The action of a mirabolic subgroup on a symmetric variety

TL;DR

This paper proves that for certain representations of GL(2n,F), invariance under a mirabolic subgroup implies invariance under the entire GL(n,E), revealing a symmetry property in the representation theory over local fields.

Contribution

It establishes a new invariance property for P-invariant linear functionals on GL(n,E)-distinguished representations of GL(2n,F).

Findings

P-invariant linear functionals are also GL(n,E)-invariant.

The result applies to irreducible smooth admissible representations.

The theorem holds over local fields of characteristic zero.

Abstract

Let F be a local field of character zero. Let E be a quadratic field extension of F. We show that any P-invariant linear functional on a GL(n,E)-distinguished irreducible smooth admissible representation of GL(2n,F) is also GL(n,E)-invariant where P is a mirabolic subgroup of GL(n,E).