On some conjectures of the unitary dual of $U(p,q)$

TL;DR

This paper introduces fundamental cases to analyze the unitary dual of $U(p,q)$ and proves two longstanding conjectures, advancing understanding of the unitary representations of this group.

Contribution

It defines fundamental cases for $U(p,q)$ and proves Salamanca-Riba and Vogan's conjectures, resolving open questions from 1998 and 2023.

Findings

Proved Salamanca-Riba and Vogan's conjectures for $U(p,q)$

Established the notion of fundamental cases for the unitary dual

Advanced the classification of unitary representations

Abstract

In this manuscript, we introduce the notion of fundamental cases to study the unitary dual of $U(p,q)$. As applications, we prove of a conjecture of Salamanca-Riba and Vogan stated in 1998, as well as the fundamental parallelepiped (FPP) conjecture of Vogan in 2023 for $U(p,q)$.