On the Stability and Gelfand Property of Symmetric Pairs

Abstract

A symmetric pair of reductive groups $(G,H,\theta )$ is called stable, if every closed double coset of $H$ in $G$ is preserved by the anti-involution $g\mapsto \theta (g^{-1})$. 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 \begin{align*} &(SL_n(F), (GL_k(F) \times GL_{n - k}(F)) \cap SL_n(F)), (U(B_1 \oplus B_2),U(B_1) \times U(B_2)),\\ &(GL_n(F),O(B)), (GL_n(F),U(B)), (GL_{2n}(F), GL_n(E)),(SL_{2n}(F), SL_n(E)), \end{align*} and the pair $(O(B_1\oplus B_2),O(B_1)\times O(B_2))$ in the real case.

47 pages, v3: made major changes from previous version. comments are welcome!

References in corpus (3)

• Galois Cohomology of Real Groups
• A partial analog of integrability theorem for distributions on p-adic spaces and applications
• Invariant distributions on non-distinguished nilpotent orbits with application to the Gelfand property of (GL(2n,R),Sp(2n,R))

Cited by in corpus (3)

• The generalized linear period
• On the Gelfand property for complex symmetric pairs
• The action of a mirabolic subgroup on a symmetric variety