Z/m-graded Lie algebras and perverse sheaves, IV

TL;DR

This paper develops a combinatorial algorithm to compute local intersection cohomology of nilpotent orbit closures in graded Lie algebras, advancing understanding of their geometric and representation-theoretic properties.

Contribution

It introduces a new combinatorial method for calculating intersection cohomology in the context of Z/m-graded Lie algebras with applications to orbit closures.

Findings

Algorithm computes intersection cohomology efficiently.

Results apply to nilpotent orbit closures in graded Lie algebras.

Enhances understanding of equivariant local systems in geometric representation theory.

Abstract

Let G be a reductive group over C. Assume that the Lie algebra g of G has a given grading (g_j) indexed by a cyclic group Z/m such that g_0 contains a Cartan subalgebra of g. The subgroup G_0 of G corresponding to g_0 acts on the variety of nilpotent elements in g_1 with finitely many orbits. We are interested in computing the local intersection cohomology of the closures of these orbits with coefficients in irreducible G-equivariant local systems which are in the "principal block". We show that these can be computed by a purely combinatorial algorithm.