an:04175287
Zbl 0714.22011
Hecht, Henryk; Mili??i??, Dragan
On the cohomological dimension of the localization functor
EN
Proc. Am. Math. Soc. 108, No. 1, 249-254 (1990).
00155211
1990
j
22E47 22E46
localization functor; Cartan subalgebra; complex semisimple Lie algebra; enveloping algebra; action; Weyl group; sheaf of twisted differential operators; flag variety; left cohomological dimension
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 \textit{A. Beilinson} and \textit{J. Bernstein} for \(\lambda\) regular [Representation theory of reductive groups, Prog. Math. 40, 35-52 (1983; Zbl 0526.22013)].
H.de Vries
Zbl 0526.22013