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

Question

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?

Answer 1

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 $2m+1$.

Answer 2

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 $(-1)^n$.

Answer 3

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

$\Gamma(X,{\mathcal O}_X)=\bigoplus \Gamma({\mathbb P}^1, {\mathcal O}_{{\mathbb P}^1}(2m))=\bigoplus V_{2m}$.

Answer 4

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.