way with respect to this structure. These facts are well-known for the moduli spaces of stable maps, and one sees immediately that their standard proofs, as given in [@Behrend], are $\ke$-independent. The general statement we will use is the one appearing as Theorem 13 in [@Getzler], with the spaces ${\overline{M}_{g,k}({W/\!\!/{{\bf G}}},\beta)}$ replaced by general ${\mathrm{Q}_{g,k}^\ke({W/\!\!/{{\bf G}}},\beta)}$.

In particular, we obtain virtual divisors covering the boundary as follows. Let $[k]:=\{1,2,\dots,k\}$. Let $$\mathcal{B}(g,k):=\{((g_1,S_1),(g_2,S_2))\;|\; g=g_1+g_2,\;  [k]=S_1\coprod S_2\;  \}.$$ be the set of ordered partitions of $(g,k)$. For each $\sigma\in \mathcal{B}(g,k)$ put $$\widetilde{D}^\ke_\sigma:=\coprod_{\beta=\beta_1+\beta_2} \mathrm{Q}^\ke_{g_1,S_1\cup\bullet}({W/\!\!/{{\bf G}}},\beta_1)\times_{{W/\!\!/{{\bf G}}}} \mathrm{Q}^\ke_{g_2,S_2\cup\star}({W/\!\!/{{\bf G}}},\beta_2),$$ where the fiber product is over the evaluation maps at the additional markings. Alternatively, $\widetilde{D}^\ke_\sigma$ is defined by the fiber product diagram $$\begin{CD}
{\widetilde {D}}^\ke_\sigma  @>>> \coprod_{\beta_1+\beta_2=\beta}\mathrm{Q}^\ke_{g_1,S_1\cup\bullet}({W/\!\!/{{\bf G}}},\beta_1)\times \mathrm{Q}^\ke_{g_2,S_2\cup\star}({W/\!\!/{{\bf G}}},\beta_2)\\
@VVV @VV ev_\bullet\times ev_\star V\\
{W/\!\!/{{\bf G}}}@ >\Delta >> {W/\!\!/{{\bf G}}}\times{W/\!\!/{{\bf G}}}.
\end{CD}$$ There is a proper gluing map $$\begin{CD}
\widetilde{D}^\ke_\sigma@>h_\sigma>> {\mathrm{Q}_{g,k}^\ke({W/\!\!/{{\bf G}}},\beta)},
\end{CD}$$ and the boundary divisor $D^\ke_\sigma$ is the stack-theoretic image of $h_\sigma$. The virtual classes of $\widetilde{D}^\ke_\sigma$ and of $D^\ke_\sigma$ are defined by $$\label{bdry1}
[\widetilde{D}^\ke_\sigma]^{\mathrm{vir}}:=\Delta^! \sum_{\beta_1+\beta_2=\beta}
[\mathrm{Q}^\ke_{g_1,S_1\cup\bullet}({W/\!\!/{{\bf G}}},\beta_1)]^{\mathrm{vir}}\otimes [\mathrm{Q}^\ke_{g_2,S_2\cup\star}({W/\!\!/{{\bf G}}},\beta_2)]^{\mathrm{vir}},$$ $$\label{bdry2}
[D^\ke_\sigma]^{\mathrm{vir}}:=(h_\sigma)_*[\widetilde{D}^\ke_\sigma]^{\mathrm{vir}}.$$

Similarly, there is a proper gluing map $$h_0:{W/\!\!/{{\bf G}}}\times_{{W/\!\!/{{\bf G}}}\times{W/\!\!/{{\bf G}}}}\mathrm{Q}^\ke_{g-1,[k]+\bullet+\star}({W/\!\!/{{\bf G}}},\beta)\lra{\mathrm{Q}_{g,k}^\ke({W/\!\!/{{\bf G}}},\beta)},$$ (fiber product over the diagonal map $\Delta$ and 