Generic Smooth Representations

  • Alexandre Pyvovarov

    Morningside Center of Mathematics, No. 55 Zhongguancun Donglu, Academy of Mathematics and Systems Science, Beijing, Haidian District, 100190 China
Generic Smooth Representations cover
Download PDF

This article is published open access.


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