Real orthogonal representations of algebraic groups

The purpose of this paper is to determine explicitly, nondegenerate real symmetric bilinear forms invariant under a real absolutely irreducible representation of a real semisimple algebraic group, G G . If G G is split, we construct an extension G ∗ {G^ \ast } containing G G and those outer automorphisms of G G fixing the highest weight of the representation. The representation is then extended to G ∗ {G^ \ast } and the form is described in terms of the character of this extension. The case of a nonsplit algebraic group is then reduced to the above. The corresponding problem for representations by matrices over the real quaternion division algebra is also considered using similar methods.