Tikaradze's analogue of the Kac–Weisfeiler conjecture

Throughout, let $\bold{k}=\bar{\bold{k}}$ be an algebraically closed field of characteristic $p>2$. Let $A$ be an affine $\bold{k}$-algebra finite over its center $\bold{Z}(A)$. For a central character $\chi\in\operatorname{Spec}\bold{Z}(A)$, write $A_{\chi}=A\otimes_{\bold{Z}(A)}\bold{k}$, where $\bold{k}$ is viewed as a $\bold{Z}(A)$-module via $\chi$, and let $i(\chi)$ be the largest power of $p$ dividing the dimensions of all simple $A_{\chi}$-modules. Let $\bigcup S_i=\operatorname{Spec}\bold{Z}(A)$ be the smooth stratification, and let $s(\chi)$ be the dimension of the smooth stratum containing $\chi$.

Tikaradze's analogue of the Kac–Weisfeiler conjecture. Suppose that $A$ is a nonnegatively filtered $\bold{k}$-algebra such that $\operatorname{gr} A$ is a finitely generated commutative domain over $\bold{k}$. Assume that $(\operatorname{gr} A)^p\subset\operatorname{gr}\bold{Z}(A)$ and that $\operatorname{Spec}\operatorname{gr} A$ is a union of finitely many symplectic leaves and is a Cohen–Macaulay variety. Then, for any central character $\chi\in\operatorname{Spec}\bold{Z}(A)$, we have

$$i(\chi)\geq\frac{1}{2}s(\chi).$$

This proposed analogue relates divisibility of dimensions of simple modules to the symplectic-leaf geometry of the associated graded algebra. The paper presents it as a motivating statement and proves a Kac–Weisfeiler-type result for rational Cherednik algebras, but the supplied text does not establish this general assertion.