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

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.

### 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