Characteristic classes in the Chow ring

Let G be a reductive algebraic group over an algebraically closed field k. An algebraic characteristic class of degree i for principal G-bundles on schemes is a function c assigning to each principal G-bundle E → X an element c(E) in the Chow group AX, natural with respect to pullbacks. These classes are analogous to topological characteristic classes (which take values in cohomology), and two natural questions arise. First, for smooth schemes there is a natural map from the Chow ring to cohomology, and we can ask if topological characteristic classes are algebraic. Second, because the notion of algebraic principal bundles on schemes is more restrictive than the notion of topological principal bundles, we can ask if there are algebraic characteristic classes which do not come from topological ones. For example, for rank n vector bundles (corresponding to G = GL(n)), the only topological characteristic classes are polynomials in the Chern classes, which are represented by algebraic cycles, but until now it was not known if these were the only algebraic characteristic classes ([V, Problem (2.4)]). In this paper we describe the ring of algebraic characteristic classes and answer these questions. One subtlety which does not occur in topology is that there are two natural notions of algebraic principal G-bundles on schemes, those which are locally trivial in the étale topology and those which are locally trivial in the Zariski topology. (Of course for groups which are special in the sense of [Sem-Chev], all principal bundles are Zariski locally trivial. Tori, GL(n), SL(n), and Sp(2n) are all examples of special groups.) Let C(G) denote the ring of characteristic classes for principal G-bundles locally trivial in the