Orbits on the affine Grassmanian, and closure ordering

Question

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 .

Answer 1

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.

Answer 2

This may be a little bit lazy but the statement you want is given on page 227 in [Lu]. The proof is given on page 228.

[Lu]=Lusztig, George Singularities, character formulas, and a q-analog of weight multiplicities. Analysis and topology on singular spaces, II, III (Luminy, 1981), 208–229, Astérisque, 101-102,