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:

 2.2e+48 Representations of Lie and real algebraic groups: algebraic methods (Verma modules, etc.) 2.2e+47 Semisimple Lie groups and their representations

Zbl 0526.22013
Full Text:

References:

 [1] Alexandre Beĭlinson and Joseph Bernstein, Localisation de \?-modules, C. R. Acad. Sci. Paris Sér. I Math. 292 (1981), no. 1, 15 – 18 (French, with English summary). · Zbl 0476.14019 [2] -, A generalization of Casselman’s submodule theorem, in ”Representation theory of reductive groups,” Birkhäuser, Boston, 1983, pp. 35-52. [3] N. Bourbaki, Algèbre commutative, Masson, Paris. · Zbl 0141.03501 [4] -, Groupes et algebres de Lie, Masson, Paris. [5] A Grothendieck, Eléments de géométrie algébrique IV, Publ. I.H.E.S. No. 20 (1964). · Zbl 0136.15901 [6] Henryk Hecht, Dragan Miličić, Wilfried Schmid, and Joseph A. Wolf, Localization and standard modules for real semisimple Lie groups. I. The duality theorem, Invent. Math. 90 (1987), no. 2, 297 – 332. · Zbl 0699.22022 [7] A. Joseph and J. T. Stafford, Modules of \?-finite vectors over semisimple Lie algebras, Proc. London Math. Soc. (3) 49 (1984), no. 2, 361 – 384. · Zbl 0543.17004 [8] D. Miličić, Localization and representation theory of reductive Lie groups,” (mimeographed notes, to appear).
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.