Weyl integration formula

Define $dQ={\pi }_{\ast }dg$. It is a G-invariant Borel probability by equivariance and [L1]. For continuous F on G, $\bar{F}(xT)={\int }_{T}F(xt)dt$ is independent of representative by Haar invariance, and is continuous by local sections [L3] and continuity on compact sets. Fubini and right invariance give ${\int }_Q\bar{F}dQ={\int }_G{\int }_{T}F(xt)dtdg(x)={\int }_{G}Fdg\mathrm{.}$ If $\nu$ is another invariant Borel probability, then for $h\in C(Q)$, Fubini and invariance give $\int hd\nu ={\int }_Q{\int }_{G}h(gxT)dgd\nu (xT)$. The inner integral is ${\int }_{G}h(gT)dg={\int }_{Q}hdQ$ by right invariance, independently of x. Hence both measures agree on continuous tests. The compact quotient is metrizable: the Ad(T)-invariant inner product of [L4] descends to a smooth invariant quotient metric via the local sections of [L3], whose distance gives the manifold topology. Thus [L7] proves uniqueness. Applying the Weil identity to $F=h\circ \pi$ likewise characterizes dQ uniquely.

Choose a bi-invariant metric as in [L4] and put $\mathfrak{m}={\mathfrak{t}}^{\perp }$. Its restriction is Ad(T)-invariant, so transporting it by the left G-action defines a smooth invariant metric on Q: at $eT$ use the isometry $d{\pi }_e:\mathfrak{m}\to T_{eT}Q$, and invariance under the isotropy T makes the transport independent of representative. Local sections from [L3] give smoothness. The metric on T is the restriction of the group metric. Write the resulting unnormalized densities and volumes as $d{v}_G,d{v}_T,d{v}_Q$ and $V_G,V_T,V_Q$. They have finite positive total masses by compactness and [L4]. The normalized densities on G and T are their Haar probabilities by invariance and [L1]; on Q the normalized density is a G-invariant Borel probability.

On the compact manifold $Q\times T$ define $q(xT,t)=xt{x}^{-1}$. This is well defined because T is abelian and smooth by local sections [L3]. Use $\mathfrak{g}=\mathfrak{m}\oplus \mathfrak{t}$ and left translation by $t^{-1}$ at the target. Differentiating $\exp (sX)t\exp (sY)\exp (-sX)$ at zero gives $d{q}_{(eT,t)}(X,Y)=(\mathrm{Ad}(t^{-1})-I)X+Y\mathrm{.}$ The first summand lies in $\mathfrak{m}$ by Ad(T)-invariance; the second lies in $\mathfrak{t}$. By [L5] the determinant on $\mathfrak{m}$ is ${\prod }_{\alpha \in \mathrm{\Phi }}(\alpha (t{)}^{-1}-1)={\prod }_{\alpha >0}\mathrm{\mid }1-\alpha (t{)}^{-1}{\mathrm{\mid }}^2=J(t)$. The real determinant equals that of its complexification; each root occurs once. Equivariance and isometries from the bi-invariant metric give the same absolute Jacobian at every $(xT,t)$ for the unnormalized product densities. Thus q is locally a diffeomorphism precisely when $t\in T_{reg}:=\{J>0\}$.

We verify the volume normalization, rather than assuming a product decomposition of Haar measure. Over a smooth local section $s:U\to G$, the map $b:U\times T\to {\pi }^{-1}U$, $b(u,t)=s(u)t$, is a diffeomorphism: its inverse is $g\mapsto (\pi g,s(\pi g{)}^{-1}g)$. At $(u,t)$, project its base tangent vectors orthogonally to the horizontal complement of the fibre tangent. Since $\pi \circ b$ is projection, their horizontal components map isometrically onto the base vectors by the definition of the quotient metric. The fibre tangent vectors are obtained by left translation from T and are isometric to its tangent vectors. The additional vertical components of base vectors give a block triangular change-of-frame matrix with identity diagonal, hence determinant 1. Therefore $b^{\ast }d{v}_G=d{v}_{Q}d{v}_T$. Take a finite section cover of compact Q and replace it by a disjoint Borel partition subordinate to the cover. The chart formulas and [L6], applied also to indicators of these sets, yield $V_G=V_Q{V}_T$. By uniqueness in step 1.1 the normalized quotient density is dQ. Consequently the normalized product measure $dQdt$ and dg have the same relative Jacobian J under q on its regular locus.

