Let $G$ be a semi-simple group over an algebraically closed field of characteristic zero. In which cases there is a unique wonderful compactification of $G$ (modulo isomorphism)?


For instance, is the wonderful compactification of $SL(n,\mathbb{C})$ unique?



Uniqueness of the wonderful compactification of a semi-simple group
I assume you mean the variety $G$ considered as a $G \times G$ variety via the action $(g,h) \cdot x = gxh^{-1}$, which is the standard interpretation in the literature. The variety $G$ is *spherical* as a $G \times G$ variety, meaning that it contains a dense $B \times B$-orbit (where $B$ is a Borel subgroup of $G$). This statement is the content of the well-known *Bruhat decomposition* of $G$.


There is a more general statement: If $X$ is a spherical $G$-variety, where $G$ is reductive algebraic group over an algebraically closed field of any characteristic, then if $X$ admits a wonderful compactification, it is unique up to $G$-isomorphism.


The variety $G$ does have a wonderful compactification, which is therefore unique up to $G \times G$-isomorphism. The existence of the wonderful compactification of $G$ goes back to the original work on wonderful compactifications, *[Complete symmetric varieties](https://link.springer.com/chapter/10.1007%2FBFb0063234)* by De Concini and Procesi ([MSN](https://mathscinet.ams.org/mathscinet-getitem?mr=718125)). I don't know where the first proof of uniqueness appears, but a good introduction to the subject that includes a full proof is Pezzini's survey article, *[Lectures on spherical and wonderful varieties](http://dx.doi.org/10.5802/ccirm.3)*.


Here's a sketch of the main ideas involved. Any wonderful compactification of $G$, which I will denote $Y$, contains a big cell $Y\_0$ with the following properties:


1. $Y\_0$ is dense in $Y$
2. $Y\_0$ is $B \times B$-invariant and isomorphic to affine space
3. $G \cdot Y\_0 = Y$


So suppose $Y$ and $Y'$ were wonderful compactifications of $G$ with big cells $Y\_0$ and $Y'\_0$. Let $g, g' \in G$ and consider the unique birational $G \times G$-equivariant map $f: Y \dashrightarrow Y'$, i.e. $f(g\_1 g g\_2^{-1}) = g\_1 g' g\_2^{-1}$. This map induces an isomorphism $Y\_0 \cong Y'\_0$ because they have isomorphic coordinate rings (an argument is needed why the map can be extended to the big cell). Then $f$ extends to an isomorphism $Y \cong Y'$ since $(G \times G) \cdot Y\_0 = Y$.


