TY - JOUR

T1 - Generalized Harish-Chandra descent, Gelfand pairs, and an Archimedean analog of Jacquet-Rallis's theorem

AU - Aizenbud, Avraham

AU - Gourevitch, Dmitry

AU - Sayag, Eitan

PY - 2009/9/1

Y1 - 2009/9/1

N2 - 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.

AB - 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.

UR - http://www.scopus.com/inward/record.url?scp=77950209828&partnerID=8YFLogxK

U2 - 10.1215/00127094-2009-044

DO - 10.1215/00127094-2009-044

M3 - Article

AN - SCOPUS:77950209828

VL - 149

SP - 509

EP - 567

JO - Duke Mathematical Journal

JF - Duke Mathematical Journal

SN - 0012-7094

IS - 3

ER -