2
$\begingroup$

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

$\endgroup$
5
$\begingroup$

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.

$\endgroup$
  • $\begingroup$ could we characterize those admetting such decomposition? $\endgroup$ – Z.A.Z.Z Jun 12 '15 at 13:33
  • 2
    $\begingroup$ 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 Jun 12 '15 at 15:11

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.