Centres of Centralizers of Unipotent Elements in Simple Algebraic Groups

作者：Lawther R; Testerman D M

来源：Memoirs of the American Mathematical Society, 2011, 210(988): 1-+.

DOI：10.1090/S0065-9266-10-00594-6

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.
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 and perform explicit calculations. For each such e we obtain not only the Lie algebra of Z(C(G)(e)), but in fact the whole upper central series of the Lie algebra of R(u)(C(G)(e)); we write each term of this series explicitly as a direct sum of indecomposable tilting modules for a reductive complement to R(u)(C(G)(e)) in C(G)(e)degrees.

• 出版日期2011-3