I hope somebody can give me a good reference for the following:


Let $G$ be a complex reductive group $H$ be a closed subgroup. Let further $R$ be any $\mathbb{C}$-algebra. Then the canonical map 
$$G(R)/H(R)\to (G/H)(R)$$
is known to be injective but in general not surjective. See for example [1].


### So now my question:


Let $Z\_e:=Z(G)\_e$ be the connected component of center of $G$ containing $e$, is then the canonical map
$$G(\mathbb{C}((t)))/Z\_e(\mathbb{C}((t)))\to (G/Z\_e)(\mathbb{C}((t))) $$
a bijection.


Or even more, does the canonical map above induce an isomorphism of ind-varietes
$$ G(\mathbb{C}( (t)))/Z\_e(\mathbb{C}((t)))\cong (G/Z\_e)(\mathbb{C}((t))) $$


### Remark:


Note that if we take $ Z$ to be the center of $G$ and $R=\mathbb{C}[[t]]$ then the map above is also surjective (this follows from the fact $\mathbb{C}[[t]]$ is strict henselian, hence by SGA III every map $Spec \ \mathbb{C}[[t]]\to G/Z$ can be lifted to a map $Spec \ \mathbb{C}[[t]]\to G $)


[1] Jantzen, Jens Carsten Representations of algebraic groups. Second edition. Mathematical Surveys and Monographs, 107. American Mathematical Society 



A question about $R$-points of an complex reductive group.
By Hilbert's Theorem 90, every torsor for a split torus over a field is trivial. Thus, as pranavk has commented, this should give surjectivity.


$\textbf{Edit.}$ The first answer I wrote (now changed) applied to the full center $Z$. I did not realize that the OP is asking about the quotient by $Z\_e$, the connected component of the identity. I have corrected my answer.


