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I am trying to understand the proof of Theorem 1 from this paper V. Kac and B. Weisfeiler (Indag. Math. 1976, DOI link).

Theorem 1. Let either $p\neq 2$ or $\varrho\in X(\mathscr{T})$. Then $\gamma(Z^\mathscr{G})=U(T)^W$ and $\gamma:Z^\mathscr{G}\to U(T)^W$ is a ring isomorphism.

Here $Z$ is the center of universal enveloping algebra $U(G)$ (here $G = \operatorname{Lie} \mathscr{G}$ ). $T$ is a Lie algebra of a maximal torus. $Z^{ \mathscr{G} }$ denote invariants (with respect to Ad-action of group $\mathscr{G}$) in $Z$.

Proof is not very short. But the last two lines are these

From 4.4, 4.6 and 5.3 we get $\overline{U(T)^W}=\overline{\gamma(Z^\mathscr{G})}$. Since both $U(T)^W$ and $\gamma(Z^\mathscr{G})$ are integrally closed we get $U(T)^W=\gamma(Z^\mathscr{G})$ as desired.

Here $\bar{A}$ means field of fraction of ring $A$. Suppose I believe that these fields are isomorphic and these rings are integrally closed.

Question How does it imply isomorphism of initial rings?

For example $ \mathbb{k} [x, xy] \subset \mathbb{k} [x, y]$

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  • $\begingroup$ The structure of the center here is far more subtle than in characteristic 0. In any case it's probably better to start with a 1999 paper which is somewhat easier to read and discusses both this and related problems. This was written by my Croatian colleague Ivan Mirkovic and our former student Dmitriy Rumynin (now at Warwick); ams.org/mathscinet-getitem?mr=1696760 $\endgroup$ – Jim Humphreys Oct 16 '16 at 17:22
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In the previous lines it is proven that $U(T)^W$ is integral over $\gamma(Z^{\mathcal G})$. Since $U(T)^W\subset Frac(U(T)^W)=Frac(\gamma(Z^{\mathcal G}))$ the equality follows from $\gamma(Z^{\mathcal G})$ being integrally closed.

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