Weyl integration formula

Intuition [Beginner]

A compact Lie group is a group that is also a smooth shape with no boundary and finite volume, like a circle or a sphere. Integrating a function over such a group means averaging over all its elements.

The Weyl integration formula says: to integrate a function that depends only on conjugacy class (i.e., $f(g)=f(hg{h}^{-1})$ for all $h$), you only need to integrate over a maximal torus — a "great circle" subgroup inside the group. But you must include a weight factor.

The weight factor is the Weyl denominator squared. It measures how many group elements are conjugate to a given torus element. Near elements with extra symmetry (where several roots vanish), the conjugacy class shrinks and the weight is small.

Visual [Beginner]

A compact Lie group drawn as a rounded shape. Inside it, a circle represents a maximal torus. Arrows fan out from each point of the torus to its conjugacy class (a "orbit" of conjugation). The Weyl denominator measures the density of these orbits: near generic points the orbits are large, near singular points they collapse.

The picture shows that the torus parametrises the conjugacy classes, and the Weyl denominator records the volume of each class.

Worked example [Beginner]

Consider the group $G=SU(2)$ of $2\times 2$ unitary matrices with determinant $1$. A maximal torus is the diagonal subgroup $T=\{\mathrm{diag}(e^{i\theta },e^{-i\theta })\}$.

Step 1. Every element of $SU(2)$ is conjugate to exactly one diagonal matrix $\mathrm{diag}(e^{i\theta },e^{-i\theta })$ with $0\le \theta \le \pi$.

Step 2. The Weyl denominator is $\Delta (\theta )=e^{i\theta }-e^{-i\theta }=2i\sin \theta$.

Step 3. The Weyl integration formula for $SU(2)$ gives the average of $f$ over the group as a weighted average over the angle $\theta$. The weight is ${\sin }^2\theta$: the total of $f$ over $SU(2)$ equals $\frac{2}{\pi }$ times the total of $f(\theta ){\sin }^2\theta$ over $0\le \theta \le \pi$.

What this tells us: the factor ${\sin }^2\theta$ vanishes at $\theta =0$ and $\theta =\pi$, reflecting that the identity and $-I$ are each their own conjugacy class.

Check your understanding [Beginner]

Formal definition [Intermediate+]

Definition (Maximal torus). A maximal torus in a compact connected Lie group $G$ is a maximal connected abelian subgroup $T$. It is isomorphic to $(S^1{)}^r$ where $r=\mathrm{rank}(G)$, and any two maximal tori are conjugate.

Definition (Weyl group). The Weyl group $W=N_G(T)/T$ is the quotient of the normaliser of $T$ by $T$ itself. It is a finite group acting on $T$ by conjugation.

Definition (Weyl denominator). Let $R^{+}$ be a choice of positive roots for $(G,T)$. The Weyl denominator is

$$\Delta (t)=\prod _{\alpha \in R^{+}}(t^{\alpha /2}-t^{-\alpha /2}),t\in T.$$

Definition (Class function). A continuous function $f:G\to \mathbb{C}$ is a class function if $f(hg{h}^{-1})=f(g)$ for all $g,h\in G$.

Counterexamples to common slips [Intermediate+]

• Forgetting the Weyl group order. The formula includes the factor $1/|W|$ because each conjugacy class intersects $T$ in $|W|$ points (counted with multiplicity). Dropping this factor gives an incorrect normalisation.
• Applying to non-class functions. The Weyl integration formula only holds for class functions. A general function on $G$ does not factor through conjugacy classes.
• Confusing the Weyl denominator with the Vandermonde determinant. They are related but not identical: the Weyl denominator is a product over roots, while the Vandermonde is a product over eigenvalues. For $U(n)$ they coincide; for other groups they differ.

Key theorem with proof [Intermediate+]

Theorem (Weyl integration formula). Let $G$ be a compact connected Lie group with maximal torus $T$ and Weyl group $W$. Normalize the Haar measures so that $\mathrm{vol}(G)=1$ and $\mathrm{vol}(T)=1/|W|$. For any continuous class function $f$ on $G$:

$${\int }_{G}f(g)dg=\frac{1}{|W|}{\int }_{T}f(t)|\Delta (t){|}^{2}dt.$$

Proof. Consider the smooth map $\varphi :G/T\times T\to G$ defined by $(gT,t)\mapsto gt{g}^{-1}$. This map is surjective: every element of $G$ is conjugate to an element of $T$ (conjugacy of maximal tori). The map is $|W|$-to-$1$: for each $w\in W$, the element $w\cdot t$ is conjugate to $t$ via an element of $N_G(T)$, giving $|W|$ preimages for generic $t$.

