Generalized Harish-Chandra Modules

Abstract

This course is an introduction to algebraic methods in the infinite-dimensional representation theory of semisimple Lie algebras over the complex numbers. In the first section we present basic definitions and theorems concerning Harish-Chandra modules, Fernando–Kac subalgebras associated to $\mathfrak{g}$-modules, generalized Harish-Chandra modules, and the special case of weight modules. Work of Kostant allows us to demonstrate that not all simple $\mathfrak{g}$-modules are generalized Harish-Chandra modules. In the second section we discuss the Zuckerman derived functors and several of their important properties. We tailor this section to the theory of algebraic constructions of generalized Harish-Chandra modules. In the third section we summarize the main results in our joint work with Ivan Penkov on the classification of generalized Harish-Chandra modules having a “generic” minimal $\mathfrak{k}$-type. This classification makes extensive use of the Zuckerman derived functors in the context of pairs $(\mathfrak{g},\mathfrak{k})$ where $\mathfrak{g}$ is a semisimple Lie algebra and $\mathfrak{k}$ is a subalgebra of $\mathfrak{g}$ which is reductive in $\mathfrak{g}$. We also utilize the theory of the cohomology of the nilpotent radical of a parabolic subalgebra with coefficients in an infinite-dimensional $(\mathfrak{g},\mathfrak{k})$-module. The crucial point of this section is that we do not assume that $\mathfrak{k}$ is a symmetric subalgebra of $\mathfrak{g}$.