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forgot one mathmode and a typo in cocharacter
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Dima Pasechnik
Dima Pasechnik
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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 $p^{-1}(K\lambda K/K)$?

Let $G$ 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 $p^{-1}(K\lambda K/K)$?

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 $p^{-1}(K\lambda K/K)$?

forgot one mathmode and a typo in cocharacter
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edit approved Jan 4, 2015 at 8:12
Thibaut Dumont
edit approved Jan 4, 2015 at 8:12
Thibaut Dumont
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Let $G$ asimplya simply connected group over $k=\bar{k}$, $B$ a Borel subgroup and $I$ the corresponding Iwahori in G(k[[t]])$G(k[[t]])$, $T$ a maximal torus and $K=G(k[[t]])$.

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

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 $p^{-1}(K\lambda K/K)$?

Let $G$ asimply 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 $\lambda\in X_{*}(T)^{+}$ a dominant cocaracter.

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 $p^{-1}(K\lambda K/K)$?

Let $G$ 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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prochet
prochet
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affine schubert cells and bruhat order

Let $G$ asimply 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 $\lambda\in X_{*}(T)^{+}$ a dominant cocaracter.

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 $p^{-1}(K\lambda K/K)$?