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On the classification of unitary highest weight modules in the exceptional cases

Pavle Pandžić, Ana Prlić, Gordan Savin, Vladimír Souček, Vít Tuček

TL;DR

This work completes the unitary highest weight module classification for the universal covers of the Hermitian groups $SO_e(2,n)$, $E_{6(-14)}$, and $E_{7(-25)}$ by deploying Dirac-inequality methods, PRV components, and Schmid modules, and by exploiting exceptional theta correspondences. It provides explicit unitarity criteria for modules with varied infinitesimal characters, including discrete, Wallach-type, and non-endpoint families, and it reveals intricate structure through dual-pair and minimal-representation constructions, notably in the $E_7$-driven framework. The analysis extends the program begun in prior work to exceptional groups, delivering a comprehensive map of the unitary dual in these Hermitian cases and highlighting connections to theta correspondences and minimal representations. Overall, the results refine our understanding of unitarizable highest-weight modules for exceptional Lie groups and illuminate the role of Dirac operators and dual-pair techniques in their classification.

Abstract

In our previous paper, we gave a complete classification of the unitary highest weight modules for the universal covers of the Lie groups $Sp(2n, \mathbb{R}), SO^{*}(2n)$ and $SU(p, q)$, using the Dirac inequality and the so called PRV product. In this paper, we complete the classification of the unitary highest weight modules for the remaining cases; i.e., universal covers of the Lie groups $SO_{e}(2, n)$, $E_{6(-14)}$ and $E_{7(-25)}$. We also describe unitary highest weight modules with given infinitesimal characters.

On the classification of unitary highest weight modules in the exceptional cases

TL;DR

This work completes the unitary highest weight module classification for the universal covers of the Hermitian groups , , and by deploying Dirac-inequality methods, PRV components, and Schmid modules, and by exploiting exceptional theta correspondences. It provides explicit unitarity criteria for modules with varied infinitesimal characters, including discrete, Wallach-type, and non-endpoint families, and it reveals intricate structure through dual-pair and minimal-representation constructions, notably in the -driven framework. The analysis extends the program begun in prior work to exceptional groups, delivering a comprehensive map of the unitary dual in these Hermitian cases and highlighting connections to theta correspondences and minimal representations. Overall, the results refine our understanding of unitarizable highest-weight modules for exceptional Lie groups and illuminate the role of Dirac operators and dual-pair techniques in their classification.

Abstract

In our previous paper, we gave a complete classification of the unitary highest weight modules for the universal covers of the Lie groups and , using the Dirac inequality and the so called PRV product. In this paper, we complete the classification of the unitary highest weight modules for the remaining cases; i.e., universal covers of the Lie groups , and . We also describe unitary highest weight modules with given infinitesimal characters.

Paper Structure

This paper contains 16 sections, 14 theorems, 142 equations.

Table of Contents

  1. Introduction
  2. Classification for $\mathfrak{so}(2, 2n-2), n\geq 3$
  3. Classification for $\mathfrak{so}(2, 2n-1), \, n \geq 2$
  4. Classification for $\mathfrak{e}_{6(-14)}$
  5. Classification for $\mathfrak{e}_{7(-25)}$
  6. An $E_6$ dual pair
  7. Minimal representation of $E_7$
  8. A branching
  9. Another dual pair
  10. An exceptional theta correspondence
  11. Application
  12. Unitary highest weight modules with fixed infinitesimal character
  13. The case of $\mathfrak{so}(2, 2n - 2), \, n \geq 3$.
  14. The case of $\mathfrak{so}(2, 2n - 1), \, n \geq 2$.
  15. The case of $\mathfrak{e}_6$.
  16. ...and 1 more sections

Key Result

Proposition 1.3

EHW With the notation as above, $L(\lambda)$ is unitary if and only if the inequality ehw di holds strictly for any $K$-type $\mu \neq \lambda$ of $L(\lambda)$.

Theorems & Definitions (22)

  • Proposition 1.3
  • Corollary 1.4
  • Lemma 2.4
  • Theorem 2.6
  • proof
  • Lemma 3.3
  • Theorem 3.5
  • Theorem 4.1
  • Theorem 5.1
  • Lemma 6.1
  • ...and 12 more