An equivariant index for proper actions III: The invariant and discrete series indices

We study two special cases of the equivariant index defined in part I of this series. We apply this index to deformations of ${\mathrm{Spin}}^{\mathrm{c}}$-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 ${\mathrm{Spin}}^{\mathrm{c}}$-Dirac operators with respect to a global slice for the action.