Homological aspects of branching laws

TL;DR

This paper explores the homological properties of how representations of p-adic groups restrict to subgroups, revealing that such restrictions are often projective modules with simple structures, especially in Bernstein blocks.

Contribution

It highlights the projective nature of restricted representations in Bernstein blocks and updates previous work with recent developments in the field.

Findings

Restriction of generic representations is often projective in Bernstein blocks.

Restricted representations have a simple structure when projective.

The work builds on and emphasizes recent research by Chan and Chan-Savin.

Abstract

In this mostly expository article, we consider certain homological aspects of branching laws for representations of a group restricted to its subgroups in the context of $p$-adic groups. We follow our earlier paper, ICM 2018 proceedings, updating it with some more recent works. In particular, following Chan and Chan-Savin, see many of their papers listed in the bibliography, we have emphasized in this work that the restriction of a (generic) representation $\pi$ of a group $G$ to a closed subgroup $H$ (most of the paper is written in the context of GGP) turns out to be a projective representation on most Bernstein blocks of the category of smooth representations of $H$. Further, once $\pi|_H$ is a projective module in a particular Bernstein block, it has a simple structure.