# Algebra of regular functions on the quadratic cone and SU(2) representations

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?

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 $$2m+1$$.
The quadratic cone is the quotient of $$\mathbb{A}^2 = \mathrm{Spec}(\mathbb{C}[u,v])$$ by the involution $$\iota \colon (u,v) \mapsto (-u,-v)$$. Consequently, $$\Gamma(\mathbb{A}^2/\iota, \mathcal{O}) = \Gamma(\mathbb{A}^2, \mathcal{O})^{\iota} = \left( \bigoplus V_n \right)^\iota = \bigoplus V_{2m}$$ since $$\iota$$ acts on $$V_n$$ by $$(-1)^n$$.
Apply the Borel-Weil theorem to the Lie group $$G={\rm SL}_2({\mathbb C})$$, with Borel $$B$$, the complexification of $${\rm SU}(2)$$. The flag variety $$G/B={\mathbb P}^1$$, and we have $$V_m=\Gamma({\mathbb P}^1, {\mathcal O}_{{\mathbb P}^1}(m))$$. On the other hand, $$X$$ is the cone over $${\mathbb P}^1\subset {\mathbb P}^2$$ embedded by $${\mathcal O}_{{\mathbb P}^1}(2)$$. So
$$\Gamma(X,{\mathcal O}_X)=\bigoplus \Gamma({\mathbb P}^1, {\mathcal O}_{{\mathbb P}^1}(2m))=\bigoplus V_{2m}$$.
This is close to Balazs's answer. $$X$$ is the nilpotent cone for $$SL(2)$$. The Springer resolution $$T^* \mathbb P^1 \to X$$ is semismall. $$T^* \mathbb P^1$$ is the total space of the line bundle $$O(-2)$$ on $$\mathbb P^1$$. So global functions on $$X$$ are the same as global functions on the total space of the bundle $$O(-2)$$ on $$\mathbb P^1$$ whose structure sheaf is $${\rm Sym}^* O(-2)^\vee$$. See this and this questions.