  Let $W_{m|n}$ be the (finite) $W$-algebra attached to the principal nilpotent
orbit in the general linear Lie superalgebra $\mathfrak{gl}_{m|n}(\mathbb{C})$.
In this paper we study the {\em Whittaker coinvariants functor}, which is an
exact functor from category $\mathcal O$ for $\mathfrak{gl}_{m|n}(\mathbb{C})$
to a certain category of finite-dimensional modules over $W_{m|n}$. We show
that this functor has properties similar to Soergel's functor $\mathbb V$ in
the setting of category $\mathcal O$ for a semisimple Lie algebra. We also use
it to compute the center of $W_{m|n}$ explicitly, and deduce some consequences
for the classification of blocks of $\mathcal O$ up to Morita/derived
equivalence.
