On the Stability and Gelfand Property of Symmetric Pairs

2015/11/04 by Shachar Carmeli, Carmeli, Shachar · 1 citation
Mathematics · #20G05 #20G25 #Advanced Algebra and Geometry #Algebraic Geometry and Number Theory #FOS: Mathematics #Geometry and complex manifolds #Representation Theory (math.RT)

paper · pdf · doi:10.48550/arxiv.1511.01381

openalex publication_date 2015/11/04 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

A symmetric pair of reductive groups (G,H,θ) is called stable, if every closed double coset of H in G is preserved by the anti-involution g↦ θ(g^-1). In this paper, we develop a method to verify the stability of symmetric pairs over local fields of characteristic 0 (Archimedean and p-adic), using non-abelian group cohomology. Combining our method with results of Aizenbud and Gourevitch, we classify the Gelfand pairs among the pairs amp;(SL_n(F), (GL_k(F) × GL_{n - k}(F)) ∩ SL_n(F)), (U(B₁ ⊕ B₂),U(B₁) × U(B₂)),
amp;(GL_n(F),O(B)), (GL_n(F),U(B)), (GL_2n(F), GL_n(E)),(SL_2n(F), SL_n(E)), and the pair (O(B₁ ⊕ B₂),O(B₁) × O(B₂)) in the real case.

Cited by

• Bound on multiplicities of symmetric pairs over p-adic fields

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