Orbit of a parahoric subgroup on a flag variety

Question

Let $G$ be a split reductive group over a nonarchimedean local field $F$ (I'm particularly interested in the case of $\operatorname{GSp}_{2n}$).

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Given a parahoric subgroup $K \subset G(F)$, and a parabolic subgroup $P$, is there a "nice" group-theoretic description of the orbits of $K$ on the flag variety $\mathcal{F}(F) = P(F) \backslash G(F)$?

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I have the following conjecture: if $T$ is a split maximal torus in $G$ and $\widetilde{W} = N_G(T)(F) / T(\mathcal{O}_F)$ is the extended affine Weyl group, then $\widetilde{W}$ is generated by a set $S$ of simple reflections, and there is a bijection between standard parahorics and proper subsets of $S$. I think that if $K_J$ is a standard parahoric corresponding to some $J \subset S$, then $$ P \backslash G / K = W_{M_P} \backslash W_G / \pi(W_J) $$ where $W_J$ is the subgroup of $\widetilde{W}$ generated by $J$, and $\pi(W_J)$ is its image in the usual Weyl group. This conjecture is true if $K$ is contained in $G(\mathcal{O}_F)$. Does anyone know if it is true in general?

Answer 1

Let $G$ a reductive group over a nonarchimedean local field $F$. Let $P_0$ be a minimal parabolic subgroup of $G$ and $A$ a maximal split torus contained in $P_0$. The normalizer $N_G(A)(F)$ acts on the apartment of $A$ via the extended affine Weyl group, which contains the affine Weyl group $W_\mathrm{aff}$ with finite index. Let $I$ be an Iwahori subgroup of $G(F)$ compatible with $P_0$.

By the Iwasawa decomposition, the double cosets in $P_0(F)\backslash G(F)/I$ are represented by the elements of the Weyl group $W := N_G(A)(F)/C_G(A)(F)$. Let $\pi$ be the natural projection $W_\mathrm{aff}\rightarrow W$. It's not hard to show that if $w\in W$ and $s\in W_\mathrm{aff}$ is a simple reflection (relative to $I$), then \begin{equation} \tag{1}\label{parabolic_mult} P_0(F) w\pi(s) I\subseteq P_0(F) w I s I \subseteq P_0(F) w\pi(s) I \sqcup P_0(F) wI, \end{equation} and if furthermore $s\in W$, then \begin{equation} \tag{2}\label{iwahori_mult} P_0(F) sw I\subseteq P_0(F) s P_0(F) w I \subseteq P_0(F) sw I\sqcup P_0(F) wI. \end{equation}

Now let $P$ be a parabolic subgroup of $G$ containing $P_0$, and let $K$ be a parahoric subgroup of $G(F)$ containing $I$. Then \begin{equation} \tag{3}\label{bruhat} P(F) = \bigsqcup_{u\in W_{M_P}}P_0(F)uP_0(F), \end{equation} where $W_{M_P}$ is the Weyl group of the standard Levi component of $P$, and \begin{equation} \tag{4}\label{iwahori} K = \bigsqcup_{v\in W_K}IvI, \end{equation} where $W_K$ is the Coxeter subgroup of $W_\mathrm{aff}$ consisting of those elements with representatives in $K$.

Given $w\in W$, consider the double coset $P(F)wK$. From (\ref{bruhat}) and (\ref{iwahori}), $$P(F)wK = \bigcup_{u\in W_{M_P},v\in W_K}P_0(F)uP_0(F)wIvI.$$ Applying (\ref{parabolic_mult}) one simple generator of $W_K$ at a time, we get that this union is equal to $$\bigcup_{u\in W_{M_P},v\in W_K}P_0(F)uP_0(F)w\pi(v)I.$$ Applying (\ref{iwahori_mult}) one simple generator of $W_{M_P}$ at a time, we find that this is equal to $$\bigcup_{u\in W_{M_P},v\in W_K}P_0(F)uw\pi(v)I.$$ It follows from the Iwasawa decomposition that $w,w'\in W$ represent the same double coset in $P(F)\backslash G(F)/K$ if and only if $w,w'$ lie in the same double coset in $W_{M_P}\backslash W/\pi(W_K)$.