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 $\hat{I}»\in {\mathfrak{h}}^{\ast }$ (not necessarily in the domninant integral cone $P^{+}$). The non-integrable simples $V=L(\hat{I}»)$ by Naito ([Trans. Amer. Soc., 1995]) are widely studied beyond integrable simple $L(\hat{I}½)s,\hat{I}½\in P^{+}$. We introduce and study: 1) A weight cone $P^{\pm }=\{\hat{I}¼\in {\mathfrak{h}}^{\ast }|\hat{I}¼(\hat{I}{\pm }_i^{\vee })\in \frac{A_{ii}}{2}{\mathbb{Z}}_{\ge 0}\text{ for all simple co-roots }\hat{I}{\pm }_i^{\vee }\}$; note Weyl vector $\ddot{I}\in P^{\pm }\setminus P^{+}$. 2) The resulting (novel) non-integrable simple $L(\hat{I}»)s,\hat{I}»\in P^{\pm }\setminus P^{+}$; their Chevalley-Serre (CS) type relations (which are, in fact, complementary to those of integrable $L(\hat{I}½)$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 $L(\hat{I}»)$s ($\forall$ $\hat{I}»\in {\mathfrak{h}}^{\ast }$) 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 $L(\hat{I}»)$ 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 $L(\hat{I}»)\text{ for }\hat{I}»\in P^{\pm }$, over negative rank-2 $\mathfrak{g}$'s; by exploring Verma module embeddings. We obtain character of every highest weight module $V$ for $\hat{I}»=\ddot{I}$ in negative $A$-type cases.

Weight-formulas for parabolic Verma modules with imaginary holes and character formulas for Weyl vectors (with sign-dominant-integral cone's study), are isolated here from arXiv:2505.08102v2 (for its longer length). Verma module structure under unique interior Kac-Kazhdan equation's solutions and full weight-formulas are strengthened in another article