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Geometric quantization and derived functor modules for semisimple Lie groups. (English) Zbl 0781.22009
Let \(G\) be a semisimple Lie group, \(\mathfrak g\) its complexified Lie algebra and \(X\) the flag variety of \(\mathfrak g\). In this paper the authors show how the geometric quantization attaches Fréchet representations of finite length to \(G\)-orbits in \(X\). These representations are maximal globalizations of Zuckerman’s standard derived functor modules. They can also be realized as local cohomology groups along \(G\)-orbits in \(X\), what leads to a geometric interpretation of the duality theorem of H. Hecht, D. Miličić and the authors [Invent. Math. 90, 297-332 (1987; Zbl 0699.22022)] as pairing given by the cup product between local cohomology groups followed by evaluation over the fundamental cycle. Some results of this paper were announced before [Bull. Am. Math. Soc., New Ser. 17, 117-120 (1987; Zbl 0649.22010)].

MSC:
22E46 Semisimple Lie groups and their representations
57T10 Homology and cohomology of Lie groups
22E60 Lie algebras of Lie groups
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