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 $SL_2$?.