Let G be a reductive linear
algebraic group defined over an algebraically closed field K. An element x ∈ G is
called regular if dimZG(x) is the rank of G—which is the smallest possible dimension
for a centralizer—otherwise x is called irregular. T. A. Springer has shown that if
x ∈ G is regular, then ZG(x)0 is abelian. In this paper, we show that when
charK is good for G, the converse is also true—if x ∈ G is irregular, then
ZG(x)0 is nonabelian. In the course of the proof of this, we show that if G is a
classical group or G2 (charK good), then dimZ(ZG(x)0) is at most rank G.
Further, if G = Sp2n(K)(charK≠2), then Z(ZG(x)0) consists of polynomials in
x.
