On Higher Multiplicity upon Restriction from GL(n) to GL(n-1)

2023/08/13 by Mohammed Saad Qadri, Qadri, Mohammed Saad
Mathematics · #11F70(primary) #22E55(secondary) #Advanced Algebra and Geometry #Advanced Topics in Algebra #Algebraic structures and combinatorial models #FOS: Mathematics #Representation Theory (math.RT)

paper · pdf · doi:10.48550/arxiv.2308.06698

openalex publication_date 2023/08/13 · openalex created_date 2023/08/16 · openalex updated_date 2026/07/28

Abstract

Let F be a non-archimedean local field. Let Π be a principal series representation of GL_n(F) induced from an irreducible cuspidal representation of a Levi subgroup. When π is an essentially square integrable representation of GL_n-1(F) we prove that Hom_GL_n-1(Π,π) = ℂ and Extⁱ_GL_n-1(Π,π) = 0 for all integers i≥ 1, with exactly one exception (up to twists), namely, when Π= ν^-((n-1)/(2)) × ν^-((n-3)/(2)) × … × ν^((n-1)/(2)) and π is the Steinberg. When Π= ν^-((n-1)/(2)) × ν^-((n-3)/(2)) × … × ν^((n-1)/(2)) and π is the Steinberg of GL_n-1(F), then dim Hom_GL_n-1(F)(Π,π)=n. We also exhibit specific principal series for which each of the intermediate multiplicities 2, 3, ⋯, (n-1) are attained. Along the way, we also give a complete list of those irreducible non-generic representations of GL_n(F) that have the Steinberg of GL_n-1(F) as a quotient upon restriction to GL_n-1(F). We also show that there do not exist non-generic irreducible representations of GL_n(F) that have the generalized Steinberg as a quotient upon restriction to GL_n-1(F).

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