Compute the Jacobian of $\varphi$ at a point $(gT,t)$. The tangent space $T_{(gT,t)}(G/T\times T)$ maps to $T_{gt{g}^{-1}}G$. Decompose the Lie algebra $\mathfrak{g}=\mathfrak{t}\oplus {⨁}_{\alpha \in R}{\mathfrak{g}}_{\alpha }$ into the Cartan decomposition. The differential $d\varphi$ maps the $T$-direction to itself (via the identity on $\mathfrak{t}$) and maps the $G/T$-directions according to the root decomposition.

For each positive root $\alpha$, the root space ${\mathfrak{g}}_{\alpha }$ and ${\mathfrak{g}}_{-\alpha }$ contribute a $2\times 2$ block to the differential. Concretely, the eigenvalues of $\mathrm{Ad}(t)$ on ${\mathfrak{g}}_{\alpha }\oplus {\mathfrak{g}}_{-\alpha }$ are $t^{\alpha }$ and $t^{-\alpha }$. The Jacobian factor from this block is $t^{\alpha /2}-t^{-\alpha /2}$ (the difference of the eigenvalues after normalising the measure on $G/T$).

Taking the product over all positive roots:

$$|\det d\varphi {|}^2=\prod _{\alpha \in R^{+}}|t^{\alpha /2}-t^{-\alpha /2}{|}^2=|\Delta (t){|}^2.$$

Now apply the change-of-variables formula for the $|W|$-to-$1$ map $\varphi$. Since $f$ is a class function, $f(gt{g}^{-1})=f(t)$ for all $g$, so:

$${\int }_{G}f(g)dg={\int }_{G/T\times T}f(gt{g}^{-1})|\det d\varphi {|}^{2}d(gT)dt=\frac{1}{|W|}{\int }_{T}f(t)|\Delta (t){|}^{2}dt.$$

The factor $1/|W|$ accounts for the $|W|$-fold covering. $\square$

Bridge. The Weyl integration formula builds toward the Weyl character formula, where the denominator $\Delta (t)$ cancels the numerator's vanishing at singular elements to produce the irreducible character. The foundational reason the formula works is that the conjugation map $\varphi :G/T\times T\to G$ has Jacobian $|\Delta (t){|}^2$, and this is exactly the bridge between the geometry of conjugacy classes and the root system of the Lie algebra. The pattern appears again in random matrix theory where the same Jacobian factor produces the sine-kernel correlations for unitary matrices. The central insight is that integrating a class function reduces to a weighted integral over the torus, and this identifies the representation theory of $G$ with harmonic analysis on $T$.

Exercises [Intermediate+]

Advanced results [Master]

Theorem 1 (Weyl character formula as a torus identity). The irreducible character ${\chi }_{\lambda }$ of $G$ with highest weight $\lambda$ satisfies

$${\chi }_{\lambda }(t)=\frac{{\sum }_{w\in W}\det (w)t^{w(\lambda +\rho )}}{\Delta (t)}$$

where $\rho =\frac{1}{2}{\sum }_{\alpha \in R^{+}}\alpha$ is the Weyl vector. This is a ratio of alternating functions on $T$, and the Weyl denominator cancels the vanishing of the numerator at the walls of the Weyl chamber. The formula is a finite Fourier series on the torus.

Theorem 2 (Character orthogonality from the integration formula). For distinct irreducible characters ${\chi }_{\lambda }$ and ${\chi }_{\mu }$:

$${\int }_G{\chi }_{\lambda }(g)\overline{{\chi }_{\mu }(g)}dg=\frac{1}{|W|}{\int }_T{\chi }_{\lambda }(t)\overline{{\chi }_{\mu }(t)}|\Delta (t){|}^{2}dt=0.$$

Substituting the character formula and using the alternating property under $W$, the integral reduces to an orthogonality relation for the exponentials $t^{\lambda +\rho }$, which are orthogonal on $T$ by Fourier analysis.

Theorem 3 (Peter-Weyl connection). The Peter-Weyl theorem states that the matrix coefficients of all irreducible unitary representations span a dense subspace of $L^2(G)$. The Weyl integration formula identifies the $L^2$-norm of class functions with a weighted $L^2$-norm on $T$, and the character formula shows that the irreducible characters form an orthonormal basis for the class functions in $L^2(G)$.

