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

TL;DR

This paper classifies irreducible representations of _n(\u00a0) distinguished by _n(\u00a0) for generic and unitary cases, constructs associated periods, and explores applications to branching problems via theta correspondence.

Contribution

It provides a complete classification of _n() representations distinguished by _n(), explicitly constructs periods, and applies results to branching problems using theta correspondence.

Findings

Complete classification of distinguished representations.

Explicit construction of associated periods.

Applications to branching problems via theta correspondence.

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.