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Generalized Harish-Chandra Modules: A New Direction in the Structure Theory of Representations

Authors:	Ivan Penkov Gregg Zuckerman

Institution:	(1) Department of Mathematics, Yale University, New Haven, CT, 06520, U.S.A.

Abstract:	Let $\mathfrak{g}$ be a reductive Lie algebra over C. We say that a $\mathfrak{g}$ -module M is a generalized Harish-Chandra module if, for some subalgebra $\mathfrak{k}$ $\mathfrak{g}$ , M is locally $\mathfrak{k}$ -finite and has finite $\mathfrak{k}$ -multiplicities. We believe that the problem of classifying all irreducible generalized Harish-Chandra modules could be tractable. In this paper, we review the recent success with the case when $\mathfrak{k}$ is a Cartan subalgebra. We also review the recent determination of which reductive in $\mathfrak{g}$ subalgebras $\mathfrak{k}$ are essential to a classification. Finally, we present in detail the emerging picture for the case when $\mathfrak{k}$ is a principal 3-dimensional subalgebra.

Keywords:	weight module Harish-Chandra module principal 3-dimensional subalgebra cohomological induction
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