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.
Highlights
1.1 Let F be a non-Archimedean local field
The Hecke algebra realization in [13,15] of the projective representations in Theorem 1.1 immediately implies that each Bernstein component of those restricted representation is indecomposable
Definition 4.1 The affine Hecke algebra Hl(q) of type A is an associative algebra over C generated by θ1, . . . , θl and Tw (w ∈ Sl) satisfying the relations: (1) θi θ j = θ j θi ; (2) Tsk θk − θk+1Tsk = (q − 1)θk, where q is a certain prime power and sk is the transposition between the numbers k and k + 1; (3) Tsk θi = θi Tsk, where i = k, k + 1 (4) (Tsk − q)(Tsk + 1) = 0; (5) Tsk Tsk+1 Tsk = Tsk+1 Tsk Tsk+1
Summary
1.1 Let F be a non-Archimedean local field. Let Gn = GLn(F). It is clear that an irreducible representation (except one-dimensional ones) restricted from Gn+1 to Gn cannot be indecomposable as it has more than one non-zero Bernstein components. The Hecke algebra realization in [13,15] of the projective representations in Theorem 1.1 immediately implies that each Bernstein component of those restricted representation is indecomposable. This is a motivation of our study in general case, and precisely we prove: Theorem 1.3 (=Theorem 6.1) Let π be an irreducible representation of Gn+1. In [12], we study some special situations of the parabolic induction that one can obtain certain indecomposability-preserving results, which have applications to the local nontempered Gan–Gross–Prasad conjectures [18]
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