UNIPOTENT ORBITAL INTEGRALS OF HECKE FUNCTIONS FOR GL(N)

Let G = GL(n, F) where F is a p-adic field, and let H(G) denote the He cke algebra of spherical functions on G. Let u1, . . . , u(p) denote a complete set of representatives for the unipotent conjugacy classes i n G. For each 1 less-than-or-equal-to i less-than-or-equal-to p, let m u(i) be the linear functional on H(G) such that mu(i)(f) is the orbita l integral of f over the orbit of u(i). Waldspurger proved that die mu (i), 1 less-than-or-equal-to i less-than-or-equal-to p, are linearly i ndependent. In this paper we give an elementary proof of Waldspurger's theorem which provides concrete information about the Hecke functions needed to separate orbits. We also prove a twisted version of Waldspu rger's theorem and discuss the consequences for SL(n, F).