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

TL;DR

This paper investigates the structure of subquotients of the category  over hyperalgebras in positive characteristic, revealing a periodicity property that relates different subquotients via shifts by multiples of p^l.

Contribution

It introduces a new periodicity result for subquotients of category , showing their equivalence under certain weight shifts, simplifying their analysis.

Findings

Subquotients _{[\u03a3]} are equivalent to _{[\u03a3 + p^l\u03b3]} under specific conditions.

The periodicity allows focusing on subquotients within the dominant chamber.

The results facilitate the study of  by reducing to more manageable subcategories.

Abstract

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.