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?



Integrality of the support of matrix coefficients?
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.)


