In these [notes](http://ccirm.cedram.org/cedram-bin/article/CCIRM_2010__1_1_33_0.pdf) 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](https://en.wikipedia.org/wiki/Borel_subgroup) 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.




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As a complex space, I believe $S^2 \simeq \widehat{\mathbb{C}}= \mathbb{C} \cup \{ \infty\}$ is called the [**Riemann sphere**](https://en.wikipedia.org/wiki/Riemann_sphere) or $\mathbb{C}P^1$ with a standard identification via [Möbius](https://en.wikipedia.org/wiki/M%C3%B6bius_transformation) transformations. If we identify $\mathbb{C}\simeq \mathbb{R}^2$ and remember the complex structure, we might be OK. 


As a [complex projective space](https://en.wikipedia.org/wiki/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 $H$.



Is $x^2 + y^2 + z^2 = 1$ a spherical variety?
Yes, $S^2$ is $PSL\_2\mathbb{C}/B$ where $B$ is the Borel subgroup, so the Riemann sphere is spherical in that sense. Spherical varieties are called spherical because the Riemann sphere is one, with action of the rotation group: see here: [Why are they called Spherical Varieties?](https://mathoverflow.net/questions/53274/why-are-they-called-spherical-varieties)


