The characters of semisimple Lie groups

If Fr^0 one can prove that there is only a finite number of possibilities for Tr (see [6(h), Theorem 3]).Lemma 1 (Mostow).Let fjo be a Cartan subalgebra of g0 and fj the complexification of h0 in g.Then there exists a compact real form u of g with the following two properties:1. »j(u)=U, 2. hf^u is a Cartan subalgebra of u.Now let u be a fixed compact real form of g such that ij(u) =u and let