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 $$ G(\mathbb{C}( (t)))/Z_e(\mathbb{C}((t)))\cong (G/Z_e)(\mathbb{C}((t))) $$

However, if we take $Z$ just to be the center of $G$, the statement above becomes false in general.

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

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