  We provide a systematic approach to obtain formulas for characters and
Kostant ${\mathfrak u}$-homology groups of the oscillator modules of the finite
dimensional general linear and ortho-symplectic superalgebras, via Howe
dualities for infinite dimensional Lie algebras. Specializing these Lie
superalgebras to Lie algebras, we recover, in a new way, formulas for Kostant
homology groups of unitarizable highest weight representations of Hermitian
symmetric pairs. In addition, two new reductive dual pairs related to the
above-mentioned ${\mathfrak u}$-homology computation are worked out.
