Stratifications and Filtrations of the Affine Grassmannian

Question

Let $G$ be a connected, simply-connected complex semisimple group. Let $$\mathcal{G}r=G(\mathcal{\mathbb{C}((t))})/G(\mathcal{\mathbb{C}[[t]]})$$ be the affine Grassmannian of $G$. We know that $\mathcal{G}r$ has a natural increasing filtration $$\mathcal{G}r_0\subseteq\mathcal{G}r_1\subseteq\ldots\subseteq\mathcal{G}r_n\subseteq\ldots\subseteq\mathcal{G}r,$$ indexed by pole order considerations. Now, fix a maximal torus $T\subseteq G$. We also know that $\mathcal{G}r$ has a stratification into the subvarieties $$\mathcal{G}r^{\lambda}:=G(\mathbb{C}[[t]])t^{\lambda},$$ where $\lambda$ ranges over the dominant coweights of $T$.

Given a fixed $n$, I am seeking a description of those dominant coweights $\lambda$ for which $\mathcal{G}r^{\lambda}\subseteq\mathcal{G}r_n$. I would appreciate any references that might provide some details concerning this description.

Answer 1

I do not really answer you question but maybe this helps:

Let $\mathcal{K} =\mathbb{C}((t))$ and $\mathcal{O}:=\mathbb{C}[[t]]$. For $n\geq 0$ denote the $\mathcal{K}_n$ the $\mathcal{O}$ ideal in $\mathcal{K}$ generated by $t^{-n}$. Choose an embedding $G\hookrightarrow GL_m$. Let $$G(\mathcal{K}_n):=\{A\in G(\mathcal{K})\vert \text{ all entries of } A \text{ and } A^{-1} \text{ are in }\mathcal{K}_n\}$$

Then $$Gr_n=G(\mathcal{K}_n)/G(\mathcal{O})$$

Via this you can in principle calculate all $\lambda$ which are in $Gr_n$. (Note that you can write every $\lambda$ in the form $\lambda: \mathbb{C}^\times \to T$, $t\mapsto \begin{pmatrix}t^{\lambda_1} & & \\ & \ddots \\ && t^{\lambda_m} \end{pmatrix}$ for some $\lambda_i\in \mathbb{Z}$ and all unspecified entries are zero.)