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A geometric construction of a resolution of the fundamental series. (English) Zbl 0723.22015

The discrete series of representations of a real semisimple Lie group are the basic building blocks from which all representations are constructed, induction from parabolic subgroups being the tool. The discrete series exists only in the presence of compact Cartan subgroups. When none exist, a close analogue is given by the fundamental series.
Different realizations of the discrete series are known, e.g. geometric (Schmid) and algebraic, i.e. infinitesimal (Enright, Varadarajan). Enright studied the fundamental series from the algebraic viewpoint and introduced his completion functors, a device which has been overshadowed by derived functor constructions. The algebro-geometric method of studying representations is in terms of the Beilinson-Bernstein localization machine and the theory of D-modules on the flag variety.
The author studies and compares resolutions of local cohomology D-modules arising from the Cousin complex with Enright’s generalization of the Bernstein-Gel’fand-Gel’fand resolution of a Verma module (described in terms of completions). The author constructs a duality functor relating the two resolutions, in the process giving a geometric description of the relevant completion modules, i.e. their localization, as so-called standard modules associated to Bruhat cells.

MSC:

22E46 Semisimple Lie groups and their representations
32C38 Sheaves of differential operators and their modules, \(D\)-modules
57T10 Homology and cohomology of Lie groups
17B10 Representations of Lie algebras and Lie superalgebras, algebraic theory (weights)
22E47 Representations of Lie and real algebraic groups: algebraic methods (Verma modules, etc.)
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