Theorem 4 (Dimension formula). Evaluating the Weyl character formula at $t=e$ (the identity) via L'Hopital gives the Weyl dimension formula:

$$\dim V_{\lambda }=\prod _{\alpha \in R^{+}}\frac{(\lambda +\rho ,\alpha )}{(\rho ,\alpha )}.$$

This is a direct consequence of the Weyl integration formula applied to ${\chi }_{\lambda }(e)=\dim V_{\lambda }$.

Theorem 5 (Connection to random matrix theory). For $G=U(n)$, the Weyl integration formula becomes

$${\int }_{U(n)}fdU=\frac{1}{n!}{\int }_{[0,2\pi {]}^n}f({\theta }_1,\dots ,{\theta }_n)\prod _{j<k}|e^{i{\theta }_j}-e^{i{\theta }_k}{|}^2\frac{d{\theta }_1\cdots d{\theta }_n}{(2\pi {)}^n}.$$

The factor ${\prod }_{j<k}|e^{i{\theta }_j}-e^{i{\theta }_k}{|}^2$ is the Vandermonde determinant squared, which controls eigenvalue correlations. This is the starting point for the Wigner-Dyson ensembles in random matrix theory.

Theorem 6 (Freudenthal's formula). The Casimir eigenvalue on $V_{\lambda }$ is $(\lambda ,\lambda +2\rho )$, which can be computed using the inner product on the weight lattice. Combined with the Weyl integration formula, this gives a purely algebraic computation of all characters.

Synthesis. The Weyl integration formula is the foundational reason that the representation theory of compact Lie groups reduces to harmonic analysis on a torus. The central insight is that class functions on $G$ are in bijection with $W$-invariant functions on $T$, and the integration formula identifies the $L^2$ inner product on class functions with a weighted inner product on $T$. This is exactly the structure that makes the Weyl character formula a finite Fourier expansion, and the bridge is that the Weyl denominator $|\Delta (t){|}^2$ is the Jacobian of the conjugation map from $G/T\times T$ to $G$. Putting these together, the Weyl character formula, the dimension formula, and character orthogonality all follow from the integration formula combined with Fourier analysis on the torus. This pattern recurs in the Langlands program where automorphic forms on reductive groups are analysed via their restrictions to maximal tori.

Full proof set [Master]

Proposition 1 (Jacobian of the conjugation map). The map $\varphi :G/T\times T\to G$, $(gT,t)\mapsto gt{g}^{-1}$, has Jacobian determinant $|\Delta (t){|}^2$ at each point $(gT,t)$.

Proof. Fix a $G$-invariant inner product on $\mathfrak{g}$ (which exists by averaging any inner product against the Haar measure on $G$). Choose an orthonormal basis of $\mathfrak{t}$ and root vectors $E_{\alpha }$ normalised so that $(E_{\alpha },E_{-\alpha })=1$.

The tangent space to $G/T$ at $gT$ is identified with ${⨁}_{\alpha \in R}{\mathfrak{g}}_{\alpha }$ (via left translation by $g$). The tangent space to $T$ at $t$ is $\mathfrak{t}$.

Compute $d\varphi$ on each piece. On $\mathfrak{t}$: $d\varphi (X)=X$ (the Lie algebra of $T$ maps to itself by the identity, since $\varphi (gT,t\exp (sX))=gt\exp (sX)g^{-1}=(gt{g}^{-1})\exp (s\cdot \mathrm{Ad}(g)X)$ and $\mathrm{Ad}(g)$ preserves $\mathfrak{t}$ when $g$ normalises $T$).

On ${\mathfrak{g}}_{\alpha }$: $d\varphi (E_{\alpha })=(t^{\alpha }-1)\cdot \mathrm{Ad}(g)E_{\alpha }$. The factor $t^{\alpha }-1$ comes from differentiating $gt\exp (s{E}_{\alpha })t^{-1}{g}^{-1}=gt{g}^{-1}\cdot \exp (s\cdot t^{\alpha }\cdot \mathrm{Ad}(g)E_{\alpha })$ and subtracting the $\mathfrak{t}$-component.

The determinant from the pair $({\mathfrak{g}}_{\alpha },{\mathfrak{g}}_{-\alpha })$ is $|t^{\alpha }-1{|}^2=|t^{\alpha /2}{|}^2|t^{\alpha /2}-t^{-\alpha /2}{|}^2=|t^{\alpha /2}-t^{-\alpha /2}{|}^2$ since $|t^{\alpha /2}|=1$ on the compact torus.

Taking the product over all positive roots:

$$|\det d\varphi {|}^2=\prod _{\alpha \in R^{+}}|t^{\alpha /2}-t^{-\alpha /2}{|}^2=|\Delta (t){|}^2.\square$$

Proposition 2 (Character orthogonality). For irreducible characters ${\chi }_{\lambda }\ne {\chi }_{\mu }$ of $G$: ${\int }_G{\chi }_{\lambda }\overline{{\chi }_{\mu }}dg=0$.

