In obtaining the spectral decomposition of $L^2(\Gamma \backslash G)$ where $G=SL_2(\mathbb{R})$, and $\Gamma$ is an arithmetic subgroup (I am satisfied with $\Gamma = SL (2,\mathbb{Z})$) we have a basis of eigenfunctions of the hyperbolic Laplacian, and orthogonal to that we have the space spanned by the incomplete Eisenstein series,
$$
E(z,\psi) = \sum_{\Gamma_\infty \backslash \Gamma} \psi (\Im(\gamma z)) = \frac{1}{2\pi i}\int_{(\sigma)} E(z,s)\tilde{\psi}(s)\mathrm{d}s
$$
where $\psi \in C_c^\infty(\mathbb{R}^+)$, $\tilde{\psi}$ is its Mellin transform, and $E(z,s) = \sum_{\gamma \in \Gamma_\infty \backslash \Gamma} \Im(\gamma(z))^s$ is the usual Eisenstein series.
My question is, where does $E(z,s)$ itself live with respect to the vector space $V = L^2(\Gamma \backslash G)$ which can be considered as the vector space of the right regular representation of $G$, and what is this parameter $s$?
A similar question of course goes for $\mathbb{R}$, where does $e^{2\pi i x}$ live with respect to $(L^2(\mathbb{R}), \rho)$?
I would appreciate a representation theoretic flavored answer, that is why I mentioned representations, but any other answer would also be an addition to my understanding of this.
In general, is there an associated space to $(V,\pi)$, an automorphic representation, such that the elements of the vector space are of moderate or rapid growth, instead of decay.
Best Answer
Surely there is not a single good answer, since the question is about how to legitimize "generalized eigenvectors", and there is no single-best notion of "legitimize".
As in other answers, one interpretation of Eisenstein series is as being in the dual to "rapidly decreasing" functions. This has various weaknesses.
"Continuous, moderate growth" is a better space for many purposes, but, note, it does not contain $L^2$ (!), but does contain suitable positively-indexed Sobolev spaces [sic].
There are interesting difficulties in understanding what "moderate growth" (of a given exponent) might mean, if/when one wants these spaces to be representation spaces for $G$ on $\Gamma\backslash G$. The most naive-and-appealing definitions of topological vector space structures are not $G$-stable, for elementary reasons, but sensible adaptations are easy, when one allows (by now 60-year-old) topological vector space notions.
In a different direction, note that the Plancherel theorem for afms does not depend upon knowing a space in which $E_{1/2+it}$ lies, any more than the usual Plancherel for Fourier transform on $\mathbb R$ depends on knowing "where $e^{i\xi x}$ lies".