Categories of induced modules for Lie algebras with triangular decomposition

We study categories of modules for Lie algebras with triangular decompositi on, which contain certain generalized Verma modules and are analogous to cl assical category O. We relate blocks of these categories to module categori es over finite-dimensional algebras, which turn out to be projectively stra tified. Moreover, we study tilting modules. Finally we show how to relate s ome of these situations to similar ones over certain proper subalgebras of the given Lie algebra.