If $G \leq GL_2(\mathbb{C})$ is generated by matrix $\begin{pmatrix} 1 & 2\\0 & -1 \end{pmatrix}$, acting on the polynomial ring $\mathbb{C}[X,Y]$, then how can we find the ring of invariants $\mathbb{C}[X,Y]^G$?
I've got it in the form $\{f \in \mathbb{C}[X,Y] : f(x+2y,-y)=f(x,y)\}$, but I think a nicer form is required. Maybe something to do with eigenvectors? Is the $G$-invariance of the zero set $Z(f)$ relevant?
Thanks for any help with this!