Symplectic reductions from compact Hamiltonian Lie group actions on symplectic manifolds are important examples in the study of symplectic topology and mirror symmetry. In late 90s, Givental introduced an equivariant Gromov-Witten theory and used it to prove the mirror conjecture under the semi-positive assumption. During the past ten years, several groups of people have been working hard in generalizing the theory using symplectic vortex equations, but unfortunately, the corresponding moduli spaces suffer serious defect in compactness for higher genus case. In my talk, I will explain my ongoing project with Bohui Chen and Bai-Ling Wang in defining a new Gromov-Witten type of invariants for the equivariant cohomology of the ambient space. Using it, we also construct a quantum Kirwan morphism for a symplectic reduction.