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.