A trace formula for symmetric spaces (original) (raw)

1993, Duke Mathematical Journal

Now there is a semisimple group G' and a maximal torus A' with Weyl group W' and root system R', and an isomorphism of A' onto A which takes W' to W and R' to R,. The usual Chevalley restriction theorem implies that the set of closed G'-conjugacy classes in G' can be identified with the spectrum of F[A'] w' that is, with the orbits of W' in A'. This gives the required bijection. We note that the isomorphic algebras F[A] w" and F[A'] w' are polynomial algebras, but the group G' need not be simply connected. This is the general principle of comparison that guides the particular example to be considered below. Of course, over a global or a local field, it must be modified suitably. Assuming for now its validity over a number field F, it is natural to postulate the existence of a trace formula identity of the form K(h, h2) dh dh2 ft K'(O', 9') do'. '(F) \G'(F,) Here K and K' are the cuspidal kernels, or rather the "discrete parts" of the respective trace formulas. They are associated to functions f and f' on G and G', respectively; the above equality is supposed to be true if f and f' have "matching orbital integrals". The only representations which contribute to the left-hand side are those which are distinguished with respect to H, that is, contain a vector such that the integral is nonzero. The equality then should characterize distinguished representations as functorial images from representations of G'. The purpose of this paper is to explore this idea in a simple case. However, one discovers quickly that such a simple scheme does not work. A more correct formula might take the form or K(h, h2)O(hx) dh dh2 ; K'(9', 9') do' K(h, h2) dhx dh2 f K'(0', O')0'(0') do', where 0 and 0' are suitable automorphic forms on H and G' respectively, which serve as "weights" in the formula.