Hesselink strata in small characteristic and Lusztig-Xue pieces

Abstract

Let $G$ be a simple algebraic group over an algebraically closed field of characteristic $p\ge 0$ and $\mathfrak{g}=\mathrm{Lie}(G)$. We show that the nilpotent pieces $\mathrm{LX}(\Delta )$ 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 $\mathrm{LX}(\Delta )$ for groups of type ${\mathrm{E}}_7$ in characteristic $2$ and for groups of type ${\mathrm{E}}_8$ in characteristic $2$ and $3$.

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