The coordinate ring of the universal centralizer via Demazure operators

We give a simple description of the coordinate ring of the universal centralizer associated to a simply connected semisimple group. To this end, we prove a general result on Weil restriction of affine schemes $X$ over the Cartan subalgebra $\mathfrak{t}$ equipped with a compatible action of the Weyl group $W$. Specifically, we show that the coordinate ring of the scheme $\mathrm{Res}^W(X)$ of $W$-fixed points of Weil restriction of $X$ to the categorical quotient $\mathfrak{t}//W$ can be obtained from the coordinate ring of $X$ by applying Demazure operators if and only if the scheme $\mathrm{Res}^W(X)$ is integral.