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

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