝔰} and 𝔑_{𝔰}. We denote by K_{H} the maximal compact subgroup of H of elements h_{g} with g ∈ G_{n}(O_{E}). Let

𝔓_{𝔥} = 𝔓_{𝔤} ∩ 𝔥, 𝔐_{𝔥} = 𝔐_{g} ∩ 𝔥, 𝔑_{𝔥} = 𝔑_{𝔤} ∩ 𝔥

and

P_{H} = 𝔓_{𝔥} ∩ H, M_{H} = 𝔐_{𝔥} ∩ H, N_{H} = I_{2n} + 𝔑_{𝔥}.

Then P_{H} = M_{H}N_{H} is a parabolic subgroup of H with M_{H} as a Levi subgroup and N_{H} as the unipotent radical. We have the Iwasawa decomposition H = K_{H}P_{H}. We normalize the Haar measure on H such that

∫_{H} f(h)dh = ∫_{K} ∫_{N} ∫_{M} f(knm)dmdndk

for an integrable function f on H. For f ∈ C_{c}^{∞}(𝔰), put

f̃(s) = ∫_{K_{H}} f(ksk^{−1})dk, s ∈ 𝔰

and

f_{𝔓}(m) = ∫_{𝔑_{𝔰}} f̃(m + s)ds, m ∈ 𝔐_{𝔰}.

Then f_{𝔓} ∈ C_{c}^{∞}(𝔐_{𝔰}).

Now let 𝔱_{𝔰} be a Cartan subspace of 𝔐_{𝔰}. Then it is also a Cartan subspace of 𝔰. For m ∈ 𝔐_{𝔤}, put

D_{𝔤/𝔐_{𝔤}}(m) = det(−adm|_{𝔤/𝔐_{𝔤}}) = \frac{D_{𝔤}(m)}{D_{𝔐_{𝔤}}(m)}.

Let 𝔑_{F} = 𝔑 ∩ M_{n}(F) and let | |_{F} be the absolute value on F. We have:

Lemma 6.3. If s = s_{α} ∈ 𝔐_{𝔰} with αᾱ a regular element in G_{n}(F), then

|D_{𝔤/𝔐_{𝔤}}(s)|_{F} = |det(adαᾱ|_{𝔑_{F}})|_{F}^{4} .

Proof. Let 𝔑_{𝔤}^{−} be the nilpotent radical of the parabolic subalgebra of 𝔤 opposite to 𝔓_{𝔤}. Then we have the decomposition

𝔤 = 𝔑_{𝔤} + 𝔐_{𝔤} + 𝔑_{𝔤}^{−}.

So for s = s_{α} ∈ 𝔐_{𝔰} with αᾱ ∈ G_{n}(F), we have

|D_{𝔤/𝔐_{𝔤}}(s)|_{E} = |det(ads|_{𝔤/𝔐_{𝔤}})|_{E} = |det(ads|_{𝔑_{𝔤}+𝔑_{𝔤}^{−}})|_{E}
= |det(ads|_{𝔑_{𝔤}})|_{E}|det(ads|_{𝔑_{𝔤}^{−}})|_{E} = |det(ads|_{𝔑_{𝔤}})|_{E}^{2}.

Now 𝔑_{𝔤} = 𝔑_{𝔥} + 𝔑_{𝔰} where ads maps 𝔑_{𝔥} into 𝔑_{𝔰} and maps 𝔑_{𝔰} into 𝔑_{𝔥}.

We first consider the map ads|_{𝔑_{𝔥}} : 𝔑_{𝔥} → 𝔑_{𝔰}. Here we regard 𝔑_{𝔥} and 𝔑_{𝔰} as vector spaces over F. Then ads|_{𝔑_{𝔥}} is a F -linear map, not E-linear. Note that for diag(β, β̄) ∈ 𝔑_{𝔥}, we have

ads(( \begin{matrix}β & 0 \\ 0 & β̄ \end{matrix} )) = ( \begin{matrix}0 & αβ̄ − βα \\ ᾱβ − β̄ᾱ & 0 \end{matrix} ) .

So ads|_{𝔑_{𝔥}} : 𝔑_{𝔥} → 𝔑_{𝔰} is equivalent to the F -linear map β ↦ αβ̄−βα from 𝔑 to 𝔑. By an elementary computation, we find that its Jacobian is |det(adαᾱ|_{𝔑_{F}})|_{F}. Similarly for s_{β} ∈ 𝔑_{𝔰} we have

ads(s_{β}) = ( \begin{matrix}αβ̄ − βᾱ & 0 \\ 0 & ᾱβ − β̄α \end{matrix} ) .

