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 $A_2$, $G_2$ and $F_4$ defined over $k$, via their rank-$2$ $k$-tori. Simple, simply connected groups of type $A_2$ 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 $A_n$ type groups, as unitary tori. We discuss conditions necessary for a rank-$2$ unitary $k$-torus to embed in simple $k$-groups of type $A_2$, $G_2$ and $F_4$ in terms of the mod-$2$ Galois cohomological invariants attached with these groups. We calculate the number of rank-$2$ $k$-unitary tori generating these algebraic groups (in fact exhibit such tori explicitly). The results in this paper and our earlier work (Invariants mod-2 and subgroups of G_2 and F_4, J. Alg. 411 (2014) 312- 336) show that the mod-$2$ invariants of groups of type $G_2,F_4$ and $A_2$ are controlled by their $k$-subgroups of type $A_1$ and $A_2$ as well as the unitary $k$-tori embedded in them.