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

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.

27 pages, v2: rearranging few materials, minor additions and changes, v3: 42 pages, expanded version