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Abstract: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 $P^+\subset \mathfrak{h}^*$ 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'' $L(\mu)$'s for $\mu$'s inside our novel signed-dominant-integral cone $P^{\pm}$ (which generalizes $P^+$). Pairings $\mu(\alpha_i^{\vee})\leq 0$ therein are multiples of $\frac{A_{ii}}{2}$ for all $i$. Nevertheless, $L(\mu)$ 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.

Submission history

From: Krishna Teja G [view email]
[v1] Thu, 18 Jun 2026 15:44:31 UTC (39 KB)