Deformation Quantization and Reduction by Stages

We discuss the Quantum-Koszul method for constructing star products on reduced phase spaces in the symplectic, regular case. It is shown that the reduction method proposed by Kowalzig, Neumaier and Pflaum for cotangent bundles is a special case of the Quantum-Koszul method. We give sufficient conditions that reduction by stages is possible in the Quantum-Koszul framework and show that the star product obtained by two steps is identical to that obtained by one step. In order to do so, we prove an equivariant version of the compatible tubular neighborhood theorem