Let $E$ be a smooth elliptic curve over algebraically closed field $k$ of characteristic zero, $\mathcal{L}$ is a line bundle over $E$, $\operatorname{deg}(\mathcal{L})=n \geq 1$. Then I define the group of translations of $\mathcal{L}$ as $H(\mathcal{L})=\{x \in E: T_x \mathcal{L} \cong \mathcal{L}\}$ and theta group (or Heisenberg group) $G(\mathcal{L})$ of line bundle $\mathcal{L}$ as central extension $$ 0 \to k^* \to G(\mathcal{L}) \to H(\mathcal{L}) \to 0. $$ Group $G(\mathcal{L})$ acts on $H^0(E, \mathcal{L})$ in an obvious way and provide an irreducible representation.

Sometimes another version of Heisenberg group is used for elliptic curves. I denote this group $\Gamma_n$. As abstract group $\Gamma_n$ has generators $\epsilon_1$, $\epsilon_2$ and $\delta$ with relations $\epsilon_1^n=\epsilon_2^n=\delta^n=1$, $\epsilon_1 \epsilon_2 =\delta \epsilon_2 \epsilon_1$. In other words it is a central extension of the group $H(\mathcal{L})$ (generated by $\epsilon_1$ and $\epsilon_2$) by group of roots of unity in $k$ (generated by $\delta$). First of all, in order to get an action on $H^0(E, \mathcal{L})$ I need to see that translation by $x \in H(\mathcal{L})$ induce an automorphism of $\mathcal{L}$ given by some root of unity. How to show this?

Where I can find a proof that $\Gamma_n$ acts irreducibly on $H^0(E, \mathcal{L})$? Is it true for higher rank vector bundles over $E$?

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