So the F -linear map ads|_{𝔑_{𝔰}} : 𝔑_{𝔰} → 𝔑_{𝔥} is equivalent to the F -linear map β ↦ αβ̄−βᾱ from 𝔑 to 𝔑. Its Jacobian is also |det(adαᾱ|_{𝔑_{F}})|_{F}. Thus we have

|D_{𝔤/𝔐_{𝔤}}(s)|_{E} = |[det(adαᾱ|_{𝔑_{F}})]^{2}|_{F}^{2} = |det(adαᾱ|_{𝔑_{F}})|_{F}^{4} .

Proposition 6.4. For s ∈ 𝔱′_{𝔰} and f ∈ C_{c}^{∞}(𝔰), we have

Φ(s; f; 𝔰/𝔱_{𝔰}) = |D_{𝔤/𝔐_{𝔤}}(s)|_{F}^{−1/4}Φ(s; f_{𝔓}; 𝔐_{𝔰}/𝔱_{𝔰}).

Proof. Let s = s_{α} be a regular element in 𝔰. Then αᾱ is a regular element in G_{n}(E). Without loss of generality, we assume that αᾱ ∈ G_{n}(F). By definition we have

Φ(s; f; 𝔰/𝔱_{𝔰}) = ∫_{H/T_{H}} f(hsh^{−1})dh

= ∫_{K_{H}} ∫_{N_{H}} ∫_{M_{H}/T_{H}} f(knmsm^{−1}n^{−1}k^{−1})dmdndk

= ∫_{N_{H}} ∫_{M_{H}/T_{H}} f̃(nmsm^{−1}n^{−1})dmdn.

Let m_{s} = msm^{−1} ∈ 𝔐_{𝔰}. Consider the map n ↦ m_{s}^{−1}nm_{s}n^{−1} from N_{H} to N_{H}. It is an isomorphism of algebraic groups over F. If we write n = h_{β}, m = h_{γ} and γ_{α} = γαγ̄^{−1} ∈ M, we have

m_{s}^{−1}nm_{s}n^{−1} = ( \begin{matrix}γ̄_{α}^{−1}β̄γ̄_{α}β^{−1} & 0 \\ 0 & γ_{α}^{−1}βγ_{α}β̄^{−1} \end{matrix} ) .

Then the above map is just the F endomorphism in 𝔑 defined by β ↦ γ̄_{α}^{−1}β̄γ̄_{α}β^{−1}. It is well known that its Jacobian is |det(I − Ad(αᾱ)^{−1}|_{𝔑_{F}})|_{F} (p. 376, [K]). So the above integral becomes

|det(I − Ad(αᾱ)^{−1}|_{𝔑_{F}})|_{F}^{−1} ∫_{M_{H}/T_{H}} ∫_{N_{H}} f̃(msm^{−1}n)dndm

= |det(I − Ad(αᾱ)^{−1}|_{𝔑_{F}})|_{F}^{−1} ∫_{M_{H}/T_{H}} ∫_{N_{H}} f̃(msm^{−1}(n − I) + msm^{−1})dmdn

= |det(I − Ad(αᾱ)^{−1}|_{𝔑_{F}})|_{F}^{−1} ∫_{M_{H}/T_{H}} ∫_{𝔑_{𝔥}} f̃(msm^{−1}n + msm^{−1})dndm.

Now consider the map n ↦ m_{s}n from 𝔑_{𝔥} to 𝔑_{𝔰}. Then its Jacobian is |det(L_{α}|_{𝔑})|_{E} = |det(L_{αᾱ}|_{𝔑_{F}})|_{F} where L_{α} is the map on 𝔑 defined by L_{α}(β) = αβ for β ∈ 𝔑 and L_{αᾱ} is defined similarly. Then L_{αᾱ}|_{𝔑_{F}} ∘ (1 − Ad(αᾱ)^{−1}|_{𝔑_{F}}) = adαᾱ|_{𝔑_{F}}. So the integral becomes

|det(I − Ad(αᾱ)^{−1}|_{𝔑_{F}})|_{F}^{−1}|det(L_{αᾱ}|_{𝔑_{F}})|_{F}^{−1} ∫_{M_{H}/T_{H}} ∫_{𝔑_{𝔰}} f̃(msm^{−1} + n)dndm

= |det(adαᾱ|_{𝔑_{F}})|_{F}^{−1} ∫_{M_{H}/T_{H}} f_{𝔓}(msm^{−1})dm

= |D_{𝔤/𝔐_{𝔤}}(s)|^{−1/4}Φ(s; f_{𝔓}; 𝔐_{𝔰}/s).

