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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 positiveroots 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 $e^{\frac{2 \pii}{h}}$ ($h$ the Coxeter number) is the Coxeter conjugacy class of the dualgroup $\widehat{G}$. This point of view gives a rather transparent proof of atheorem of Kostant on the character values of irreducible finite dimensionalrepresentations of $G$ at the Coxeter element: the proof amounting to the factthat in $\widehat{G}_{sc}$, the simply connected cover of $\widehat{G}$, thereis a unique regular conjugacy class whose image in $\widehat{G}$ has order $h$(which is the Coxeter conjugacy class).

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