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

Abstract

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