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

Abstract

Let G G be a reductive group over C \mathbf {C} . Assume that the Lie algebra g \frak g of G G has a given grading ( g j ) (\frak g_j) indexed by a cyclic group Z / m \mathbf {Z}/m such that g 0 \frak g_0 contains a Cartan subalgebra of g \frak g . The subgroup G 0 G_0 of G G corresponding to g 0 \frak g_0 acts on the variety of nilpotent elements in g 1 \frak g_1 with finitely many orbits. We are interested in computing the local intersection cohomology of closures of these orbits with coefficients in irreducible G 0 G_0 -equivariant local systems in the case of the principal block. We show that these can be computed by a purely combinatorial algorithm.