Remark 3

Assume that G is simply-connected. Let A be an alcove of $\mathfrak{t}$, and $dx$ the Haar measure on $\mathfrak{t}$ such that $\int_Adx= 1$. Then the measure $\nu$ can also be obtained by transporting the measure $\frac{1}{w(G)}\prod_{\alpha\in R_+(G,T)}4$ sin$^2\pi \widehat{\alpha}(x)dx$ on $\overline{A}$ by

the homeomorphism $\overline{A}\rightarrow T/W ($§5, no. 2, Cor. 1 of Prop. 2).