# Reference request: Affine Grassmannian and G-bundles

Let $G$ be an affine algebraic group over an algebraically closed field $k$ of zero characteristic. The set of cosets $X_G=G(k((t))/G(k[[t]])$ is called the Affine Grassmannian of $G$ and can be given the structure of an ind-$k$-variety, so that for a closed embedding of groups $H\to G$, we get a natural morphism $X_H\to X_G$, which is a closed embedding if $G/H$ is affine. I am interested in a (as detailed as possible) reference for this, but from a specific perspective as will be explain in the following.

There are several possible (equivalent) constructions. A direct approach, in the language of ind-varieties, can be found in Kumar's "Infinite grassmannians and Moduli spaces of G-bundles" for example (for a reductive $G$, which is fine for me), which describes the ind-variety structure explicitly. Apart of using some representation theory, that I don't know well enough, it also lacks an explicit universal property which makes it difficult to operate with and in particular to construct morphsims to and from it.

A more abstract approach is to describe a functor $\operatorname{Gr}_G:kAlg\to Set$ for which $X_G$ is the set of $k$-points. Here is were the $G$-bundles (torsors) appear. There are the "global" and "local" approaches, in which roughly, $\operatorname{Gr}_G (A)$ classifies $A$-families of $G$-Bundles on a curve or a formal disc resp. together with a trivializaion away from a point. Now that one has a functor, it is possible to show that it is an ind-scheme. It is this approach that I would like to have a reference for.

I would like to mention, that this approach is outlined in Gaitsgory's seminar notes, and a more competent student would probably be able to fill in the details by herself, but unfortunately I find it difficult, so I was hopping there might be a more thorough treatment available.

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If you provide contact information (e.g., a webpage which lists your email address) then I can send you something. But perhaps someone else will provide a more detailed literature reference (I am not aware of any). –  user28172 Jun 10 '13 at 11:17
@nosr, thank you for your help. you can send it to kotelkanim1@gmail.com. –  KotelKanim Jun 10 '13 at 11:37
I sympathize greatly with your last sentence and I wish I'd done something to formalize all the exercises I did over the years while in grad school, but alas, when it came time to write the thesis it wasn't worth making it even longer to include something "standard"...therein, I suspect, lies the problem. –  Ryan Reich Jun 10 '13 at 11:52
I believe your second approach is Proposition 2 (p.505) of Heinloth's "Uniformization of G-bundles." (link.springer.com/article/10.1007%2Fs00208-009-0443-4) –  expz Jun 10 '13 at 12:12
There is a description in Mirkovic-Vilonen, but it is missing some details: arxiv.org/abs/math/0401222 –  S. Carnahan Jun 10 '13 at 12:36

I believe your second approach is Proposition 2 (p.6) of Heinloth's Uniformization of $G$-bundles available from Heinloth's website: http://staff.science.uva.nl/~heinloth/Uniformization_17-8-09.pdf

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Another reference with a slightly different approach is Drinfeld & Beilinsons Quantization of Hitchin Integrable Systems and Hecke Eigensheaves available online here link text. The affine Grassmannian is defined in 5.3.11 and it is explained in the previous remark 5.3.10 c. why this is represented by a formally smoooth ind-proper ind-scheme.

Here, the authors keep track of the points away from which the $G$-bundle has a trivialization, this is denoted by $Gr_{X^I}$ and then take the colimit $colim_{I}Gr_{X^I}$, where the colimit is taken over the category $\mathbf{fSet}^{op}$ of all finite sets $I$ with surjections between them.

In the language of Gaitsgory in Contractibility of the Space of Rational Maps this means that the affine Grassmannian is a pseudo ind-proper pseudo ind-scheme link text. The main point, I think, is that it is possible to define a well-behaved category of D-modules over such pseudo ind-schemes.

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The following papers might also be helpful:

G. Faltings, Algebraic loop groups and moduli spaces of bundles, J. Eur. Math. Soc. 5 (2003), 41-68.

G. Pappas, M. Rapoport, Twisted Loop groups and their affine flag varieties, Adv. Math. 219 (2008), no. 1, 118-198

G. Pappas, X. Zhu, Local models of Shimura varieties and a conjecture of Kottwitz, http://arxiv.org/abs/1110.5588v4, to appear in Invent. math.

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