I would like to bound uniformly in $\pi$ the $p$-adic Harisch-Chandra characters $\Theta_p$ 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 :

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

I would like to do the same for division quaternion algebras. By Jacquet-Langlands, we can tranfert to the $GL_2$ setting. My question follows : what are the representations obtained from that tranfert like ? Could we write them as direct sums of supercuspidal (discrete series) representations of $SL_2$ ?

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 welcomed.

Best regards

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