In these notes of Pezzini and the obvious question arises, is a sphere, a "spherical variety". I'm interested in $L^2(X)$ for the space: $$ X = \{ x^2 + y^2 + z^2 = 1\} \subseteq \mathbb{R}^3 $$ In order to become a spherical variety I need a couple of things:
- $G = $ reductive connected linear algebraic group. I'm guessing $G = \text{SO}_3$.
- The Borel subgroup $B$ of orthogonal matrices would be the upper triangular orthogonal matrices.
- The maximal torus would be $T \simeq \{ (e^{i\theta_1}, e^{i\theta_2}, e^{i\theta_3}): \theta_1 + \theta_2 + \theta_3 = 0 \}$ except those are complex. And I'm getting that $B = T$.
- Unsure about the correct choice of $H$. It just says that $X$ is some $G$-variety, possible that $X = G/H$.
Wikipedia also has discussion of Homogeneous spaces. It has that:
- $S^2 = \text{O}(3)/\text{O}(2)$
- $\text{Gr}(k,n) = \text{O}(n) / \text{O}(k) \times \text{O}(n-k) $
Then maybe I'd have that $L^2(X)$ is just the spherical haromnics, even if I pass into positive characteristic.
As a complex space, I believe $S^2 \simeq \widehat{\mathbb{C}}= \mathbb{C} \cup \{ \infty\}$ is called the Riemann sphere or $\mathbb{C}P^1$ with a standard identification via Möbius transformations. If we identify $\mathbb{C}\simeq \mathbb{R}^2$ and remember the complex structure, we might be OK.
As a complex projective space we might consider the Hopf fibration $\mathbb{C}P^1 = S^3/S^1$ or possibly $U(2)/\big( U(1) \times U(1) \big)$, perhaps leading to appropriate choices of $G$ and $H$.
