On cocharacters associated to nilpotent elements of reductive groups

Abstract

Let $G$ be a connected reductive linear algebraic group defined over an algebraically closed field of characteristic $p$. Assume that $p$ is good for $G$. In this note we consider particular classes of connected reductive subgroups $H$ of $G$ and show that the cocharacters of $H$ that are associated to a given nilpotent element $e$ in the Lie algebra of $H$ are precisely the cocharacters of $G$ associated to $e$ that take values in $H$. In particular, we show that this is the case provided $H$ is a connected reductive subgroup of $G$ of maximal rank; this answers a question posed by J. C. Jantzen.