I was reading the paper "[Short Star-Products for Filtered Quantizations](https://www.emis.de/journals/SIGMA/2020/014)" by Pavel Etingof and Douglas Stryker ([MSN](https://mathscinet.ams.org/mathscinet-getitem?mr=4074404)), 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?



Algebra of regular functions on the quadratic cone and SU(2) representations
The ring $\Bbb C[x,y,z]/(xy-z^2)$ is $\Bbb N$-graded because $xy-z^2$ is homogeneous. Its $m$-th component has dimension $2m+1$, because a basis is given by $\{ x^ay^b \mid a+b=m \} \cup \{ x^ay^bz \mid a+b=m-1 \}$. 


It's easy to see that the representation is irreducible (I assume the action is the adjoint action of $SU(2)$ on $\mathfrak{su}(2) \otimes \Bbb C \cong \Bbb C^3$) giving the results since $V\_{2m}$ is the unique irreducible representation of $SU(2)$ with dimension $2m+1$. 


