ORBITAL INTEGRALS ON SYMMETRICAL SPACES AND SPHERICAL CHARACTERS

In the first section of this paper we obtain an asymptotic expansion n ear semi-simple elements of orbital integrals mu((x) over bar)((f) ove r tilde) of C-c(infinity)-functions (f) over tilde on symmetric spaces G/N. Here G is a reductive p-adic group, and H is the group of fixed points of an involution a on G. This extends the germ expansion of J. Shalika and M.-F. Vigneras in the group case. The main part of the pap er studies examples of groups G with involution sigma, which have the property that the spherical characters associated with their spherical admissible representations are not identically zero on the regular se t of G/H. These include G = GL(n + m), H = GL(n) x GL(m) for n = m = 1 or 2, and n = 1, m greater than or equal to 3. More general results h ad been obtained by J. Sekiguchi in the case of real symmetric spaces, generalizing Harish-Chandra's work in the group case, over archimedea n and non-archimedean fields. Our interest is in the p-adic case. Ther e the techniques are entirely different from Sekiguchi's. In fact we u se the recent work of J. Hakim and C. Rader and S. Rallis who showed t hat the spherical character is smooth on the regular set, and has asym ptotic expansion in terms of Fourier transforms of invariant distribut ions on the nilpotent cone, as found by Harish-Chandra in the group ca se. Our study of the nonvanishing of some spherical characters uses a construction of an explicit basis of the space of invariant distributi ons on the nilpotent cone. This is done on regularizing spherical orbi tal integrals, and taking suitable linear combinations. This local wor k is motivated by concrete applications to the theory of Deligne-Kazhd an lifting of spherical automorphic representations. In some other exa mples, concerning G = GL(3n) and H = GL(n) x GL(2n), and G = O(3, 2), H = O(2, 2), we explicitly construct invariant distributions on the ni lpotent 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 t hat mu((x) over bar)((f) over tilde) is locally constant on the regula r set of (x) over tilde, uniformly in (f) over tilde, in some cases. F ollowing D. Kazhdan's proof of his density theorem we show that an (f) over tilde which annihilates all spherical characters has mu((x) over bar)((f) over tilde) = 0 on the regular elliptic set. (C) 1996 Academ ic Press, Inc.