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Special nilpotent orbits in simple Lie algebras play an important role inrepresentation theory. The nilpotent cone of a reductive Lie algebra ispartitioned into locally closed subvarieties called special pieces, eachcontaining exactly one special orbit. Lusztig conjectured that each specialpiece is the quotient of some smooth variety by a precise finite group $H$, aresult proved for the classical types by Kraft and Procesi. The present work isabout exceptional types. Our main result is a local version of Lusztig'sconjecture: the intersection of a special piece with a Slodowy slice transverseto the minimal orbit in the piece is isomorphic to the quotient of a vectorspace by $H$. Along the way, we complete our previous work on the genericsingularities of nilpotent orbit closures, by providing proofs for the last two`exotic' singularities. Two further, non-isolated, exotic singularities arestudied: we show that quotients $\overline{{\mathcal O}_{\rmmini}(\mathfrak{so}_8)}/\mathfrak{S}_4$ and $\overline{{\mathcal O}_{\rmmini}(\mathfrak{sl}_3)}/\mathfrak{S}_4$ occur as Slodowy slice singularitiesbetween nilpotent orbits in types $F_4$ and $E_8$ respectively. We also extend,to fields other than ${\mathbb C}$, the results of Brylinski and Kostant onshared orbit pairs. In the course of our analysis, we discover a shared pairwhich is missing from Brylinski and Kostant's classification.

Local geometry of special pieces of nilpotent orbits