Irreducible dual of p-adic U(5)

TL;DR

This paper analyzes the reducibility and subquotients of parabolically induced representations of the p-adic unitary group U(5), identifying key points and lines where reducibility occurs.

Contribution

It provides a detailed classification of reducibility points and subquotients for induced representations of U(5) over p-adic fields, including from various Levi subgroups.

Findings

Identifies reducibility points for induced representations from different Levi subgroups.

Classifies irreducible subquotients arising from these induced representations.

Maps the structure of reducibility lines and points in the representation space.

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 $ M_0 $ and from non-cuspidal, not fully-induced representations of $ M_1 $ and $ M_2. $ We determine the points and lines of reducibility and the irreducible subquotients of these representations.