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 $\fL=\bigoplus_{(\alpha, a) \in \Gamma \times \Z_2} {\fL}_{(\alpha, a)}$ be a ($\Gamma\times\Z_{2}$)-graded Lie superalgebra. In this paper, by applying formal differential operators and the Laplacian to the denominator identity of $\fL$, we derive a new recursive formula for the dimensions of homogeneous subspaces of $\fL$. 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.