Generalized Harish-Chandra descent and applications to Gelfand pairs

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 arbitrary local field F of characteristic zero. Our main tool is Luna slice theorem. In the second part of the paper we apply this technique to symmetric pairs. In particular we prove that the pair (GL(n,C),GL(n,R)) is a Gelfand pair. We also prove that any conjugation invariant distribution on GL(n,F) is invariant with respect to transposition. For non-archimedean F the later is a classical theorem of Gelfand and Kazhdan. We use the techniques developed here in our subsequent work [AG3] where we prove an archimedean analog of the theorem on uniqueness of linear periods by H. Jacquet and S. Rallis.

34 pages, 1 figure. v2: A proof of a version of localization principle inserted. v3: minor changes. v4: definition of symmetric pair slightly changed. v5: minor changes + lemma D.0.3 added for clarification. v6: minor changes - see Theorem 4.0.5. v7:minor changes. Appendix A by Avraham Aizenbud, Dmitry Gourevitch and Eitan Sayag

References in corpus (4)

• Schwartz functions on Nash manifolds
• (GL(n+1,F),GL(n,F)) is a Gelfand pair for any local field F
• Multiplicity one Conjectures
• An archimedean analog of Jacquet - Rallis theorem

Cited by in corpus (4)

• A partial analog of integrability theorem for distributions on p-adic spaces and applications
• Invariant distributions on non-distinguished nilpotent orbits with application to the Gelfand property of (GL(2n,R),Sp(2n,R))
• An archimedean analog of Jacquet - Rallis theorem
• (GL(2n,C),SP(2n,C)) is a Gelfand Pair