of $\omega(\xi,A')$: $$\textstyle \sum_{\alpha\in G}\alpha\cdot\omega(\xi,A')=\sum_{\alpha\in G}\omega(\alpha\cdot\xi,A')\in{\operatorname{Ext}}^1_C({{\mathcal C}}_{C/X},{{\mathscr O}}_C)^G$$ is realized as the obstruction to lifting the $G$-equivariant deformation $\times_{\alpha\in G}\alpha\cdot\xi$ over $\otimes^{|G|}A$ of the embedding $C\subset X$ in a $G$-equivariant fashion.

For this we use an interpretation of $({\operatorname{Sym}}^2 F)^{G,{L({{\mathscr B}})}\not= 1}$ as a conormal space of moduli spaces. We first note that the restriction map $$\Phi: ({\operatorname{Sym}}^2F)^G=({\operatorname{Sym}}^2 H^0(C, \Omega_C))^G \to H^0(C, \Omega^{\otimes 2}_C)^G$$ can be understood as follows: the dual of $H^0(C, \Omega^{\otimes 2}_C)^G$ is the space of first order deformations of $C$ as a $G$-curve and the dual of $({\operatorname{Sym}}^2F)^G$ is the space of first order deformations of deformations of the Jacobian $J(C)$ as a ppav with $G$-action, and $\Phi$ is then the obvious map. (By Max Noether’s theorem, the natural map ${\operatorname{Sym}}^2F={\operatorname{Sym}}^2H^0(C, \Omega_C)\to H^0(C, \Omega^{\otimes 2}_C)$ is onto and hence the same is true for $\Phi$.) As mentioned above, the dual of $({\operatorname{Sym}}^2F)^{L({{\mathscr B}})}$ has the interpretation of the space of first order deformations of $J(C)$ as a ppav endowed with an involution preserving homomorphism of ${{\mathscr B}}$ to the endomorphism algebra tensored with ${{\mathbb R}}$. In other words, $({\operatorname{Sym}}^2 F)^{G,{L({{\mathscr B}})}\not= 1}$ vanishes on the first order deformations of $J(C)$ as $G$-ppav which retain ${{\mathscr B}}$ 