The coordinate ring of the universal centralizer via Demazure operators

TL;DR

The paper describes the coordinate ring of the universal centralizer for simply connected semisimple groups using Demazure operators, linking Weil restriction and Weyl group actions.

Contribution

It establishes a general result connecting Weil restriction, Weyl group actions, and Demazure operators for affine schemes over the Cartan subalgebra.

Findings

Coordinate ring of the universal centralizer is characterized via Demazure operators.

A criterion for when the Weil restriction scheme's coordinate ring can be obtained by Demazure operators.

The scheme's integrality is key for applying Demazure operators in this context.

Abstract

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.