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Let $G$ be a compact connected semisimple Lie group. The algebro-geometric definition of the affine Grassmannian is the coset space $$\mathcal{G}r=G_{\mathbb{C}}(\mathcal{\mathbb{C}((t))}/G_{\mathbb{C}}(\mathbb{C}[[t]]).$$ Technically, $\mathcal{G}r$ is an ind-scheme over $\mathbb{C}$, realized as an inductive limit of schemes.

On the other hand, we have a differential-geometric version of the affine Grassmannian described in Chapters 7 and 8 of Pressley-Segal. In particular, they construct $\mathcal{G}r_0^{\mathfrak{g}}$. This is homotopy-equivalent to $\Omega G$.

I am looking for a reference that explicitly relates these two notions of the affine Grassmannian. In particular, I am seeking answers to some of the questions below.

  1. In what categories is it reasonable to compare these versions of the affine Grassmannian?

  2. In these categories, are the two versions isomorphic?

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    $\begingroup$ I don't know references, but I think the basic thing to understand is the third space $G($Laurent polynomials$)/G($polynomials$)$. I believe the statement is that this is the same as $\mathcal Gr$ if $G$ is semisimple but not otherwise. Definitely they're different for $G=GL(1)$. $\endgroup$
    – Allen Knutson
    Commented Nov 28, 2013 at 1:15
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    $\begingroup$ @AllenKnutson: Aren't they both $\mathbb Z$ for $GL(1)$? $\endgroup$
    – Will Sawin
    Commented Nov 28, 2013 at 3:01
  • $\begingroup$ Hmm. I will ask around and try to figure out what I'm misremembering. $\endgroup$
    – Allen Knutson
    Commented Nov 28, 2013 at 4:15
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    $\begingroup$ There's a discussion of the Laurent polynomial point of view in chapter 13.2 of Kumar's Kac-Moody Groups book (he gives a relation between $G(\mathbb C[t,t^{-1}])$ and the group denoted by $\mathcal G^{min}$). He relates the homogeneous spaces associated to $G(\mathbb C[t,t^{-1}])$ and $G(\mathbb C((t)))$ there, although I don't know much about the details. $\endgroup$
    – Chuck Hague
    Commented Nov 28, 2013 at 16:19
  • $\begingroup$ Have no idea how to make it rigorous, but elements of $G_{\mathbb C}(\mathbb C((t)))$ give rise to maps from the punctured disk to $G(\mathbb C)$, and $G_{\mathbb C}(\mathbb C[[t]])$ corresponds to maps that can be extended to the origin. Hence $\mathcal Gr$ maps (bijectively?) onto the set of homotopy classes of free loops of $G(\mathbb C)$ modulo contractible free loops. This probably means that one should not expect an isomorphism. Rather, a quotient of $\mathcal Gr^{\mathfrak g}_0$ should embed in $\mathcal Gr$ as classes of based loops modulo contractible based loops. $\endgroup$
    – მამუკა ჯიბლაძე
    Commented Aug 1, 2020 at 12:48

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