How does the restriction of representations change under translations? A story for the general linear groups and the unitary groups

TL;DR

This paper introduces a new framework for understanding how symmetry breaking in representations of general linear groups changes under translations, using fences for interleaving patterns to refine Weyl chamber walls.

Contribution

It develops the concept of fences for interleaving patterns, providing a refined analysis of how translation functors affect symmetry breaking between group representations.

Findings

Multiplicity remains constant unless fences are crossed.

Established a new vanishing theorem for symmetry breaking.

Presented a non-vanishing theorem for period integrals.

Abstract

We present a new approach to symmetry breaking for pairs of real forms of $(GL(n, \mathbb{C}), GL(n-1, \mathbb{C}))$. Translation functors are powerful tools for studying families of representations of a single reductive group $G$. However, when applied to a pair of groups $G \supset G'$, they can significantly alter the nature of symmetry breaking between the representations of $G$ and $G'$, even within the same Weyl chamber of the direct product group $G \times G'$. We introduce the concept of "fences for the interleaving pattern", which provides a refinement of the usual notion of walls of Weyl chambers. We then establish a theorem stating that the multiplicity remains constant unless these "fences" are crossed, together with a new general vanishing theorem for symmetry breaking. These general results are illustrated with examples involving both tempered and non-tempered representations. In addition, we present a new non-vanishing theorem for period integrals for pairs of reductive symmetric spaces, which is further strengthened by this approach.