sl(2)ˆ decomposition of denominator formulae of some BKM Lie superalgebras

We study a family of Siegel modular forms that are constructed using Jacobi forms that arise in Umbral moonshine. All but one of them arise as the Weyl-Kac-Borcherds denominator formula of some Borcherds-Kac-Moody (BKM) Lie superalgebras. These Lie superalgebras have a sl(2)ˆ subalgebra which we use to study the Siegel modular forms. We show that the expansion of the Umbral Jacobi forms in terms of sl(2)ˆ characters leads to vector-valued modular forms. We obtain closed formulae for these vector-valued modular forms. In the Lie algebraic context, the Fourier coefficients of these vector-valued modular forms are related to multiplicities of roots appearing on the sum side of the Weyl-Kac-Borcherds denominator formulae.