A question about the affine Grassmanian

For $SL(2, \mathbf{C} ((t)))$ the affine Grassmanian is defined as: $$SL(2, \mathbf{C}((t))) / SL(2, \mathbf{C} [[t]])$$ Now that is fine but $SL(2, \mathbf{C} ((t)))$ has smaller parabolic subgroups. Define: $$B = \begin{pmatrix} \mathcal{O}^{\times} & \mathcal{O} \\\ t\mathcal{O} & \mathcal{O}^{\times} \end{pmatrix}$$ where $\mathcal{O} = \mathbf{C}[[t]]$. Then: $$SL(2, \mathbf{C} [[t]]) = B \left \{ 1, \begin{pmatrix} & 1 \\\ -1 & \end{pmatrix} \right \} B$$

Question: Why are people so much interested in the affine Grassmanian when it actually sits in a larger, equally well behaved projective ind-scheme (in our example above $SL(2, \mathbf{C}((t)))/ B$)? Both are flag varieties for the loop group so it seems natural to go with the full flag variety.

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I think Scott and David's answer bring this out a bit, but let me emphasize: lots of people have studied this flag variety (try googling "affine flag variety" and "affine Springer fiber"); they aren't ignoring it out of spite, or something. It's just that the affine Grassmannian is so darn interesting. – Ben Webster Jan 23 '11 at 6:42
One thing that seems worth mentioning: the affine Grassmannian is analogous to spaces studied a lot by number theorists; replace $\mathbb C$ by a finite field $\mathbb F$ and suddenly what you're looking at is the group over the local field $\mathbb F((t))$ modulo its maximal compact. That's a lot like if you look at $G(\mathbb Q_p)$ and mod out by $G(\mathbb Z_p)$, which plays an important role in the theory of Hecke operators. – Ben Webster Jan 23 '11 at 6:52

Just to elaborate on Scott's answer: the affine Grassmannian is an algebra in an approrpiate sense, which endows sheaves on it with a rich structure which does not exist for the affine flag variety -- in fact the affine flag variety is naturally viewed as a module for this algebra. The easiest (and most fundamental) manifestation of this algebra structure comes from the identification of the affine Grassmannian with based loops in the compact form of G. As such it is a group (not algebraic) -- but even more importantly, it's a double loop space (double based loops in BG -- which is the topologist's version of the algebraic identification of the Grassmannian as moduli of G-bundles trivialized away from a point).

Double loop spaces are "slightly commutative homotopy groups" (aka $E_2$ spaces, or spaces with a braided multiplication), while affine flags don't admit this structure. There's a precise analogy

$E_2$ algebra: topological field theory :: vertex algebra: conformal field theory,

and indeed Beilinson and Drinfeld gave an algebraic form of this $E_2$ structure known as "factorization space" ( a nonlinear version of a vertex algebra). In physics $E_2$ is the analog of saying the structure underlying operator product expansions for local operators in a two-dimensional quantum field theory. This structure is the most basic ingredient in the geometric Langlands program. It's in particular the reason for the commutativity of Hecke operators in the story, which is the first step towards even imagining there should be a geometric Langlands story.

As for the module structure of affine flags over the affine Grassmannian you can read about it in Gaitsgory's paper.

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One should add that the affine flag variety is extremely important and well studied in representation theory, most spectacularly perhaps in the works of Bezrukavnikov with Ginzburg, Arkhipov, Mirkovic, Finkelberg and Ostrik proving numerous conjectures of Lusztig (see his ICM address for a great overview). – David Ben-Zvi Jan 23 '11 at 4:30
Could you be a bit more specific about what I should look into? It seems every non-empty subset of the people you mentioned have written a paper together! Also in which ICM did Lusztig talk about this? – Najdorf Jan 23 '11 at 6:29
One more point, Gaitsgory is working $\mathbf{F}_q$. One immediate advantage is that he has a Haar measure on the loop group. But loop groups over $\mathbf{C}$ are not locally compact hence have no Haar measure. So I wonder how the Hecke algebras are defined in that case ... – Najdorf Jan 23 '11 at 7:45
I was referring to Bezrukavnikov's ICM address arXiv:math/0604445, though Lusztig's 1990 ICM is a beautiful must-read in the area. As for $F_q$ vs $C$ that's interesting but orthogonal to the flags vs Grassmannian question (let me just note Gaitsgory's not using measures and his paper could equally have been written over $C$ - which is eg where Bezrukavnikov applies his result). – David Ben-Zvi Jan 23 '11 at 16:08

People are interested in the affine Grassmannian because it has nice properties that let you prove theorems. Sometimes, the full flag variety does not have the same nice properties.

Perhaps most importantly, the affine Grassmannian for $SL_2$ has a multiplication structure that endows the category of $SL_{2,\mathbb{C}[[t]]}$-equivariant perverse sheaves with a symmetric monoidal structure. Mirkovic and Vilonen showed that this category is in fact the category of complex algebraic representations of $PGL_2$. More generally, they showed that there is an equivalence of categories between representations of an algebraic group $G$ over any ring $R$ and $R$-valued equivariant perverse sheaves on the affine Grassmannian for the Langlands dual group ${}^L G$.

The full flag variety does not seem to admit such a multiplication rule, and appears to be somewhat more unwieldy in general.

I should mention that the object you defined appears to be a set of points, and the affine Grassmannian has a richer structure as an ind-scheme.

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1. Where is good place to start reading about Vilonen's work? 2. The full flag variety I mentioned is a projective ind-variety and not just a set of points. – Najdorf Jan 23 '11 at 3:12
1. front.math.ucdavis.edu/9911.5050 2. I agree, but you presented both as set-theoretic or topological quotients. – S. Carnahan Jan 23 '11 at 3:23