Unitary Dual 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. Let $Gal(E:F)=\{1,\sigma \}.$ We write $\sigma (x)=\overline{x}\forall x\in E.$ Let $E^{\ast }:=E\setminus \{0\}$ and let $E^1:=\{x\in E\mid x\overline{x}=1\}$. $U(5)$ has three proper standard Levi subgroups, the minimal Levi subgroup $M_0\cong E^{\ast }\times E^{\ast }\times E^1$ and the two maximal Levi subgroups $M_1\cong GL(2,E)\times E^1$ and $M_2\cong E^{\ast }\times U(3)$. 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.

27 pages