Casselman, L-groups - philosophy of cusp forms

1.
Recall Parabolic induction in representation of Lie groups. Consider ($G=SL(2,\mathbb R)$) and ($P( \mathbb  R)=M(\mathbb R)\ltimes N(\mathbb R)$). 
Let ($\sigma$)  be a representation of the levi-component ($M(\mathbb R)$). We inflate it to be a representation of ($P(\mathbb R)$) by letting ($N(\mathbb R)$) acting trivially,  and normalized parabolic induced to be a representation of ($G(\mathbb R)$) via
($$
i_{M,P}^G \sigma=ind_{P}^G(\sigma\otimes\delta^{1/2})
$$)

2.
Let ($\Gamma$) be a principal  congruence  subgroup of ($SL(2,\mathbb Z)$). How to obtain an automorphic forms on ($G(\mathbb R)$) w.r.t. ($\Gamma$) from automomrphic forms on ($M(\R)$) via the above philosophy?

2-1.
Assume we have an automorphic form on ($M(\mathbb R)$) with respect to ($M_\Gamma=M(\mathbb R)\cap \Gamma$), i.e. a function
($$
I_s:M_\Gamma\backslash M(\mathbb R)\rightarrow \mathbb C,\quad I_s\left(\begin{pmatrix} \pm y^{\frac{1}{2}}\\&\pm y^{-1/2}\end{pmatrix} \right)= |y|^s
$$)

2-2.
We inflate it to be a function on ($P(\mathbb R)$) via letting ($N(\mathbb R)$) acts trivially,
and obtains an automorphic forms on ($P$),
($$
I_s:P_\Gamma N(\mathbb R)\backslash P(\mathbb R)\rightarrow \mathbb C,\quad I_s\left(\begin{pmatrix} \pm y^{\frac{1}{2}}&x\\&\pm y^{-1/2}\end{pmatrix}\right)= |y|^s
$$)

2-3.
We use the normalized induction to obtain a function on ($G(\mathbb R)$), 
($$
\varphi_s: P_\Gamma N(\mathbb R)\backslash G(\mathbb R)\rightarrow\mathbb C,\quad \varphi_s\left(\begin{pmatrix}\pm y^{1/2}&x\\&\pm y^{-1/2}\end{pmatrix} g\right)=|y|^{s+\frac{1}{2}}\varphi_s(g).
$$)
Note that we also need the condition that ($\varphi_s$) is right ($K_\R$)-finite.

2-4.
($\varphi_s$) is defined on ($G(\mathbb R)$), but only invariant under the action of ($P_\Gamma$). To obtain an automorphic form on ($G(\mathbb Z)\backslash G(\mathbb R)$), we finally define
($$
\mathcal E(g,\varphi_s):=\sum_{\gamma\in P_\Gamma\backslash \Gamma}\varphi_s(\gamma g)
$$)
It is an automorphic forms on ($\Gamma\backslash G(\R)$).