Generic Smooth Representations

Abstract

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