Half the sum of positive roots, the Coxeter element, and a theorem of Kostant

Interchanging the character and co-character groups of a torus T over a held k introduces a contravariant functor T -> T-V. Interpreting rho : T(C) -> C-x, half the sum of positive roots for T, a maximal torus in a simply connected semi-simple group G (over C) using this duality, we get a co-character rho(V) : C-x -> T-V(C) for which rho(v)(e(2 pi i/h)) (h the Coxeter number) is the Coxeter conjugacy class of the dual group G(v)(C). This point of view gives a rather transparent proof of a theorem of Kostant on the character values of irreducible finite-dimensional representations of G(C) at the Coxeter conjugacy class: the proof amounting to the fact that in G(sc)(V)(C), the simply connected cover of G(V)(C), there is a unique regular conjugacy class whose image in G(V)(C) has order h (which is the Coxeter conjugacy class).