  We show, in full generality, that Lusztig's $\mathbf{a}$-function describes
the projective dimension of both indecomposable tilting modules and
indecomposable injective modules in the regular block of the BGG category
$\mathcal{O}$, proving a conjecture from the first paper. On the way we show
that the images of simple modules under projective functors can be represented
in the derived category by linear complexes of tilting modules. These
complexes, in turn, can be interpreted as the images of simple modules under
projective functors in the Koszul dual of the category $\mathcal{O}$. Finally,
we describe the dominant projective module and also projective-injective
modules in some subcategories of $\mathcal{O}$ and show how one can use
categorification to decompose the regular representation of the Weyl group into
a direct sum of cell modules, extending the results known for the symmetric
group (type $A$).
