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

Abstract

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