On reducibility of induced representations of odd unitary groups: the depth zero case

TL;DR

This paper investigates when certain induced representations of odd unitary groups over p-adic fields are reducible, focusing on depth zero supercuspidal representations and employing Hecke algebra techniques.

Contribution

It provides a criterion for reducibility of parabolic inductions in odd unitary groups using Hecke algebra methods for depth zero supercuspidal representations.

Findings

Determines reducibility conditions for induced representations

Uses Hecke algebra techniques for analysis

Focuses on depth zero supercuspidal representations

Abstract

We study a problem concerning parabolic induction in certain $p$-adic unitary groups. More precisely, for $E/F$ a quadratic extension of $p$-adic fields the associated unitary group $G=\mathrm{U}(n,n+1)$ contains a parabolic subgroup $P$ with Levi component $L$ isomorphic to $\mathrm{GL}_n(E) \times \mathrm{U}_1(E)$. Let $\pi$ be an irreducible supercuspidal representation of $L$ of depth zero. We use Hecke algebra methods to determine when the parabolically induced representation $\iota_P^G \pi$ is reducible.