We show that every torsion free weight module with finite dimensional weight spaces over a symplectic complex Lie algebra, which is different from sp(2, C), is completely reducible.
Abstract. We study the structure of generalized Verma modules over a semi-simple complex finitedimensional 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-GelfandGelfand category 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.
We introduce a notion of a category with full projective functors. It encodes certain common properties of categories appearing in representation theory of Lie groups, Lie algebras and quantum groups. We describe the left or right exact functors which naturally commute with projective functors and provide a unified approach to the verification of relations between such functors. 2000 Mathematics Subject Classification 17B10.
We calculate the finitistic dimension of certain stratified algebras in terms of the projective dimension of the characteristic tilting module. This includes, in particular, quasi-hereditary algebras, whose Koszul dual is again quasi-hereditary; stratified algebras, which are quotients of quasi-hereditary algebras over complete local commutative rings; and stratified algebras associated with HarishChandra bimodules for complex semi-simple finite-dimensional Lie algebras.
With each generalized Verma module induced from a "well-embedded" parabolic subalgebra of a Lie algebra with triangular decomposition, we associate a Verma module over the same algebra in a natural way. In the case when the semisimple part of the Levi factor of the parabolic subalgebra is isomorphic to sl 2 and the generalized Verma module is induced from an infinite-dimensional simple module, we prove that the associated Verma module is simple if and only if the original generalized Verma module is simple.
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