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Dirac series of $E_{8(-24)}$

Yi-Hao Ding, Chao-Ping Dong, Chengyu Du, Yong-Zhi Luan, Liang Yang

TL;DR

This work achieves a near-complete classification of Dirac series for the real exceptional group $E_{8(-24)}$, combining refined bounds on the Helgason–Johnson inequality with Vogan's fundamental parallelepiped framework. By leveraging atlas and cohomological induction from Levi subgroups, it dissects Dirac series into 211 FS-scattered representations and 3766 strings, organized via spin-lowest $K$-types and $u$-small convex hulls. A sharpened HP-integral criterion and a detailed Certs/Ω-based pruning led to a small set of unitary candidates, including the minimal representation, and provided a robust atlas-based computational pipeline. The results deepen understanding of the unitary dual for this quaternionic-type form and have potential implications for automorphic construction via Dirac series, while the appendix offers a complete, citable catalog of the FS-scattered pieces.

Abstract

This paper classifies the Dirac series of $E_{8(-24)}$, the linear quaternionic real form of complex $E_8$. One tool for us is a further sharpening of the Helgason-Johnson bound in 1969. Our calculation continues to support Vogan's fundamental parallelepiped conjecture.

Dirac series of $E_{8(-24)}$

TL;DR

This work achieves a near-complete classification of Dirac series for the real exceptional group , combining refined bounds on the Helgason–Johnson inequality with Vogan's fundamental parallelepiped framework. By leveraging atlas and cohomological induction from Levi subgroups, it dissects Dirac series into 211 FS-scattered representations and 3766 strings, organized via spin-lowest -types and -small convex hulls. A sharpened HP-integral criterion and a detailed Certs/Ω-based pruning led to a small set of unitary candidates, including the minimal representation, and provided a robust atlas-based computational pipeline. The results deepen understanding of the unitary dual for this quaternionic-type form and have potential implications for automorphic construction via Dirac series, while the appendix offers a complete, citable catalog of the FS-scattered pieces.

Abstract

This paper classifies the Dirac series of , the linear quaternionic real form of complex . One tool for us is a further sharpening of the Helgason-Johnson bound in 1969. Our calculation continues to support Vogan's fundamental parallelepiped conjecture.
Paper Structure (15 sections, 5 theorems, 51 equations, 2 figures, 29 tables)

This paper contains 15 sections, 5 theorems, 51 equations, 2 figures, 29 tables.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. A brief introduction to the software atlas
  4. Presenting the infinite set $\widehat{G}^d$
  5. The structure of $E_{8(-24)}$
  6. Positive root systems and the set $W(\mathfrak{g}, \mathfrak{t}_f)^1$
  7. Lambda norm and infinitesimal character
  8. Spin norm and Dirac inequality
  9. Vogan pencil and the u-small convex hull
  10. Sharpening the the Helgason-Johnson bound for $E_{8(-24)}$
  11. Dirac series of $E_{8(-24)}$
  12. FS-scattered representations of $E_{8(-24)}$
  13. Counting the strings in $\widehat{E_{8(-24)}}^d$
  14. Examples
  15. Appendix

Key Result

Theorem 1.1

(HP) Let $\pi$ be an irreducible $(\mathfrak{g}, K)$ module with infinitesimal character $\Lambda$. Suppose that $H_D(\pi)$ is non-zero. Then for any $\widetilde{K}$-type $E_{\gamma}$ in $H_D(\pi)$, there exists $w\in W(\mathfrak{g}, \mathfrak{t}_f)$ such that $w(\gamma+\rho_c)=\Lambda$.

Figures (2)

  • Figure 1: The Vogan diagram for $E_{8(-24)}$
  • Figure 2: The Dynkin diagram for $\Delta^+(\mathfrak{k}, \mathfrak{t}_f)$

Theorems & Definitions (14)

  • Theorem 1.1
  • Theorem 1.2
  • Example 3.1
  • Remark 3.2
  • Proposition 4.1
  • Example 4.2
  • Proposition 4.3
  • proof
  • Lemma 5.1
  • Example 6.1
  • ...and 4 more