Let G = GL(n, F) where F is a p-adic field, and let 9J{G) denote the Hecke algebra of spherical functions on G. Let u\,..., u p denote a complete set of representatives for the unipotent conjugacy classes in G. For each 1 < / < p, let \ii be the linear functional on 9~[{G) such that //;(/") is the orbital integral of/ over the orbit of «,-. Waldspurger proved that the /z,-, 1 < i < p, are linearly independent. In this paper we give an elementary proof of Waldspurger's theorem which provides

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... rete information about the Hecke functions needed to separate orbits. We also prove a twisted version of Waldspurger's theorem and discuss the consequences for SL(«, F).