Differential operators on homogeneous spaces. I

Summary

In this paper, we extend recent work of one of us [Br] to investigate an old problem of the other one [B2]. Given a connected semisimple complex Lie-groupG with Lie-algebrag, we study the representation$\psi _X :U(\mathfrak{g}) \to D(X)$ of the enveloping algebra of$\mathfrak{g}$ by global differential operators on a complete homogeneous spaceX=G/P. It turns out that the kernelI _x of ψ _X is the annihilator of a generalizedVerma-module. On the other hand, we study the associated graded ideal grI _x , and relate it to the geometry of a generalizedSpringer-resolution, that is a map$\pi _X :T^* (X) \to \mathfrak{g}$ of the cotangent-bundle ofX onto a nilpotent variety in$\mathfrak{g}$, as studied e.g. in [BM1]. We prove, for instance, that grI _x is prime if and only if π _X is birational with normal image. In general, we show that$\sqrt {grI_X }$ is prime. Equivalently, the associated variety ofI _x in$\mathfrak{g}$ is irreducible: In fact, it is the closure of theRichardson-orbit determined byP. For the homogeneous spaceY=G/(P, P), we prove that the analogous idealI _y has for associated variety the closure of theDixmier-sheet determined byP. From this main result, we derive as a corollary, that for any module induced from a finitedimensional LieP-module the associated variety of the annihilator is irreducible, proving an old conjecture [B2], 2.5. Finally, we give some applications to the study of associated varieties of primitive ideals.