Let $G$ a split connected reductive group over $\mathbb{C}$. $F=\mathbb{C}((t))$ and $\mathcal{O}$ the ring of integers.

Let $B$ a Borel subgroup and $I$ the corresponding Iwahori.

Let $\hat{\Delta}=\{\alpha_{0},\alpha_{1},\dots,\alpha_{r}\}$ the set of affine simple roots with $\alpha_{i}\in W$ for $i\geq 1$ and $\alpha_{0}$ the affine root.

For each $\alpha\in\hat{\Delta}$, I have a parahoric $\hat{P}_{\alpha}=I\cup Is_{\alpha}I$.

How we describe the unipotent radical of $\hat{P}_{\alpha}$? If, $\alpha$ is a finite root, we just have to consider the inverse image of $R_{u}(P_{\alpha})$, where $P_{\alpha}$ is the reduction of $\hat{P}_{\alpha}$ mod t.

So it remains to describe it for the affine root $\alpha_{0}$.

  • $\begingroup$ and also, what is the quotient $\hat{P}_{\alpha}/R_{u}(\hat{P}_{\alpha})$ $\endgroup$ – prochet Jan 5 '14 at 15:04
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    $\begingroup$ Parahorics are not algebraic groups, but are "just" totally disconnected topological groups, so please define what you mean by "unipotent radical" of a parahoric. For instance, what is your definition of "unipotent radical" of $I$? Do you mean to ask about what is often called the "pro-unipotent radical"? $\endgroup$ – user76758 Jan 5 '14 at 16:37
  • $\begingroup$ for I, it's just the inverse image of the unipotent radical of B, so what you call the pro-unipotent radical. $\endgroup$ – prochet Jan 5 '14 at 17:13

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