2003/10/09 by Ivan Penkov, Penkov, Ivan, Gregg Zuckerman +1
Mathematics · #17B10 #FOS: Mathematics #Representation Theory (math.RT) #math.RT #msc:17B10
paper · pdf · doi:10.48550/arxiv.math/0310140
arxiv created 2003/10/09 · arxiv updated 2009/12/01
Let \frak g be a reductive Lie algebra over \bold C. We say that a \frak g-module M is a generalized Harish-Chandra module if, for some subalgebra \frak k ⊂\frak g, M is locally \frak k-finite and has finite \frak 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 \frak k is a Cartan subalgebra. We also review the recent determination of which reductive in \frak g subalgebras \frak k are essential to a classification. Finally, we present in detail the emerging picture for the case when \frak k is a principal 3-dimensional subalgebra.