2017/08/14 by Paul Sobaje, Sobaje, Paul
Mathematics · #20G05 (Secondary) #20G07 (Primary) #FOS: Mathematics #Group Theory (math.GR) #Representation Theory (math.RT) #math.GR #math.RT #msc:20G05 #msc:20G07
paper · pdf · doi:10.48550/arxiv.1708.04153
28 pages, comments welcome!
arxiv created 2017/08/14 · arxiv updated 2017/08/15
Let G be a simple and simply connected algebraic group over an algebraically closed field \Bbbk of characteristic p>0. Assume that p is good for the root system of G and that the covering map Gsc → 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 resticted 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 CG(u) that is isomorphic to a truncated Witt group. We also develop a number of other explicit and new computations for \mathfrakg 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.