Intégrales orbitales sphériques pour GL(N) sur un corps p-adique. (Spherical orbital integrals for GL(N) on a p-adic field). (French) Zbl 0706.22015

Orbites unipotentes et représentations. II: Groupes p-adiques et réels, Astérisque 171-172, 279-337 (1989).
[For the entire collection see Zbl 0694.00012.]
The spherical orbital integrals referred to in the title are of the form \(\int f(g^{-1}\gamma g)dg\), where f belongs to the Hecke algebra H(G) of compactly supported K-bi-invariant functions on \(G=GL(n,F)\), K is the subgroup of elements of G with integral coordinates in F, \(\gamma\) is a regular elliptic element of G, and the integration is over G modulo its center. The purpose of this article is to give an explicit description of the distribution \(f\to I(f,\gamma)=\int f(g^{-1}\gamma g)dg\); by way of Howe’s conjecture, one also obtains an explicit expression for “Shalika’s germs” for GL(n), albeit “only” in the spherical (unramified) case.
More precisely, fix \(\Sigma =[{\mathbb{R}}/(2\pi i/\log q){\mathbb{Z}}]^ n\), where q is the number of elements in the residue class field of F, and fix a locally constant function \(\phi\) on G with compact support inside the set of elliptic regular elements; then there exists a distribution \(D(s,\phi)\) on \(\Sigma\) such that \[ \int_{G}I(f,\gamma)\phi (\gamma)d\gamma =\int_{\Sigma}\tilde f(s)D(s,\phi)ds, \] with \(\tilde f(s)\) the Satake transform of f in H(G). The author’s main result is that \(D(s,\phi)\) is equal to an explicitly computable constant times a certain function \(R(s,\phi)\) involving the trace of intertwining operators attached to the spherical induced representations \(\pi(s,\cdot)\); since L. Clozel’s recent proof of Howe’s conjecture [Ann. Math., II. Ser. 129, No.2, 237-251 (1989; Zbl 0675.22007)] makes it possible to exhibit \(f\to I(f,\gamma)\) as a linear combination of unipotent orbital integrals, one obtains finally an expression for Shalika’s germs about an elliptic regular point \(\gamma\) in terms of residues of the functions \(R(s,\phi)\), \(\phi\) being the characteristic function of a small neighborhood of \(\gamma\). Like previous works of the author, this one blends technical power with originality, and unearths concrete, unexpected phenomena in the midst of a difficult general theory; how these results can be generalized remains to be seen.
Reviewer: S.Gelbart


22E50 Representations of Lie and linear algebraic groups over local fields
43A80 Analysis on other specific Lie groups
11F70 Representation-theoretic methods; automorphic representations over local and global fields