Orbital variety closures and the convolution product in Borel–Moore homology

Abstract

This paper settles a twenty-year-old conjecture describing the inclusion relations between orbital variety closures. The solution is in terms of the top Borel–Moore homology of the Steinberg variety and mirrors the way in which the Verma module multiplicities determine the inclusion relations of primitive ideals. It thus gives a link between geometry and representation theory which is more precise than what one would obtain by a naive application of the orbit method. Unlike the primitive ideal case which uses Duflo involutions, the geometric result exploits a link between correspondences and the moment map pertaining to the cotangent bundle on the flag variety. Krull equidimensionality is needed to ensure that all correspondences are recovered from the homology convolution product.