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> .



Orbits on the affine Grassmanian, and closure ordering
My knowledge about this is cumming from p-adic groups and not from ind/prog groups, so the following (especially part III) might be inaccurate or incomplete.


I. Proof that $\bigcup Gr\_\lambda=Gr$ for the $GL\_n$ (or similarly $SL\_n$) case. 


This is basically Gauss elimination posses.
We have to proof that any matrix can bring to a diagonal form with ascending powers of $t$ on the diagonal by multiplying it from both side by a matrix in G(\mathcal{O}). in other words we are allowed to make the following elementary operations:


a) switching 2 rows or colons


b) multiply a row or a colon be an element in $\mathcal O^\times$.


c) adding one row (or colon) to another after multiplying in by element in $\mathcal O$


Using (a) we can put the entry with the lowest valuation (i.e. with the lowest power of $t$ ) to be $a\_{11}$. then using (c) we can kill all the rest of the first row and colon. Using (b) we can normalize $a\_{11}$ to be a power of $t$. 


We continue by induction.


II. Proof that $ Gr\_\lambda \cap Gr\_\mu=\emptyset$ for the case of $SL\_2$.


consider the action of $G(\mathcal K)$ on the set of $\mathcal O$ submodulus in $\mathcal K^2$. let $f(g)$ be the maximal valuation of $g(\mathcal O^2)$. it is easy to see that
$f$ separate the $Gr\_\lambda$. 


A similar proof is poseble for $GL\_n$. I believe you can find in <http://www.math.uchicago.edu/~mitya/langlands.html>
(Course on representations of p-adic groups)


III. description of $\overline {Gr\_\lambda}$ for the case of $SL\_2$.


as we sow above $G\_{\lambda}=f^{-1}(\lambda)$. We need to show that the closure of $f^{-1}(\lambda)$ is $f^{-1}(\{\mu| \mu\geq \lambda\})$. If instead of $f$ we consider the valuation on $\mathcal K$ this statement is obvious. So I believe it easily follows from the ind-variety structure on the affine grassmannian


I think that the general case of (III) can be deduce from the $SL\_2$ case. I think also that (I) and (II) are easily generalized to general groups if you know some structure theory, but I'll have to think about it. 


