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 $k$. 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.

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.