Characters of modules over negative rank-2 Borcherds-Kac-Moody Lie algebras

Let $\mathfrak{g}=\mathfrak{g}(A)$ be the Borcherds-Kac-Moody Lie algebra (BKM LA), corresponding to a BKM Cartan matrix $A$ filled by negative integers. Let $P^{+}\subset {\mathfrak{h}}^{\ast }$ the classical dominant integral cone (wherein pairings are non-negative). The non-integrable simple highest weight modules $L(\mu )$‘s widely studied were broadly those by Naito ([Trans. Amer. Soc., 1995]), for $\mu$’s dot-linked to $P^{+}$-translates of sums $-{\sum }_{j\in J}{\alpha }_j$ of mutually orthogonal and imaginary simple roots ${\alpha }_j$’s. Recently, we computed weights of all highest weight $\mathfrak{g}$-modules $V$’s, and characters of $L(\rho )$ for Weyl vector $\rho$ in negative type-$A$. These needed a family of integrable'' $L(μ)$'s for $μ$'s inside our novel signed-dominant-integral cone $P^{\pm}$ (which generalizes $P^+$). Pairings $μ(α_i^{\vee})\leq 0$ therein are multiples of $\frac{A_{ii}}{2}$ for all $i$. Nevertheless, $L(μ)$ contain Chevalley-Serre relations’’ $f_i^{\frac{2}{A_{ii}}\mu ({\alpha }_i^{\vee })+1}L(\mu {)}_{\mu }=0$; which differ from relations in $L(\lambda )$ for all $\lambda \in P^{+}$, and are seemingly unstudied earlier (also by Naito). This paper initiates the study in rank-2, of the module structures and maximal vectors (or Verma embeddings) in the Verma covers $M(\mu )$ of $L(\mu )$'s for $\mu \in P^{\pm }$. In this, our goal is to explore in weight spaces of those Verma covers, the strictness (or otherwise, an uniform equality) of lower bounds by Kac and Kazhdan ([Adv. Math., 1979]) for count of linearly independent maximal vectors. We obtain presentations and characters of all $V$’s when Kac-Kazhdan equation has unique solution in the interior of root-cone. This builds on the unique solution case in Lemma 3.1 from that paper.