  We introduce the $SU(N)$ Casson-Lin invariants for links $L$ in $S^3$ with
more than one component. Writing $L = \ell_1 \cup \cdots \cup \ell_n$, we
require as input an $n$-tuple $(a_1,\ldots, a_n) \in {\mathbb Z}^n$ of labels,
where $a_j$ is associated with $\ell_j$. The $SU(N)$ Casson-Lin invariant,
denoted $h_{N,a}(L)$, gives an algebraic count of certain projective $SU(N)$
representations of the link group $\pi_1(S^3 \smallsetminus L)$, and the family
$h_{N,a}$ of link invariants gives a natural extension of the $SU(2)$
Casson-Lin invariant, which was defined for knots by X.-S. Lin and for
2-component links by Harper and Saveliev. We compute the invariants for the
Hopf link and more generally for chain links, and we show that, under mild
conditions on the labels $(a_1, \ldots, a_n)$, the invariants $h_{N,a}(L)$
vanish whenever $L$ is a split link.
