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

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 $G\supset G^{\prime }$, they can significantly alter the nature of symmetry breaking between the representations of $G$ and $G^{\prime }$, even within the same Weyl chamber of the direct product group $G\times G^{\prime }$. 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.

50 pages