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 ∈ G be unipotent. We study the centralizer CG(u), especially its centre Z(CG(u)). We calculate the Lie algebra of Z(CG(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(CG(u)) in terms of the labelled diagram associated to the conjugacy class containing u. We proceed by using the existence of a Springer map to replace u by a nilpotent element lying in the Lie algebra of G. The bulk of the work concerns the cases where G is of exceptional type. For these we produce a set of nilpotent orbit representatives e e and perform explicit calculations. For each such e we obtain not only the Lie algebra of Z(CG(e)), but in fact the whole upper central series of the Lie algebra of Ru(CG(e)); we write each term of this series explicitly as a direct sum of indecomposable tilting modules for a reductive complement to Ru(CG(e)) in CG(e)∘.