Definition of Affine Grassmannian
We give two explanations. Pick which one you like better.
There's a proof that the two definitions you give are equivalent for $\text{GL}_n$ in the article "Affine Springer Fibers and Affine Deligne-Lusztig Varieties" by U. Görtz here.
Görtz, Ulrich. Affine Springer fibers and affine Deligne-Lusztig varieties. Affine flag manifolds and principal bundles, 1–50, Trends Math., Birkhäuser/Springer Basel AG, Basel, 2010.
The claim you want is Proposition 2.10. To see why your Definition 2 is the same as the quotient $LG/L^+G$ (using Görtz's notation) notice that the trivialization $\beta$ gives you an element of $LG$ (i.e. an automorphism of the trivial $G$-torsor on the formal punctured disc) once you trivialize the torsor $\epsilon$. However, there are $L^+G$ many ways of doing that trivialization, so the set of pairs $(\epsilon, \beta)$ is the same as the quotient, You can make that precise by doing it in $R$-families.
You can find this in the paper "Conformal blocks and generalized theta functions" by Beauville and Laszlo (Section 1) here, together with the paper "Un lemme de descente" by the same authors here.
Beauville, Arnaud; Laszlo, Yves. Conformal blocks and generalized theta functions. Comm. Math. Phys. 164 (1994), no. 2, 385–419.
Beauville, Arnaud; Laszlo, Yves. Un lemme de descente. C. R. Acad. Sci. Paris Sér. I Math. 320 (1995), no. 3, 335–340.
If $R=k$, the idea is that every vector bundle on either the affine line, or the formal disk is trivial, so every vector bundle on the projective line arises by gluing the trivial bundle on the affine line and the trivial bundle on the formal disk by specifying a gluing isomorphism. This gluing isomorphism can be given by an element of the loop group, but is not uniquely determined. If we pass from vector bundles to pairs of vector bundles and a trivialization as in the original post, then the non-uniqueness corresponds exactly to taking the quotient of the loop group by the positive loop group (if one stayed with vector bundles, one would instead arrive at a double quotient of the loop group).
If $R$ is any $k$-algebra, one has to work more in order to carry this out "in families", in particular if $R$ is not Noetherian. See the above-mentioned papers.
May I ask, in addition, how to establish the equivalence between $\text{GL}_n$-torsors and vector bundles of rank $n$?
Again, we give two explanations. Pick which one you like better.
If you're familiar with the "structure" group (in terms of gluing charts) perspective on vector bundles and torsors, then you can see this because they have the same structure group.
Let $V$ be the standard $n$-dimensional $\text{GL}_n$-representation. Given a $\text{GL}_n$-torsor over $X$, say $P \to X$, we can take the associated vector bundle $(P \times V)/G$ where $G$ acts on $P \times V$ by the diagonal action (i.e. for $\rho \in P$ and $v \in V$, $g * (\rho, v) = (g * \rho, g^{-1} * v)$ — the twisting by $^{-1}$ is not particularly important but makes things look nicer, in particular you can think about this quotient as "allowing you to move the $G$-action across the product"). It turns out to be an equivalence; the inverse is given by the "frame bundle" construction. That is, given a vector bundle $V$ on $X$, we associate to it a $\text{GL}_n$-torsor $P$ over $X$ which, as a set, is $(x, b_1, \ldots, b_n)$ where $x \in X$ and $b_1, \ldots, b_n$ is a basis of the fiber. There are some details on topologizing this which I'm not sure of offhand.
I'm not sure what the structure group is offhand, but you can think about a torsors and as some element in some kind of cohomology, specifically $H^2(X, G)$ (we have to be a little careful about what category we mean here — i.e. if $X$ is a topological space then $G$ is the constant sheaf. If $X$ is a scheme, you probably have to choose a Grothendieck topology, etc.). That is, a torsor has to satisfy a cocycle condition and two torsors are conditions if they are related by a coboundary. On the other hand, you can do the same for vector bundles: they're locally given by trivializations with transition maps sections to $\text{GL}_n$, i.e. an element of $H^2(X, \text{GL}_n)$. They key thing here is that the transition maps for $\text{GL}_n$-torsors and for vector bundles are both $\text{GL}_n$-valued maps from $X$, so the "data" for the two are the same, in a "natural" way (i.e. the cocycle conditions and coboundaries should match up, and what it means to be a morphism as well).
Also, the construction in the paragraph "Let $V$ be..." can in fact be done with any group $G$ and any representation $V$; that is, we have a way to turn a representation and a torsor into a vector bundle. This construction is also functorial. There are two "variables" on the left hand side of this functor. If we fix a chosen representation $V$, we obtain a functor from $G$-torsors to vector bundles. For a good enough representation (I'm not familiar offhand with the details on what kind of representation would suffice, but certainly, for example, the trivial representation would not), we can repeat the above for other groups $G$ (and construct inverse equivalences, e.g. orthonormal frame bundle, I'm not sure). We can also fix instead of the representation, fix the torsor; this is the "Tannakian" point of view where I think.
If you look at the last paragraph on page 6 here, you can find the translation between $\text{GL}_n$-torsors and vector bundles.
- X. Zhu. An introduction to affine Grassmannians and the geometric Satake equivalence. arXiv:1603.05593.
In addition, in algebraic geometry, a vector bundle on an affine scheme $\text{Spec}\,A$ is the same as a finite projective $A$-module. So to give a $\text{GL}_n$-torsor on $D_R$ is the same as to give a rank $n$ vector bundle on $\text{Spec}\,R[[t]]$, which is the same as to give a finite projective $R[[t]]$-module $L$ of rank $n$. Now the trivialization on $D_R^*$ means an isomorphism$$L \otimes_{R[[t]]} R((t)) \cong R((t))^n.$$Via this isomorphism, we may regard $L$ as a submodule of $R((t))^n$. This is exactly the definition of a lattice.