Phase Space Reduction of Star Products on Cotangent Bundles.

In this paper we construct star products on Marsden-Weinstein
reduced spaces in case both the original phase space and the reduced phase
space are (symplectomorphic to) cotangent bundles. Under the assumption
that the original cotangent bundle $T^*Q$ carries a symplectic structure
of form $\omega_{B_0}$ = \omega_0 + \pi^*B_0$ with $B_0$ a closed two-form
on $Q$, is equipped by the cotangent lift of a proper and free Lie group
action on $Q$ and by an invariant star product that admits a
$G$-equivariant quantum momentum map, 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 in terms of the initial star
product on $T^*Q$ and prove that our reduction scheme is independent of
the characteristic class of the initial star product. Unlike other
existing reduction schemes we are thus able to reduce not only strongly
invariant star products. Furthermore in this article, we establish a
relation between the characteristic class of the original star product and
the characteristic class of the reduced star product and provide a
classification up to $G$-equivalence of those star products on $(T^*Q,
\omega_{B_0})$, which are invariant with respect to a lifted Lie group
action. Finally, we investigate the question under which circumstances
`quantization commutes with reduction' and show that in our examples
non-trivial restrictions arise.