The coordinate ring of the universal centralizer equals the result of applying Demazure operators to the coordinate ring of X precisely when the W-fixed points of the Weil restriction of X is an integral scheme.
Vakil.The Rising Sea: Foundations of Algebraic Geometry
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The coordinate ring of the universal centralizer via Demazure operators
The coordinate ring of the universal centralizer equals the result of applying Demazure operators to the coordinate ring of X precisely when the W-fixed points of the Weil restriction of X is an integral scheme.