## 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

### MSC:

 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

### Citations:

Zbl 0694.00012; Zbl 0675.22007