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

Abstract

Let $G$ be a simple simply-connected algebraic group over an algebraically closed field $k$ of characteristic $p>0$ with $\mathfrak{g}=\mathrm{Lie}(G)$. 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.

revised version, many typos corrected, 25 pages

References in corpus (3)

• Nilpotent orbits over ground fields of good characteristic
• Induced nilpotent orbits of the simple Lie algebras of exceptional type
• Adjoint Jordan Blocks

Cited by in corpus (5)

• The Jacobson--Morozov theorem and complete reducibility of Lie subalgebras
• Masses, Sheets and Rigid SCFTs
• Jordan blocks of nilpotent elements in some irreducible representations of classical groups in good characteristic
• On the first restricted cohomology of a reductive Lie algebra and its Borel subalgebras
• A Morita theorem for modular finite W-algebras