Periodicity for subquotients of the modular category $\mathcal{O}$

In this paper we study the category $\mathcal{O}$ over the hyperalgebra of a reductive algebraic group in positive characteristics. For any locally closed subset $\mathcal{K}$ of weights we define a subquotient $\mathcal{O}_{[\mathcal{K}]}$ of $\mathcal{O}$. It has the property that its simple objects are parametrized by elements in $\mathcal{K}$. We then show that $\mathcal{O}_{[\mathcal{K}]}$ is equivalent to $\mathcal{O}_{[\mathcal{K}+p^l\gamma]}$ for any dominant weight $\gamma$ if $l>0$ is an integer such that $\mathcal{K}\cap (\mathcal{K}+p^l\eta)=\emptyset$ for all dominant weights $\eta$. This allows one, for example, to restrict attention to subquotients inside the dominant (or the antidominant) chamber.