Dirac inequality for highest weight Harish-Chandra modules I

TL;DR

The paper develops a direct Dirac-inequality approach to the unitarity problem for highest weight Harish-Chandra modules over classical Hermitian-type groups. By formulating explicit Dirac inequalities for each reductive case through Schmid modules and generalized PRV components, it translates unitarity into concrete weight-bounds on the highest weights $oldsymbol{rac{}}$, and uses these to characterize when the generalized Verma module $N(oldsymbol{rac{}})$ is irreducible and when its irreducible quotient $L(oldsymbol{rac{}})$ is unitarizable. The work provides case-by-case criteria for $rak{sp}(2n,b{R})$, $rak{so}^*(2n)$, $rak{su}(p,q)$, $rak{so}(2,2n-2)$, and $rak{so}(2,2n-1)$, setting up a more direct path to the EHW classification and paving the way for the sequel PPST2 to complete the program. The methods also hint at applications to convergence and analysis of $K$-type decompositions in these unitary representations.

Abstract

Let $G$ be a connected simply connected noncompact classical simple Lie group of Hermitian type. Then $G$ has unitary highest weight representations. The proof of the classification of unitary highest weight representations of $G$ given by Enright, Howe and Wallach is based on the Dirac inequality of Parthasarathy, Jantzen's formula and Howe's theory of dual pairs where one group in the pair is compact. In this paper we focus on the Dirac inequality which can be used to prove the classification in a more direct way.

Theorems & Definitions (30)

• Theorem 1.1
• Lemma 2.1
• Lemma 2.2
• proof
• Theorem 3.6
• proof
• Theorem 3.23
• Theorem 3.31
• proof
• Theorem 3.47