The generalized linear period

Abstract

Let $F$ be a non-archimedean local field of characteristic zero. We study the linear period problem for the pair $(G,H_{p,p+1})=(G{L}_{2p+1}(F),G{L}_p(F)\times G{L}_{p+1}(F))$ and we prove that any bi-$(H_{p,p+1},\mu )$-invariant generalized function on $G$ is invariant under the matrix transpose when μis a good character. We also show that any $P\cap H_{p,p+1}$-invariant linear functional on an $H_{p,p+1}$-distinguished irreducible smooth representation of $G$ is also $H_{p,p+1}$-invariant when F is nonarchimedean, where $P$ is a standard mirabolic subgroup of $G$ with last row vector $(0,\cdots ,0,1)$.

References in corpus (1)

• (GL(n+1,F),GL(n,F)) is a Gelfand pair for any local field F

Cited by in corpus (1)

• The action of a mirabolic subgroup on a symmetric variety