SPRINGER ISOMORPHISMS IN CHARACTERISTIC p

Abstract

Let G be a simple algebraic group over an algebraically closed field of characteristic p, and assume that p is a separably good prime for G. Let P be a parabolic subgroup whose unipotent radical U P has nilpotence class less than p. We show that there exists a particularly nice Springer isomorphism for G which restricts to a certain canonical isomorphism Lie(U P ) ∼ − → U P defined by J.-P. Serre. This answers a question raised both by G. McNinch in [M2], and by J. Carlson et. al in [CLN]. For the groups SLn, SOn, and Sp 2n , viewed in the usual way as subgroups of GLn or GL 2n , such a Springer isomorphism can be given explicitly by the Artin-Hasse exponential series.

Unipotent elements and generalized exponential maps

“…These issues, and much more, are resolved in this paper, which significantly improves upon the results in [22], and contains new techniques for moving back and forth between characteristics 0 and p. Some of the clearest and best insights here have been contributed by Deligne, who has graciously allowed us to include several of his arguments along with our own (see "Acknowledgments," Remark 3.3.3, and the beginning of Section 4.3).…”

“…We considered in [22] the problem of finding Springer isomorphisms for G that were "even more like" the exponential map, with our focus on what these isomorphisms should do upon restriction to the subvariety N 1 (g) ⊆ N (g) of all X such that X [p] = 0. In particular, we proved the existence of Springer isomorphisms that, when restricted to the unipotent radicals of certain parabolic subgroups of G, agreed with the isomorphism coming from the exponential map in characteristic 0 (see [20]).…”

“…This definition is motivated by the well-known method of embedding truncated Witt groups into GL n with the Artin-Hasse exponential series (see Section 4.2). In fact, we used the Artin-Hasse exponential series in [22] to obtain generalized exponential maps (not yet formally named as such) for groups of type A − D. The key new idea here is that this can be generalized in a natural way to groups of exceptional type. We prove the following theorems, the second of which is the main theorem of this paper.…”

“…We have adapted our own proofs in light of his, opting for the clearest and strongest results, and this paper is significantly better because of the ideas he has shared. We are very grateful to Professor Deligne for allowing us to include his work, for helpful conversations about this material, and for his interest in [22].…”

“…The Artin-Hasse Exponential For Classical Groups. The definition above (and the work in [22]) was motivated by the following well-known fact for GL n . First, recall that the Artin-Hasse exponential series for p is the formal power series…”