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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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