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Structure of Modules Induced from Simple Modules with Minimal Annihilator

Published online by Cambridge University Press:  20 November 2018

Oleksandr Khomenko  and
Volodymyr Mazorchuk
Oleksandr Khomenko
Affiliation:
Mathematisches Institut der Universität Freiburg, Eckerstrasse 1, D-79104, Freiburg im Breisgau, FRG email: oleksandr.khomenko@math.uni-freiburg.de
Volodymyr Mazorchuk
Affiliation:
Department of Mathematics, Uppsala University, Box 480, SE 751 06, Uppsala, Sweden email: mazor@math.uu.se
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Abstract

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We study the structure of generalized Verma modules over a semi-simple complex finite-dimensional Lie algebra, which are induced from simple modules over a parabolic subalgebra. We consider the case when the annihilator of the starting simple module is a minimal primitive ideal if we restrict this module to the Levi factor of the parabolic subalgebra. We show that these modules correspond to proper standard modules in some parabolic generalization of the Bernstein-Gelfand-Gelfand category $\mathcal{O}$ and prove that the blocks of this parabolic category are equivalent to certain blocks of the category of Harish-Chandra bimodules. From this we derive, in particular, an irreducibility criterion for generalized Verma modules. We also compute the composition multiplicities of those simple subquotients, which correspond to the induction from simple modules whose annihilators are minimal primitive ideals.

Keywords

17B1022E47parabolic inductiongeneralized Verma modulesimple moduleHarish-Chandra bimoduleequivalent categories
Type
Research Article
Information
Canadian Journal of Mathematics , Volume 56 , Issue 2 , 01 April 2004 , pp. 293 - 309
Copyright
Copyright © Canadian Mathematical Society 2004

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