  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).
