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Let $F$ be a number field, and $G$ be the group of units of a quaternion algebra $D$ over $F$. At a certain ramified place $v$, for $\gamma_v \in G(F_v)$, could we bound the orbital integral $$\mathcal{O}_{\gamma_v}(MC(\pi_v)) = \int_{G_{\gamma_v}(F_v) \backslash G(F_v)} MC(\pi_v)(u^{-1}\gamma_v u) \mathrm{d}u$$

where $MC(\pi_v)$ is the matrix coefficient attached to $\pi_v$? I am seeking for a bound in terms of the Plancherel measure of $\pi_v$.

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  • $\begingroup$ Can you please define $\mathcal{O}_{\gamma_v}(MC(\pi_v))$? I am not familiar with the notations. $\endgroup$
    – Subhajit Jana
    Dec 12, 2018 at 17:55
  • $\begingroup$ @SubhajitJana, the definition is the right-hand side of the equality. $\endgroup$
    – LSpice
    Dec 13, 2018 at 2:40
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    $\begingroup$ This is just a remark. If $\pi_\nu$ is supercuspidal and $\gamma_\nu$ is a regular elliptic element of $G(F_\nu )$ then the RHS is the value of the Harish-Chandra character of $\pi_\nu$ at $\gamma_\nu$. So in that case you're trying to bound the character of $\pi_\nu$ in terms of the Plancherel measure. $\endgroup$
    – Paul Broussous
    Dec 13, 2018 at 10:06
  • $\begingroup$ @PaulBroussous Indeed, however is there something standard to do that? I can think of character formulas to bound in general (Weyl's character formula for instance, at archimedean places for instance), but without using $\pi_v$ and making its Plancherel measure appear $\endgroup$
    – TheStudent
    Dec 16, 2018 at 0:53

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