Differential operators on the flag varieties. (English) Zbl 0537.14010

Brylinski, J. L.

Astérisque 87-88, 43-60 (1981).

Let \(G\) be a connected semisimple 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 \(U(\mathfrak G)/I\).
[For the entire collection see Zbl 0468.00006.]

Reviewer: Franz Pauer (Innsbruck)

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

Keywords:

algebra of global differential operators on the flag variety; Verma module

Citations:

Zbl 0468.00006