Subgroups of polynomial automorphisms with diagonalizable fibers

Let $R$ be an integral domain over a field $k$, and $G$ a subgroup of the automorphism group of the polynomial ring $R[x_1,..., x_n]$ over $R$. In this paper, we discuss when $G$ is diagonalizable under the assumption that $G$ is diagonalizable over the field of fractions of $R$. We are particularly interested in the case where $G$ is a finite abelian group. Kraft-Russell (2014) implies that every finite abelian subgroup of ${\rm Aut}_R(R[x_1,x_2])$ is diagonalizable if $R$ is an affine PID over $k={\bf C}$. One of the main results of this paper says that the same holds for a PID $R$ over any field $k$ containing enough roots of unity.