Let $\mathcal{K} = \mathbb{C}((t)), \mathcal{O}=\mathbb{C}[[t]]$, $G=SL_2$ (or any semisimple group), and $\text{Gr}_G=G(\mathcal{K})/G(\mathcal{O})$; there is a left action of $G(\mathcal{O})$ on $\text{Gr}_G$. Let $X_*(T)=\text{Hom}(\mathbb{C}^{\times},T)$ (note that there is a natural embedding of $X_*(T)$ inside $G(\mathcal{K})$). Let $B$ be a Borel subgroup. Let $X^*(T) = \text{Hom}(T, \mathbb{C}^{\times})$.
Choose $\lambda \in X_*(T)$ to be dominant, and (abusing notation), let $\lambda$ also denote the image of $\lambda \in X_*(T) \subset G(\mathcal{K})$ in the quotient $\text{Gr}_G$. Define $\text{Gr}^{\lambda} = G(\mathcal{O}) \cdot L_{\lambda}$.
(1) How can we prove that $\text{Gr}_G$ is the disjoint union of $\text{Gr}^{\lambda}$ (as $\lambda$ ranges across the dominant weights)?
This is stated on the fourth paragraph of page $4$, http://arxiv.org/pdf/math/0401222v4.pdf .
(2) Let $\mu \in X_*(T)$ be another dominant weight. Why is the following statement true: $\text{Gr}^{\mu} \subset \overline{\text{Gr}^{\lambda}}$ if and only if $\lambda - \mu$ is a sum of positive co-roots?
I'm guessing we need to construct a set of limit points (to show that $L_{\mu} \in \overline{\text{Gr}^{\lambda}} \Leftrightarrow \lambda - \mu$ is a sum of positive co-roots) - but I'm having trouble.
This is stated in Remark $2.2$ on pg $4$ of http://arxiv.org/pdf/math/0401222v4.pdf .