K-theory of D-modules and primitive factors of enveloping algebras of semisimple Lie algebras. (English) Zbl 0672.17008

Hodges, Timothy J.

Bull. Sci. Math., II. Sér. 113, No. 1, 85-88 (1989).

Let X be a smooth quasi-projective variety over an algebraically closed field k of characteristic 0, \({\mathcal D}^ a \)sheaf of twisted differential operators over X, \({\mathcal M}({\mathcal D})\) the category of coherent left \({\mathcal D}\)-modules. It is shown that \(K_ i({\mathcal M}({\mathcal D}))\cong K_ i(X)\) for all \(i\geq 0\). It follows that if \({\mathfrak g}\) is a semisimple Lie algebra and F is a minimal primitive ideal of its enveloping algebra U(\({\mathfrak g})\) with regular central character, then \(K_ i(U({\mathfrak g})/P)\cong K_ i(G/B)\), where G/B is the associated flag variety.

Reviewer: L.Vaserstein and E.Wheland

MSC:

17B35	Universal enveloping (super)algebras
14C35	Applications of methods of algebraic \(K\)-theory in algebraic geometry
16D60	Simple and semisimple modules, primitive rings and ideals in associative algebras
17B20	Simple, semisimple, reductive (super)algebras

Keywords:

twisted differential operators; semisimple Lie algebra; minimal primitive ideal; enveloping algebra; flag variety