# Homological branching law for $(\mathrm{GL}_{n+1}(F), \mathrm{GL}_n(F))$: projectivity and indecomposability

## TL;DR

This paper investigates the homological properties of irreducible smooth representations of general linear groups over non-Archimedean fields, establishing indecomposability of Bernstein components and classifying projective restrictions.

## Contribution

It introduces a new framework using derivatives to analyze the restriction of representations, proving indecomposability and classifying projective cases in the homological branching law.

## Key findings

- Bernstein components are indecomposable upon restriction.
- All projective irreducible representations are classified.
- The branching law is determined in both directions.

## Abstract

Let $F$ be a non-Archimedean local field. This paper studies homological properties of irreducible smooth representations restricted from $\mathrm{GL}_{n+1}(F)$ to $\mathrm{GL}_n(F)$. A main result shows that each Bernstein component of an irreducible smooth representation of $\mathrm{GL}_{n+1}(F)$ restricted to $\mathrm{GL}_n(F)$ is indecomposable. We also classify all irreducible representations which are projective when restricting from $\mathrm{GL}_{n+1}(F)$ to $\mathrm{GL}_n(F)$. A main tool of our study is a notion of left and right derivatives, extending some previous work joint with Gordan Savin. As a by-product, we also determine the branching law in the opposite direction.