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

Abstract

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

Preliminary version, 9 pages, all comments welcome