Let $G$ be a connected, reductive group over a number field $k$. Let $\mathbb A$ be the ring of adeles of $k$, $\omega$ be unitary character of $Z_G(\mathbb A)/Z_G(k)$, and $V = L^2(G(k) \backslash G(\mathbb A); \omega)$ the space of functions $f: G(\mathbb A) \rightarrow \mathbb C$, modulo functions vanishing almost everywhere, such that:
$f(\gamma zg) = \omega(z) f(g)$ for all $\gamma \in G(k), z \in Z_G(\mathbb A), g \in G(\mathbb A)$.
$\int\limits_{Z_G(\mathbb A) G(k) \backslash G(\mathbb A)} |f(g)|^2dg < \infty.$
Then $V$ is a Hilbert space with inner product $\langle f_1, f_2 \rangle = \int\limits_{Z_G(\mathbb A) G(k) \backslash G(\mathbb A)} f_1(g) \overline{f_2(g)} dg$, and $G(\mathbb A)$ acts continuously on $V$ by right translation.
Let $V_{\textrm{disc}}$ be the closure of the subspace linearly spanned by closed, irreducible $G(\mathbb A)$-invariant subspaces. A function $f \in V$ is called cuspidal if $$\int\limits_{N(k) \backslash N(\mathbb A)} f(ng)dn = 0$$ for all unipotent radicals $N$ of all proper parabolic subgroups of $G$ and almost all $g \in G(\mathbb A)$. The set $V_{\textrm{cusp}}$ of cuspidal functions in $V$ is a closed subspace of $V_{\textrm{disc}}$.
We let $V_{\textrm{res}}$ be the orthogonal complement of $V_{\textrm{cusp}}$ in $V_{\textrm{disc}}$, and $V_{\textrm{cont}}$ the orthogonal complement of $V_{\textrm{disc}}$ in $V$. Then $V$ decomposes into a direct sum of subrepresentations: $$V = V_{\textrm{disc}} \oplus V_{\textrm{cont}} = V_{\textrm{cusp}} \oplus V_{\textrm{res}} \oplus V_{\textrm{cont}}. \tag{1}$$ I have the following questions:
$V_{\textrm{res}}$ is said to consist of "residues of Eisenstein series." What does this mean precisely? I understand that an Eisenstein series can be associated to a function in the induced space $\operatorname{Ind}_{P(\mathbb A)}^{G(\mathbb A)} \sigma$, where $P$ is a parabolic subgroup of $G$, and $\sigma$ is a cuspidal automorphic representation of $P/N(\mathbb A)$, where $N$ is the unipotent radical of $N$.
$V_{\textrm{cont}}$ is said to be "spanned by Eisenstein series." What does this mean exactly?
There is apparently an orthogonal direct sum decomposition of $V$:
$$V = \bigoplus\limits_{(P,\sigma)} V_{P,\sigma} \tag{2}$$ where $(P,\sigma)$ are equivalence classes of pairs, where $P$ is a parabolic subgroup of $G$, $M$ is a Levi subgroup of $P$, and $\sigma$ is a cuspidal representation of $M(\mathbb A)$. These equivalence classes are called "cuspidal automorphic data." If I understand correctly, the sum of the spaces $V_{P,\sigma}$ with $P = G$ constitute the space $V_{\textrm{cusp}}$. Whatever this decomposition means, is supposed to tell us that understanding all the cuspidal representations of $G$ and its Levi subgroups is the same as understanding $V$ itself.
What is the connection between the decompositions (1) and (2)? If cuspidal representations of Levi subgroups of $G$ show up in $V$ somehow, how do we determine whether they appear in $V_{\textrm{cont}}$ or in $V_{\textrm{res}}$?
