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.
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