Generic Smooth Representations

摘要

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 (ext{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.