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Structure of modules induced from simple modules with minimal annihilator

  Published:2004-04-01
 Printed: Apr 2004
  • Oleksandr Khomenko
  • Volodymyr Mazorchuk
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Abstract

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 $\Oo$ 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: parabolic induction, generalized Verma module, simple module, Ha\-rish-\-Chand\-ra bimodule, equivalent categories parabolic induction, generalized Verma module, simple module, Ha\-rish-\-Chand\-ra bimodule, equivalent categories
MSC Classifications: 17B10, 22E47 show english descriptions Representations, algebraic theory (weights)
Representations of Lie and real algebraic groups: algebraic methods (Verma modules, etc.) [See also 17B10]
17B10 - Representations, algebraic theory (weights)
22E47 - Representations of Lie and real algebraic groups: algebraic methods (Verma modules, etc.) [See also 17B10]
 

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