  Using Howe duality we compute explicitly Kostant-type homology groups for a
wide class of representations of the infinite-dimensional Lie superalgebra
$\hat{\frak{gl}}_{\infty|\infty}$ and its classical subalgebras at positive
integral levels. We also obtain Kostant-type homology formulas for the Lie
algebra $ widehat{\frak{gl}}_\infty$ at negative integral levels. We further
construct resolutions in terms of generalized Verma modules for these
representations.
