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Let $\Phi$ be the root system of a split group $G$ over a field $k$. The differentials $d\alpha$ of the roots define a polynomial called the discriminant $$\prod_{\alpha\in\Phi}d\alpha$$ on $\mathfrak t$, the Cartan subalgebra of $\mathfrak g$. The polynomial is invariant under the action of the Weyl group, and gives a function on $\mathfrak c=\text{Spec}(k[\mathfrak t]^W)$. The divisor $\mathfrak D_G$ of this polynomial is called the discriminant divisor.

If $H$ is a split subgroup of $G$ equal to a centralizer of a semisimple element of $G$, there is a morphism $$\nu:\mathfrak c_H=\mathfrak t_H/W\to\mathfrak c=\mathfrak t/W$$ that comes from an isomorphism of Cartan subalgebras $\mathfrak t_H\to\mathfrak t$ and an inclusion of Weyl groups $W_H\subset W$.

The claim is that there exists a resultant divisor $\mathfrak R$ defined by $$\prod_{\alpha\in\Psi}d\alpha$$ where $\Psi$ is the subset of positive roots in $\Phi-\Phi_H$ such that the following is true: $$\nu^*\mathfrak D_G=\mathfrak D_H+2\mathfrak R$$ This claim is made in Ngo B.C., 1.10.3 but I don't see how the proof of it works.

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