7. The nilpotent orbits (II)

We continue with our study of the nilpotent orbital integrals on 𝔰. Recall that for any nilpotent orbit in 𝔰, there is an element s_{α_{0}} in it such that α_{0} is a nilpotent matrix in M_{n}(F). We denote by 𝔓 the parabolic subalgebra of M_{n}(E) stabilizing the flag {kerα_{0}^{n_{i}}} in E^{n}. Let Λ be the distribution defined by the orbital integral on the H orbit of s_{α_{0}} in 𝔰. Our purpose is to prove that the Fourier transform Λ̂ of Λ is represented by a locally integrable function on 𝔰, i.e. there is a locally integrable function Θ on 𝔰 such that

Λ̂(f) = ∫_{𝔰} f(s)Θ(s)ds, f ∈ C_{c}^{∞}(𝔰).

For this we will follow the arguments in [vD] as well as in the proof of Lemma 4.7.6 in [S].

We denote by C_{𝔰} (resp. C_{𝔐_{𝔰}} ) the set of the conjugacy classes of Cartan subspaces in 𝔰 (resp. 𝔐_{𝔰} ) under H (resp. M_{H} ). Both C_{𝔰} and C_{𝔐_{𝔰}} are finite sets. Given a Cartan subspace 𝔱 of 𝔐_{𝔰}, we write C_{𝔱} as a subset of C_{𝔐_{𝔰}} consisting of all the M_{H} conjugacy classes of Cartan subspaces of 𝔐_{𝔰} which are H conjugate to 𝔱. Let Y_{𝔱} = {y ∈ H|y^{−1}𝔱y ⊂ 𝔐_{𝔰}}. Then it is easily seen that the map y ↦ y^{−1}𝔱y defines a one to one correspondence between T_{H}^{∗}\Y_{𝔱}/M_{H} and C_{𝔱}. Here T_{H}^{∗} is the normalizer of 𝔱 in H. Let y_{1}, ..., y_{l_{𝔱}} be a set of representatives for T_{H}^{∗}\Y_{𝔱}/M_{H}. Set 𝔱_{i} = y_{i}^{−1}𝔱y_{i}. Then 𝔱_{1}, ..., 𝔱_{l_{𝔱}} is a set of representatives of C_{𝔱}. For s ∈ 𝔱′, define

Θ(s) = \frac{1}{|D_{𝔤}(s)|^{1/4}} ∑_{i=1}^{l_{𝔱}} ∑_{h} |D_{𝔐_{𝔤}}(hs_{i}h^{−1})|^{1/4}, h ∈ W_{H}(𝔱_{i})/W_{M_{H}} (𝔱_{i})

where we set s_{i} = y_{i}^{−1}sy_{i} ∈ 𝔱_{i} for i = 1, 2, ..., l_{𝔱}. Then for a general element s in 𝔰, we define

Θ(s) = \{ \begin{matrix}Θ(s_{0}), & if s = hs_{0}h^{−1}, h ∈ H and s_{0} ∈ 𝔐′_{𝔰} \\ 0, & otherwise. \end{matrix}

Here Θ is a well defined function on 𝔰 which is locally constant on the regular set 𝔰′ and is supported in ⋃_{h∈H} h𝔐′_{𝔰}h^{−1}. It also satisfies Θ(hsh^{−1}) = Θ(s) for h ∈ H and s ∈ 𝔰.

Proposition 7. The function Θ is locally integrable on 𝔰.

Proof. To prove Θ is locally integrable, it is enough to show that for any function f ∈ C_{c}^{∞}(𝔰), we have

∫_{𝔰} |f(s)Θ(s)|ds < +∞.

Or equivalently

I_{f} = ∑_{{𝔱}∈C_{𝔰}} |W_{H}(𝔱)|^{−1} ∫_{𝔱′} |D_{𝔤}(s)|^{1/2}Φ(s; |f|; 𝔰/𝔱)Θ(s)ds < +∞

by the Weyl integration formula. For convenience, we assume f ≥ 0 from now on. Because Θ(s) = 0 unless there is h ∈ H such that hsh^{−1} ∈ 𝔐_{𝔰}, any class in C_{𝔰} which contributes to the above sum must be H-conjugate to a Cartan subspace of 𝔐_{𝔰}. Let X_{𝔰} = {𝔱} with 𝔱 ⊂ 𝔐_{𝔰} be a set of representatives for such classes in C_{𝔰}. So we get

I_{f} = ∑_{𝔱∈X_{𝔰}} |W_{H}(𝔱)|^{−1} ∫_{𝔱′} |D_{𝔤}(s)|^{1/2}Φ(s; f; 𝔰/𝔱)Θ(s)ds.

