Hesselink strata in small characteristic and Lusztig-Xue pieces

TL;DR

This paper proves that Lusztig-Xue nilpotent and unipotent pieces coincide with Hesselink strata, providing a partition of the nilpotent cone in algebraic groups over fields of small characteristic, especially for exceptional types.

Contribution

It establishes the equivalence of Lusztig-Xue pieces and Hesselink strata, confirming their partition property for groups of type E7 and E8 in small characteristics.

Findings

Lusztig-Xue pieces coincide with Hesselink strata.

The pieces form a partition of the nilpotent cone.

Results confirmed for E7 and E8 in characteristic 2 and 3.

Abstract

Let $G$ be a simple algebraic group over an algebraically closed field of characteristic $p\ge 0$ and $\mathfrak{g}={\rm Lie}(G)$. We show that the nilpotent pieces ${\rm LX}(\Delta)$ introduced by Lusztig coincide with the corresponding Hesselink strata $\mathcal{H}(\Delta)$ and hence form a partition of the nilpotent cone of $\mathfrak{g}$. Similar results are obtained for the unipotent pieces of $G$. Here $\Delta$ runs through the set of all weighted Dynkin diagrams of $G$. Thanks to the results obtained by Lusztig, Xue and Voggesberger this boils down to establishing the partition property of the pieces ${\rm LX}(\Delta)$ for groups of type ${\rm E_7}$ in characteristic $2$ and for groups of type ${\rm E_8}$ in characteristic $2$ and $3$.