The irreducible characters of the Lie superalgebras of type \(A(n,m)\) and filtrations of their Kac modules (Q677461)

The author considers the Lie superalgebra \(G=\text{ sl}(n,m)\) and its Kac modules [\textit{V. G. Kac}, Lect. Notes Math. 676, 597-626 (1978; Zbl 0388.17002)]. These Kac modules are highest weight modules \(K_\lambda\) which are in general indecomposable, unless the highest weight \(\lambda\) is typical, in which case they are simple. The author studies the case when \(\lambda=\phi\), where the highest weight \(\phi\) is trivial with respect to \(\text{ sl}(n)\). An expression for the contravariant form on \(K_\phi\) is given, and the filtration associated to this form is shown to be semisimple. Following this, a recursion formula is obtained for obtaining the contravariant form and the characters of each composition factor in the Kac module \(K_\lambda\) for arbitrary \(\lambda\).