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

Avraham Aizenbud, Dmitry Gourevitch, Eitan Sayag

Research output: Contribution to journalArticlepeer-review

53 Scopus citations

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 (GLn+k(F), GLn(F) × GLk(F)) and (GLn(E), GLn(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 GLn(F) is invariant with respect to transposition. For non-Archimedean F, the latter is a classical theorem of Gelfand and Kazhdan.

Original languageEnglish
Pages (from-to)509-567
Number of pages59
JournalDuke Mathematical Journal
Volume149
Issue number3
DOIs
StatePublished - 1 Sep 2009

ASJC Scopus subject areas

  • Mathematics (all)

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