Gromov-Witten invariants for abelian and nonabelian quotients

Let X be a smooth projective variety over C with the (linearized) action of a complex reductive group G, and let T ⊂ G be a maximal torus. In this setting, there are two geometric invariant theory (GIT) quotients, X//T and X//G, with a rational map Φ : X//T − −> X//G between them. We will further assume that “stable = semistable” in the GIT and that all isotropy of stable points is trivial, so X//T and X//G are smooth projective varieties, and Φ is a G/T fibration.