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Differential operators on the flag varieties. (English) Zbl 0537.14010
Astérisque 87-88, 43-60 (1981).
[For the entire collection see Zbl 0468.00006.]
Let G be a connected semi-simple algebraic group over a field of characteristic 0 and let X be the flag variety of G. The author determines the algebra structure of $$\Gamma$$ (X,$${\mathcal D}_ X)$$, the algebra of global differential operators on X: Let $$U({\mathfrak G})$$ be the enveloping algebra of the Lie-algebra of G, Z be the center of $$U({\mathfrak G})$$ and I the ideal $$U({\mathfrak G})\cdot Z(\cap(U({\mathfrak G})\cdot {\mathfrak G}))$$. Then $$\Gamma$$ (x,$${\mathcal D}_ X)$$ is isomorphic to $$U({\mathfrak G})/I$$.
Reviewer: F.Pauer

MSC:
 14F10 Differentials and other special sheaves; D-modules; Bernstein-Sato ideals and polynomials 14M15 Grassmannians, Schubert varieties, flag manifolds 14L35 Classical groups (algebro-geometric aspects) 14L30 Group actions on varieties or schemes (quotients) 20G15 Linear algebraic groups over arbitrary fields