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On corank 4 unitary representations of classical groups

Baiying Liu, Chi-Heng Lo, Brian Wen

Abstract

In this paper, we explicitly classify the corank 4 unitary representations of symplectic or split odd special orthogonal groups over non-Archimedean local fields of characteristic zero, by classifying Arthur representations of corank 4 and verifying the corresponding unitary dual conjecture recently proposed by Hazeltine-Jiang-Liu-Lo-Zhang in [HJLLZ24].

On corank 4 unitary representations of classical groups

Abstract

In this paper, we explicitly classify the corank 4 unitary representations of symplectic or split odd special orthogonal groups over non-Archimedean local fields of characteristic zero, by classifying Arthur representations of corank 4 and verifying the corresponding unitary dual conjecture recently proposed by Hazeltine-Jiang-Liu-Lo-Zhang in [HJLLZ24].
Paper Structure (29 sections, 213 theorems, 425 equations, 5 figures)

This paper contains 29 sections, 213 theorems, 425 equations, 5 figures.

Table of Contents

  1. Introduction
  2. Notations and preliminary results
  3. Local Langlands classification
  4. Arthur's parameterization of the tempered spectrum
  5. Supercuspidal representations and reducibility points
  6. Representations of good parity and critical type
  7. Local Arthur packets and reduction to good parity
  8. Extended multi-segments and operators
  9. Preservation of irreducibility and unitarizability
  10. The unitary dual conjecture
  11. Classification of corank 3 tempered representations of good parity
  12. Classification of corank 4 non-tempered representations of good parity
  13. involving supercuspidal representations
  14. Classification of corank 4 non-tempered representations of good parity
  15. The case
  16. ...and 14 more sections

Key Result

Theorem 1.2

Figures (5)

  • Figure 1: Unitarizability for $\pi_{(x,y)}$ (case $\alpha = 3$)
  • Figure 2: Unitarizability for $\pi_{(x,y)}$ (case $\alpha=2$)
  • Figure 3: Unitarizability for $\pi_{(x,y)}$ (case $\alpha=\frac{3}{2}$)
  • Figure 4: Unitarizability for $\pi_{(x,y)}$ (case $\alpha=1$)
  • Figure 5: Unitarizability for $\pi_{(x,y)}$ (case $\alpha=\frac{1}{2}$)

Theorems & Definitions (346)

  • Conjecture 1.1: Unitary Dual Conjecture, HJLLZ24, see also Conjecture \ref{['unitary dual conjecture']}
  • Theorem 1.2
  • Theorem 1.3
  • Theorem 2.1: Art13
  • Theorem 2.2: Moe11a, Xu17a
  • Definition 2.3
  • Definition 2.4
  • Conjecture 2.5: HJLLZ24
  • Theorem 2.6: AM25HLL25
  • Definition 2.7
  • ...and 336 more