Certain holomorphic symplectic varieties of interest, such as Hilbert scheme of points on $C^2$ and more general moduli of sheaves on surfaces have a singular affine Poisson blowdown $X_0$ (which for Hilbert schemes is the symmetric power of the surface). Even more classically, such blowdowns exist for cotangent bundles to homogeneous varieties $G/P$. Equivariant quantum cohomology of such symplectic resolutions show a particularly close connections to classical structures in representation theory. There is an ongoing project to better understand them pursued by several groups from several directions. I will explain the general shape of this program and some of the more interesting examples worked out so far.