# 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?

• For your $x$ we have $f(1)\neq 0$. Because of continuity there exists an open ball around $1$ which is contained in the support of $f$. – Subhajit Jana Apr 18 at 10:44
• $G$ means what? $D^\times(F_v)$ for a ramified nonarchimedean $v$? – Kimball Apr 18 at 12:45
• @Kimball Yes this is the case. – Gory Apr 18 at 14:20
• @SubhajitJana yes however this gives the reverse inclusion, isn't it? I would like to get a support included in a congruence subgroup and your argument provides a congruence subgroup included in it (since it is a basis of neighborhood). – Gory Apr 18 at 14:22

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.)