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On the cohomological dimension of the localization functor. (English) Zbl 0714.22011
The localization functor of the title is defined by \(\Delta_{\lambda}(V)={\mathcal D}_{\lambda}\otimes_{{\mathcal U}_{\lambda}}V\), where \(\lambda\) is an element of the dual of a fixed Cartan subalgebra of a complex semisimple Lie algebra \({\mathfrak g}\), \({\mathcal U}_{\lambda}\) the quotient of the enveloping algebra of \({\mathfrak g}\) by the ideal generated by the maximal ideal of its centre corresponding in the appropriate way to the orbit of \(\lambda\) under the (shifted) action of the Weyl group, V a \({\mathcal U}_{\lambda}\)-module, and \({\mathcal D}_{\lambda}\) the sheaf of twisted differential operators corresponding to \(\lambda\) on the flag variety of \({\mathfrak g}\) as defined by A. Beilinson and J. Bernstein. The main theorem says that for \(\lambda\) singular the left cohomological dimension of \(\Delta_{\lambda}\) is infinite, which contrasts with a result proved by A. Beilinson and J. Bernstein for \(\lambda\) regular [Representation theory of reductive groups, Prog. Math. 40, 35-52 (1983; Zbl 0526.22013)].
Reviewer: H.de Vries

MSC:
22E47 Representations of Lie and real algebraic groups: algebraic methods (Verma modules, etc.)
22E46 Semisimple Lie groups and their representations
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