I recently learned about automorphic spectral decomposition from the book "Spectral decomposition and Eisenstein series" by Moeglin and Waldspurger. (Let me call it M-W)

I have a question about the characterization of discrete spectra.

Let me explain the basic notation as in M-W.

Let $G$ be a connected reductive group over algebraic field $k$ and $\xi$ be a unitary character of $Z_G(A)$.

Let $L^2(G(k) \setminus G(A))_\xi$ be $L^2$-functions on $G(k)\setminus G(A)$ with central character $\xi$.

Then, $L^2(G(k) \setminus G(A))_\xi$ decomposes into the space generated by iterated residues of Eisenstein series and its complement, that is described by direct integrals of Eisenstein series.(M-W, IV 2.1)

Let me call the first space $L^2_d$.

(I think that $L^2_d$ is the closure of span of $L^2$ automorphic forms in $L^2(G(k) \setminus G(A))_\xi$.)

Let me call the semi-simple part i.e. Hilbert direct sum of topologically irreducible subrepresentations of $L^2(G(k) \setminus G(A))_\xi$, by a name $L^2_{ss}$.

Definition of discrete spectrum and continuous and basic properties

In the article above, it is called discrete spectrum.

My questions are

- Are $L^2_d$ and $L^2_{ss}$ the same?
- If so, how to prove it? Can we prove it by means of elementary functional analysis (e.g. the knowledge of the book "Functional Analysis" by Walter Rudin) like the proof of the theorem of Gelfand-Graev-Patetski-Shapiro i.e. like in cuspidal case?

I think that it is obvious that $L^2_d$ contains $L^2_{ss}$, but I wonder whether the converse is true. I would appreciate any clues to resolve this question. Thanks!

Edited: I added one more question and definition of $L^2_{ss}$ in line with the comments. Thanks for the comments!