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

Abstract

Let $\mathfrak{g}=\mathfrak{g}(A)$ be the Borcherds-Kac-Moody Lie algebra (BKM LA) for a BKM Cartan matrix $A$ that is filled by negative integers. Fix a Cartan subalgebra $\mathfrak{h}$ of $\mathrm{g}$ and the classical cone of dominant integral weights $P^{+}\subset {\mathfrak{h}}^{\ast }$. The non-integrable simple highest weight $\mathfrak{g}$-modules $L(\mu )$'s widely studied were 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 (over all BKM LA's), and character of $L(\rho )$ for Weyl vector $\rho$. 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 $\mu ({\alpha }_i^{\vee })\le 0$ for $\mu \in P^{\pm }$ are multiples of $\frac{A_{ii}}{2}$ for all $i$. Nevertheless, $L(\mu )$ contains ``Chevalley-Serre relations'' $f_i^{\frac{2}{A_{ii}}\mu ({\alpha }_i^{\vee })+1}L(\mu {)}_{\mu }=0$, which seem to be previously unstudied and even in Naito's works. This paper initiates in rank-2, the study of module structures and maximal vectors (or Verma embeddings) in Verma covers $M(\mu )$'s of $L(\mu )$'s for $\mu \in P^{\pm }$. Our goal in this is to explore in weight spaces of those Vermas, the strictness, or else a 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 results of Kac and Kazhdan in crucial unique solution case.

27 Pages, 8 Figures. We could verify the count of maximal vectors in Verma modules (for negative "Cartan matrices'') equalling Kac-Kazhdan's lower bound in some cases. A natural question that arises is, does one always have quality therein?