Let $\mathcal K = k((t))$ be the field of formal Laurent series over $k$, and by $\mathcal O = k[[t]]$ the ring of formal power series over $k$.
Let $G$ be an algebraic group over $k$. The affine Grassmannian $Gr_G$ is the functor that associates to a $k$-algebra $A$ the set of isomorphism classes of pairs $(E, \varphi)$, where $E$ is a principal homogeneous space for $G$ over $Spec A[[t]]$ and $\varphi$ is an isomorphism, defined over $Spec A((t))$, of $E$ with the trivial $G$-bundle $G \times Spec A((t))$.
By choosing a trivialization of $E$ over all of $Spec \mathcal O$, the set of $k$-points of $Gr_G$ is identified with the coset space $G(\mathcal K)/G(\mathcal O)$.
Let $V$ be an $n$-dimensional vector space. Then $GL(V)$ acts transitively on the set of all $r$-dimensional subspaces of $V$. Let $H$ be the stabilizer of this action. Then the usual Grassmannian is $GL(V)/H$.
What are the relations between affine Grassmannian and Grassmannian? Why it is important to study affine Grassmannian? Thank you very much.
Best Answer
I am not an expert, but the affine Grassmannian is intimately related to the representation theory of the Langlands dual group $G^{\vee}$. Assume that $G$ is complex semisimple and simply-connected (for simplicity, so to speak). Recall that the dominant weights of $G^{\vee}$ index the finite-dimensional irreducible complex $G^{\vee}$-modules. The dominant weights of $G^{\vee}$ are the dominant coweights of $G$.
Note also that the dominant coweights of $G$ index the strata in a stratification of $Gr_G$. Given a dominant coweight $\lambda$ of $G$, let $Gr^{\lambda}$ denote the corresponding stratum. It turns out that the intersection homology $IH_*(\overline{Gr^{\lambda}})$ is naturally a $G^{\vee}$-module, and is isomorphic to the irrep of $G^{\vee}$ of highest coweight $\lambda$. This description also yields some nice bases for irreps of $G^{\vee}$, called MV cycles. You might read some papers by Kamnitzer and Mirkovic-Vilonen on this subject.