Orbital integrals on symmetric spaces and spherical characters
type: Item
instance of: scholarly article
title: Orbital integrals on symmetric spaces and spherical characters
zbMATH Open document ID: 0856.22022
author: Yuval Z. Flicker
published in: Journal of Algebra
publication date: 1996-11-12T00:00:00Z
review text: The first section of this paper obtains an asymptotic expansion near semisimple elements of orbital integrals \(\mu_{\widetilde x}(\widetilde f)\) of \(C^\infty_c\)-functions \(\widetilde f\) on symmetric spaces \(G/H\). Here \(G\) is a reductive \(p\)-adic group, and \(H\) is the group of fixed points of an involution \(\theta\) on \(G\). This extends the germ expansion of \textit{J. A. Shalika} [Ann. Math., II. Ser. 95, 226-242 (1972; Zbl 0281.22011)] and \textit{M.-F. Vigneras} [J. Fac. Sci. Univ. Tokyo Sect. I A 28, 945-961 (1981; Zbl 0499.22011)] in the group case. The main part of the paper studies examples of groups \(G\) with involution \(\theta\), which have the property that the spherical characters associated with its spherical admissible representations are not identically zero on the regular set of \(G/H\). These include \(G= GL(n+ m)\), \(H= GL(n)\times GL(m)\) for \(n= m= 1\) or 2, and \(n= 1\), \(m\geq 3\). More general results were obtained by \textit{J. Sekiguchi} [Adv. Stud. Pure Math. 6, 83-126 (1985; Zbl 0578.22011)] in the case of real symmetric spaces. These generalized Harish-Chandra's work in the group case, which had dealt not only with Archimedean but also with non-Archimedean fields. The present paper concerns the \(p\)-adic case. It uses the work of \textit{C. Rader} and \textit{S. Rallis} [Am. J. Math. 118, 91-178 (1996)], which shows 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 \textit{Harish-Chandra} [Queen's Pap. Pure Appl. Math. 48, 281-346 (1978; Zbl 0433.22012)] in the group case. To study the non-vanishing of some spherical characters, the paper constructs an explicit basis of the space of invariant distributions on the nilpotent cone, on regularizing spherical orbital integrals, and taking suitable linear combinations. This local work is motivated by concrete applications to the theory of Delign…
reviewed by: Yuval Z. Flicker
Mathematics Subject Classification ID: 22E50
Mathematics Subject Classification ID: 11F70
Mathematics Subject Classification ID: 22E35
Mathematics Subject Classification ID: 43A85
zbMATH DE Number: 930164
zbMATH Keywords: orbital integrals
zbMATH Keywords: symmetric spaces
zbMATH Keywords: \(p\)-adic group
zbMATH Keywords: germ expansion
zbMATH Keywords: spherical characters
zbMATH Keywords: spherical admissible representations
zbMATH Keywords: Fourier transforms
zbMATH Keywords: invariant distributions
MaRDI profile type: Publication
OpenAlex ID: W2063263215
DOI: 10.1006/JABR.1996.0282
description: scientific article; zbMATH DE number 930164