2013/10/30 by Ivan Penkov, Penkov, Ivan, Gregg J. Zuckerman +2
Mathematics · #Advanced Algebra and Geometry #Advanced Topics in Algebra #Algebraic structures and combinatorial models #math.RT #msc:17B10 #msc:17B55
paper · pdf · doi:10.48550/arxiv.1310.8058
Keywords : generalized Harish-Chandra module, (g,k)-module of finite type, minimal k-type, Fernando-Kac subalgebra, eligible subalgebra; Pages no. : 13; Bibliography : 21 items
arxiv created 2013/10/30 · arxiv updated 2013/10/31
This paper is a review of results on generalized Harish-Chandra modules in the framework of cohomological induction. The main results, obtained during the last 10 years, concern the structure of the fundamental series of (\mathfrakg,\mathfrakk)-modules, where \mathfrakg is a semisimple Lie algebra and \mathfrakk is an arbitrary algebraic reductive in \mathfrakg subalgebra. These results lead to a classification of simple (\mathfrakg,\mathfrakk)-modules of finite type with generic minimal \mathfrakk-types, which we state. We establish a new result about the Fernando-Kac subalgebra of a fundamental series module. In addition, we pay special attention to the case when \mathfrakk is an eligible r-subalgebra (see the definition in section 4) in which we prove stronger versions of our main results. If \mathfrakk is eligible, the fundamental series of (\mathfrakg,\mathfrakk)-modules yields a natural algebraic generalization of Harish-Chandra's discrete series modules.