$3$-punctured sphere. It is also equivalent to $H^0(C, \Omega_C^{\otimes 2})^G$ being nonzero.

The hypothesis that $G$ does not contain a hyperelliptic involution in its center is needed in order to ensure that some point of ${{\mathscr M}}^\phi$ is defined by a nonhyperelliptic curve (this will actually imply that every point of ${{\mathscr M}}^\phi$ has that property).

Observe that if a nontrivial automorphism of $C$ acts trivially on its canonical image in $\check{{\mathbb P}}(H^0(C,\Omega_C))$, then $C$ is hyperelliptic and this automorphism is the hyperelliptic involution. The latter acts as $-1$ on $H^0(C,\Omega_C)$, and so $G$ always acts faithfully on $H^1(C,{{\mathbb C}})$.

The hypothesis on the $G$-module structure of $H^0(C,\Omega_C)$ comes from the use of the following lemma in the proof.

\[lemma:quadratic\] Let $F$ be a finite dimensional ${{\mathbb C}}G$-module with the property that

1.  every irreducible symplectic representation occurring in $F$ occurs with multiplicity $\ge 3$ and

2.  every non-self-dual representation occurring in $F$ occurs along with its dual and both do so with multiplicity $\ge 2$.

Let $L$ be a compact Lie subgroup of ${\operatorname{GL}}(F)$ which contains the image of $G$ and has the property that $({\operatorname{Sym}}^2F)^L= ({\operatorname{Sym}}^2F)^G$. Then $L$ is contained in the image of ${{\mathbb R}}G$ in ${\operatorname{End}}(F)$.

