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 $({\rm GL}_{n+k}(F),{\rm GL}_n(F) \times {\rm GL}_k(F))$ and $({\rm GL}_n(E),{\rm GL}_n(F))$ 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 ${\rm GL}_n(F)$ is invariant with respect to transposition. For non-Archimedean $F$ , the latter is a classical theorem of Gelfand and Kazhdan