BGG categories in prime characteristics

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