I have the group $O(2)$ acting on $\mathbb{C}^2$ with the action generated by \begin{equation} \theta(z_1, z_2) = (e^{i\theta}z_1, e^{i\theta}z_2), \\ \kappa(z_1,z_2) = (\bar{z_1}, \bar{z_2}), \end{equation} and I want to compute the group orbit representatives and isotropy subgroups.
We have defined the isotropy subgroup of x as the group $\Sigma_x = \{\gamma \in \Gamma \vert \gamma x = x\}$, and the group orbit of a point $x$ is the set $\Gamma x = \{\gamma x \vert \gamma \in \Gamma\}$, where in this case we have $\Gamma = O(2)$.
If I take the point $0$, then I get the group orbit representative is just the set containing $0$, and the isotropy subgroup is $O(2)$. I'm not sure how to deal with any other cases though.
I think I'm most confused about how to actually use the action. If I took a point in $\mathbb{C}^2$, and some $\gamma \in O(2)$, how would I actually compute the action if there are two components to it?