Let $F=\mathbb{C}((t))$ and $O=\mathbb{C}[[t]]$. Take $G=SL_n$, the affine Grassmannian of $G$, $Gr_G=G(K)/G(O)$. Then we have Cartan decomposition: $$ Gr_G=\bigsqcup_{\lambda}G(O)t^{\lambda}G(O)/G(O), $$ where $\lambda$ runs over the dominant coweight, i.e., the sequence $(\lambda_1, \lambda_2,\cdots, \lambda_n) \in \mathbb{Z}^n$ such that $\sum_i \lambda_i=0$, and $$t^{\lambda}=\pmatrix{t^{\lambda_1}&&&& \\ &t^{\lambda_2}&&&\\ &&\ddots &&\\ &&&t^{\lambda_n} }.$$
Actually I am confused when study affine Grassmannian slice $G_1[t^{-1}]G(O)\cap \overline{G(O)t^{\lambda}G(O)}$.
If we take $n=3$, then $g\in G_1[t^{-1}]$ means there is nilpotent matrix $A\in Mat_{3\times 3}(\mathbb{C}[t^{-1}])$ such that $g=I+At^{-1}$ and $$t^{3\varpi_1}=\pmatrix{t^2&& \\ &t^{-1}&\\ &&t^{-1}\\}, \quad t^{\varpi_1+\varpi_2}=\pmatrix{t&& \\ &1&\\ &&t^{-1}\\},$$ where $\varpi_1, \varpi_2$ are fundamental coweight.
Then I was told that $$ G_1[t^{-1}]G(O)\cap \overline{G(O)t^{3\varpi_1}G(O)}\cong \mathcal{N}_{sl_3}, $$ $\mathcal{N}_{sl_3}$ is nilpotent cone of lie algebra $sl_3$, and $$ G_1[t^{-1}]G(O)\cap \overline{G(O)t^{\varpi_1+\varpi_1}G(O)}\cong \Theta_{2,1}, $$ $\Theta_{2,1}$ means the set of all nilpotent matrices that has Jordan canonical form of (2,1), i.e., a $2\times 2$ block and a $1\times 1$ block.
What make me confused is: In both case $G_1[t^{-1}]$ are the same, so I think the difference is in the component of affine Grassmannian. But the elements in the component will be $3\times 3$ matrix with entries $\in tO+O+t^{-1}O$ in both case.
I can't tell why the latter situation will lose some nilpotent matrices? how can I see the different precisely when I take two different coweight?