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Algebraic Methods in the Theory of Generalized Harish-Chandra Modules

  • Ivan PenkovEmail author
  • Gregg Zuckerman
Chapter
Part of the Developments in Mathematics book series (DEVM, volume 38)

Abstract

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 \((\mathfrak{g},\mathfrak{k})\)-modules, where \(\mathfrak{g}\) is a semisimple Lie algebra and \(\mathfrak{k}\) is an arbitrary algebraic reductive in \(\mathfrak{g}\) subalgebra. These results lead to a classification of simple \((\mathfrak{g},\mathfrak{k})\)-modules of finite type with generic minimal \(\mathfrak{k}\)-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 \(\mathfrak{k}\) is an eligible r-subalgebra (see the definition in Sect. 4) in which we prove stronger versions of our main results. If \(\mathfrak{k}\) is eligible, the fundamental series of \((\mathfrak{g},\mathfrak{k})\)-modules yields a natural algebraic generalization of Harish-Chandra’s discrete series modules.

Key words

Generalized Harish-Chandra module \((\mathfrak{g}\mbox{,}\,\mathfrak{k})\)-module of finite type Minimal \(\mathfrak{k}\)-type Fernando–Kac subalgebra Eligible subalgebra 

Mathematics Subject Classification (2010):

17B10 17B55. 

Notes

Acknowledgements

I. Penkov thanks Yale University for its hospitality and partial financial support during the spring of 2012 when this paper was conceived, as well as the Max Planck Institute for Mathematics in Bonn where the work on the paper was continued. G. Zuckerman thanks Jacobs University for its hospitality.

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Copyright information

© Springer International Publishing Switzerland 2014

Authors and Affiliations

  1. 1.Jacobs University BremenBremenGermany
  2. 2.Department of MathematicsYale UniversityNew HavenUSA

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