On the annihilator variety of a highest weight Harish-Chandra module

Abstract

Let $G$ be a Hermitian type Lie group with maximal compact subgroup $K$. Let $L(\hat{I}»)$ be a highest weight Harish-Chandra module of $G$ with the infinitesimal character $\hat{I}»$. By using some combinatorial algorithm, we obtain a description of the annihilator variety of $L(\hat{I}»)$. As an application, when $L(\hat{I}»)$ is unitarizable, we prove that the Gelfand-Kirillov dimension of $L(\hat{I}»)$ only depends on the value of $z=(λ,β^{\vee})$, where ${\hat{I}}^2$ is the highest root.

29 pages