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I can give an additional point of view, which is coming from the theory of parabolic induction. Parabolic induction plays a prominent role in representation theory and gives you a better intuition for the higher rank situation. This point of view is often better stressed in the adelic theory than in the classical picture.

Let $G =PSL_2(\mathbb{R})$ with standard parabolic $B$ and $\Gamma$ a cofinite lattice.

My intuition is that te analytic Eisenstein series on $\Gamma \backslash G$ are vectors of the induced representation:

$$Ind_{\Gamma N}^G 1,$$

but there is one major issue with this, namely that $\Gamma N$ is not a group.

Rigorously seen they are given as $P$ series, i.e.

$$E: C_c^\infty(N \backslash G) \rightarrow L^2(\Gamma \backslash G),$$

by defining the $B$ series $$E(f) (g)= \sum\limits_{B \cap \Gamma \backslash \Gamma} f(\gamma g).$$

The image of $E$ generates a dense subspace of the orthocomponent to the cuspidal forms.

Now, we one notices that $C_c^\infty(N \backslash G)$ is a dense subspace of $Ind_N^G 1$. Induction by steps gives a decomposition $$Ind_N^G 1 \cong Ind_B^G Ind_N^B 1.$$ Now for $Ind_N^B 1 \cong L^2(B/N) = L^2(M)$, where $M$ are the diagonal matrices. Pontryagin duality gives you a direct integral decomposition of $L^2(M)$, and we have as a result

$$Ind_N^G 1 \cong \int\limits_{\Re s = 0}^{\oplus} Ind_B^G | \cdotp |^s.$$

Certainly one hopes that $E$ extends to $Ind_B^G | \cdotp |^s$, but convergence only happens $\Re s >1/2$, and the operators has to be defined by analytic continuation to make sense on $\Re s = 0$.

Perhaps it useful to give at least one definition here: functions $f \in Ind_B^G | \cdotp |^s$ are defined as $f(bg) = |b_{1,1} / b_{2,2}|^{s+1/2} f(g)$ for $b \in B$ with $f|_K= f|_K\in L^2(K)$ for $K= PSO(2)$.

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The

Let $G =PSL_2(\mathbb{R})$ with standard parabolic $B$ and $\Gamma$ a cofinite lattice.

My intuition is that te analytic Eisenstein series on $\Gamma \backslash G$ are vectors of the induced representation:

$$Ind_{\Gamma N}^G 1 = \bigoplus_{\chi: B / \Gamma 1,$$

but there is one major issue with this, namely that $\Gamma N$ is not a group.

Rigorously seen they are given as $P$ series, i.e.

$$E: C_c^\infty(N \cap B backslash G) \rightarrow \mathbb{C}^\times} \int^\oplus_{\Re s = 0} Ind_{\Gamma B}^G L^2(\Gamma \chi(s),$$

where backslash G),$$by defining the B is the Borel subgroup and series \chi: B / E(f) (g)= \Gamma\cap B sum\limits_{B \rightarrow cap \mathbb{C}^\times Gamma \backslash \Gamma} f(\gamma g).$$

The image of $E$ generates a one dimensional representation, which is necessary constant on the upper triangular matrices.

As you point out, dense subspace of the Eisenstein series fail orthocomponent to be square integrable, so the induced cuspidal forms.

Now, we one notices that $C_c^\infty(N \backslash G)$ is not a dense subspace , but we can project (of $Ind_N^G 1$. Induction by steps gives a densely defined operatordecomposition$$Ind_N^G 1 \cong Ind_B^G Ind_N^B 1.$$Now for $Ind_N^B 1 \cong L^2(B/N) onto spaces of the form = L^2(M)$, where $Ind_{\Gamma N}^G 1,$ via M$are the operatordiagonal matrices. Pontryagin duality gives you a direct integral decomposition of$L^2(M)$, and we have as a result $$Mf(g) Ind_N^G 1 \cong \int\limits_{\Re s = 0}^{\oplus} Ind_B^G | \int_N f(ng) d n.$$ You might perhaps notice, cdotp |^s. $$Certainly one hopes that E extends to Ind_B^G | \cdotp |^s, but convergence only happens \Re s >1/2, and the cuspidal live exactly operators has to be defined by analytic continuation to make sense on \Re s = 0. Perhaps it useful to give at least one definition here: functions f \in the (closed) kernel of the above operatorInd_B^G | \cdotp |^s are defined as f(bg) = |b_{1,1} / b_{2,2}|^{s+1/2} f(g) for b \in B with f|_K= L^2(K) for K= PSO(2). 4 added 16 characters in body; deleted 6 characters in body I can give an additional point of view, which is coming from the theory of parabolic induction. Parabolic induction plays a prominent role in representation theory and gives you a better intuition for the higher rank situation. This point of view is often better stressed in the adelic theory than in the classical picture. The analytic Eisenstein series on \Gamma \backslash G are vectors of the induced representation:$$ Ind_{\Gamma N}^G 1 = \bigoplus_{\chi: B / \Gamma \cap B \rightarrow \mathbb{C}^\times} \int^\oplus_{\Re s = 0} Ind_{\Gamma B}^G \chi(s),$$where B is the Borel subgroup and \chi: \Gamma B / \cap Gamma\cap B \rightarrow \mathbb{C}^\times a one dimensional representation, which is necessary constant on the upper triangular matrices. As you point out, the Eisenstein series fail to be square integrable, so the induced is not a subspace, but we can project (by a densely defined operator) onto spaces of the form Ind_{\Gamma N}^G 1, via the operator$$ Mf(g) = \int_N f(ng) d n.$\$

You might perhaps notice, that the cuspidal live exactly in the (closed) kernel of the above operator.