  We formulate a quantization commutes with reduction principle in the setting
where the Lie group $G$, the symplectic manifold it acts on, and the orbit
space of the action may all be noncompact. It is assumed that the action is
proper, and the zero set of a deformation vector field, associated to the
momentum map and an equivariant family of inner products on the Lie algebra
$\mathfrak{g}$ of $G$, is $G$-cocompact. The central result establishes an
asymptotic version of this quantization commutes with reduction principle.
Using an equivariant family of inner products on $\mathfrak{g}$ instead of a
single one makes it possible to handle both noncompact groups and manifolds, by
extending Tian and Zhang's Witten deformation approach to the noncompact case.
