Weights and characters of highest weight modules

Abstract

Let $\mathfrak{g}=\mathfrak{g}(A)$ be any Borcherds-Kac-Moody $\mathbb{C}$-Lie algebra (BKM LA) for BKM-Cartan matrix $A$, with Cartan subalgebra $\mathfrak{h}$. Let $V$ denote a highest weight $\mathfrak{g}$-module, with top weight $\lambda\in \mathfrak{h}^*$ (not necessarily in the domninant integral cone $P^+$). The non-integrable simples $V= L(\lambda)$ by Naito ([Trans. Amer. Soc., 1995]) are widely studied beyond integrable simple $L(\nu)s,\ \nu \in P^+$. We introduce and study: 1) A weight cone $P^{\pm}=\big\{\mu\in \mathfrak{h}^*\ \big|\ \mu(\alpha_i^{\vee})\in \frac{A_{ii}}{2}\mathbb{Z}_{\geq 0}\text{ for all simple co-roots }\alpha_i^{\vee}\big\}$; note Weyl vector $\rho\in P^{\pm}\setminus P^+$. 2) The resulting (novel) non-integrable simple $L(\lambda)s, \ \lambda \in P^{\pm}\setminus P^{+}$; their Chevalley-Serre (CS) type relations (which are, in fact, complementary to those of integrable $L(\nu)$s); 3) Higher length CS type relations in any highest weight module under the name ``holes". Using these, we obtain explicitly and uniformly, (notably) Weyl-orbit typed formulas for weight-sets of: all simples $L(\lambda)$s ($\forall$ $\lambda\in \mathfrak{h}^*$) and all quotients of parabolic Verma modules along imaginary directions. This generalizes and extends in one stroke, such formulas over Kac-Moody (KM) $\mathfrak{g}$, of all $L(\lambda)$ by Khare ([Trans. Amer. Math. Soc. 2017]), and Dhillon and Khare ([Adv. Math., 2017], and also of all $V$ by Khare and Teja recently; which used parabolic and higher order Verma modules. We obtain Weyl-Kac-Borcherds type character formulas for $L(\lambda) \text{ for } \lambda\in P^{\pm}$, over negative rank-2 $\mathfrak{g}$'s; by exploring Verma module embeddings. We obtain character of every highest weight module $V$ for $\lambda=\rho$ in negative $A$-type cases.