Duke Mathematical Journal

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

Avraham Aizenbud, Dmitry Gourevitch, and Eitan Sayag

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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

Article information

Duke Math. J., Volume 149, Number 3 (2009), 509-567.

First available in Project Euclid: 24 August 2009

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Zentralblatt MATH identifier

Primary: 22E50: Representations of Lie and linear algebraic groups over local fields [See also 20G05] 46F10: Operations with distributions
Secondary: 20C99: None of the above, but in this section 20G05: Representation theory 22E45: Representations of Lie and linear algebraic groups over real fields: analytic methods {For the purely algebraic theory, see 20G05} 14L24: Geometric invariant theory [See also 13A50] 14L30: Group actions on varieties or schemes (quotients) [See also 13A50, 14L24, 14M17]


Aizenbud, Avraham; Gourevitch, Dmitry; Sayag, Eitan. Generalized Harish-Chandra descent, Gelfand pairs, and an Archimedean analog of Jacquet-Rallis's theorem. Duke Math. J. 149 (2009), no. 3, 509--567. doi:10.1215/00127094-2009-044. https://projecteuclid.org/euclid.dmj/1251120011

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