Homological branching law for (GL_n+1(F), GL_n(F)): projectivity and indecomposability

2019/05/05 by Kei Yuen Chan, Chan, Kei Yuen
Mathematics · #11F70 #20C08 #22E50 #Advanced Algebra and Geometry #Algebraic Geometry and Number Theory #Algebraic structures and combinatorial models #FOS: Mathematics #Number Theory (math.NT) #Representation Theory (math.RT) #math.NT #math.RT #msc:11F70 #msc:20C08 #msc:22E50

paper · pdf · doi:10.48550/arxiv.1905.01668

27 pages, v2: rearranging few materials, minor additions and changes, v3: 42 pages, expanded version

openalex publication_date 2019/05/05 · arxiv created 2020/09/28 · arxiv updated 2020/09/29 · openalex created_date 2022/07/29 · openalex updated_date 2026/07/28

Abstract

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