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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

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    $\begingroup$ I think that it is much better to use explicit $\mathrm{GL}_2$ character formulæ than to try to contort to use the $\mathrm{SL}_2$ results. I believe that Silberger has some such formulæ, but I'm not familiar. As long as you are willing to assume that $p \ne 2$, DeBacker's thesis (part of which was excerpted in DeBacker and Sally, 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.) $\endgroup$
    – LSpice
    Commented Mar 24, 2016 at 14:07
  • $\begingroup$ @desideriusseverus $1+\dots\ll1$? $\endgroup$
    – Turbo
    Commented Jan 14, 2021 at 7:51

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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 $St$.

  1. 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.

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  • $\begingroup$ Another helpful reference for decomposition is Moy and Sally, Supercuspidal representations of $\mathrm{SL}_n$ over a $p$-adic field: the tame case (MR). Are you sure that your statement that your $n$ divides $4$ is true even for $p = 2$? (I don't know that it isn't, only that I am automatically suspicious.) $\endgroup$
    – LSpice
    Commented Mar 24, 2016 at 14:03
  • $\begingroup$ @ L Spice: I think it holds for $p=2$. you are right, the reference I mentioned does assume $p=2$, however in Labesse and Langlands, L-Indistinguishability for SL(2), $p$ is arbitrary. Also if you believe that the size of the $L$-packet is in bijection with the component group of the centralizer of the associated projective Weil-Deligne representation it is an exercise to compute that the only possible sizes are of 1,2 or 4. $\endgroup$
    – Judith Ludwig
    Commented Mar 24, 2016 at 14:36
  • $\begingroup$ @JudithLudwig Thank you for those explanation, they seems to do the job for my purposes for now. The main point was the uniformly bounded number of representations in the decomposition of the restriction, and it is donne by Shelstad as mentioned in Lansky and Raghuram's article ! $\endgroup$
    – Desiderius Severus
    Commented Mar 24, 2016 at 22:53
  • $\begingroup$ @LSpice Thanks for the references, especially leading me towards DeBacker's thesis. $\endgroup$
    – Desiderius Severus
    Commented Mar 24, 2016 at 22:54
  • $\begingroup$ @JudithLudwig Could we lift a bound on the character associated to $\pi_{SL_2(\mathbf{Q}_p)}$ to a bound on the character associated to $\pi$, representation on $GL_2$? $\endgroup$
    – Desiderius Severus
    Commented Dec 18, 2016 at 9:58

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