In this talk, we will study the Gelfand-Kirillov dimensions and associated varieties of highest weight modules for simple Lie algebras.

By using a conjecture of Tanisaki, we will show that there is a bijection between the Kazhdan-Lusztig right cells and associated varieties of highest weight modules with trivial infinitesimal character. When L is a simple integral highest weight module of sl(n,C) with the minimal Gelfand-Kirillov dimension n-1, we will show that its associated variety is irreducible. In particular, its associated variety will be given in the information of its highest weight. When L is a simple highest weight module in a given parabolic category O with maximal Gelfand-Kirillov dimension, we will show that its associated variety is also irreducible and such weights will be characterized explicitly in the case of type A.

This is a joint work with Wei Xiao and Xun Xie.