On the category $\mathcal{O}$ for generalized Weyl algebras

Let $H(R,\phi ,z)$ be a generalized Weyl algebra associated with a ring $R$, its central element $z\in Z(R)$ and an automorphism $\phi ,$ such that for some $l\ge 1$, ${\phi }^l(z)-z$ is nilpotent and $(z,{\phi }^i(z))=R$ for all $0<i<l$. We prove that the category $\mathcal{O}$ over $H(R,z,\phi )$ is equivalent to the category $\mathcal{O}$ over its $l$-th twist the generalized Weyl algebra $H(R,z,{\phi }^l)\mathrm{.}$ This result is significantly more general than the corresponding one for the Weyl algebra over $\mathbb{Z}\mathrm{/}{p}^n\mathbb{Z}\mathrm{.}$