$\DeclareMathOperator\SL{SL}$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 $\SL(2,\mathbb{R})$. As far as we know, the multiplicities of discrete series or principle series are just the dimensions of holomorphic cusp forms or Maass cusp forms (see Theorem 2.10 of Gelbart's book Automorphic Forms on Adele Groups).

Is there any further (complete) result for the decomposition or for any specific/general case? The corresponding results for the adèle groups and automorphic representations are also welcome!

$\DeclareMathOperator\SL{SL}$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!

$\DeclareMathOperator\SL{SL}$It is well-known that the cupsidal (or discrete) part $L^2(\SL(2,\mathbb{Z})\backslash{\SL(2,\mathbb{R})})$ decomposes into irreducible representations of $\SL(2,\mathbb{R})$. As far as we know, the multiplicities of discrete series or complementary series are just the dimensions of cusp forms or wave forms (see Theorem 2.10 of S. Gelbart's book in 1975).

Is there any further (complete) result for the decomposition or for any specific/general case? The corresponding results for the adèle groups and automorphic representations are also welcome!

$\DeclareMathOperator\SL{SL}$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 $\SL(2,\mathbb{R})$. As far as we know, the multiplicities of discrete series or principle series are just the dimensions of holomorphic cusp forms or Maass cusp forms (see Theorem 2.10 of Gelbart's book Automorphic Forms on Adele Groups).

Is there any further (complete) result for the decomposition or for any specific/general case? The corresponding results for the adèle groups and automorphic representations are also welcome!

It is well-known that the cupsidal (or discrete) part $L^2(SL(2,\mathbb{Z})\backslash SL(2,\mathbb{R}))$ decomposes into irreducible representations of $SL(2,\mathbb{R})$. As far as we know, the multiplicities of discrete series or complementary series are just the dimensions of cusp forms or wave forms (See Theorem 2.10 of S. Gelbart's book in 1975  ).

Is there any further (complete) result for the decomposition or for any specific/general case? The corresponding results for the adele groups and automorphic representations are also welcome!

$\DeclareMathOperator\SL{SL}$It is well-known that the cupsidal (or discrete) part $L^2(\SL(2,\mathbb{Z})\backslash{\SL(2,\mathbb{R})})$ decomposes into irreducible representations of $\SL(2,\mathbb{R})$. As far as we know, the multiplicities of discrete series or complementary series are just the dimensions of cusp forms or wave forms (see Theorem 2.10 of S. Gelbart's book in 1975).

Is there any further (complete) result for the decomposition or for any specific/general case? The corresponding results for the adèle groups and automorphic representations are also welcome!