Graded Lie algebras and intersection cohomology

Abstract

Let i be a homomorphism of the multiplicative group into a connected reductive algebraic group over C. Let G^i be the centralizer of the image i. Let LG be the Lie algebra of G and let L_nG (n integer) be the summands in the direct sum decomposition of LG determined by i. Assume that n is not zero. For any G^i-orbit O in L_nG and any irreducible G^i-equivariant local system L on O we consider the restriction of some cohomology sheaf of the intersection cohomology complex of the closure of O with coefficients in L to another orbit O' contained in the closure of O. For any irreducible G^i-equivariant local system L' on O' we would like to compute the multiplicity of L' in that restriction. We present an algorithm which helps in computing that multiplicity.