Let $G$ be a simply connected algebraic group over $\mathbb{C}$, $W$ be the Weyl group for $G$ and $W_{aff}$ be the affine Weyl group for the loop group $G(\mathbb{C}((t)))$, $\Phi$ be the coweight lattice of $G$. $W_{aff} \cong W \ltimes \Phi.$

There is a simple reflection $s_0$ in $W_{aff}$ that does not correspond to any simple reflections in $W$.

So how to write $s_0$ in terms of the presentation $W_{aff} \cong W \ltimes \Phi?$

What is the answer for type A? How to embed $s_0$ into $SL_n(\mathbb{C}((t)))$?

What is the answer for all types?

My question is remotely related to this MO question earlier: affine schubert cells and bruhat order

Let $G$ be a simply connected algebraic group over $\mathbb{C}$, $W$ be the Weyl group for $G$ and $W_{aff}$ be the affine Weyl group for the loop group $G(\mathbb{C}((t)))$, $\Phi$ be the coweight lattice of $G$. $W_{aff} \cong W \ltimes \Phi.$

There is a simple reflection $s_0$ in $W_{aff}$ that does not correspond to any simple reflections in $W$.

So how to write $s_0$ in terms of the presentation $W_{aff} \cong W \ltimes \Phi?$

What is the answer for type A? How to embed $s_0$ into $SL_n(\mathbb{C}((t)))$?

What is the answer for all types?

My question is remotely related to this MO question earlier: affine schubert cells and bruhat order