Centres of Centralizers of Unipotent Elements in Simple Algebraic Groups

Let $G$ be a simple algebraic group defined over an algebraically closed field $k$ whose characteristic is either $0$ or a good prime for $G$, and let $u\in G$ be unipotent. The authors study the centralizer $C_G(u)$, especially its centre $Z(C_G(u))$. They calculate the Lie algebra of $Z(C_G(u))$, in particular determining its dimension; they prove a succession of theorems of increasing generality, the last of which provides a formula for $\dim Z(C_G(u))$ in terms of the labelled diagram associated to the conjugacy class containing $u$.