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Bernstein-Gelfand-Gelfand sequences. (English) Zbl 1159.58309
Summary: This paper is devoted to the study of geometric structures modeled on homogeneous spaces \(G/P\), where \(G\) is a real or complex semisimple Lie group and \(P\subset G\) is a parabolic subgroup. We use methods from differential geometry and very elementary finite-dimensional representation theory to construct sequences of invariant differential operators for such geometries, both in the smooth and the holomorphic category. For \(G\) simple, these sequences specialize on the homogeneous model \(G/P\) to the celebrated (generalized) Bernstein-Gelfand-Gelfand resolutions in the holomorphic category, while in the smooth category we get smooth analogs of these resolutions. In the case of geometries locally isomorphic to the homogeneous model, we still get resolutions, whose cohomology is explicitly related to a twisted de Rham cohomology. In the general (curved) case we get distinguished curved analogs of all the invariant differential operators occurring in Bernstein-Gelfand-Gelfand resolutions (and their smooth analogs). On the way to these results, a significant part of the general theory of geometrical structures of the type described above is presented here for the first time.

58J10 Differential complexes
17B10 Representations of Lie algebras and Lie superalgebras, algebraic theory (weights)
53C28 Twistor methods in differential geometry
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