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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?

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  • $\begingroup$ 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 $f$. $\endgroup$
    – Subhajit Jana
    Commented Apr 18, 2018 at 10:44
  • $\begingroup$ $G$ means what? $D^\times(F_v)$ for a ramified nonarchimedean $v$? $\endgroup$
    – Kimball
    Commented Apr 18, 2018 at 12:45
  • $\begingroup$ @Kimball Yes this is the case. $\endgroup$
    – Gory
    Commented Apr 18, 2018 at 14:20
  • $\begingroup$ @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). $\endgroup$
    – Gory
    Commented Apr 18, 2018 at 14:22

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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.)

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