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

Abstract

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