The generalized linear period

TL;DR

This paper investigates the invariance properties of certain functions and linear functionals related to the linear period problem for the pair (GL_{2p+1}(F), GL_{p}(F)×GL_{p+1}(F)) over non-archimedean local fields, establishing new invariance results.

Contribution

It proves that bi-invariant generalized functions are invariant under transpose and that certain invariant linear functionals are also invariant, advancing understanding of the linear period problem.

Findings

Bi-invariant generalized functions are transpose-invariant.

H_{p,p+1}-invariant linear functionals are also invariant.

Results hold for non-archimedean local fields.

Abstract

Let $F$ be a non-archimedean local field of characteristic zero. We study the linear period problem for the pair $(G,H_{p,p+1})=(GL_{2p+1}(F), GL_{p}(F)\times GL_{p+1}(F))$ and we prove that any bi-$(H_{p,p+1},\mu)$-invariant generalized function on $G$ is invariant under the matrix transpose when \mu is 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 $(0,\cdots,0,1)$.