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

TL;DR

This paper extends Harish-Chandra's descent method to reductive groups over local fields, applies it to symmetric pairs, and establishes new Gelfand pair results and invariance properties of distributions.

Contribution

It generalizes Harish-Chandra's descent technique using Luna Slice Theorem and proves new Gelfand pair properties for specific symmetric pairs over local fields.

Findings

Proved pairs (GL(n+k,F), GL(n,F) x GL(k,F)) are Gelfand pairs.

Proved pairs (GL(n,E), GL(n,F)) are Gelfand pairs.

Established invariance of conjugation-invariant distributions under transposition.

Abstract

In the first part of the paper 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 of the paper 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 proven earlier by Jacquet and Rallis and the second by Flicker. 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.