  We study a class of localized indices for the Dirac type operators on a
complete Riemannian orbifold, where a discrete group acts properly,
co-compactly and isometrically. These localized indices, generalizing the
$L^2$-index of Atiyah, are obtained by taking certain traces of the higher
index for the Dirac type operators along conjugacy classes of the discrete
group. Applying the local index technique, we also obtain an $L^2$-version of
the Lefschetz fixed point formula for orbifolds. These cohomological formulae
for the localized indices give rise to a class of refined topological
invariants for the quotient orbifold.
