Graded Lie Algebras and Intersection Cohomology

Abstract

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

To Toshiaki Shoji on the occasion of his 60th birthday

Mathematics Subject Classifications (2000): 20G99