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A pretty naïve question: Which meaning has the term "affine" in the notion of affine Grassmanian. Especially, I do not see any immediate connection to the concept of an "affine scheme" in the classical sense, so seemingly the terminology arose from another concept. But which one? Is there a reasonable "justification" known for the choice of this name?

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    $\begingroup$ Surely it comes from affine Lie algebras and affine Lie groups, which act on the affine Grassmanian. In that case "affine" comes from the Weyl group being a group of affine transformations. $\endgroup$
    – Will Sawin
    Commented Oct 9, 2023 at 18:09
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    $\begingroup$ I'd agree that it's misleading terminology, if one does not know the context that @WillSawin mentions. That is, "affine" groups/algebras have non-finite Weyl groups, which, in many useful cases, still do have models that act by affine transformations on a plane (rather than on a sphere, as with finite Weyl groups). $\endgroup$
    – paul garrett
    Commented Oct 9, 2023 at 18:46
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    $\begingroup$ Geometrically speaking, an affine Grassmannian is a subset of vertices (of fixed type) of an affine building. Just like an ordinary Grassmannian is a subset of vertices of a spherical building. The ordinary, Grassmannian has structure of a scheme. An affine Grassmannian, similarly, has structure of an ind-scheme. $\endgroup$
    – Moishe Kohan
    Commented Oct 10, 2023 at 3:38

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