Homological branching law for (GLn+1(F),GLn(F)): projectivity and indecomposability

Abstract

Let F be a non-Archimedean local field. This paper studies homological properties of irreducible smooth representations restricted from GL n+1(F) to GL n(F). A main result shows that each Bernstein component of an irreducible smooth representation of GL n+1(F) restricted to GL n(F) is indecomposable. We also classify all irreducible representations which are projective when restricting from GL n+1(F) to 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.