Infinite-dimensional Lie superalgebras and hook Schur functions

Making use of a Howe duality involving the infinite-dimensional Lie super-algebra gl_∞|∞ and finite-dimensional group GL_l of [CW3] we derive a character formula for a certain class of irreducible quasi-finite representations of gl_∞|∞ in terms of hook Schur functions. We use the reduction procedure of gl_∞|∞ to gl_n|n to derive a character formula for a certain class of level 1 highest weight irreducible representations of gl_n|n, the affine Lie superalgebra associated to the finite-dimensional Lie superalgebra gl_n|n. These modules turn out to form the complete set of integrable gl_n|n-modules of level 1. We also show that the characters of all integrable level 1 highest weight irreducible gl_m|n-modules may be written as a sum of products of hook Schur functions.