A generalization of Duflo's conjecture

TL;DR

This paper extends Duflo's conjecture to include non-discrete series representations, providing a geometric framework for understanding restriction laws of unitary representations of GL(n) to the mirabolic subgroup.

Contribution

It offers a unified geometric description of restriction laws for a broader class of representations, generalizing previous conjectures.

Findings

Unified geometric description of restriction laws

Extension of Duflo's conjecture to non-discrete series

Application to representations of GL(n) over real and complex fields

Abstract

In this article, we generalize Duflo's conjecture to understand the branching laws of non-discrete series. We give a unified description on the geometric side about the restriction of an irreducible unitary representation $\pi$ of $\mathrm{GL}_n(k)$, $k=\mathbb{R}$ or $\mathbb{C}$, to the mirabolic subgroup, where $\pi$ is attached to a certain kind of coadjoint orbit.