Orbital integrals on symmetric spaces and spherical characters (Q1922904)

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 Deligne-Kazhdan lifting of spherical automorphic representations [the author, J. Algebra 174, 678-697 (1995; Zbl 0828.11029); Math. Nachr. (1996)]. In some other examples, concerning \(G= GL(3n)\) and \(H= GL(n)\times GL(2n)\), and \(G= O(3, 2)\), \(H= O(2, 2)\), the paper explicitly constructs invariant distributions on the nilpotent cone which are equal to their Fourier transform. Such examples do not exist in Harish-Chandra's group case.