  The positive-energy unitary irreducible representations of the $q$-deformed
conformal algebra ${\cal C}_q = {\cal U}_q(su(2,2))$ are obtained by
appropriate deformation of the classical ones. When the deformation parameter
$q$ is $N$-th root of unity, all these unitary representations become
finite-dimensional. For this case we discuss in some detail the massless
representations, which are also irreducible representations of the $q$-deformed
Poincar\'e subalgebra of ${\cal C}_q$. Generically, their dimensions are
smaller than the corresponding finite-dimensional non-unitary representation of
$su(2,2)$, except when $N=2$, $h=0$ and $N = 2 \vert h\vert +1$, where $h$ is
the helicity of the representations. The latter cases include the fundamental
representations with $h = \pm 1/2$.
