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 is an element of G be unipotent. We study the centralizer C-G(u), especially its centre Z(C-G(u)). We calculate the Lie algebra of Z(C-G(u)), in particular determining its dimension; we 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.