Orbital Integrals on Symmetric Spaces and Spherical Characters

Abstract

In the first section of this paper we obtain an asymptotic expansion near semi-simple elements of orbital integrals μx(f) ofCc∞-functionsfon symmetric spacesG/H. HereGis a reductivep-adic group, andHis the group of fixed points of an involution σ onG. This extends the germ expansion of J. Shalika and M.-F. Vigneras in the group case.The main part of the paper studies examples of groupsGwith involution σ, which have the property that the spherical characters associated with their spherical admissible representations are not identically zero on the regular set ofG/H. These includeG=GL(n+m),H=GL(n)×GL(m) forn=m=1 or 2, andn=1,m≥3. More general results had been obtained by J. Sekiguchi in the case of real symmetric spaces, generalizing Harish-Chandra's work in the group case, over archimedean and non-archimedean fields. Our interest is in thep-adic case. There the techniques are entirely different from Sekiguchi's. In fact we use the recent work of J. Hakim and C. Rader and S. Rallis who showed that the spherical character is smooth on the regular set, and has asymptotic expansion in terms of Fourier transforms of invariant distributions on the nilpotent cone, as found by Harish-Chandra in the group case.Our study of the nonvanishing of some spherical characters uses a construction of an explicit basis of the space of invariant distributions on the nilpotent cone. This is done on regularizing spherical orbital integrals, and taking suitable linear combinations. This local work is motivated by concrete applications to the theory of Deligne-Kazhdan lifting of spherical automorphic representations. In some other examples, concerningG=GL(3n) andH=GL(n)×GL(2n), andG=O(3,2),H=O(2,2), we explicitly construct invariant distributions on the nilpotent cone which are equal to their Fourier transform. Such examples do not exist in Harish-Chandra's group case.In the last two sections, following Harish-Chandra's simple proof in the group case, we show that μx(f) is locally constant on the regular set ofx, uniformly inf, in some cases. Following D. Kazhdan's proof of his density theorem we show that anfwhich annihilates all spherical characters has μx(f)=0 on the regular elliptic set.