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

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.

A merge of arXiv:0803.3395 and arXiv:0803.3397 (with no additional material). Appendix D by Avraham Aizenbud, Dmitry Gourevitch and Eitan Sayag. v2,v3: minor changes. v4: correction in the speciality criterion (7.3.7). v5: minor correction in the proof of Proposition 7.2.1

References in corpus (3)

• Schwartz functions on Nash manifolds
• (GL(n+1,F),GL(n,F)) is a Gelfand pair for any local field F
• (O(V+F), O(V)) is a Gelfand pair for any quadratic space V over a local field F

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