I would like to bound uniformly in $\pi$ the $p$-adic Harisch-Chandra characters $\Theta_\pi$ for division quaternion algebras. By the Jacquet-Langlands correspondence, it is sufficient to bound it on $GL_2(\mathbf{Q}_p)$.

An idea, for instance appearing in Kin, Shin and Templier [1], is to use the Sally-Shalika formula giving explicit calculations for the characters of $SL_2$, providing a bound for all supercuspidal representations :

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

I would like to do the same for division quaternion algebras. By Jacquet-Langlands, we can tranfer to the $GL_2$ setting. By Labesse and Langlands and the bound above we can bound any $\Theta_{\tilde{\pi}}$ where $\tilde{\pi}$ is the restriction of $\pi$ to $SL_2$. My question follows: **could we lift this bound obtained for restrictions to $SL_2$ to a bound on the characters for the whole representations on $GL_2$?**

Perhaps my question only betray my deep ununderstanding of the relations between representations of $GL_2$ and those of $SL_2$. Anyway, every enlightening comment or answer will be warmly welcome.

Best regards

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

Germs, characters, and the Fourier transforms of nilpotent orbits(MR)) has probably the most readable discussion. (He reduces to the LCE near the identity, but Murnaghan computes the relevant coefficients.) $\endgroup$