Generalized Harish-Chandra descent, Gelfand pairs, and an Archimedean analog of Jacquet-Rallis's theorem

Abstract

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