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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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##### References:
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