Parabolic induction for Springer fibres

TL;DR

This paper explores the relationship between Springer fibres associated with nilpotent orbits in reductive groups, introducing a combinatorial description of Lusztig--Spaltenstein induction via a new stacking rule for standard tableaux.

Contribution

It provides a new combinatorial framework for understanding Lusztig--Spaltenstein induction on Springer fibres using a novel stacking operation on standard tableaux.

Findings

Established a closed morphism between Springer fibres of induced orbits.

Proved the injection on irreducible components induced by the morphism.

Introduced a new associative stacking rule for standard tableaux.

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 $G = \operatorname{GL}_N$ 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.