Let \(G\) be a linear algebraic group over an algebraically closed field \(\Bbbk\) of characteristic \(p>0\) and let \(\mathfrak{g}\) be its associated Lie algebra. This paper defines and studies ‘generalized exponential maps,’ which are Springer isomorphisms between the unipotent elements of \(G\) and the nilpotent elements of \(\mathfrak{g}\) which behave similarly to the usual exponential map. In particular, it is demonstrated that such maps exist over \(\mathbb{Z}_{(p)}\). The paper uses these maps to address the “saturation problem” for certain unipotent elements of \(G\). Furthermore, a full parameterization of all generalized exponential maps for \(G\) is given.
After addressing preliminaries in Section 2, Section 3 constructs all Springer isomorphisms for \(G\). Using \(\mathrm{SL}_n\) as a model, where every Springer isomorphism is given by \(X \mapsto I_n + a_1 X + \dots + a_{n-1}X^{n-1}, a_1 \in \Bbbk^\times\), the author shows that the variety of Springer isomorphisms for \(G\) identifies with the variety \(\Bbbk^\times \times \Bbbk^{r-1}\), where \(r\) is the rank of \(G\). To obtain an operation analogous to \(X\mapsto X^i\), the next construction is of “multiplicative power” morphisms. Using representations (natural for classical type and adjoint for exceptional type) which have the feature that the trace form on the matrix algebra restricts to a non-degenerate symmetric bilinear form on the embedded Lie algebra, the author obtains multilinear maps \((\mathfrak{g}_{\mathbb{Z}_{(p)}})^i \rightarrow \mathfrak{g}_{\mathbb{Z}_{(p)}}\) and uses them to define the multiplicative power maps on \(\mathfrak{g}_{\mathbb{Z}_{(p)}}\).
Section 4 defines and constructs generalized exponential maps for \(G\) of all types. First, using a closed embedding of affine group schemes \(G_{\mathbb{Z}_{(p)}}\rightarrow \mathrm{GL}_{n,\mathbb{Z}_{(p)}}\) and the Artin-Hasse exponential power series applied to \(\mathfrak{gl}_{n,\mathbb{Z}_{(p)}}\), the author builds a generalized exponential map for \(G\) of classical type. The argument is then modified to obtain generalized exponential maps for types E, F, and G.
Section 5 takes a nilpotent element \(X \in \mathfrak{g}\) and using any generalized exponential map obtains a decomposition of the connected center of the centralizer, \(Z(C_G(X))^0\), into a product of truncated Witt groups. This decomposition holds as an isomorphism of algebraic varieties thus strengthening a result of G. M. Seitz [J. Algebra 279, No. 1, 226–259 (2004; Zbl 1078.20051)]. The author also discusses how if the root system of \(G\) is not of type \(\mathrm{D}_{p^n+1}\) then the Witt groups occurring in the decomposition do not depend on the chosen generalized exponential map.
Section 6 parameterizes all generalized exponential maps for \(G\) by fixing one such map and showing how any other can be expressed in terms of the fixed map. The parameterization then arises from analyzing conditions on the coefficients that appear in the expression. Here, again groups \(G\) with root system of type D\(_{p^n+1}\) are an exceptional case.
Lastly, Section 7 discusses the case when the covering map \(G_{\mathrm{sc}} \rightarrow G\) (where \(G_{\mathrm{sc}}\) is the simply connected group isogenous to \(G\)) is not separable, which only occurs in type A. Using observations of Serre and Jay Taylor, the author demonstrates how there are no Springer isomorphisms for \(\mathrm{PGL}_2\) in characteristic 2. The paper concludes with an appendix by Jay Taylor containing computations related to the \(\mathrm{PGL}_2\) example.