Let $G=SL_2$. Then the Weyl group is generated by $s_1$. On page 3 of the lecture notes, it is said that the affine Weyl group is generated by $s_0, s_1$.
(1) The element $s_0s_1$ can be identified with $e^{\alpha}$, the translation over the root $\alpha$.
(2) The extended Weyl group allows translations over integer multiples of the fundamental weight $\frac{1}{2}\alpha$.
I don't understand (1) and (2) very much. Why element $s_0s_1$ can be identified with $e^{\alpha}$? Why the extended Weyl group allows translations over integer multiples of the fundamental weight $\frac{1}{2}\alpha$? Thank you very much.