Unicity of types for supercuspidal representations of $ {\rm GL}_N$

Abstract

Let F be a non-Archimedean local field, with the ring of integers oF. Let G = GLN(F), K = GLN (oF), and π be a supercuspidal representation of G. We show that there exists a unique irreducible smooth representation τ of K, such that the restriction to K of a smooth irreducible representation π′ of G contains τ if and only if π′ is isomorphic to π ⊗ χ ° det, where χ is an unramified quasicharacter of F×. Moreover, we show that π contains τ with the multiplicity 1. As a corollary we obtain a kind of inertial local Langlands correspondence. 2000 Mathematics Subject Classification 22E50.