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15 September 2009 Generalized Harish-Chandra descent, Gelfand pairs, and an Archimedean analog of Jacquet-Rallis's theorem
Avraham Aizenbud, Dmitry Gourevitch, Eitan Sayag
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Duke Math. J. 149(3): 509-567 (15 September 2009). DOI: 10.1215/00127094-2009-044


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


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Avraham Aizenbud. Dmitry Gourevitch. Eitan Sayag. "Generalized Harish-Chandra descent, Gelfand pairs, and an Archimedean analog of Jacquet-Rallis's theorem." Duke Math. J. 149 (3) 509 - 567, 15 September 2009.


Published: 15 September 2009
First available in Project Euclid: 24 August 2009

zbMATH: 1221.22018
MathSciNet: MR2553879
Digital Object Identifier: 10.1215/00127094-2009-044

Primary: 22E50 , 46F10
Secondary: 14L24 , 14L30 , 20C99 , 20G05 , 22E45

Rights: Copyright © 2009 Duke University Press


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Vol.149 • No. 3 • 15 September 2009
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