Let X a curve over an algebraically closed k. Fix $x$ and $y$ two distinct closed points of X. Let G be a connected reductive group over k.

We denote Spec $\hat{\mathcal{O}}_{X,x}$ the formal neighborhood around $x$.

Let $I_{x}\subset G(\hat{\mathcal{O}}_{X,x})$ be the Iwahori subgroup on x.

Let $g\in G(X-y)$. We have in particular that $G(X-y)\subset G(\hat{\mathcal{O}}_{X,x})$

Can we find an element $k\in G(X-x)\cap I_{y}$ such that $kg\in I_{x}$?