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




---



> 
> 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 $\mathcal{F}(F) = P(F) \backslash G(F)$?
> 
> 
> 




---


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?



Orbit of a parahoric subgroup on a flag variety
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 $W\_{M\_P}\backslash W/\pi(W\_K)$.