Let $C_G(t)$ be the closed centralizer. Its Lie algebra is $\ker (\mathrm{Ad}(t)-I)$: differentiating commutation gives one inclusion and exponential naturality gives the converse by one-parameter subgroups. By [L5], for $t\in T_{reg}$ this is $\mathfrak{t}$. Since $T\subseteq C_G(t{)}^0$ has the same Lie algebra, exponential charts and connectedness imply $C_G(t{)}^0=T$. Any torus containing t lies in this identity component, so T is the unique maximal torus containing t. More generally the dimension of this fixed algebra is $\dim T$ exactly for $t\in T_{reg}$, and is larger for singular t. Dimension is invariant under conjugation. By [L2] every element is conjugate into T, so $G_{reg}:=q(Q\times T_{reg})$ is the complement of $q(Q\times (T\setminus T_{reg}))$. The latter compact set is precisely the critical-value set of q by step 2.1; it is null by [L7], hence Haar-null by the smooth positive density chart formula [L4]. Thus $G_{reg}$ is open and of full Haar measure. No null-set assertion is transported through a singular local map.

Fix $q(xT,t)$ in $G_{reg}$. If $q(yT,t^{\mathrm{\prime }})=q(xT,t)$, put $m=x^{-1}y$; then $m^{-1}tm=t^{\mathrm{\prime }}$. Conjugation preserves regularity by step 3.2, and the uniqueness of the maximal torus containing t shows $mT{m}^{-1}=T$. Conversely each $m\in N_G(T)$ gives the preimage $(xmT,m^{-1}tm)$, and two such pairs agree exactly when the cosets mT agree. Thus every fibre of $q_r:Q\times T_{reg}\to G_{reg}$ has exactly $\mathrm{\mid }W\mathrm{\mid }$ points. To obtain an evenly covered neighborhood of any g, choose disjoint local-diffeomorphism neighborhoods at its finitely many preimages, and intersect their open images. After restriction each supplies one preimage of every point of this intersection; the constant fibre count just proved leaves no other preimages. These are the required $\mathrm{\mid }W\mathrm{\mid }$ sheets. This proves the covering directly and does not assert that its open source is compact.

Choose a countable cover of $G_{reg}$ by evenly covered coordinate neighborhoods small enough that each of their finitely many sheets lies in a coordinate neighborhood of the source; second countability permits this refinement. Subtract preceding sets to obtain a disjoint Borel partition. On each set and each sheet apply [L6] to the nonnegative measurable function, or to the four nonnegative parts of an integrable complex function. The normalized density Jacobian is J by step 3.1. Summing the countable disjoint pieces and the $\mathrm{\mid }W\mathrm{\mid }$ sheets gives $\mathrm{\mid }W\mathrm{\mid }{\int }_{G_{reg}}f(g)dg={\int }_{Q\times T_{reg}}f(xt{x}^{-1})J(t)dQdt\mathrm{.}$ For continuous f on compact G all integrals are absolutely finite because f and J are bounded and the normalized source measure is finite.

The target complement is Haar-null by step 3.2; on the omitted source $Q\times (T\setminus T_{reg})$ one has $J=0$ pointwise. Hence step 5.1 already extends to integration over all of G and $Q\times T$, without any uniform approximation by functions supported in $G_{reg}$. Fubini [L6] puts the torus integral outside and gives the formula in the statement with the unique probability of step 1.1. For a class function the integrand is independent of xT, giving the stated specialization; independence of positive roots follows from [L5]. If there are no roots, the adjoint decomposition makes $\mathfrak{g}=\mathfrak{t}$, hence connected G equals T by exponential charts, Q is a point, W is trivial, and J is the empty product 1. This includes the trivial group. Choice supplies the assumptions of the stated Lie, Haar, density and countable-chart interfaces.