Parahoric subgroup over a local field

$$\DeclareMathOperator\SL{SL}$$Let $$F$$ be a local field and $$\mathcal{O}_{F}$$ its valuation ring. Let $$\pi\in \mathcal{O}_{F}$$ be a uniformizer and $$\mathfrak{p}=\pi\mathcal{O}_{F}$$. Let $$G$$ be a split semisimple algebraic group over $$F$$. I think the case $$G=\SL_{3}$$ as an example. Let us consider a generator system $$S=\{s_{1},s_{2},w_{1}\}$$ of the affine Weyl group of $$\SL_{3}(F)$$, where $$s_{1}= \left( \begin{array}{ccc} 0 & 1 & 0 \\ -1 & 0 & 0 \\ 0 & 0 & 1 \end{array} \right),\quad s_{2}= \left( \begin{array}{ccc} 1 & 0 & 0 \\ 0 & 0 & 1 \\ 0 & -1 & 0 \end{array} \right),\quad w_{1}= \left( \begin{array}{ccc} 0 & 0 & -\pi^{-1} \\ 0 & 1 & 0 \\ \pi & 0 & 0 \end{array} \right).$$ It is well-known that holds the affine Bruhat decomposition $$\SL_{3}(F)=B\langle S\rangle B$$ and $$\SL_{3}(\mathcal{O}_{F})=B\langle s_{1},s_{2}\rangle B$$ where $$B$$ is the standard Iwahori subgroup of $$\SL_{3}(F)$$.

Question: What is the remaining component $$B\langle w_{1}\rangle B=B\cup Bw_{1}B$$? I want to know the explicit form of this subgroup.

• By the way, the more usual term than "affine parabolic" is "parahoric". Apr 6, 2022 at 16:43
• What do you mean by "remaining component"? We don't have $\langle s_1, s_2 , w_1 \rangle = \langle s_1, s_2 \rangle \cup \langle w_1 \rangle$. Apr 6, 2022 at 19:05
• Sorry. The term of "remaining component" had no special meaning. I understand that $\langle S\rangle \neq \langle s_{1},s_{2} \rangle\cup \langle w_{1}\rangle$. Apr 7, 2022 at 1:42

$$\DeclareMathOperator\SL{SL}\newcommand\O{\mathcal O_F}\newcommand\P{\pi\mathcal O_F}\newcommand\Pi{\pi^{-1}\mathcal O_F}$$I assume that the standard Iwahori is the group of matrices in $$\SL_3(\O)$$ that are upper triangular modulo $$\pi$$. Then $$B \cup Bw_1 B$$ is $$K \mathrel{:=} \begin{pmatrix} \O & \O & \Pi \\ \P & \O & \O \\ \P & \P & \O \end{pmatrix} \cap \SL_3(F)$$.
Indeed, it is clear that this is a subgroup, that it contains (hence is stable under left- and right-multiplication by) $$B$$, and that it contains $$w_1$$. Therefore, it contains $$B \cup Bw_1 B$$.
On the other hand, suppose $$\gamma = \begin{pmatrix} a & b & \pi^{-1} c \\ \pi d & e & f \\ \pi g & \pi h & i \end{pmatrix}$$ belongs to $$K$$ (so that $$a, \dotsc, i$$ belong to $$\O$$). If $$c$$ belongs to $$\P$$, then $$\gamma$$ belongs to $$B$$. Otherwise, $$b \mathrel{:=} \begin{pmatrix} 1 \\ & 1 \\ \pi i/c && 1 \end{pmatrix}$$ and $$(b w_1)^{-1}\gamma$$ both belong to $$B$$, so $$\gamma$$ belongs to $$B w_1 B$$.
• Did you mean to intersect with $SL_3(F)$ rather than $SL_3(\mathcal O_F)$? Apr 6, 2022 at 19:06