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$\DeclareMathOperator\SL{SL}$It is well-known that the cuspidal (or discrete) part of $L^2(\SL(2,\mathbb{Z})\backslash{\SL(2,\mathbb{R})})$ decomposes into irreducible representations of $\SL(2,\mathbb{R})$. One can see Theorem 2.6 of Gelbart's book Automorphic Forms on Adele Groups.

$L^2_{\text{cusp}}(\SL(2,\mathbb{Z})\backslash{\SL(2,\mathbb R)})=\bigoplus_{\pi\in\widehat{SL(2,\mathbb{R})}}m_{\pi}\cdot\pi$.

My question is which $m_\pi$ is nonzero and what is the formula?

In the Gelbart's book (Theorem 2.10), for the discrete series $\pi_k$, $m_{\pi_k}=\dim S_k(\SL(2,\mathbb{Z}))$, the dimension of cusp forms of weight $k$. How about the other $m_{\pi}$'s? The principal series may also be related to the dimension of wave forms.

Is there any further (complete) result for the decomposition or for a general pair of $\Gamma\subset G$, a lattice in a real Lie group? The corresponding results for the adèle groups and automorphic representations are also welcome!

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    $\begingroup$ Which 1975 Gelbart book? $\endgroup$
    – LSpice
    Commented Jan 7, 2022 at 22:25
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    $\begingroup$ There's some imprecision in the premises of your question: yes, the (holomorphic or antiholomorphic) discrete series occur exactly as holo or anti-holo cuspforms, but wavefore-cuspforms generate principal_series repns of $SL_2(\mathbb R)$. Possibly your premise was just a mis-statement? $\endgroup$
    – paul garrett
    Commented Jan 7, 2022 at 22:34
  • $\begingroup$ Are you asking whether these representations occur with multiplicity one? The space of modular forms of weight $k$ has dimension $\approx \frac{k}{12}$, so the discrete series representation of weight $k$ occurs with high multiplicity. For principal series representations, it is conjectured that the multiplicity is one (see e.g. cds.cern.ch/record/260472/files/P00022028.pdf). See also my answer here: mathoverflow.net/questions/320171/… $\endgroup$
    – Peter Humphries
    Commented Jan 7, 2022 at 23:06
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    $\begingroup$ Even after Peter's edit, I have no idea what the OP is asking. Please ask a specific, precise question. $\endgroup$
    – Kimball
    Commented Jan 7, 2022 at 23:50
  • $\begingroup$ @paulgarrett I edited the question. Thank you! $\endgroup$
    – Jun Yang
    Commented Jan 8, 2022 at 2:34

1 Answer 1

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You are asking what is known about the dimension of weight $k$ holomorphic cusp forms for $\mathrm{SL}_2(\mathbb{Z})$, and the multiplicities of Laplace eigenvalues of weight $0$ and weight $1$ Maass forms for $\mathrm{SL}_2(\mathbb{Z})$. This question is very open ended, similar to asking what is known about the distribution of prime numbers. Well, for holomorphic cusp forms the dimension is known explicitly, and can be found in any introductory textbook (for a more general formula see e.g. Shimura's book Introduction to the arithmetic theory of automorphic functions). For Maass forms, the multiplicities are usually studied by the Selberg trace formula (see e.g. Hejhal's books The Selberg trace formula for $\mathrm{PSL}(2,\mathbb{R})$, Volumes I-II).

I should add that finding which $m_\pi$'s are nonzero is a computationally difficult task. For example, finding the 20th Laplace eigenvalue up to 100 decimal digits is quite challenging (see Booker-Strömbergsson-Venkatesh: Effective computation of Maass cusp forms).

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  • $\begingroup$ Thank you! Are the $m_{\pi}$'s known when $\Gamma\backslash G$ is compact? $\endgroup$
    – Jun Yang
    Commented Jan 8, 2022 at 21:03
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    $\begingroup$ @JunYang No. Even when $\Gamma$ is a co-compact subgroup of $G=\mathrm{SL}_2(\mathbb{R})$, the Laplace eigenvalues occurring in $L^2_{\text{cusp}}(\Gamma\backslash G)$ are elusive (e.g. difficult to compute). In fact the "transcendental nature" of the Laplace eigenvalues (and of Hecke eigenvalues) is one of the attractive features in the subject (at least to me). We cannot put our hands on these eigenvalues directly, hence they are more mysterious, more interesting. $\endgroup$
    – GH from MO
    Commented Jan 8, 2022 at 21:26

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