Character formulas in Category $\mathcal O_p$

Let $\mathcal O_p$ denote the characteristic $p>0$ version of the ordinary category $\mathcal O$ for a semisimple complex Lie algebra. In this paper we give some (formal) character formulas in $\mathcal O_p$. First we concentrate on the irreducible characters. Here we give explicit formulas for how to obtain all irreducible characters from the characters of the finitely many restricted simple modules as well as the characters of a small number of infinite dimensional simple modules in $\mathcal O_p$ with specified highest weights. We next prove a strong linkage principle for Verma modules which allow us to split $\mathcal O_p$ into a finite direct sum of linkage classes. There are corresponding translation functors and we use these to further cut down the set of irreducible characters needed for determining all others. Then we show that the twisting functors on $\mathcal O$ carry over to twisting functors on $\mathcal O_p$, and as an application we prove a character sum formula for Jantzen-type filtrations of Verma modules with antidominant highest weights. Finally, we record formulas relating the characters of the two kinds of tilting modules in $\mathcal O_p$.