Generalized Harish-Chandra descent and applications to Gelfand pairs

Details Generalized Harish-Chandra descent and applications to Gelfand pairs Generalized Harish-Chandra descent and applications to Gelfand pairs

2008-03-24

arxiv.org/abs/0803.3395v7

Mathematics

Representation Theory

34 pages, 1 figure. v2: A proof of a version of localization principle inserted. v3: minor changes. v4: definition of symmetri

Scientific paper

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.

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