Proof. By the Weyl integration formula:

$${\int }_G{\chi }_{\lambda }\overline{{\chi }_{\mu }}dg=\frac{1}{|W|}{\int }_T{\chi }_{\lambda }(t)\overline{{\chi }_{\mu }(t)}|\Delta (t){|}^{2}dt.$$

Substitute the Weyl character formula: ${\chi }_{\lambda }=A_{\lambda +\rho }/\Delta$ where $A_{\nu }={\sum }_{w\in W}\det (w)t^{w\nu }$. Then ${\chi }_{\lambda }\overline{{\chi }_{\mu }}|\Delta {|}^2=A_{\lambda +\rho }\overline{A_{\mu +\rho }}$. Both $A_{\lambda +\rho }$ and $A_{\mu +\rho }$ are alternating functions under $W$. Their product is $W$-invariant. Expanding:

$$\frac{1}{|W|}{\int }_T{A}_{\lambda +\rho }(t)\overline{A_{\mu +\rho }(t)}dt={\int }_T{t}^{\lambda +\rho }\overline{t^{\mu +\rho }}dt$$

by $W$-invariance (the cross terms cancel). For $\lambda \ne \mu$, the characters $t^{\lambda +\rho }$ and $t^{\mu +\rho }$ are distinct monomials on $T$, hence orthogonal by Fourier orthogonality. $\square$

Connections [Master]

• Weyl complete reducibility 07.06.22. The Weyl integration formula depends on the existence of a bi-invariant Haar measure on compact Lie groups, which is the analytic input behind Weyl's complete reducibility proof (the "unitary trick"). Complete reducibility ensures the irreducible characters span the class functions, and the integration formula makes them orthogonal.

• Non-abelian Fourier transform 07.01.09. The Fourier transform on a compact group decomposes functions into matrix coefficients of irreducible representations. The Weyl integration formula reduces the non-abelian Fourier inversion formula to Fourier analysis on the torus, where the Weyl denominator provides the correct measure. The bridge is that the Peter-Weyl decomposition of $L^2(G)$ restricts to a decomposition of the class functions indexed by characters on $T$.

• Peter-Weyl theorem 07.07.02. The Peter-Weyl theorem provides the abstract framework: matrix coefficients span $L^2(G)$. The Weyl integration formula is the concrete computational tool for the class-function sector, giving explicit formulas for the characters and their orthogonality. Together they identify the representation theory of $G$ with Fourier analysis on $T/W$.

Historical & philosophical context [Master]

Hermann Weyl introduced the integration formula in his 1925-26 four-part paper ^{[Weyl 1925]} as the central analytic tool in his derivation of the character formula for compact semisimple Lie groups. The formula reduced the computation of characters from an integration over the full group to a finite combinatorial calculation on the maximal torus.

The connection to the Vandermonde determinant for unitary groups was recognised by Weyl himself and later became foundational in random matrix theory through the work of Wigner, Dyson, and Mehta. The integration formula is the bridge between representation theory and probability: it describes the joint eigenvalue distribution of random unitary matrices.

The modern exposition, emphasising the Jacobian of the conjugation map and the root-theoretic structure of the Weyl denominator, is due to Chevalley and Helgason. Helgason's Groups and Geometric Analysis ^[Helgason] remains the canonical reference for the formula in full generality.

Bibliography [Master]

@article{weyl1925,
  author = {Weyl, Hermann},
  title = {Theorie der Darstellung kontinuierlicher halb-einfacher Gruppen durch lineare Transformationen},
  journal = {Math. Zeitschrift},
  volume = {23, 24},
  year = {1925--1926}
}

@book{helgason-gga,
  author = {Helgason, Sigurdur},
  title = {Groups and Geometric Analysis},
  publisher = {Academic Press},
  year = {1984}
}

@book{knapp-lie,
  author = {Knapp, Anthony W.},
  title = {Lie Groups Beyond an Introduction},
  publisher = {Birkh{\"a}user},
  year = {1996}
}

@book{brocker-tomdieck,
  author = {Br{\"o}cker, Theodor and tom Dieck, Tammo},
  title = {Representations of Compact Lie Groups},
  publisher = {Springer},
  year = {1985}
}

@book{hall-lie,
  author = {Hall, Brian C.},
  title = {Lie Groups, Lie Algebras, and Representations},
  publisher = {Springer},
  year = {2015}
}