Could someone please provide a precise reference in the literature where the following well-known fact is proved. Also if someone could write out the proof that would be great.
$G(\mathbb{C}((t)))/G(\mathbb{C}[[t]])$ is in bijection with the set of $G$-bundles on the disk together with a trivialisation away from the center.
This is correct as long as you say "up to isomorphism over the disk." An element of $G(\mathbb{C}((t)))$ is the same as a trivialization away from the center of the trivial $G$-bundle on the disk. Two such trivializations differ by an automorphism of the trivial $G$-bundle over the disk if and only if the elements of $G(\mathbb{C}((t)))$ differ by an element of $G(\mathbb{C}[[t]])$. This shows that the quotient injects into the set of isomorphism classes we're considering.
Surjectivity is just the statement that any $G$-bundle on the disk is trivial. To do this choose a trivialization over the center, then use the fact that $G$ is smooth, hence formally smooth, to extend to a trivialization over the disk (basically Hensel's lemma).
