Unipotent elements and generalized exponential maps

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 CG(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.