Non-Abelian Reduction in Deformation Quantization

We consider a G-invariant star-product algebra A on a symplectic manifold (M,) obtained by a canonical construction of deformation quantization. Under assumptions of the classical Marsden–Weinstein Theorem we define a reduction of the algebra A with respect to the G-action. The reduced algebra turns out to be isomorphic to a canonical star-product algebra on the reduced phase space B. In other words, we show that the reduction commutes with the canonical G-invariant deformation quantization. A similar statement within the framework of geometric quantization is known as the Guillemin–Sternberg conjecture (by now, completely proved).