Subgroups of simple algebraic groups containing elements of fundamental subgroups

Abstract

Let G be a simple algebraic group over an algebraically closed field K of characteristic p. If Σ is the root system of G and U_α is the root subgroup of G corresponding to a long root α∈Σ, then 〈U_α, U_−α〉 is an image of SL₂ and any G-conjugate of this subgroup is called a fundamental subgroup of G. In [LS1], the closed connected semisimple subgroups of G generated by long root elements were determined. Of course, long root elements are unipotent elements of fundamental subgroups. In this paper we consider subgroups of G which are generated by semisimple elements lying in fundamental subgroups.

Our first three results (Theorems 1–3) concern semisimple connected subgroups of G which contain a maximal torus of a fundamental subgroup; we call such a torus a fundamental torus. Notice that for classical groups, the elements in fundamental tori have fixed spaces of small codimension in the natural module; indeed, for the groups SL(V) and Sp(V), the elements of fundamental tori are precisely those semisimple elements with fixed space of codimension 2, while for SO(V), the codimension is 4.

As consequences of these results, we obtain information on subgroups (finite or infinite) of classical groups which are generated by conjugates of a single element of a fundamental torus of order at least 5 (see Theorems 4, 5 and Corollary 6); for example, Theorem 5 determines those finite irreducible subgroups containing such an element which are quasisimple and of Lie type in characteristic p.

We now state our results in detail. Recall from [LS1] that a subsystem subgroup of G is a connected semisimple subgroup which is invariant under a maximal torus of G.