Unitarity of highest weight Harish-Chandra modules and smoothness of Schubert varieties

TL;DR

This paper establishes a link between the unitarity of highest weight Harish-Chandra modules for Hermitian Lie groups and the smoothness of associated Schubert varieties, providing a combinatorial classification of unitary modules.

Contribution

It introduces a bijection connecting Dynkin subdiagrams to unitary modules and characterizes unitarity via Schubert variety smoothness, advancing understanding of representation theory.

Findings

Unitary modules correspond to smooth Schubert varieties.

Cardinality of unitary modules is determined by Kazhdan-Lusztig cells.

A bijection links Dynkin subdiagrams to unitary modules.

Abstract

Let $G_{\mathbb{R}}$ be a Lie group of Hermitian type, and $L(\lambda)$ a highest weight Harish-Chandra module of $G_{\mathbb{R}}$ with highest weight $\lambda$. In this article, we exhibit a bijection between the set of connected Dynkin subdiagrams containing the noncompact simple root and the set of unitary highest weight modules $L(-w\rho-\rho)$, where $\rho$ is half the sum of positive roots. We find that $L(-w\rho-\rho)$ is unitary if and only if the Schubert variety $X(w)$ is smooth. We also give the cardinality of the set of unitary highest weight modules $L(-w\rho-\rho)$ for each Kazhdan-Lusztig right cell.