Now for a given Cartan subspace 𝔱 of 𝔐_{𝔰}, we have

∫_{𝔱′} |D_{𝔤}(s)|^{1/2}Φ(s; f; 𝔰/𝔱)Θ(s)ds

= ∑_{i=1}^{l_{𝔱}} ∑_{h∈W_{H}(𝔱_{i})/W_{M_{H}} (𝔱_{i})} ∫_{𝔱′} |D_{𝔤}(s)|^{1/2}Φ(s; f; 𝔰/𝔱)|D_{𝔤}(s)|^{−1/4}|D_{𝔐_{𝔤}}(hs_{i}h^{−1})|^{1/4}ds.

For any s ∈ 𝔱′ and h ∈ W_{H}(𝔱_{i}), it is routine to verify that

Φ(s; f; 𝔰/𝔱) = Φ(s_{i}; f; 𝔰/𝔱_{i}) = Φ(hs_{i}h^{−1}; f; 𝔰/𝔱_{i}).

So the above sum of integrals becomes

|W_{H}(𝔱)| ∑_{i=1}^{l_{𝔱}} |W_{M_{H}} (𝔱_{i})|^{−1} ∫_{𝔱′_{i}} |D_{𝔤}(s)|^{1/4}|D_{𝔐_{𝔤}}(s)|^{1/4}Φ(s; f; 𝔰/𝔱_{i})ds

= |W_{H}(𝔱)| ∑_{i=1}^{l_{𝔱}} |W_{M_{H}} (𝔱_{i})|^{−1} ∫_{𝔱′_{i}} |D_{𝔐_{𝔤}}(s)|^{1/2}Φ(s; f_{𝔓}; 𝔐_{s}/𝔱_{i})ds.

Here we used the descent formula for the orbital integrals as proved in the last section. Thus we obtain

I_{f} = ∑_{𝔱∈X_{𝔰}} ∑_{i=1}^{l_{𝔱}} |W_{M_{H}} (𝔱_{i})|^{−1} ∫_{t′_{i}} |D_{𝔐_{𝔤}}(s)|^{1/2}Φ(s; f_{𝔓}; 𝔐_{𝔰}/𝔱_{i})ds

= ∑_{{𝔱}∈C_{𝔐𝔰}} |W_{M_{H}} (𝔱)|^{−1} ∫_{𝔱′} |D_{𝔐_{𝔤}}(s)|^{1/2}Φ(s; f_{𝔓}; 𝔐_{𝔰}/𝔱)ds

= ∫_{𝔐_{𝔰}} f_{𝔓}(m)dm

by the Weyl integration formula for f_{𝔓} on 𝔐_{𝔰}. Since f_{𝔓} is in C_{c}^{∞}(𝔐_{𝔰}), we have

I_{f} = ∫_{𝔐_{𝔰}} f_{𝔓}(m)dm < ∞.

This ends the proof of the proposition.

We actually proved that for f ∈ C_{c}^{∞}(𝔰),

∫_{𝔰} f(s)Θ(s)ds = ∫_{𝔐_{𝔰}} f_{𝔓}(m)dm.

On the other hand, by proposition 5.1, we have

Λ̂(f) = ∫_{𝔓_{𝔰}}(∫_{K_{H}} f(kpk^{−1})dk)dp = ∫_{𝔓_{𝔰}} f̃(p)dp
= ∫_{𝔐_{𝔰}} ∫_{𝔑_{𝔰}} f̃(m + n)dndm = ∫_{𝔐_{𝔰}} f_{𝔓}(m)dm.

So we obtained the following proposition. Proposition 8. For f ∈ C_{c}^{∞}(𝔰), we have

Λ̂(f) = ∫_{𝔰} f(s)Θ(s)ds.

Hence Λ̂ is locally integrable.

Theorem 2.3 follows. Theorem 1.1 is then a consequence of theorem 2.1, theorem 2.3 and proposition 4.

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Max-Planck-Institut für Mathematik, Gottfried-Claren-Strasse 26, 53225 Bonn, Germany

   E-mail address: jguo@mpim-bonn.mpg.de