$\DeclareMathOperator\SL{SL}$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!



$\DeclareMathOperator\SL{SL}$Multiplicities of irreducible representations in discrete part of $L^2(\SL(2,\mathbb{Z})\backslash{\SL(2,\mathbb R)})$
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).


