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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?

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    $\begingroup$ I think a lot of times wonderful compactifications are considered only for groups of adjoint type. $\endgroup$
    – GTA
    Jun 28 '20 at 9:21
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    $\begingroup$ @GTA But for a semisimple Lie group $G$ with maximal compact $K$ and center $Z$ we have $G/K=(G/Z)/(KZ/Z)$. $\endgroup$
    – YCor
    Jun 28 '20 at 9:25
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    $\begingroup$ What are these "some papers"? $\endgroup$
    – abx
    Jun 28 '20 at 10:01
  • $\begingroup$ 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 $\endgroup$
    – user125056
    Jun 28 '20 at 13:23
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    $\begingroup$ The wonderful compactifications in the non-adjoint case are best considered as Deligne-Mumford stacks, cf. the articles of Martens -- Thaddeus. $\endgroup$ Jun 28 '20 at 13:57

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