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

Kayue Daniel Wong

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 $U(p,q)$.

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

Abstract

In this manuscript, we introduce the notion of fundamental cases to study the unitary dual of . 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 .
Paper Structure (18 sections, 19 theorems, 73 equations, 1 algorithm)

This paper contains 18 sections, 19 theorems, 73 equations, 1 algorithm.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Langlands classification of irreducible $(\mathfrak{g},K)$-modules
  4. Hermitian forms and unitarity
  5. Salamanca-Riba and Vogan's conjecture
  6. Vogan's FPP conjecture
  7. Methods for detecting non-unitarity
  8. Parthasarathy's Dirac inequality
  9. Jantzen filtration
  10. Representations of $U(p,q)$
  11. $\lambda_a$-blocks
  12. $\lambda_a$-data and $K$-types
  13. Langlands classification
  14. Fundamental Case
  15. Proof of Theorem \ref{['thm-upq']} - Case (a)
  16. ...and 3 more sections

Key Result

Theorem 1.1

Conjecture 5.7 of SRV98 and the FPP conjecture hold for $G = U(p,q)$.

Theorems & Definitions (53)

  • Theorem 1.1
  • Theorem 2.1: SRV98 Theorem 2.9
  • Theorem 2.2: SRV98 Proposition 4.1(c)
  • Theorem 2.4: KZ76
  • Corollary 2.5
  • proof
  • Definition 2.6
  • Theorem 2.7: SRV98, Proposition 2.10
  • Remark 2.8
  • Definition 2.9
  • ...and 43 more