On $sl_2$ -triples in Lie algebras of reductive algebraic groups in positive characteristic

Pengelly, Rachel ORCID: 0000-0003-4640-4943 (2023). On $sl_2$ -triples in Lie algebras of reductive algebraic groups in positive characteristic. University of Birmingham. Ph.D.

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Abstract

In this thesis we consider $sl_2$ -triples in $g$ = Lie( $G$ ), the Lie algebra of a connected reductive algebraic group over a field of positive characteristic $p>2$ .

We focus on good primes for $G$ which are smaller than the coxeter number of $G$ , and determine to what extent the theorems of Jacobson--Morozov and Kostant hold in this setting. To do so, we determine the maximal $G$ -stable closed subvariety $V$ of the nilpotent cone $N$ of $g$ such that the $G$ -orbits in $V$ are in bijection with the $G$ -orbits of $sl_2$ -triples ( $e,h,f$ ) with $e,f$ $\in$ $V$ .

We also determine the maximal $G$ -stable closed subvariety $V$ of the nilpotent cone $N$ of $g$ such that any subalgebra $h$ = $⟨e,h,f⟩ \cong$ $sl_2$ ( $k$ ) with $e$ , $f$ ∈ $V$ is $G$ -completely reducible