A weight-formula for all highest weight modules, and a higher order parabolic category $\mathcal{O}$

. Let g be a complex Kac–Moody algebra, with Cartan subalgebra h . Also fix a weight λ ∈ h ∗ . For M ( λ ) ։ V an arbitrary highest weight g -module, we provide a cancellation-free, non-recursive formula for the weights of V . This is novel even in finite type, and is obtained from λ and a collection H = H V of independent sets in the Dynkin diagram of g that are associated to V . Our proofs use and reveal a finite family (for each λ ) of “higher order Verma modules” M ( λ, H ) – these are all of the universal modules for weight-considerations. They (i) generalize and subsume parabolic Verma modules M ( λ, J ), and (ii) have pairwise distinct weight-sets, which exhaust the weight-sets of all modules M ( λ ) ։ V . As an application, we explain the sense in which the modules M ( λ ) of Verma and M ( λ, J V ) of Lepowsky are respectively the zeroth and first order upper-approximations of every V , and continue to higher order upper-approximations M k ( λ, H V ) (and to lower-approximations). We also determine the k th order integrability of V , for all k > 0. We then introduce the category O H ⊆ O , which is a higher order parabolic analogue that contains the higher order Verma modules M ( λ, H ). We show that O H has enough projectives, and also initiate the study of BGG reciprocity, by proving it for all O H over g = sl ⊕ n 2 . Finally, we provide a BGG resolution for the universal modules M ( λ, H ) in certain cases; this yields a Weyl-type character formula for them, and involves the action of a parabolic Weyl semigroup.