INTEGRALS OVER COMPONENTS OF THE SPRINGER FIBER FOR sl(n,C)

The main result in [BaZ, Section 6] is an algorithm to compute the cohomology of a certain class of irreducible components of the Springer fiber for sl(n,C). As explained, for instance, in [J3, Corollary 6.7], this is related to the computation of the integrals over such components of exponentiated Chern classes of homogeneous line bundles on the flag variety. In turn, [Ch, Section 2] implies results about multiplicities in associated cycles of irreducible discrete series representations of SU(p, q). The algorithm relies crucially on the geometric description of the relevant class of components given in [BaZ, Section 4] (which is of independent interest). The most computationally intensive portion of the algorithm involves a classical branching problem from GL(n,C) to GL(m,C). The purpose of this note is to describe an algorithm to compute the relevant integrals over any component of the Springer fiber for sl(n,C). We do this in two steps. First we present an algorithm to compute the multiplicity in the associated cycle of an arbitrary irreducible Harish-Chandra module for SU(p, q) with regular integral infinitesimal character in the block of a finite-dimensional representation1. (The argument does applies with superficial changes to SL(n,C).) This algorithm has been known to a handful of experts for some time