Differential operators on homogeneous spaces. III

Abstract

LetG be a semi-simple complex algebraic group with Lie algebra ϱ and flag varietyX=G/B. For each primitive idealJ with trivial central character in the enveloping algebraU(ϱ) we define a characteristic variety in the cotangent bundle ofX, which projects under the Springer resolution mapT * X→ϱ onto the closure of a nilpotent orbit. We prove that this characteristic variety is theG-saturation of the characteristic variety of a highest weight module with annihilatorJ. We conjecture that it is irreducible forG=SL n . Our conjecture would provide a geometrical explanation for the classification of primitive ideals in terms of Weyl group representations, as achieved by A. Joseph. The presentation of these ideas here is simultaneously used to some extent as an opportunity to continue our more general systematic discussion of differential operators on a complete homogeneous space, and to study more generally characteristic varieties of Harish-Chandra modules.