  We introduce the_inertial cohomology ring_ NH^*_T(Y) of a stably almost
complex manifold carrying an action of a torus T. We show that in the case that
Y has a locally free action by T, the inertial cohomology ring is isomorphic to
the Chen-Ruan orbifold cohomology ring H_{CR}^*(Y/T) of the quotient orbifold
Y/T.
  For Y a compact Hamiltonian T-space, we extend to orbifold cohomology two
techniques that are standard in ordinary cohomology. We show that NH^*_T(Y) has
a natural ring surjection onto H_{CR}^*(Y//T), where Y//T is the symplectic
reduction of Y by T at a regular value of the moment map. We extend to
NH^*_T(Y) the graphical GKM calculus (as detailed in e.g.
[Harada-Henriques-Holm]), and the kernel computations of [Tolman-Weitsman,
Goldin].
  We detail this technology in two examples: toric orbifolds and weight
varieties, which are symplectic reductions of flag manifolds. The Chen-Ruan
ring has been computed for toric orbifolds, with \Q coefficients, in
[Borisov-Chen-Smith]); symplectic toric orbifolds obtained by reduction by a
connected torus (though with different computational methods), and extend them
to \Z coefficients in certain cases, including weighted projective spaces.
