Characters of (relatively) integrable modules over affine Lie superalgebras

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 g . The problem consists of two parts. First, it is the reduction of the problem to the g ¯ -module F ( L ), where g ¯ is the associated to L integral Lie superalgebra and F ( L ) is an integrable irreducible highest weight 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 g -modules when g is basic, and all maximally atypical non-critical integrable g -modules when g is affine with non-zero dual Coxeter number.