Characters of (relatively) integrable modules over affine Lie superlagebras

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 problems 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.