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nt.quantity principle – Bounding $p$-adic characters and Jacquet-Langlands transfert

I’d love to sure uniformly in $pi$ the $p$-adic Harisch-Chandra characters $Theta_pi$ for division quaternion algebras. By the Jacquet-Langlands correspondence, it’s adequate to sure it on $GL_2(mathbf{Q}_p)$.

An thought, as an example showing in Kin, Shin and Templier [1], is to employ the Sally-Shalika components giving categorical calculations for the characters of $SL_2$, offering a sure for all supercuspidal representations :

$$|Theta_pi(gamma_p)| leqslant 1 + 2D(gamma_p)^{-1/2} ll 1$$

I’d love to do the identical for division quaternion algebras. By Jacquet-Langlands, we will tranfer to the $GL_2$ setting. By Labesse and Langlands and the sure above we will sure any $Theta_{tilde{pi}}$ the place $tilde{pi}$ is the restriction of $pi$ to $SL_2$. My query follows: **may we raise this sure obtained for restrictions to $SL_2$ to a sure on the characters for the all representations on $GL_2$?**

Perhaps my query solely betray my abysmal ununderstanding of the relations between representations of $GL_2$ and people of $SL_2$. Anyway, each enlightening observation or respond will breathe warmly welcome.

Best regards

[1] Kim, Shin and Templier, Asymptotics and Local Constancy of Characters of $p$-adic Groups, 2015

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