Simple supermodules over Lie superalgebras

We show that, for many Lie superalgebras admitting a compatible $\mathbb{Z}$-grading, the Kac induction functor gives rise to a bijection between simple supermodules over a Lie superalgebra and simple supermodules over the even part of this Lie superalgebra. This reduces the classification problem for the former to the one for the latter. Our result applies to all classical Lie superalgebras of type $I$, in particular, to the general linear Lie superalgebra $\mathfrak{gl}(m\vert n)$. In the latter case we also show that the rough structure of simple $\mathfrak{gl}(m\vert n)$-supermodules and also that of Kac supermodules depends only on the annihilator of the $\mathfrak{gl}(m)\oplus \mathfrak{gl}(n)$-input and hence can be computed using the combinatorics of BGG category $\mathcal {O}$.