= Conjugation Jacobian for a compact Lie group
{title2=$J(t)=\prod_{\alpha>0}|1-\alpha(t)|^2$}
For a connected <compact Lie group> with <maximal torus> $T$, the map $q:G/T\times T\to G$, $q(xT,t)=xtx^{-1}$, 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 $(X,Y)\mapsto(\operatorname{Ad}(t^{-1})-I)X+Y$. 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 $\operatorname{vol}G=\operatorname{vol}(G/T)\operatorname{vol}T$. The map covers regular elements $|W|$ times, giving the <Weyl integration formula> with conjugacy averaging for general functions.
Back to article page
