Unitary representations of p-adic U(5)

Abstract

We study the parabolically induced complex representations of the unitary group in 5 variables, U (5), defined over a p-adic field. Let F be a p-adic field. Let E : F be a field extension of degree two. U (5) has three proper standard Levi subgroups, the minimal Levi subgroup M 0 ∼ = E * × E * × E 1 and the two maximal Levi subgroups M 1 ∼ = GL(2, E) × E 1 and M 2 ∼ = E * × U (3). We consider representations induced from M 0 , representations induced from non-cuspidal, not fully-induced representations of M 1 and M 2 and representations induced from cuspidal representations of M 1. We determine the points and lines of reducibility and the irreducible subquotients of these representations. Further we describe-except several particular cases-the unitary dual in terms of Langlands quotients. Contents 1. Introduction 2. Definitions 3. Previous Results 4. The representations of U (3) 4.1. The irreducible representations of U (3) 4.2. The irreducible unitary representations of U (3) 5. The irreducible representations of U (5) 5.1. Levi decomposition for U (5) 5.2. Representations with cuspidal support in M 0 , fully-induced 5.3. Representations induced from M 1 and M 2 , with cuspidal support in M 0 6. 'Special' Reducibility points of representations of U (5) with cuspidal support in M 0