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

Abstract

Let $\mathfrak{g}$ be a complex Kac-Moody algebra, with Cartan subalgebra $\mathfrak{h}$. Also fix a weight $\lambda\in\mathfrak{h}^*$. For $M(\lambda)\twoheadrightarrow V$ an arbitrary highest weight $\mathfrak{g}$-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 $\lambda$ and a collection $\mathcal{H}=\mathcal{H}_V$ of independent sets in the Dynkin diagram of $\mathfrak{g}$ that are associated to $V$. Our proofs use and reveal a finite family (for each $\lambda$) of "higher order Verma modules" $\mathbb{M}(\lambda,\mathcal{H})$ - these are all of the universal modules for weight-considerations. They (i) generalize and subsume parabolic Verma modules $M(\lambda,J)$, and (ii) have pairwise distinct weight-sets, which exhaust the weight-sets of all modules $M(\lambda)\twoheadrightarrow V$. As an application, we explain the sense in which the modules $M(\lambda)$ of Verma and $M(\lambda,J_V)$ of Lepowsky are respectively the zeroth and first order upper-approximations of every $V$, and continue to higher order upper-approximations $\mathbb{M}_k(\lambda,\mathcal{H}_V)$ (and to lower-approximations). We determine every $k$th order integrability of $V$. We then introduce the category $\mathcal{O}^\mathcal{H}\subset\mathcal{O}$, which is a higher order parabolic analogue that contains the higher order Verma modules $\mathbb{M}(\lambda,\mathcal{H})$. We show that $\mathcal{O}^\mathcal{H}$ has enough projectives, and also initiate the study of BGG reciprocity, by proving it for all $\mathcal{O}^\mathcal{H}$ over $\mathfrak{g}=\mathfrak{sl}_2^{\oplus n}$. Finally, we provide a BGG resolution for our universal modules $\mathbb{M}(\lambda,\mathcal{H})$ in certain cases including all rank-3 $\mathfrak{g}$; this yields their Weyl-type character formulas, with the actions of parabolic Weyl semigroups.