Unitary Dual of p-adic U(5)

TL;DR

This paper classifies the unitary dual of the p-adic group U(5) by analyzing parabolically induced representations from various Levi subgroups, providing a detailed description in terms of Langlands-quotients.

Contribution

It offers a comprehensive description of the unitary dual of U(5) over p-adic fields, including induced representations from different Levi subgroups and their Langlands-quotients.

Findings

Complete classification of the unitary dual of U(5)

Descriptions of induced representations from various Levi subgroups

Most cases described in terms of Langlands-quotients

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. Let $Gal(E : F ) = \{ 1, \sigma \}.$ We write $ \sigma(x) = \overline{x} \; \forall x \in E. $ Let $ E^* := E \setminus \{ 0 \} $ and let $ E^1 := \{x \in E \mid x \overline{x} = 1 \}$. $U(5) $ has three proper standard Levi subgroups, the minimal Levi subgroup $ M_0 \cong E^* \times E^* \times E^1 $ and the two maximal Levi subgroups $ M_1 \cong GL(2, E) \times E^1 $ and $ M_2 \cong E^* \times U(3)$. We consider representations induced from the minimal Levi subgroup $ M_0, $ representations induced from non-cuspidal, not fully-induced representations of the two maximal Levi subgroups $ M_1 $ and $ M_2, $ and representations induced from cuspidal representations of $ M_1.$ We describe - except several particular cases - the unitary dual in terms of Langlands-quotients.