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In this paper we construct star products on Marsden-Weinstein reduced spacesin case both the original phase space and the reduced phase space are(symplectomorphic to) cotangent bundles. Under the assumption that the originalcotangent bundle $T^*Q$ carries a symplectique structure of form$\omega_{B_0}=\omega_0 + \pi^*B_0$ with $B_0$ a closed two-form on $Q$, isequipped by the cotangent lift of a proper and free Lie group action on $Q$ andby an invariant star product that admits a $G$-equivariant quantum momentummap, we show that the reduced phase space inherits from $T^*Q$ a star product.Moreover, we provide a concrete description of the resulting star product interms of the initial star product on $T^*Q$ and prove that our reduction schemeis independent of the characteristic class of the initial star product. Unlikeother existing reduction schemes we are thus able to reduce not only stronglyinvariant star products. Furthermore in this article, we establish a relationbetween the characteristic class of the original star product and thecharacteristic class of the reduced star product and provide a classificationup to $G$-equivalence of those star products on $(T^*Q,\omega_{B_0})$, whichare invariant with respect to a lifted Lie group action. Finally, weinvestigate the question under which circumstances `quantization commutes withreduction' and show that in our examples non-trivial restrictions arise

Phase Space Reduction of Star Products on Cotangent Bundles