  We prove that the tensor product of a simple and a finite dimensional
$\mathfrak{sl}_n$-module has finite type socle. This is applied to reduce
classification of simple $\mathfrak{q}(n)$-supermodules to that of simple
$\mathfrak{sl}_n$-modules. Rough structure of simple
$\mathfrak{q}(n)$-supermodules, considered as $\mathfrak{sl}_n$-modules, is
described in terms of the combinatorics of category $\mathcal{O}$.

Representation Theory