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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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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 $G=GL(1)$. – Allen Knutson Nov 28 '13 at 1:15
@AllenKnutson: Aren't they both $\mathbb Z$ for $GL(1)$? – Will Sawin Nov 28 '13 at 3:01
Hmm. I will ask around and try to figure out what I'm misremembering. – Allen Knutson Nov 28 '13 at 4:15
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. – Chuck Hague Nov 28 '13 at 16:19

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