Let $G$ be a simply connected group over $k=\bar{k}$, $B$ a Borel subgroup and $I$ the corresponding Iwahori in $G(k[[t]])$, $T$ a maximal torus and $K=G(k[[t]])$.

Let $\lambda\in X_{*}(T)^{+}$ be a dominant cocharacter.

We know that : $\overline{I\lambda I}/I=\coprod\limits_{w\leq\lambda}IwI$ where $\leq$ is the Bruhat order in the group $W_{aff}=W\rtimes X_{*}(T)$.

but can we have a more explicit description of the set $\{w\in W_{aff}\vert w\leq\lambda\}$?

Moreover, we have a map from $p:\overline{I\lambda I}/I\rightarrow\overline{K\lambda K}/K$ given by the projection, how can we describe $p^{-1}(K\lambda K/K)$?


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