In this article we consider quantum phase space reduction when zero is a regular value of the momentum map. By analogy with the classical case we define the BRST cohomology in the framework of deformation quantization. We compute the quantum BRST cohomology in terms of a 'quantum' Chevalley-Eilenberg cohomology of the Lie algebra on the constraint surface. To prove this result, we construct an explicit chain homotopy, both in the classical and quantum case, which is constructed out of a prolongation of functions on the constraint surface. We have observed the phenomenon that the quantum BRST cohomology cannot always be used for quantum reduction, because generally its zero part is no longer a deformation of the space of all smooth functions on the reduced phase space. But in case the group action is 'sufficiently nice', e.g. proper (which is the case for all compact Lie group actions), it is shown for a strongly invariant star product that the BRST procedure always induces a star product on the reduced phase space in a rather explicit and natural way. Simple examples and counter examples are discussed. *

Abstract. We use the method of homological quantum reduction to construct a deformation quantization on singular symplectic quotients in the situation, where the coefficients of the moment map define a complete intersection. Several examples are discussed, among others one where the singularity type is worse than an orbifold singularity.

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Let [Formula: see text] be a finite-dimensional representation of the group [Formula: see text] of [Formula: see text] matrices with complex coefficients and determinant one. Let [Formula: see text] be the algebra of [Formula: see text]-invariant polynomials on [Formula: see text]. We present a calculation of the Hilbert series [Formula: see text] as well as formulas for the first four coefficients of the Laurent expansion of [Formula: see text] at [Formula: see text].

Let K → U(V ) be a unitary representation of the compact Lie group K. Then there is a canonical moment mapping ρ :We show that the Koszul complex is a resolution of the smooth functions on ρ −1 (0) if and only if G → GL(V ) is 1-large, a concept introduced in [11,12]. Now let M be a symplectic manifold with a Hamiltonian action of K. Let ρ be a moment mapping and consider the Koszul complex given by the component functions of ρ. We show that the Koszul complex is a resolution of the smooth functions on Z = ρ −1 (0) if and only if the complexification of each symplectic slice representation at a point of Z is 1-large.