A characterization of unitarity of some highest weight Harish-Chandra modules

TL;DR

This paper characterizes when certain highest weight Harish-Chandra modules are unitary based on a simple criterion involving their highest weight, and provides a uniform formula for their Gelfand--Kirillov dimension.

Contribution

It offers a new criterion for unitarity of non-maximal associated variety modules and generalizes the Gelfand--Kirillov dimension formula for these modules.

Findings

Unitarity determined by a simple condition on the highest weight parameter z.

Provides a uniform Gelfand--Kirillov dimension formula for all such modules.

Uses distinguished antichains of positive noncompact roots in proofs.

Abstract

Let $L(\lambda)$ be a highest weight Harish-Chandra module with highest weight $\lambda$. When the associated variety of $L(\lambda)$ is not maximal, that is, not equal to the nilradical of the corresponding parabolic subalgebra, we prove that the unitarity of $L(\lambda)$ can be determined by a simple condition on the value of $z = (\lambda + \rho, \beta^{\vee})$, where $\rho$ is half the sum of positive roots and $\beta$ is the highest root. In the proof, certain distinguished antichains of positive noncompact roots play a key role. By using these antichains, we are also able to provide a uniform formula for the Gelfand--Kirillov dimension of all highest weight Harish-Chandra modules, generalizing our previous result for the case of unitary highest weight Harish-Chandra modules.