GENERALIZED HARISH-CHANDRA DESCENT, GELFAND PAIRS, AND AN ARCHIMEDEAN ANALOG OF JACQUET-RALLIS'S THEOREM

In the first part of this article, we generalize a descent technique due to Harish-Chandra to the case of a reductive group acting on a smooth affine variety both defined over an arbitrary local field F of characteristic zero. Our main tool is the Luna slice theorem. In the second part, we apply this technique to symmetric pairs. In particular, we prove that the pairs (GL(n+k) (F), GL(n) (F) x GL(k)(F)) and (GL(n) (E), GL(n) (F)) are Gelfand pairs for any local field F and its quadratic extension E. In the non-Archimedean case, the first result was proved earlier by Jacquet and Rallis [JR] and the second result was proved by Flicker [F]. We also prove that any conjugation-invariant distribution on GL(n) (F) is invariant with respect to transposition. For non-Archimedean F, the latter is a classical theorem of Gelfand and Kazhdan.