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

Abstract

Let $G_{\mathbb{R}}$ be a Lie group of Hermitian type, and $L(λ)$ a highest weight Harish-Chandra module of $G_{\mathbb{R}}$ with highest weight $λ$. 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 $L(-wρ-ρ)$, where $ρ$ is half the sum of positive roots. We find that $L(-wρ-ρ)$ 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 $L(-wρ-ρ)$ for each Kazhdan-Lusztig right cell.