Distinguished representations of $GL(n,\mathbb{C})$

TL;DR

The paper proves that for certain distinguished representations of complex general linear groups, invariance under a real mirabolic subgroup implies invariance under the entire real general linear group, revealing a symmetry property.

Contribution

It establishes a new invariance property for distinguished representations of $GL(n,\mathbb{C})$, linking subgroup invariance to full group invariance.

Findings

Invariance under the real mirabolic subgroup implies $GL_n(\mathbb{R})$-invariance.

The result applies to irreducible, admissible, $GL_n(\mathbb{R})$-distinguished representations.

The proof advances understanding of symmetry properties in representation theory.

Abstract

Let $V$ be a $GL_n(\mathbb{R})$-distinguished, irreducible, admissible representation of $GL_n(\mathbb{C})$. We prove that any continuous linear functional on $V$, which is invariant under the action of the real mirabolic subgroup, is automatically $GL_n(\mathbb{R})$-invariant.