Parabolic induction for Springer fibres

Abstract

Let $G$ be a reductive group satisfying the standard hypotheses, with Lie algebra $\mathfrak{g}$. For each nilpotent orbit ${\mathcal{O}}_0$ in a Levi subalgebra ${\mathfrak{g}}_0$ we can consider the induced orbit $\mathcal{O}$ defined by Lusztig and Spaltenstein. We observe that there is a natural closed morphism of relative dimension zero from the Springer fibre over a point of ${\mathcal{O}}_0$ to the Springer fibre over $\mathcal{O}$, which induces an injection on the level of irreducible components. When $G={\mathrm{GL}}_N$ the components of Springer fibres was classified by Spaltenstein using standard tableaux. Our main results explains how the Lusztig--Spaltenstein map of Springer fibres can be described combinatorially, using a new associative composition rule for standard tableaux which we call stacking.

15 pages, comments welcome