Sub-principal Homomorphisms in Positive Characteristic

LetG be a reductive group over an algebraically closed field of characteristic p, and let u ∈ G be a unipotent element of order p. Suppose that p is a good prime for G. We show in this paper that there is a homomorphism φ : SL2/k → G whose image contains u. This result was first obtained by D. Testerman (J. Algebra, 1995) using case considerations for each type of simple group (and using, in some cases, computer calculations with explicit representatives for the unipotent orbits). The proof we give is free of case considerations (except in its dependence on the Bala-Carter theorem). Our construction of φ generalizes the construction of a principal homomorphismmade by J.-P. Serre in (Invent. Math. 1996); in particular, φ is obtained by reduction modulo p from a homomorphism of group schemes over a valuation ring A in a number field. This permits us to show moreover that the weight spaces of a maximal torus of φ(SL2/k) on Lie(G) are “the same as in characteristic 0”; the existence of a φ with this property was previously obtained, again using case considerations, by Lawther and Testerman (Memoirs AMS, 1999) and has been applied in some recent work of G. Seitz (Invent. Math. 2000).