Rigid orbits and sheets in reductive Lie algebras over fields of prime characteristic

Let $G$ be a simple simply-connected algebraic group over an algebraically closed field $k$ of characteristic $p>0$ with $\mathfrak{g}={\rm Lie}(G)$. We discuss various properties of nilpotent orbits in $\mathfrak{g}$, which have previously only been considered over $\mathbb{C}$. Using a combination of theoretical and computational methods, we extend to positive characteristic various calculations of de Graaf with nilpotent orbits in exceptional Lie algebras. In particular, we classify those orbits which are reachable, those which satisfy a certain related condition due to Panyushev, and determine the codimension in the centraliser $\mathfrak{g}_e$ of its the derived subalgebra $[\mathfrak{g}_e,\mathfrak{g}_e]$. Some of these calculations are used to show that the list of rigid nilpotent orbits in $\mathfrak{g}$, the classification of sheets of $\mathfrak{g}$ and the distribution of the nilpotent orbits amongst them are independent of good characteristic, remaining the same as in the characteristic zero case. We also give a comprehensive account of the theory of sheets in reductive Lie algebras over algebraically closed fields of good characteristic.