  In this paper we construct a bivariant Chern character for the equivariant
KK-theory of a totally disconnected group with values in bivariant equivariant
cohomology in the sense of Baum and Schneider. We prove in particular that the
complexified left hand side of the Baum-Connes conjecture for a totally
disconnected group is isomorphic to cosheaf homology. Moreover, it is shown
that our transformation extends the Chern character defined by Baum and
Schneider for profinite groups.
