The $\mathbb C((z))$-rational points of a complex semi-simple group $G$

Question

By definition, if $R$ is a $\mathbb C$-algebra and $G$ is a $\mathbb C$-scheme then the set of $R$-valued points on $G$ is $G(R)=\hom_{\text{Sch}_{\mathbb C}}(\operatorname{Spec} R, G)$

In Ginzburg's paper Perverse sheaves on a loop group and Langlands' duality, the following (paraphrased) statement is made:

If $G$ is a complex semi-simple group and $\mathbb K = \mathbb C((z))$, a choice of embedding $G \hookrightarrow GL_n$ presents $G(\mathbb K)$ as a subgroup of $\mathbb K$-valued $n\times n$ matrices and the group $G(\mathbb C[z,z^{-1}])$ as the group of maps $$ f(z)= \sum_{i=-m}^m A_i z^i$$ where the $A_i$ are $n\times n$ matrices and $f(z) \in G(\mathbb C)$ for all $z \in \mathbb C^\times$.

I believe that I understand the second statement regarding $G(\mathbb C[z,z^{-1}])$ but it required use of the equivalence of the category of finite type integral affine schemes over $\mathbb C$ with the category of affine varieties. This may not be the best way of dealing with this, but I do not know another way to nicely characterize those scheme morphisms. Unfortunately, $G(\mathbb K)$ is not amenable to this approach because it is not of finite type. (Edit: By this I mean $\mathbb K$ is not of finite type.)

How do we realize the first statement regarding $G(\mathbb K)$?

The (closed?) embedding $G \hookrightarrow GL_n$ should realize $G$ as a closed subscheme of $GL_n$, so that we are looking at $G(\mathbb K) \subseteq GL_n(\mathbb K)$. Despite the nomenclature "$\mathbb K$-valued points of $G$ ($GL_n$)" suggesting that the answer should be what it is, I do not understand the naming convention, nor how to see it directly. Any insight would be greatly appreciated.

Edit: Let's see if I can clarify where I think my issue is occurring. Let $G$ be a (smooth) complex semisimple linear algebraic group. From what I can see, we have three ways of thinking about $G$: as a manifold, as a scheme in terms of the spectrum of an algebra, and as a group scheme given by a functor of points.

1. Given an embedding $G \hookrightarrow GL_n(\mathbb C)$ (as say, manifolds), how does this correspond to an embedding $G \hookrightarrow GL_n$ as affine group schemes (thought of as functors of points)? Does $G \hookrightarrow GL_n(\mathbb C)$ correspond to some map on closed points $G(\mathbb C) \hookrightarrow GL_n(\mathbb C)$ which lifts to a map of affine group schemes?
2. Taking as the definition $GL_n(\mathbb C) = \operatorname{Spec}(\mathbb C[x_{ij}]_{i,j=1}^{n+1}/(\det \cdot y -1) )$ and $R$ a $k$-algebra, can one recover the functor $GL_n: \text{Alg}_k \to \text{Grp}, R \mapsto GL_n(R)$ from this?

I would posit that I'm really just having trouble reconciling all of the different ways of thinking of $G$ (or $GL_n$).

Answer 1

I'm not sure I understand what is being asked. A closed immersion $j:G\hookrightarrow\mathrm{GL}_n$ of $\mathbf{C}$-groups yields, for any $\mathbf{C}$-algebra $R$, an injective homomorphism $G(R)\rightarrow\mathrm{GL}_n(R)$. Explicitly this map sends an $R$-point $g:\mathrm{Spec}(R)\rightarrow G$ to the composite $j\circ g:\mathrm{Spec}(R)\rightarrow\mathrm{GL}_n$. The map is injective because closed immersions are monomorphisms.

Note that, for any $\mathbf{C}$-algebra $R$, $\mathrm{GL}_n(R)=\mathrm{GL}_{n/R}(R)$, where the subscript $R$ denotes base change along $\mathbf{C}\rightarrow R$. This is true whether or not $R$ is of finite type over $\mathbf{C}$.