Wonderful compactification of $\mathrm{SL}(2)/\mathrm{SO}(2)$

Let $$\mathbb{P}^2 = \mathbb{P}(\operatorname{Sym}^2\mathbb{C}^2)$$ be the projective space of $$2\times 2$$ symmetric matrices over $$\mathbb{C}$$ modulo scalar.

Define an $$\mathrm{SL}(2)$$-action on $$\mathbb{P}^2$$ by $$(A,Z)\mapsto AZA^t$$, where $$A\in \mathrm{SL}(2)$$ and $$Z\in \mathbb{P}^2$$. The stabilizer of the identity is then $$S = \{A\in \mathrm{SL}(2)\: | \: AA^t = \lambda I\}$$ for some $$\lambda\in\mathbb{C}^{*}$$. However, some papers refer to $$\mathbb{P}^2$$ as the wonderful compactification of $$\mathrm{SL}(2)/\mathrm{SO}(2)$$ even though it seems to me that it would be more correct to say that $$\mathbb{P}^2$$ is the wonderful compactification of $$\mathrm{SL}(2)/S$$.

For instance, via the map $$\mathrm{SL}(2)\rightarrow\mathbb{P}^2,\: A\mapsto AA^t$$ the matrix $$\left( \begin{array}{cc} -i & 0 \\ 0 & i \end{array} \right)$$ is mapped to $$-\left( \begin{array}{cc} 1 & 0 \\ 0 & 1 \end{array} \right) = \left( \begin{array}{cc} 1 & 0 \\ 0 & 1 \end{array} \right)$$ Note that $$\left( \begin{array}{cc} -i & 0 \\ 0 & i \end{array} \right)\in S\setminus \mathrm{SO}(2)$$ so the morphism $$\mathrm{SL}(2)/\mathrm{SO}(2)\rightarrow\mathbb{P}^2$$ is not of degree one.

What am I misunderstanding here?

• I think a lot of times wonderful compactifications are considered only for groups of adjoint type.
– GTA
Jun 28 '20 at 9:21
• @GTA But for a semisimple Lie group $G$ with maximal compact $K$ and center $Z$ we have $G/K=(G/Z)/(KZ/Z)$.
– YCor
Jun 28 '20 at 9:25
• What are these "some papers"?
– abx
Jun 28 '20 at 10:01
• For instance this (end of page 13) Compactifications of Symmetric and Locally Symmetric Spaces Armand Borel, Lizhen Ji and this (page 20) arxiv.org/pdf/1401.8050.pdf Jun 28 '20 at 13:23
• The wonderful compactifications in the non-adjoint case are best considered as Deligne-Mumford stacks, cf. the articles of Martens -- Thaddeus. Jun 28 '20 at 13:57