  We study two special cases of the equivariant index defined in part I of this
series. We apply this index to deformations of Spin$^c$-Dirac operators,
invariant under actions by possibly noncompact groups, with possibly noncompact
orbit spaces. One special case is an index defined in terms of multiplicities
of discrete series representations of semisimple groups, where we assume the
Riemannian metric to have a certain product form. The other is an index defined
in terms of sections invariant under a group action. We obtain a relation with
the analytic assembly map, quantisation commutes with reduction results, and
Atiyah-Hirzebruch type vanishing theorems. The arguments are based on an
explicit decomposition of Spin$^c$-Dirac operators with respect to a global
slice for the action.
