Characters of (relatively) integrable modules over affine Lie superalgebras

M. Gorelik and V.G. Kac

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.