from residues on the [*unpointed*]{} graph spaces $QG^{0+}_{0,0,\beta}({W/\!\!/{{\bf G}}})$. These residues have been explicitly calculated in (almost) all interesting examples, giving closed formulas for the small $I$-function.

Also define, for any $\gamma\in H^*({W/\!\!/{{\bf G}}})$, $$\label{S operator}
S^\ke_t(\gamma):= \sum_{i=1}^s\gamma_i {\langle\!\langle}\frac{\gamma^i}{z-\psi},\gamma{\rangle\!\rangle}^\ke_{0,2}(t).$$ The convention $${\langle}\frac{\gamma^i}{z-\psi},\gamma{\rangle}^\ke_{0,2,0}={\langle}\gamma^i,\gamma{\rangle}$$ is made for the unstable term (corresponding to $m=0,\beta=0$) in ${\langle\!\langle}... {\rangle\!\rangle}^\ke_{0,2}$. Directly from definitions, for every $i=1,\dots, s$, $$\label{derivatives}
z\frac{\partial}{\partial t_{0i}} J^\ke(t)= S^\ke_t(\gamma_i).$$ It is shown in [@CK0] that the formula defines a family (with parameter $t$) of symplectic transformations on the symplectic space $\mathcal{H}=H^*({W/\!\!/{{\bf G}}},\Lambda)\{\!\{z,z^{-1}\}\!\}$ appearing in Givental’s formalism of Gromov-Witten theory, [@Givental-symplectic].

The most general $\ke$-wall-crossing formula in genus zero applies to the operators $S^\ke_t$, see Conjecture 6.1.1 and Theorem 7.3.1 in [@CK0]. We state here a special case, as formulated in [@CK0], Theorem 1.2.2.

\[mirror\] Assume that the ${{\bf T}}$-action on ${W/\!\!/{{\bf G}}}$ has isolated fixed points. Then for every $\ke\geq 0+$ we have $$S^\ke_t(\one)=S^\infty_{\tau^\ke(t)}(\one)$$ where the (invertible) transformation $\tau^\ke(t)$ is the following series of primary $\ke$-quasimap invariants $$\label{mirror map}
\begin{split}
&\tau^\ke(t)=\sum_{i=1}^s \gamma_i{\langle\!\langle}\gamma^i,\one{\rangle\!\rangle}(t)-\one=\\
&t+\sum_{i=1}^s\gamma_i\sum_{\beta\neq 0}\sum_{m\geq 0}\frac{q^\beta}{m!}{\langle}\gamma^i,\one,t,\dots,t{\rangle}^\ke_{0,2+m,\beta}.
\end{split}$$ Moreover, the same is true for $E$-twisted theories, where $E$ is a convex ${{\bf G}}$-representation.

No positivity assumptions are made in Theorem \[mirror\] on $(W,{{\bf G}},\theta)$, or 