Characters of (relatively) integrable modules over affine Lie superalgebras

Abstract

In the paper we consider the problem of computation of characters of relatively integrable irreducible highest weight modules L over finite-dimensional basic Lie superalgebras and over affine Lie superalgebras \({\mathfrak{g}}\). The problem consists of two parts. First, it is the reduction of the problem to the \({\overline{\mathfrak{g}}}\)-module F(L), where \({\overline{\mathfrak{g}}}\) is the associated to L integral Lie superalgebra and F(L) is an integrable irreducible highest weight \({\overline{\mathfrak{g}}}\)-module. Second, it is the computation of characters of integrable highest weight modules. There is a general conjecture concerning the first part, which we check in many cases. As for the second part, we prove in many cases the KW-character formula, provided that the KW-condition holds, including almost all finite-dimensional \({\mathfrak{g}}\)-modules when \({\mathfrak{g}}\) is basic, and all maximally atypical non-critical integrable \({\mathfrak{g}}\)-modules when \({\mathfrak{g}}\) is affine with non-zero dual Coxeter number.