Let $G$ be a compact connected Lie group with [Lie algebra](/page/Lie%20Algebra) $\mathfrak g$, let $T\le G$ be a maximal torus with Lie algebra $\mathfrak t$, and fix an $\operatorname{Ad}(G)$-invariant real [inner product](/page/Inner%20Product) $(\cdot,\cdot)_{\mathfrak g}$ on $\mathfrak g$. Let $\mathfrak t^\perp\subset \mathfrak g$ denote the orthogonal complement of $\mathfrak t$, so that
Equip $G$ and $T$ with the Riemannian volume densities induced by $(\cdot,\cdot)_{\mathfrak g}$ and its restriction to $\mathfrak t$, and equip $G/T$ with the quotient Riemannian volume density for which the differential of the quotient map identifies $\mathfrak t^\perp$ isometrically with $T_{eT}(G/T)$.
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Let
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\begin{align*}
\mathfrak g_{\mathbb C}:=\mathfrak g\otimes_{\mathbb R}\mathbb C
\end{align*}
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and
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\begin{align*}
\mathfrak t_{\mathbb C}:=\mathfrak t\otimes_{\mathbb R}\mathbb C.
\end{align*}
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Let $R\subset \mathfrak t_{\mathbb C}^*$ be the root system of $\mathfrak g_{\mathbb C}$ with respect to $\mathfrak t_{\mathbb C}$, choose a positive system $R^+\subset R$, and write
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\begin{align*}
\mathfrak g_{\mathbb C,\alpha}:=\{Z\in \mathfrak g_{\mathbb C}:[H,Z]=\alpha(H)Z\text{ for every }H\in \mathfrak t_{\mathbb C}\}
\end{align*}
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for the root space of $\alpha\in R$. For each $\alpha\in R$, let
denote the corresponding torus character, so that $\operatorname{Ad}_t$ acts on $\mathfrak g_{\mathbb C,\alpha}$ by multiplication by $e^\alpha(t)$. Define the regular subset of $T$ by
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\begin{align*}
T_{\mathrm{reg}}:=\{t\in T:e^\alpha(t)\neq 1\text{ for every }\alpha\in R\}.
\end{align*}
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Define the smooth conjugation map
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\begin{align*}
\Phi:G/T\times T\to G
\end{align*}
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by
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\begin{align*}
\Phi(gT,t):=gtg^{-1}.
\end{align*}
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Then, for every $gT\in G/T$ and every $t\in T_{\mathrm{reg}}$, the absolute Jacobian of $d\Phi_{(gT,t)}$ with respect to the above volume densities is