Let $G$ be a linear algebraic group over $\mathbb C$ (say $SL\_r$) consider a formal power series $$g(t)\in G(\mathbb C((t)))$$ 
My question is: Is it possible to decompose $g$ as $$g=ha$$ with $h\in G(\mathbb C[1/t]) $ and $a\in G(\mathbb C[[t]])$


N.B: the important case that I need is $SL\_r$ and $GL\_r$.


Thanks



Formal Power series decomposition
Let's try for $G = GL\_1$. Then $G(\mathbb{C} [ 1/t]) = (\mathbb{C} [1/t])^\times = \mathbb{C}^\times$. Then $g = t^{-1}$ does not decompose in the way you wish.


Edit: But if you want to replace $\mathbb{C} [1/t]$ by $\mathbb{C}(t)$ in your question, then the answer is yes, at least for $GL\_r$. See Harbater, Formal Patching and Adding Branch Points (1993), Lemma 2.


