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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

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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.

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  • $\begingroup$ could we characterize those admetting such decomposition? $\endgroup$
    – Z.A.Z.Z
    Commented Jun 12, 2015 at 13:33
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    $\begingroup$ I would start by finding out how exactly $GL_2 (\mathbb{C} [1/t])$ looks like. Using matrix multiplication. Then you should be able to give a description of your elements in that case. And so on. I doubt there is a general theory giving you a nice description. $\endgroup$
    – Helene Sigloch
    Commented Jun 12, 2015 at 15:11

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