Generalized Harish-Chandra descent, Gelfand pairs, and an Archimedean analog of Jacquet-Rallis's theorem
By Avraham Aizenbud, D. Gourevitch, Eitan Sayag. 2008
102 citations recorded by the source corpus.
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.
Subfields: Algebra and Number Theory, Geometry and Topology, Mathematical Physics
View Paper: https://www.semanticscholar.org/paper/2fc1abe4575e75ae4dff2edb1fa00509882b8437
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Citations in the corpus
- (GLn+1(F), GLn(F)) is a Gelfand pair for any local field F (2007, methodology): “ext theorem shows that in certain cases it is enough to show that there are no equivariant Schwartz distributions. This will allow us to use Fourier transform. We will need the following theorem from [AG3], Theorem 4.0.2. Theorem 4.2.4. Let a real reductive group Gact on a smooth affine real algebraic”
- On linear periods (2013, methodology): “We refer the reader to [1] and [2] for more details.”