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 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 and are Gelfand pairs for any local field and its quadratic extension . 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 is invariant with respect to transposition. For non-Archimedean , the latter is a classical theorem of Gelfand and Kazhdan
Avraham Aizenbud. Dmitry Gourevitch. Eitan Sayag. "Generalized Harish-Chandra descent, Gelfand pairs, and an Archimedean analog of Jacquet-Rallis's theorem." Duke Math. J. 149 (3) 509 - 567, 15 September 2009. https://doi.org/10.1215/00127094-2009-044