Leading Term Cycles of Harish-chandra Modules and Partial Orders on Components of the Springer Fiber

We use the geometry of characteristic cycles of Harish-Chandra modules for a real semisimple Lie group GR to prove an upper triangularity relationship between two bases of each special representations of a classical Weyl group. One basis consists of Goldie rank polynomials attached to primitive ideals in the enveloping algebra of the complexified Lie algebra g; the other consists of polynomials that measure the Euler characteristic of the restriction of an equivariant line bundle on the flag variety for g to an irreducible component of the Springer fiber. While these two bases are defined only using the structure of the complex Lie algebra g, the relationship between them is closely tied to the real group GR. More precisely the order leading to the upper triangularity result is a suborder of closure order for the orbits of the complexification of a maximal compact subgroup of GR on the flag variety for g.