Intersections of Deligne--Lusztig varieties and Springer fibres

TL;DR

This paper establishes a direct geometric link between Deligne--Lusztig varieties and Springer fibres in type A, revealing how unipotent elements relate to Weyl group components and providing new geometric insights into classical formulas.

Contribution

It introduces a geometric relation connecting Deligne--Lusztig varieties and Springer fibres, using combinatorial tools to interpret unipotent centralizer dimensions.

Findings

Each Springer fibre component corresponds to a unique Deligne--Lusztig variety component.

The Springer fibre component is an open dense subset of a Deligne--Lusztig component.

Provides a geometric interpretation of a classical dimension formula.

Abstract

In this paper we prove a direct geometric relation between Deligne--Lusztig varieties and Springer fibres in type $\mathsf{A}$: For any rational unipotent element, the Springer fibre cuts out a unique component of a specific Deligne--Lusztig variety; moreover, this component forms an open dense subset of a component of the Springer fibre. This boils down to a map from the unipotent variety to the Weyl group, and combines several constructions with a combinatorial flavour (like Weyr normal forms, Robinson--Schensted correspondence, and Spaltenstein's and Steinberg's labellings); it also provides a geometric interpretation of a classical dimension formula of unipotent centralisers.