Kazhdan-Lusztig polynomials and character formulae for the Lie superalgebra gl(m|n)

The problem of computing the characters of the finite dimensional irreducible representations of the Lie superalgebra $\mathfrak{gl}(m|n)$ over $\C$ was solved a few years ago by V. Serganova. In this article, we present an entirely different approach. One consequence is a direct and elementary proof of a conjecture made by van der Jeugt and Zhang for the composition multiplicities of Kac modules. This does not seem to follow easily from Serganova's formula, since that involves certain alternating sums. We also compute Ext's between simple modules in the category of finite dimensional representations, and formulate for the first time a conjecture for the characters of the infinite dimensional irreducible representations in the analogue of category $\mathcal O$ for the Lie superalgebra $\mathfrak{gl}(m|n)$.