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.

令F为非阿基米德本地域。本文研究了从\（{\ mathrm {GL}} _ {n + 1}（F）\）到\（{\ mathrm {GL}} _ n（F）\）的不可约光滑表示的同源性。主要结果表明\（{\ mathrm {GL}} _ {n + 1}（F）\）的不可约光滑表示的每个Bernstein分量都限于\（{\ mathrm {GL}} _ n（F）\ ）是不可分解的。我们还对从\（{\ mathrm {GL}} _ {n + 1}（F）\）限制为\（{\ mathrm {GL}} _ n（F）\）时投射的所有不可约表示进行分类。。我们研究的主要工具是左右导数的概念，它扩展了先前与Gordan Savin的工作。作为副产品，我们还确定相反方向的分支规律。