Orbifold index and equivariant K-homology

We consider a invariant Dirac operator D on a manifold with a proper and cocompact action of a discrete group G. It gives rise to an equivariant K-homology class [D]. We show how the index of the induced orbifold Dirac operator can be calculated from [D] via the assembly map. We further derive a formula for this index in terms of the contributions of finite cyclic subgroups of G. According to results of W. Lueck, the equivariant K-homology can rationally be decomposed as a direct sum of contributions of finite cyclic subgroups of G. Our index formula thus leads to an explicit decomposition of the class [D].