  The purpose of the present paper is two-fold. First, we obtain a non-abelian
localization theorem when M is any even dimensional compact manifold :
following an idea of E. Witten, we deform an elliptic symbol associated to a
Clifford bundle on M with a vector field associated to a moment map. Second, we
use this general approach to reprove the [Q,R] = 0 theorem of
Meinrenken-Sjamaar in the Hamiltonian case, and we obtain mild generalizations
to almost complex manifolds. This non-abelian localization theorem can be used
to obtain a geometric description of the multiplicities of the index of general
spin^c Dirac operators (see preprint arXiv:1411.7772).
