Let $\pi:X\rightarrow Z$ a double cover of an elliptic curve with genus $g\geq 3$. Choose a general rank 2 and degree -1 vector bundle $F$ on $Z$, let $E=\pi^*F$ and fix $x\in X$. The involution $i$ of the cover induces an involution on $H^0(X,E)$. Consider the multiplication $$H^0(X,\mathcal O(x+i(x)))\otimes H^0(X,E(-x-i(x))\rightarrow H^0(X,E)$$ My question: Does the image of this multiplication lies in the invariant part of the action of $i$?

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