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