Mathematics – Representation Theory
Scientific paper
2008-12-30
Duke Mathematical Journal, Volume 149, Number 3 (2009)
Mathematics
Representation Theory
A merge of arXiv:0803.3395 and arXiv:0803.3397 (with no additional material). Appendix D by Avraham Aizenbud, Dmitry Gourevitc
Scientific paper
10.1215/00127094-2009-044
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.
Aizenbud Avraham
Gourevitch Dmitry
Sayag Eitan
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