Compact Lie group

A compact Lie group is a Lie group whose underlying manifold is compact. Averaging any positive inner product over its Haar measure supplies an invariant inner product on each finite-dimensional representation, including the Adjoint representation.

A real torus is the compact Lie group obtained by identifying real vectors differing by integer vectors, with addition induced from Euclidean addition. It is a finite product of circle groups and has normalized Haar measure. Dimension two is the familiar surface torus, but toral dynamics and Fourier characters naturally use arbitrary finite dimension.

Let $G$ be a connected compact Lie group, $T$ a maximal torus, and $W$ its Weyl group. With normalized Haar measures and $D(t)={\prod }_{\alpha >0}(1-e^{-\alpha }(t))$, every continuous class function satisfies ${\int }_{G}f(g)dg=|W{|}^{-1}{\int }_{T}f(t)|D(t){|}^{2}dt$. For a general continuous function, replace $f(t)$ by its average ${\int }_{G/T}f(gt{g}^{-1})d(gT)$. Conjugation from $G/T\times T$ covers the regular elements $|W|$ times, and its real Jacobian determinant is $|D(t){|}^2$.

For a connected compact Lie group with maximal torus $T$, the map $q:G/T\times T\to G$, $q(xT,t)=xt{x}^{-1}$, has the displayed real Jacobian determinant. Choose an invariant inner product and identify the tangent space of $G/T$ with ${\mathfrak{t}}^{\perp }$. Translation of the derivative at $(T,t)$ to the identity gives $(X,Y)\mapsto (\mathrm{Ad}(t^{-1})-I)X+Y$. Each pair of opposite roots of a root system corresponds to a real two-plane on which the determinant is $|1-\alpha (t){|}^2$. The torus direction has determinant one. Normalized quotient and Haar measures have matching volume constants because the Riemannian submersion gives $\mathrm{vol}G=\mathrm{vol}(G/T)\mathrm{vol}T$. The map covers regular elements $|W|$ times, giving the Weyl integration formula with conjugacy averaging for general functions.

For the unitary group, the diagonal maximal torus has coordinates $z_i=e^{i{\theta }_i}$ and Weyl group $S_n$. The Weyl integration formula weights independent uniform angles by $n{!}^{-1}{\prod }_{i<j}|z_i-z_j{|}^2$. The Vandermonde determinant $\det (z_i^{n-j})$ has squared torus integral $n!$, by Fourier orthogonality. Multiplying an irreducible character of highest weight $\lambda$ by this determinant gives an alternating Laurent polynomial. Its leading alternant has coefficient one; character orthogonality for compact groups forces every other alternant coefficient to vanish. This yields the quotient $\det (z_i^{{\lambda }_j+n-j})/\det (z_i^{n-j})$.