Local geometry of special pieces of nilpotent orbits

The nilpotent cone of a simple Lie algebra is partitioned into locally closed subvarieties called special pieces, each containing exactly one special orbit. Lusztig conjectured that each special piece is the quotient of some smooth variety by a precise finite group $H$, a result proved for the classical types by Kraft and Procesi. The present work is about exceptional types. Our main result is a local version of Lusztig's conjecture: the intersection of a special piece with a Slodowy slice transverse to the minimal orbit in the piece is isomorphic to the quotient of a vector space by $H$. Along the way, we complete our previous work on the generic singularities of nilpotent orbit closures, by providing proofs for the last two `exotic' singularities. Four further, non-isolated, exotic singularities are studied: we show that quotients $\overline{{\mathcal 0}_{\text{mini}}(\mathfrak{so}_8)}/\mathfrak{S}_4$, $S^2({\mathbb C}^2/\mu_3)$, $S^3({\mathbb C}^2/\mu_2)$ and $\overline{{\mathcal 0}_{\text{mini}}(\mathfrak{sl}_3)}/\mathfrak{S}_4$ occur as Slodowy slice singularities between nilpotent orbits in types $F_4$, $E_6$, $E_7$ and $E_8$ respectively. We also extend, to fields other than ${\mathbb C}$, the results of Brylinski and Kostant on shared orbit pairs. In the course of our analysis, we discover a shared pair which is missing from Brylinski and Kostant's classification.