Integrality of the support of matrix coefficients? Consider a division quaternion algebra $D$ over a number field $F$. For an automorphic representation $\pi$ of $D$, I am interested in the associated matrix coefficients
$$f : \gamma \in G \longmapsto \langle \pi(\gamma)x, x \rangle, $$
where $x$ is a suitably normalized vector (namely of norm $1/d_\pi$ where $d_\pi$ is the formal degree). Since $D$ is a division quaternion algebra, $f$ is compactly supported modulo the center. 
Embedding in $GL(2)$, does the support lie in a congruence subgroup, or at least in integer points?
 A: According to the comments, I understand you to mean the following local question: Say $D$ is the quaternion division algebra over an $p$-adic field $F$,
and $\pi$ is a smooth representation of $D^\times$.  Can we regard its matrix coefficients as having support in $Z \cdot$ GL(2,$\mathcal O_E$) for a quadratic extension $E/F$?  (Here $Z \simeq F^\times$ is the center.)
No this is not true.  Here $\pi$ is a finite dimensional representation, so you can take the formal degree to be the dimension.  The simplest case is $\pi$ is 1-dimensional, and thus of the form $\mu \circ N_{D/F}$ where $\mu$ is a character of $F^\times$ and $N_{D/F}$ denotes the reduced norm.  This character is its own matrix coefficient, and the support is all of $D^\times$.
For general $\pi$, here is the reason for unramified $E/F$.  If $\varpi_D$ is a uniformizer for $D$, then it will not embed in $Z \cdot$ GL(2,$\mathcal O_E$), but $\pi(\varpi_D)$ must have nonzero matrix coefficients.  (For ramified $E/F$, similarly choose a $g \in D^\times$ which does not map into $Z \cdot$ GL(2,$\mathcal O_E$) for your choice of embedding.)
