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

Witten non abelian localization for equivariant K-theory, and the [Q,R]=0 theorem