p-adic典型群上轨道积分的若干性质

俞小祥

（江苏师范大学 数学与统计学院，江苏 徐州 221116）

0 Introduction

The study of intertwining operators plays an important role in Langlands program and harmonic analysis，it is raised from the study of Eisenstein series［1－3］and Eisenstein integrals［4－5］.Recent breakthrough to study those operators attached to non－generic supercuspidal representations is to study the orbit integrals associated to them［6－8］，since these integrals don't require to bear a Whittaker model，by doing so，one may relate the harmonic analysis on non－generic representations to generic ones.As the theories on generic representations［2，7］are fairly well known，this will find a way to approach the local Langlands program.

To save context，we'll freely use notations from［6－8］，particularly from those in［9－11］.However，we are not trying to expose the whole frame work mentioned above，but rather，only demonstrate some elementary and interesting properties of these orbit integrals.

Suppose Fis a p－adic field，let Glbe a quasi－split connected reductive classical group of rank l defined over F.Let M＝GLn（F）×Gmbe the Levi subgroup of a maximal parabolic subgroup Pof Gl，where Gmis a subgroup of the same type of Gl，with rank m（l＝n＋m）.Letτ′andτbe unitary irreducible supercuspidal representations of GLn（F）and Gm，respectively，withτ′ being ramified.Then the value of，A（s，τ′⊗τ，w0）h（e）〉at s＝0is proportional to［10］

Here his a function which lies in the induction space IndGlP（τ′⊗τ⊗｜det｜s⊗1）；ψτ′and fτare matrix coeffi－cients ofτ′andτ，respectively；Λcis the set of compact parameters；Uois the quotient of the twisted cen－；（ ）1（ ）tralizer of the orbit parametrized byλmodula that of the centralizerYdgand D2adhprovide inλλvariant measures on theε－conjugacy class of Yλand conjugacy class of aλ，respectively；and finally，Yλis related to aλunder norm correspondence.

1 Orbit integrals

Lemma 1Suppose G′is a reductive linear algebraic group defined over F，Γis a closed unimodular subgroup of G′，Kis a compact open subgroup of G′.If f∈Cc（G′／Γ）such that

Then it's not hard to see that∈Cc（G′）and（gγ）＝（g）＝f）for any g∈G′，γ∈Γ′，wheregis the representative of gin G′／Γ.Normalize the measure onΓ′so that meas（Γ′）＝1.Then

Here，dgis a left invariant measure on G′，the last equation is obtained by applying Lemma 1.2.5in［5］.

Since Kis open compact and~fis compactly supported，

Therefore，there is at least one xisuch that

Let V′be the representation space ofτ′.Supposeψτ′（g）is defined by

Theorem 1Let A（0，τ′⊗τ）be the expression in（1），then there is a functionalφon Cc（GLn（F）），so that

where Zis the center of GLn（F），G（ε，In）is the twisted centralizer of Inin GLn（F），dis a left invariant measure on GLn／ZGε，In.

We'll also define a right representationσof GLn（F）on Cc（GLn（F））by

Define a functional Pon Cc（GLn（F）／Z）by

for any f∈Cc（GLn（F））.Then by the definition of P，it's directly checked that Pis linear on Cc（GLn（F））and

Applying Corollary 1.8.3in［5］toψτ′，there is one functionalφon Cc（GLn（F）／Z）so that（2）holds.

Let Kbe an open compact subgroup of GLn（F）.Let ZK＝K∩Z，G（ε，K）＝G（ε，In）∩K.Then

Proposition 1（1）is nonzero only if for a matrix coefficientφ（x）of V′，

LetΓ＝ZG（ε，In），thenΓis obviously closed and unimodular.One may regardψτ′（ge（g））as a function of GLn（F）／Γ，defined by

Then by Theorem 1and Lemma 1，（1）is nonzero only if

But

Then（4）can be rewritten as（3）.

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