\delta_1\bar\psi_1^{a_1},\dots,\delta_k\bar\psi_k^{a_k}
\rangle^\ke_{g,k,\beta}:=\int_{[{\mathrm{Q}_{g,k}^\ke({W/\!\!/{{\bf G}}},\beta)}]^{\mathrm{vir}}} \prod_{j=1}^k ev_j^*(\delta_j)\bar\psi_j^{a_j},$$ with $\bar\psi_j:=f^*\psi_{j,k}$ and $\psi_{j,k}$ the $\psi$ class at the $j^{\mathrm{th}}$ marking on ${\overline{M}_{g,k}}$.

### genus zero TRR

For $m\geq 0, 2g-2+k>0$, consider the forgetful map $$f: {\mathrm {Q}}^\ke_{g,k+m}({W/\!\!/{{\bf G}}},\beta)\lra{\overline{M}_{g,k}}.$$ Fix $j\in[k]$. We have the following basic comparison of the classes $\psi_j$ and $f^*\psi_{j,k}$.

\[psi comparison\] Let $A_j\subset\mathcal{B}(g,k+m)$ be the subset consisting of ordered partitions $\sigma=((0,g), (S_1,S_2))$ with $j\in S_1$ and $ [k]\setminus\{ j\}\subset S_2$. Then $$\label{psi comparison eqn}
(\psi_j-f^*\psi_{j,k})\cap [{\mathrm{Q}_{g,k}^\ke({W/\!\!/{{\bf G}}},\beta)}]^{\mathrm{vir}}=\sum_{\sigma\in A_j} [D^\ke_\sigma]^{\mathrm{vir}}.$$

The argument in Gromov-Witten theory is not $\ke$-dependent, so will work in general.

In Gromov-Witten theory, Lemma \[psi comparison\] is used to express descendant invariants in terms of ancestors. This can be done for quasimap invariants too, but we will not deal with it in this paper. Instead, we note that another consequence of the Lemma, the genus zero Topolgical Recursion Relation (TRR) holds for all stability parameters.

\[TRR\] (Genus zero TRR) For all $\ke\geq 0+$, all $a_1,a_2,a_3\geq 0$ and all $\delta_1,\delta_2,\delta_3\in H^*({W/\!\!/{{\bf G}}},\QQ)$ we have

$${\langle\!\langle}\delta_1\psi_1^{a_1},\delta_2\psi_2^{a_2},\delta_3\psi_3^{a_3}{\rangle\!\rangle}_{0,3}^\ke=
\sum_{i=1}^s {\langle\!\langle}\delta_1\psi_1^{a_1-1},\gamma_i{\rangle\!\rangle}_{0,2}^{{\varepsilon }}{\langle\!\langle}\gamma^i, \delta_2\psi_2^{a_2},\delta_3\psi_3^{a_3}{\rangle\!\rangle}_{0,3}^\ke.
$$

Apply Lemma \[psi comparison\] for $g=0$, $k=3$, and use the fact that $\psi_1$ vanishes on $\overline{M}_{0,3}={{\mathrm{Spec}}}(\CC)$.

\[string etc\] Gromov-Witten invariants satisfy additional structures encoded in the string, dilaton, and divisor 