Generic Smooth Representations

  • Alexandre Pyvovarov

    Morningside Center of Mathematics, No. 55 Zhongguancun Donglu, Academy of Mathematics and Systems Science, Beijing, Haidian District, 100190 China
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Abstract

Let FF be a non-archimedean local field. In this paper we explore genericity of irreducible smooth representations of GLn(F)GL_n(F) by restriction to a maximal compact subgroup KK of GLn(F)GL_n(F). Let (J,λ)(J, \lambda) be a Bushnell-Kutzko type for a Bernstein component Ω\Omega. The work of Schneider-Zink gives an irreducible KK-representation σmin(λ)\sigma_{min}(\lambda), which appears with multiplicity one in IndJKλ\text{Ind}_J^K \lambda. Let π\pi be an irreducible smooth representation of GLn(F)GL_n(F) in Ω\Omega. We will prove that π\pi is generic if and only if σmin(λ)\sigma_{min}(\lambda) is contained in π\pi, in which case it occurs with multiplicity one.

Cite this article

Alexandre Pyvovarov, Generic Smooth Representations. Doc. Math. 25 (2020), pp. 2473–2485

DOI 10.4171/DM/804