---
tag: QMH530N27M
kind: statement
role: theorem
name: Weyl integration formula
msc: ["22E30", 22C05, 43A77]
uses: [9TFC6ADC1C, 3JVGZVVTB0, EZGTEK3HQN, VY0HXD3GC4, P6ZP0TE07M]
about: [3JVGZVVTB0, VY0HXD3GC4]
sources:
  - url: https://en.wikipedia.org/wiki/Weyl_integration_formula
same_as:
  wikidata: Q96414618
  wikipedia: Weyl integration formula
---

## Statement

Let $G$ be a connected [[compact Lie group|9TFC6ADC1C]] with Lie algebra $\mathfrak{g}$.
Let $T$ be a [[maximal torus|3JVGZVVTB0]] with Lie algebra $\mathfrak{t}$, and let $W$ be the [[Weyl group|EZGTEK3HQN]].
Give $G$, $T$ and $G/T$ their invariant probability measures $dg$, $dt$ and $d\dot g$ (normalised [[Haar measures|VY0HXD3GC4]]).
For $t \in T$ let
$$\delta(t) = \det\left(\id - \operatorname{Ad}(t^{-1})\big|_{\mathfrak{g}/\mathfrak{t}}\right),$$
where $\operatorname{Ad}$ is the [[adjoint representation|P6ZP0TE07M]].
Then for every continuous function $f \colon G \to \C$,
$$\int_G f(g) \, dg = \frac{1}{\abs{W}} \int_T \delta(t) \int_{G/T} f(g t g^{-1}) \, d\dot g \, dt.$$
In particular, if $f$ is a class function, then $\int_G f(g) \, dg = \frac{1}{\abs{W}} \int_T f(t)\,\delta(t) \, dt$.

## 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 $\frac{1}{\pi} \cdot \pi = 1$.

## Explanations

### L1 human

To integrate a function over a compact Lie group, one can first average over conjugates and then integrate over a maximal torus.
The torus is much smaller, but its points must be weighted: points where many conjugates pile up get more weight.
The weight is an explicit function $\delta$, and the formula divides by the number of times the torus is covered.

### L2 human

The formula is the main tool for computing integrals of characters.
With the [[Schur orthogonality relations|ETNDTECY97]] 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|230F5FXPMQ]].
In random matrix theory, the case $G = U(n)$ gives the joint density of eigenvalues of a Haar-random unitary matrix, proportional to $\prod_{j < k} \abs{z_j - z_k}^2$.