On $A$-parameters containing unitary lowest weight representations of $\mathrm{U}(p, q)$

TL;DR

This paper classifies all Arthur packets containing irreducible unitary lowest weight representations of the real unitary group U(p, q), providing explicit formulas for their lowest K-types and demonstrating uniqueness within each packet.

Contribution

It determines all Arthur packets with irreducible unitary lowest weight representations of U(p, q), including non-scalar cases, using Barbasch-Vogan parametrization and Trapa's algorithm.

Findings

Arthur packets contain at most one such lowest weight representation

Explicit formulas for lowest K-types are provided

Classification includes non-scalar cases

Abstract

In this paper, we determine all the Arthur packets containing an irreducible unitary lowest weight representation $\pi$ of real unitary group $G = \mathrm{U}(p, q)$, including non-scalar cases. Our methods are the Barbasch-Vogan parametrization of representations of $G$ and Trapa's algorithm to calculate the cohomologically induced representations. In particular, we show that an Arthur packet has at most one irreducible unitary lowest weight representation of $G$. As a consequence, if an irreducible unitary lowest weight representation $\pi$ exists in the Arthur packet of $\psi$, we give an explicit formula of the lowest $K$-type of $\pi$.