Howe duality and Kostant Homology Formula for infinite-dimensional Lie superalgebras

Abstract

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