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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