Noetherian Centralizing Hopf Algebra Extensions and Finite Morphisms of Quantum Groups

Let $A\subset H$ be a finite centralizing extension of noetherian Hopf algebras, and let $X$ denote the group of characters of $H$ that restrict to the augmentation map on $A$. Our main results provide necessary and sufficient conditions for the fibers of the canonical surjection from $specH$ onto $spec A$tocoincidewiththe$X$-orbitsin$spec H$. In particular, all of the fibers are $X$-orbits if and only if the fiber over the augmentation ideal of $A$ is an $X$-orbit. An application to the representation theory of quantum function algebras, at roots of unity, is presented.