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There is a well established theory of embeddings $G/H \to X$ into a normal variety $X$ where $G$ is a reductive group over an algebraically closed field and the image of any Borel $B$ is open in $X$; that standard reference is Knop.

I wonder if there is similar theory when $G$ is replaced by a Kac-Moody group. I'm most interested in the case the group is a loop group.

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    $\begingroup$ Spherical Kac-Moody varieties in general I do not know, but there is a nice theory of (the subclass of) Kac-Moody symmetric spaces. See e.g. arXiv:0712.2320v1 [math.RA] and arXiv:1003.4435v1 [math.DG] as starting points. $\endgroup$
    – Claudio Gorodski
    Commented Jun 18, 2011 at 17:12

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