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

Abstract

Abstract Interchanging the character and co-character groups of a torus T over a field k introduces a contravariant functor T → T ∨. Interpreting ρ : T(ℂ) → ℂ×, 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 ρ∨ : ℂ× → T ∨(ℂ) for which ρ∨(e 2πi/h ) (h the Coxeter number) is the Coxeter conjugacy class of the dual group G ∨(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(ℂ), the simply connected cover of G ∨(ℂ), there is a unique regular conjugacy class whose image in G ∨(ℂ) has order h (which is the Coxeter conjugacy class).