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3 contains vs is contained in

Let $G$ be a connected, simply connected, semi-simple algebraic group defined and split over a local non-arch field $k$ with integer ring $R$. Let $B$ be the corresponding reduced building. Fix an apartment $A$ and corresponding root system $\Phi$ and split torus $T$; then for each $x \in A$ and $r>0$, we have a corresponding Moy-Prasad filtration subgroup $G_{x,r}$. Our hypotheses give $G_x=G_{x,0}$.

Define $\Omega_{A,r}$ as the set $${ y \in A : \forall \alpha \in \Phi, \vert \alpha(x-y) \vert \leq r }.$$

Is it true that $T(R)G_{x,r} = \cap_{y\in \Omega_{A,r}} G_y$ ?

(With $r>0$ the product gives a group which contains is contained in the given intersection. Equality seems to be about whether all hyperplanes $H_{\alpha,r}$ meet $\Omega_{A,r}$.)

Let $A(x)$ be the set of all apartments of $B$ containing $x$. Let $Z$ be the (finite) center of $G$.

Is it true that $ZG_{x,r} = \cap_{A \in A(x)}\cap_{y\in \Omega_{A,r}}G_y$ ?
Equiv by (1): Is $ZG_{x,r} =\cap_{g \in G_x} (gT(R)g^{-1})G_{x,r}$?

This is part of the larger question: I would like to understand the stabilizers of certain subsets of $B$ (which are not contained within any apartment $A$). Pointers to any literature much appreciated.

2 Improved formatting.

Let $G$ be a connected, simply connected, semi-simple algebraic group defined and split over a local non-arch field $k$ with integer ring $R$. Let $B$ be the corresponding reduced building. Fix an apartment $A$ and corresponding root system $\Phi$ and split torus $T$; then for each $x \in A$ and $r>0$, we have a corresponding Moy-Prasad filtration subgroup $G_{x,r}$. Our hypotheses give $G_x=G_{x,0}$.

Define $\Omega_{A,r}$ as the set $${ y \in A : \forall \alpha \in \Phi, \vert \alpha(x-y) \vert \leq r }.$$

Is it true that $T(R)G_{x,r} = \cap_{y\in \Omega_{A,r}} G_y$ ?

(With $r>0$ the product gives a group which contains the given intersection. Equality seems to be about whether all hyperplanes $H_{\alpha,r}$ meet $\Omega_{A,r}$.)

Let $A(x)$ be the set of all apartments of $B$ containing $x$. Let $Z$ be the (finite) center of $G$.

Is it true that $ZG_{x,r} = \cap_{A \in A(x)}\cap_{y\in \Omega_{A,r}}G_y$ ?
Equiv by (1): Is $ZG_{x,r} =\cap_{g \in G_x} (gT(R)g^{-1})G_{x,r}$?

This is part of the larger question: I would like to understand the stabilizers of certain subsets of $B$ (which are not contained within any apartment $A$). Pointers to any literature much appreciated.

1

# When is a Moy-Prasad filtration subgroup the stabilizer of a subset of the building (up to center)?

Let $G$ be a connected, simply connected, semi-simple algebraic group defined and split over a local non-arch field $k$ with integer ring $R$. Let $B$ be the corresponding reduced building. Fix an apartment $A$ and corresponding root system $\Phi$ and split torus $T$; then for each $x \in A$ and $r>0$, we have a corresponding Moy-Prasad filtration subgroup $G_{x,r}$. Our hypotheses give $G_x=G_{x,0}$.

Define $\Omega_{A,r}$ as the set $${ y \in A : \forall \alpha \in \Phi, \vert \alpha(x-y) \vert \leq r }.$$

1. Is it true that $T(R)G_{x,r} = \cap_{y\in \Omega_{A,r}} G_y$ ?

(With $r>0$ the product gives a group which contains the given intersection. Equality seems to be about whether all hyperplanes $H_{\alpha,r}$ meet $\Omega_{A,r}$.)

Let $A(x)$ be the set of all apartments of $B$ containing $x$. Let $Z$ be the (finite) center of $G$.

1. Is it true that $ZG_{x,r} = \cap_{A \in A(x)}\cap_{y\in \Omega_{A,r}}G_y$ ?
Equiv by (1): Is $ZG_{x,r} =\cap_{g \in G_x} (gT(R)g^{-1})G_{x,r}$?

This is part of the larger question: I would like to understand the stabilizers of certain subsets of $B$ (which are not contained within any apartment $A$). Pointers to any literature much appreciated.
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