${{\bf {T}}}$ satisfying the assumptions in §\[equivariant\] and will consider the ${{\bf {T}}}$-equivariant $\ke$-quasimap theories (the torus is allowed to be trivial if ${W/\!\!/{{\bf G}}}$ is projective). We write simply $H^*({W/\!\!/{{\bf G}}})$ for the appropriate ${{\bf {T}}}$-equivariant (localized) cohomology group.

In addition, some of the results in this section are stated for both untwisted and twisted theories of ${W/\!\!/{{\bf G}}}$. All arguments we give are identical whether the twisting is present or not. Hence we do not provide separate proofs, nor do we include the twisting in the notation for brackets, double brackets etc.

Summary of results from [@CK0]
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### Graph spaces, $J^\ke$-functions, and $S^\ke$-operators

Let $0+\leq\ke$ be a stability parameter. For each $k\geq 0$, we have the [*graph space*]{} ${QG^\ke_{0,k,\beta}({W/\!\!/{{\bf G}}})}$, see [@CKM], §7.2 and [@CK0], §2.6. It is the moduli space of genus zero, $k$-pointed, $\ke$-stable quasimaps whose domain curve contains an irreducible component which is a parametrized $\PP^1$. The $\CC^*$-action on the parametrized component lifts to an action on ${QG^\ke_{0,k,\beta}({W/\!\!/{{\bf G}}})}$. The fixed loci for this action are described e.g., in §4 of [@CK0] and their geometry has played an important role in the study of genus zero $\ke$-wall-crossings in quasimap theory undertaken in [@CK0] (they 