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  "body": "\n## Statement\n\nLet $G$ be a connected [[compact Lie group|9TFC6ADC1C]] with Lie algebra $\\mathfrak{g}$.\nLet $T$ be a [[maximal torus|3JVGZVVTB0]] with Lie algebra $\\mathfrak{t}$, and let $W$ be the [[Weyl group|EZGTEK3HQN]].\nGive $G$, $T$ and $G/T$ their invariant probability measures $dg$, $dt$ and $d\\dot g$ (normalised [[Haar measures|VY0HXD3GC4]]).\nFor $t \\in T$ let\n$$\\delta(t) = \\det\\left(\\id - \\operatorname{Ad}(t^{-1})\\big|_{\\mathfrak{g}/\\mathfrak{t}}\\right),$$\nwhere $\\operatorname{Ad}$ is the [[adjoint representation|P6ZP0TE07M]].\nThen for every continuous function $f \\colon G \\to \\C$,\n$$\\int_G f(g) \\, dg = \\frac{1}{\\abs{W}} \\int_T \\delta(t) \\int_{G/T} f(g t g^{-1}) \\, d\\dot g \\, dt.$$\nIn particular, if $f$ is a class function, then $\\int_G f(g) \\, dg = \\frac{1}{\\abs{W}} \\int_T f(t)\\,\\delta(t) \\, dt$.\n\n## Remarks\n\nThe 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 $\\abs{W}$.\nIts Jacobian at $(gT, t)$ is $\\delta(t)$, which gives the formula by change of variables.\nIn terms of roots, $\\delta(t) = \\prod_{\\alpha} (1 - t^{-\\alpha})$, the product over all roots $\\alpha$, which equals $\\abs{\\prod_{\\alpha \u003e 0} (t^{\\alpha/2} - t^{-\\alpha/2})}^2$ when this makes sense, so $\\delta \\ge 0$.\nExample: 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 $\\abs{W} = 2$.\nSo 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 $\\frac{1}{\\pi} \\cdot \\pi = 1$.\n\n## Explanations\n\n### L1 human\n\nTo integrate a function over a compact Lie group, one can first average over conjugates and then integrate over a maximal torus.\nThe torus is much smaller, but its points must be weighted: points where many conjugates pile up get more weight.\nThe weight is an explicit function $\\delta$, and the formula divides by the number of times the torus is covered.\n\n### L2 human\n\nThe formula is the main tool for computing integrals of characters.\nWith the [[Schur orthogonality relations|ETNDTECY97]] it shows that an irreducible character times the Weyl denominator is an alternating sum of characters of $T$.\nThis is the core of the proof of the [[Weyl character formula|230F5FXPMQ]].\nIn random matrix theory, the case $G = U(n)$ gives the joint density of eigenvalues of a Haar-random unitary matrix, proportional to $\\prod_{j \u003c k} \\abs{z_j - z_k}^2$.\n",
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