Let \(M\) be a closed manifold with an action of a compact Lie group \(K\) with Lie algebra \(\mathfrak{k}\) and \(\Phi : M \to \mathfrak{k}\) a \(K\)-equivariant map. In this memoir, the authors firstly prove a localization theorem that localize the study of the equivariant index of any \(\sigma \in \mathrm{K}^{0}_{K}(T^{\ast}M)\) to the study of the equivariant index of the deformed element on a neighborhood of the zeros of the map \(\Phi\). The theorem is a generalization of a \(K\)-theoretical analogue of the Witten deformation by the first author [P.-E. Paradan, J. Funct. Anal. 187, No. 2, 442–509 (2001; Zbl 1001.53062)].
Secondly, the authors reprove the \([Q,R] = 0\) theorem (quantization commutes with reduction theorem) of E. Meinrenken and S. Sjamaar [Topology 38, No. 4, 699–762 (1999; Zbl 0928.37013)] by using their \(K\)-theoretical localization theorem. They also prove a generalization of the \([Q,R] = 0\) theorem in the context of almost complex manifolds and a \([Q,R] = 0\) theorem for Clifford bundles in an asymptotic sense. The asymptotic \([Q,R] = 0\) theorem plays an essential role in [P.-E. Paradan, J. Symplectic Geom. 17, No. 5, 1389–1426 (2019; Zbl 1445.22003)].
Finally, the authors describe in detail some applications of their \([Q,R] = 0\) theorem. In particular, they give a unified proof on the piecewise quasi-polynomial behaviour of the multiplicity function \(\mathrm{m}_{\mu}(L)\) via Hamiltonian geometry and the \([Q,R] = 0\) theorem. Here, \(\mathrm{m}_{\mu}(L)\) is the multiplicity function in the geometric quantization \(RR_{K}(M,L) = \sum_{\mu \in \hat{K}}\mathrm{m}_{\mu}(L)V_{\mu}\) for a symplectic manifold \(M\) and the Kostant line bundle \(L\).