Leading-term cycles of Harish-Chandra modules and partial orders on components of the Springer fiber

Abstract

We use the geometry of characteristic cycles of Harish-Chandra modules for a real semi-simple Lie group $G_{\mathbb{R}}$ to prove an upper triangularity relationship between two bases of each special representation 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 ${\mathfrak{g}}$; the other consists of polynomials that measure the Euler characteristic of the restriction of an equivariant line bundle on the flag variety for ${\mathfrak{g}}$ to an irreducible component of the Springer fiber. While these two bases are defined only using the structure of the complex Lie algebra ${\mathfrak{g}}$, the relationship between them is closely tied to the real group $G_{\mathbb{R}}$. 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 $G_{\mathbb{R}}$ on the flag variety for ${\mathfrak{g}}$.