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I'm learning basic stuff about automorphic forms, please, if anything I say is not true, excuse me.

Let $X = \Gamma\setminus \mathcal{H}$ be a modular curve and let $f(\tau)$ be a Maass cusp form. Since $$\mathcal{H} = Z_{GL}(\mathbb{R})O_2(\mathbb{R})\setminus GL_2(\mathbb{R}),$$ we can "adelize" $f(\tau)$ as follows. By strong approximation, $GL_2(\mathbb{A}) = GL_2(\mathbb{Q})GL_2(\mathbb{R})K_f$, with $K_f$ compact open subgroup of $GL_2(\mathbb{A}_f)$, therefore we denote by $g = g_{\mathbb{Q}}g_{\mathbb{R}}k$ every element $g\in GL_2(\mathbb{A})$. We define $$\phi_f(g) = f(g_{\mathbb{R}}\cdot i),$$ the adelization of $f$. We know some properties about $\phi_f$, such as $2-$integrability and that it transforms by a character for the compact subgroup. The automorphic representations of $GL_2(\mathbb{A})$ are classified in Bump's book and depending of the properties of $f$, $\phi_f$ belongs to determined representation (Discrete series, principal series...).

My question is the following, we can also express the Poincare Half plane as $$Z_{SL}(\mathbb{R})SO_2(\mathbb{R})\setminus SL_2(\mathbb{R}).$$ Then, if we carry out the previous construction and we obtain an adelization of $f$ for $SL_2$, i.e. $$\tilde{\phi}_f:\;SL_2(\mathbb{A})\to \mathbb{C},$$ does it satisfy the same properties of $\phi_f$? the representation of the right translates of $\tilde{\phi}_f$ are classified depending on the properties of $f$?

I have found some paper where they review the real place, "Representations of $SL_2(\mathbb{R})$ and nearly holomorphic modular forms". Does it work similar for $p-$ adic groups?

Extra question: Does the previous picture change a lot if we use $Mp_2$ the metaplectic group instead of $SL_2$?.

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  • $\begingroup$ It seems $\phi_f$ and $\tilde{\phi}_f$ are compatible in the sense that the restriction of $\phi_f$ to $SL_2(\mathbb A)$ is $\tilde{\phi}_f$. That means the $SL_2(\mathbb A)$ representation is obtained by restricting the $GL_2(\mathbb A)$ representation to $SL_2(\mathbb A)$ and taking a quotient (think about how the subgroup acts by translation). Irreducible representations of $GL_2(\mathbb Q_p)$ restricted to $SL_2(\mathbb Q_p)$ split into one or two irreducible factors so one needs to check when these share the properties. $\endgroup$
    – Will Sawin
    Feb 4, 2021 at 4:14
  • $\begingroup$ Thank you, do you know some references about this splitting? $\endgroup$
    – Aersk
    Feb 4, 2021 at 9:11
  • $\begingroup$ Labesse and Langlands' paper "L-indistinguishability for SL2" is the canonical reference. $\endgroup$ Feb 7, 2021 at 12:40

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