Published online by Cambridge University Press: 22 January 2016
Let be a free abelian group of finite rank, let Γ be a sub-semigroup of
satisfying certain finiteness conditions, and let
be a (Γ × Z2)-graded Lie superalgebra. In this paper, by applying formal differential operators and the Laplacian to the denominator identity of
, we derive a new recursive formula for the dimensions of homogeneous subspaces of
. 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.