I was reading the paper "Short Star-Products for Filtered Quantizations" by Pavel Etingof and Douglas Stryker (MSN), where in the introduction they claim that the algebra of regular functions on the quadratic cone $X$ in $\mathbb{C}^3$ is $$\mathbb{C}[x,y,z]/(xy-z^2)=\bigoplus_{m\geq 0}{V_{2m}}$$ where $V_m$ is the irreducible representation of $\operatorname{SU}(2)$ with highest weight $m$.

Why is that true?