Embeddings of rank-2 tori in algebraic groups

Abstract

Let k be a field of characteristic different from 2 and 3. In this paper we study connected simple algebraic groups of type A2, G2 and F4 defined over k, via their rank-2 k-tori. Simple, simply connected groups of type A2 play a pivotal role in the study of exceptional groups and this aspect is brought out by the results in this paper. We refer to tori, which are maximal tori of An type groups, as unitary tori. We discuss conditions necessary for a rank-2 unitary k-torus to embed in simple k-groups of type A2, G2 and F4 in terms of the mod-2 Galois cohomological invariants attached with these groups. The results in this paper and our earlier paper ([6]) show that the mod-2 invariants of groups of type G2,F4 and A2 are controlled by their k-subgroups of type A1 and A2 as well as the unitary k-tori embedded in them.