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A duality theorem for extensions of induced highest weight modules. (English) Zbl 0733.17005
Let ($${\mathfrak g},{\mathfrak p})$$ be a pair consisting of a complex semisimple Lie algebra $${\mathfrak g}$$ and its parabolic subalgebra $${\mathfrak p}$$. Let also $${\mathfrak p}={\mathfrak l}\oplus {\mathfrak n}$$ be a Levi decomposition of $${\mathfrak p}$$ with $${\mathfrak l}$$ semisimple. The main result is a duality theorem for extensions of induced modules in the category of $${\mathfrak g}$$- modules which are $${\mathfrak l}$$-semisimple and $${\mathfrak p}$$-locally finite. For the proof, the authors transfer the problem into the smooth vetor bundle category where there is a natural duality, the adjoint.

##### MSC:
 17B10 Representations of Lie algebras and Lie superalgebras, algebraic theory (weights) 17B66 Lie algebras of vector fields and related (super) algebras
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