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I apologize if this is too basic for MO.

I have an embarrassing admission to make: I don't know the actual definition of the discrete/continuous spectrum of a reductive group $G/\mathbb{Q}$ (in the context of automorphic forms if there is some ambiguity I'm unaware of).In particular, I know what the discrete series representations are for various groups $G$ (e.g. $\text{GL}_2$), and I assume that the discrete spectrum is made up of the discrete series (pluse extra! see the comments below), but I don't know how to define them in a general, 'elegant' way.

I have the, possibly wildly incorrect, impression that the discrete spectrum of $G$ is the largest semisimple $(\mathfrak{g},K)\times G(\mathbb{A}^\infty)$-subrepresentation of $L^2(G(\mathbb{Q})\backslash G(\mathbb{A}))$ and that the continuous spectrum is the orthogonal complement of this which should be some sort of 'direct integral' over continuous parameters. But, to be honest, in the searching that I've done I haven't seen this explicitly stated.

So, with this being said, I have the following three basic questions:

Question 1: What is the rigorous, 'elegant' definition of the discrete and continuous spectrum of a reductive group $G/\mathbb{Q}$?

Question 2: Why are these called 'discrete' and 'continuous'?

Question 3: Can we explicitly describe the discrete and/or continuous spectrum in any reasonable way for general $G$? If not, which $G$ can we describe it for?

NB: I am a relative neophyte, so I would appreciate if any answer could be in as simple language as possible.

Thanks!

EDIT: There's also the following 'bonus question' if anyone feels up to it:

Question 4: How do Eiseinstein representations fit into this? Why do they appear in both the continuous and discrete spectrum?

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    $\begingroup$ I might be wrong but I am confused with your "impression". Take $G=SL_2(\mathbb{R})$. The discrete series representations basically consist of various (non-zero) weight modular forms, where as the principal series representations contain Maass forms which are part of the discrete spectrum of $L^2(\Gamma\backslash G)$. So "discrete series representation" is somewhat smaller than "discrete spectrum in $L^2$". $\endgroup$ May 4, 2016 at 20:33
  • $\begingroup$ @SubhajitJana Ah yes, you are one-hundred percent correct. This is more a function of my ignorance than of your misunderstanding, although I do think I knew that deep down. :) Thanks! I will edit. $\endgroup$ May 4, 2016 at 22:21

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A few comments.

Question 1: When working with reductive groups, it is better to work with the subgroup $G(\mathbb{A})^1\subset G(\mathbb{A})$ or to work with the subspace of $L^2(G(\mathbb{Q})\backslash G(\mathbb{A}))$ transforming under the center according to a fixed character. If you care about the representations occurring in these $L^2$ spaces, then you can define the discrete spectrum as the subspace of $L^2(G(\mathbb{Q})\backslash G(\mathbb{A})^1)$ (say) which decomposes as a direct sum of irreducible sub representations of the action of $G(\mathbb{A})$ by right translation. Then the continuous spectrum is the orthogonal complement of the discrete spectrum as you said, so $$ L^2(G(\mathbb{Q})\backslash G(\mathbb{A})^1) = L^2_\mathrm{disc}(G(\mathbb{Q})\backslash G(\mathbb{A})^1)\oplus L^2_\mathrm{cont}(G(\mathbb{Q})\backslash G(\mathbb{A})^1). $$ The cuspidal spectrum is a subspace of the discrete spectrum, but in general there are discrete spectrum representations which are not cuspidal. The definition of an automorphic representation is more general than these representations "occurring in" $L^2$. For this, see the articles of Borel-Jacquet and Langlands in Corvallis.

Question 3: It depends on what you would like to know. For example, the Moeglin-Waldspurger classification for $\mathrm{GL}(N)$ describes the discrete spectrum in terms of the cuspidal spectrum of Levi subgroups. Namely, it says that there is a bijection between pairs $(\mu,m)$ and the discrete spectrum of $\mathrm{GL}(N)$, where $m$ divides $N$ and $\mu$ is a cuspidal representation of $\mathrm{GL}(m)$. Recent work of Arthur describes the discrete spectrum of quasi-split orthogonal and symplectic groups in terms of the discrete spectrum for $\mathrm{GL}(N)$. If $G$ is such a classical group, he provides a decomposition of the space $L^2_\mathrm{disc}(G(\mathbb{Q})\backslash G(\mathbb{A}))$ into a direct sum parametrized by representations of $\mathrm{GL}(N)$. This work has been extended by Mok to quasi-split unitary groups.

As far as the continuous spectrum, one place to start might be Section 7 of Arthur's "An Introduction to the Trace Formula," where he describes Langlands' main theorem on Eisenstein series. He also discusses all the things mentioned above. I won't try to say much else about this.

Question 4: For classical groups, I do not think that a representation occurring in the continuous spectrum can also occur in the discrete spectrum. I believe this is the problem of "embedded eigenvalues." For this, I would check out Arthur's note "Eigenfamilies, characters and multiplicities" and "The embedded eigenvalue problem for classical groups."

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  • $\begingroup$ Hey Mike. Thanks for the answer! It'll take me a day or so to process it/look up these references. One in the interim: what is $G(\mathbb{A})^1$? The only definition I can think of seems like it would depend on a choice of a faithful representation of $G$. Moreover, I have not seen anyone discuss eschewing the central character using this $G(\mathbb{A})^1$. Do you have a reference? $\endgroup$ May 7, 2016 at 0:29
  • $\begingroup$ Try section 3 of the "Introduction to the Trace Formula." For example, $\mathrm{GL}(N,\mathbb{A})^1$ is the subgroup of matrices whose determinant has absolute value 1. $\endgroup$
    – Mike B
    May 8, 2016 at 16:13

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