Example

Take G to be the group $\mathbf{S}\mathbf{U}(2,\mathbf{C})$ and T to be the subgroup of diagonal matrices (§3, no. 6); identify $\mathfrak{t}$ with $\mathbf{R}$ by the choice of basis $\{iH\}$ of $\mathfrak{t}($loc. cit.). Put $A = 0, \pi$; this is an alcove of $\mathfrak{t}$. The interval $A = 0, \pi$ can be identified with the space of conjugacy classes of G, the element $\theta$ of

$(e^{i\theta}$ 0 $)$

$\overline{A}$ corresponding to the conjugacy class of $-_{i\theta}$ . Let $d\theta$ be Lebesgue

0 $e$

measure on $0, \pi$; it follows from the preceding that the image on $\overline{A}$ of the Haar measure on G is the measure $\frac{2}{\pi}$ sin$^2\theta d\theta$.