On unipotent and nilpotent pieces for classical groups

TL;DR

This paper proves the equivalence of Lusztig's geometric definition of unipotent and nilpotent pieces for classical groups with a combinatorial approach, and provides a formula relating unipotent classes across characteristics.

Contribution

It establishes the coincidence of geometric and combinatorial definitions of unipotent and nilpotent pieces, and introduces a formula connecting classes in different characteristics.

Findings

Geometric and combinatorial definitions coincide.

A closed formula maps unipotent classes from characteristic 2 to characteristic 0.

Fibers of the map correspond to unipotent and nilpotent pieces.

Abstract

We show that the definition of unipotent (resp. nilpotent) pieces for classical groups given by Lusztig (resp. Lusztig and the author) coincides with the combinatorial definition using closure relations on unipotent classes (resp. nilpotent orbits). Moreover we give a closed formula for a map from the set of unipotent classes (resp. nilpotent orbits) in characteristic 2 to the set of unipotent classes in characteristic 0 such that the fibers are the unipotent (resp. nilpotent) pieces.