(GL(2n,C),SP(2n,C)) is a Gelfand Pair

2008/05/19 by Eitan Sayag, Sayag, Eitan
Mathematics · #20G05 #20G25 #22E #22E45 #46F99 #Advanced Algebra and Geometry #Algebraic Geometry and Number Theory #FOS: Mathematics #Finite Group Theory Research #Number Theory (math.NT) #Representation Theory (math.RT)

paper · pdf · doi:10.48550/arxiv.0805.2625

openalex publication_date 2008/05/19 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We prove that (GL_2n(C),Sp_2n(C)) is a Gelfand pair. More precisely, we show that for an irreducible smooth admissible Frechet representation (π,E) of GL_2n(C) the space of continuous functionals Hom_Sp_2n(\cc)(E,C) is at most one dimensional. For this we show that any distribution on GL_2n(C) invariant with respect to the double action Sp_2n(C) × Sp_2n(C) is transposition invariant. Such a result was previously proven for p-adic fields by M. Heumos and S. Rallis.

Citations

• Generalized Harish-Chandra descent and applications to Gelfand pairs

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