Classification of ${\mathrm{GL}}_n(\mathbb{C})$-Representations Distinguished by ${\mathrm{GL}}_n(\mathbb{R})$

Abstract

This paper provides a complete classification of ${\mathrm{GL}}_n(\mathbb{R})$-distinguished irreducible representations of ${\mathrm{GL}}_n(\mathbb{C})$ when the representations are either generic or unitary. Additionally, for each such ${\mathrm{GL}}_n(\mathbb{R})$-distinguished representation, we explicitly construct the associated period and prove its non-vanishing on the distinguished minimal $K$-type. Furthermore, we offer some applications to the branching problem using theta correspondence.

We add some new results and modify some proofs