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Normal form of this group action?

Let $d\in\mathbb{N}$. We consider the vector space $V=\mathbb{C}^2\otimes\mathbb{C}[x_0,x_1]_d$ where $\mathbb{C}[x_0,x_1]_d$ is the space of homogeneous binary forms of degree $d$. We have a natural ...