Let $G$ be a complex, connected, simply connected, semisimple group. I'm trying to compare the following two spaces: The free loop space $LG$ of $G$, and the $\mathbb C((z))$-valued points of $G$, $G(\mathbb C((z)))$, $$LG = \{ f:S^1 \to G : f \substack{\text{is the restriction of a} \\ \text{ holomorphic map }\mathbb C^\times \to G}\}.$$ $$ G( \mathbb C((z))) = \hom_{\text{Sch}_{\mathbb C}}(\operatorname{Spec} \mathbb C((z)), G).$$

In the way they are currently defined, they seem like they might be troublesome to work with, so let's fix an embedding $G \hookrightarrow GL_n(\mathbb C)$. The elements of $f\in LG$ then look like functions with Fourier series $$ f(z) = \sum_{k=-\infty}^\infty A_k z^k, \qquad A_k \in M_{n\times n}\mathbb C, \text{ and } f(z) \in G \text{ for all }z \in \mathbb C^\times;\tag{1}$$ that is, there is an analytic convergence condition on the element $f(z)$.

Is there a sense in which the (Lie group?) embedding $G \hookrightarrow GL_n(\mathbb C)$ corresponds to a map of affine group schemes $G \hookrightarrow GL_n$ which agrees with $G(\mathbb C) \hookrightarrow GL_n(\mathbb C)$?

If this is the case, then we should get a corresponding map $G(\mathbb C((z))) \hookrightarrow GL_n(\mathbb C((z)))$, and we can then write elements $f\in GL_n(\mathbb C((z)))$ in a similar sense to those in (1); namely, $$f(z) = \sum_{k=-\infty}^\infty A_k z^k, \qquad A_k \in M_{n\times n}\mathbb C. \tag{2}$$ Such sums in this context seem to be taken in a strictly formal context. Nonetheless, it seems as though there should be some ``spiritual'' notion of convergence to guarantee that the above sum lives in not only $GL_n(\mathbb C((z)))$ but actually in $G(\mathbb C((z)))$ rather than simply $M_{n\times n}\mathbb C((z))$.

Is there a notion of convergence for elements of the form (2)? How does one write this down? Does it agree with that of (1)?

Edit: @S. Carnahan has been good enough to point out an algebraic condition which implies precisely when elements of $GL_n(\mathbb C((z)))$ are in $G(\mathbb C((z)))$.

However, I still have a strange suspicion there should be some sort of analytic condition. To corroborate this, let us consider Ginzburg's paper Perverse sheaves on a loop group and Langlands' duality. On page 6, he discusses a finite dimensional analogue $G(\mathbb C[z,z^{-1}])$ where elements, given the embedding above, have the form $$f(z) = \sum_{k=-m}^m A_k z^k, \qquad A_k \in M_{n\times n}\mathbb C, \text{ and } f(z) \in G(\mathbb C) \text{ for all }z \in \mathbb C^\times.$$ How are these (finite) sums still endowed with a convergence property? In particular, we note that $z$ is taken to be in $\mathbb C^\times$ rather than just a formal sum satisfying some polynomial conditions which come from the embedding. Is it really the finiteness of the summation which somehow makes this possible? Why would that break in an infinite sum?