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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
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