Peterson-type dimension formulas for graded Lie superalgebras

Abstract

Let $\widehat {\Gamma}$ be a free abelian group of finite rank, let $\Gamma$ be a sub-semigroup of $\widehat {\Gamma}$ satisfying certain finiteness conditions, and let $\mathfrak{L}=\bigoplus_{(\alpha, a) \in \Gamma \times \mathbb{Z}_2} \mathfrak{L}_{(\alpha, a)}$ be a ($\Gamma\times\mathbb{Z}_{2}$)-graded Lie superalgebra. In this paper, by applying formal differential operators and the Laplacian to the denominator identity of $\mathfrak{L}$, we derive a new recursive formula for the dimensions of homogeneous subspaces of $\mathfrak{L}$. When applied to generalized Kac-Moody superalgebras, our formula yields a generalization of Peterson's root multiplicity formula. We also obtain a Freudenthal-type weight multiplicity formula for highest weight modules over generalized Kac-Moody superalgebras.