  We construct an element, which we call the Kirillov projector, that connects
the topics of the title: on the one hand, it is defined using the Yangian of
$\mathfrak{gl}_N$, on the other hand, it gives a canonical projection onto the
space of Whittaker vectors for any Whittaker module over the mirabolic
subalgebra. Using the Kirillov projector, we deduce some categorical properties
of Whittaker modules, for instance, we prove a mirabolic analog of Skryabin's
theorem. We also show that it quantizes a rational version of the
Cremmer-Gervais $r$-matrix. As application, we construct a universal vertex-IRF
transformation from the standard dynamical $R$-matrix to this constant one in
categorical terms.
