Orthogonal groups containing a given maximal torus

Abstract

Let k be a field of characteristic different from 2 and let T be a fixed k-torus of dimension n. In this paper we study faithful k-representations ρ : T → SO (A,σ) , where ( A, σ) is a central simple algebra of degree 2 n with orthogonal involution σ. Note that in this case ρ( T ) is a maximal torus in SO (A,σ) . We are interested in describing the pairs ( A, σ) for which there is such a representation. We compute invariants for these algebras (discriminant and Clifford algebra), which are sufficient to determine their isomorphism class when I 3( k)=0 by a theorem of Lewis and Tignol. The first part of the paper is devoted to the case where A is split over k and an application to a theorem of Feit on orthogonal groups over Q is given.