  Let $\mathfrak g$ be a simple complex Lie algebra. In this paper we study the
BGG category $\mathcal O_q$ for the quantum group $U_q(\mathfrak g)$ with $q$
being a root of unity in a field $K$ of characteristic $p >0$. We first
consider the simple modules in $\mathcal O_q$ and prove a Steinberg tensor
product theorem for them. This result reduces the problem of determining the
corresponding irreducible characters to the same problem for a finite subset of
finite dimensional simple modules. Then we investigate more closely the Verma
modules in $\mathcal O_q$. Except for the special Verma module, which has
highest weight $-\rho$, they all have infinite length. Nevertheless, we show
that each Verma module has a certain finite filtration with an associated
strong linkage principle. The special Verma module turns out to be both simple
and projective/injective. This leads to a family of projective modules in
$\mathcal O_q$, which are also tilting modules. We prove a reciprocity law,
which gives a precise relation between the corresponding family of characters
for indecomposable tilting modules and the family of characters of simple
modules with antidominant highest weights. All these results are of particular
interest when $q = 1$, and we have paid special attention to this case.
