Unipotent elements and generalized exponential maps

Let G be a simple and simply connected algebraic group over an algebraically closed field k of characteristic p>0. Assume that p is good for the root system of G and that the covering map G_sc→G is separable. In previous work we proved the existence of a (not necessarily unique) Springer isomorphism for G that behaved like the exponential map on the restricted nullcone of G. In the present paper we give a formal definition of these maps, which we call ‘generalized exponential maps.’ We provide an explicit and uniform construction of such maps for all root systems, demonstrate their existence over Z_(p), and give a complete parameterization of all such maps. One application is that this gives a uniform approach to dealing with the “saturation problem” for a unipotent element u in G, providing a new proof of the known result that u lies inside a subgroup of C_G(u) that is isomorphic to a truncated Witt group. We also develop a number of other explicit and new computations for g and for G. This paper grew out of an attempt to answer a series of questions posed to us by P. Deligne, who also contributed several of the new ideas that appear here.