$SL_2(\mathbb C)$ action on an affine quadric

Question

What is an example of an affine quadric $Q\subset \mathbb C^3$ such that $SL_2(\mathbb C)$ acts transitively and $Q=SL_2(\mathbb C)/\mathbb C^*$?

Answer 1

We can do this by two steps:

1. The group $$ \mathrm{SO}_{3}(\mathbb{C}) = \{A\in M_{3}(\mathbb{C}) \,|\, AA^{T} = I\} $$ naturally acts on the quadric $$ Q = \{(x, y, z)\in \mathbb{C}^{3}\,|\, x^{2}+y^{2}+z^{2} =1\} $$ by $(x, y, z)^{T} \mapsto A(x, y, z)^{T}$. This action is transitive, and stabilier of $e_{1} = (1, 0, 0)^{T}$ is $$ \begin{pmatrix} 1 & \mathbf{0}^{T} \\ \mathbf{0} & g \end{pmatrix} $$ with $g\in \mathrm{SO}_{2}(\mathbb{C})$. So $Q \simeq \mathrm{SO}_{3}(\mathbb{C}) / \mathrm{SO}_{2}(\mathbb{C})$.
2. We have an isomorphism $$ \mathrm{PSL}_{2}(\mathbb{C}) \simeq \mathrm{SO}_{3}(\mathbb{C}). $$ This can be obtained by considering adjoint action of $\mathrm{SL}_{2}(\mathbb{C})$, and observing that for $v, w\in \mathfrak{sl}_{2}(\mathbb{C})$, $(v, w) \mapsto \mathrm{Tr}(vw)$ is a bilinear pairing which is invariant under the adjoint action. (For details, see my note.) Hence we have a surjective homomorphism $\mathrm{SL}_{2}(\mathbb{C}) \to \mathrm{SO}_{3}(\mathbb{C})$, which can be explicitly described. Under this map, $\mathrm{SL}_{2}(\mathbb{C})$ acts transitively on $Q$, and (although I didn't check rigorously) the stabilizer of $e_{1}$ is isomorphic to $\mathbb{C}^{\times}$).