Parabolic induction for Springer fibres

Abstract

Let G G be a reductive group satisfying the standard hypotheses, with Lie algebra g \mathfrak {g} . For each nilpotent orbit O 0 \mathcal {O}_0 in a Levi subalgebra g 0 \mathfrak {g}_0 we can consider the induced orbit O \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 O 0 \mathcal {O}_0 to the Springer fibre over O \mathcal {O} , which induces an injection on the level of irreducible components. When G = G L N G = GL_N the components of Springer fibres were classified by Spaltenstein using standard tableaux. Our main result 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.