Unipotent elements and generalized exponential maps

Abstract

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 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 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.

Integration questions in separably good characteristics

“…This has been observed by Sobaje in [Sob15]. Furthermore, the non-separably good characteristics case is addressed in [Sob18, § 7]. The author explains there why Springer isomorphisms fail to exist without this assumption.…”

“…When is simple, Sobaje reminds the reader of the existence of Springer isomorphisms in separably good characteristics (see [Sob15, Theorem 1.1 and Remark 2]). Moreover, in [Sob18, § 7] the author investigates the non-separably good characteristic case. He also emphasises that in separably good characteristics one can always find an isomorphism that restricts to an isomorphism of reduced schemes for any Borel subgroup .…”

“…Note that here is endowed with the group law induced by the Baker–Campbell–Hausdorff law (which is well-defined, see the preamble of [Sei00, § 5]). When , Sobaje explains in [Sob18, Theorem 6.0.2] that there exists a unique Springer isomorphism that restricts to , whose tangent map is the identity and that is compatible with the -power. Note that when , maps satisfying the three aforementioned requirements still exist, but this time, there are many of such maps.…”

“…Note that when , maps satisfying the three aforementioned requirements still exist, but this time, there are many of such maps. Sobaje study and classify in [Sob18, Theorem 6.0.2] a specific class of such maps, the so-called generalised exponential maps.…”