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




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



Bounding $p$-adic characters and Jacquet-Langlands transfert
First some remarks on the Jacquet-Langlands correspondence. The image of the local Jacquet-Langlands transfer $\mathrm{JL}$ from $G$, the group of units in a non-split quaternion algebra, to $\mathrm{GL}\_2(\mathbb{Q}\_p)$ is the set of discrete series representations.
This set contains the supercuspidal representations as well as the Steinberg representation $St$ and all its twists by characters.


Facts: 1. The image $\mathrm{JL}(\mathbf{1}\_G)$ of the trivial 1-dimensional representation $\mathbf{1}\_G$ of $G$ is given by $St$.


2. For any character $\chi:\mathbb{Q}\_p^\*\rightarrow \mathbb{C}^\*$ and any irreducible smooth representation $\pi$ of $G$, 
$$\mathrm{JL}(\pi \otimes (\chi\circ \mathrm{Nrd})) = \mathrm{JL}(\pi)\otimes (\chi \circ \det).$$


Together, 1 and 2 give that $\mathrm{JL}(\pi)$ is supercuspidal, iff $\dim(\pi)>1$.


Regarding the relation between representations of $\mathrm{GL}\_2(\mathbb{Q}\_p)$ and $\mathrm{SL}\_2(\mathbb{Q}\_p)$:  

Given an irreducible smooth representation $\pi$ of $\mathrm{GL}\_2(\mathbb{Q}\_p)$, the restriction to $\mathrm{SL}\_2(\mathbb{Q}\_p)$ decomposes into a finite direct sum of irreducible smooth representations $\pi\_i$ of $\mathrm{SL}\_2(\mathbb{Q}\_p)$
$$\pi|\_{\mathrm{SL}\_2(\mathbb{Q}\_p)}\cong \bigoplus\_{i=1}^n\pi\_i,$$ 
each $\pi\_i$ occurring with multiplicity one. The number $n$ is either 1,2 or 4. 


If $\pi$ is supercuspidal, then so are all the $\pi\_i$. 
The restriction of the Steinberg representation is the Steinberg representation of $\mathrm{SL}\_2(\mathbb{Q}\_p)$. 


For more details I think Section 2.3 in 
<http://msp.org/pjm/2007/231-1/pjm-v231-n1-p08-s.pdf> 
and the references mentioned there should be helpful.


