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Dirac inequality for highest weight Harish-Chandra modules II

Pavle Pandžić, Ana Prlić, Vladimír Souček, Vít Tuček

TL;DR

The paper addresses the problem of classifying unitarizable highest weight modules for a noncompact Hermitian simple Lie group of type $\mathfrak{e}_6$ or $\mathfrak{e}_7$ using the Dirac inequality. It develops the Dirac-operator framework on $(\mathfrak{g},K)$, presents the Dirac square formula, and derives explicit Dirac inequalities for the basic Schmid modules of $\mathfrak{e}_6$ and $\mathfrak{e}_7$, aided by generalized PRV components. The results organize the unitarity criteria into finite case analyses (three basic cases for $\mathfrak{e}_6$ and three for $\mathfrak{e}_7$), enabling determinations of when $L(\lambda)$ is unitary based on strict vs non-strict inequalities. These findings provide concrete, checkable criteria for unitarity in the representation theory of exceptional Hermitian groups and advance the classification program for real reductive Lie groups. The work sets the stage for further development (to be completed in PPST2) and illustrates the effectiveness of Dirac-inequality methods in this context.

Abstract

Let $G$ be a connected simply connected noncompact exceptional simple Lie group of Hermitian type. In this paper, we work with the Dirac inequality which is a very useful tool for the classification of unitary highest weight modules.

Dirac inequality for highest weight Harish-Chandra modules II

TL;DR

The paper addresses the problem of classifying unitarizable highest weight modules for a noncompact Hermitian simple Lie group of type or using the Dirac inequality. It develops the Dirac-operator framework on , presents the Dirac square formula, and derives explicit Dirac inequalities for the basic Schmid modules of and , aided by generalized PRV components. The results organize the unitarity criteria into finite case analyses (three basic cases for and three for ), enabling determinations of when is unitary based on strict vs non-strict inequalities. These findings provide concrete, checkable criteria for unitarity in the representation theory of exceptional Hermitian groups and advance the classification program for real reductive Lie groups. The work sets the stage for further development (to be completed in PPST2) and illustrates the effectiveness of Dirac-inequality methods in this context.

Abstract

Let be a connected simply connected noncompact exceptional simple Lie group of Hermitian type. In this paper, we work with the Dirac inequality which is a very useful tool for the classification of unitary highest weight modules.
Paper Structure (4 sections, 14 theorems, 209 equations, 2 tables)

This paper contains 4 sections, 14 theorems, 209 equations, 2 tables.

Table of Contents

  1. Introduction
  2. Dirac inequalities
  3. Dirac inequality for $\mathfrak{e}_6$
  4. Dirac inequality for $\mathfrak{e}_7$

Key Result

Theorem 1.1

Let us assume that $\mathfrak{g}, \rho, \lambda, s$ are as in tables tab:table1 and tab: table2. (1) Let $s_0$ be a Schmid module such that the strict Dirac inequality holds for any Schmid module $s$ of strictly lower level than $s_0$, and such that Then $L(\lambda)$ is not unitary. (2) If holds for all Schmid modules $s$, then $N(\lambda)$ is irreducible and unitary.

Theorems & Definitions (26)

  • Theorem 1.1
  • Lemma 1.4
  • Theorem 2.1
  • proof
  • Theorem 2.2
  • proof
  • Lemma 2.3
  • proof
  • Lemma 2.4
  • proof
  • ...and 16 more