Weyl integration formula

Remarks

The map $G/T \times T \to G$, $(gT, t) \mapsto gtg^{-1}$, is surjective by the conjugacy of maximal tori, and over the dense set of regular elements it is a covering of degree $\abs{W}$. Its Jacobian at $(gT, t)$ is $\delta(t)$, which gives the formula by change of variables. In terms of roots, $\delta(t) = \prod_{\alpha} (1 - t^{-\alpha})$, the product over all roots $\alpha$, which equals $\abs{\prod_{\alpha > 0} (t^{\alpha/2} - t^{-\alpha/2})}^2$ when this makes sense, so $\delta \ge 0$. Example: for $G = SU(2)$ and $t = \operatorname{diag}(e^{i\theta}, e^{-i\theta})$, $\delta(t) = \abs{e^{i\theta} - e^{-i\theta}}^2 = 4\sin^2\theta$ and $\abs{W} = 2$. So for a class function $\int_{SU(2)} f \, dg = \frac{1}{\pi}\int_0^{2\pi} f(t_\theta) \sin^2\theta \, d\theta$; for $f = 1$ this gives $\frac{1}{\pi} \cdot \pi = 1$.

Going deeper

For a student

The formula is the main tool for computing integrals of characters. With the Schur orthogonality relations it shows that an irreducible character times the Weyl denominator is an alternating sum of characters of $T$. This is the core of the proof of the Weyl character formula. In random matrix theory, the case $G = U(n)$ gives the joint density of eigenvalues of a Haar-random unitary matrix, proportional to $\prod_{j < k} \abs{z_j - z_k}^2$.