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I am stuck with the following statement in the study of modular forms:

$$ M_k(\Gamma_1(N)) = \bigoplus_{\chi \mod N} M_k(N, \chi), $$ where $\Gamma_1(N) := \left\{\begin{pmatrix}a & b\\ c & d\end{pmatrix} \in SL_2(\mathbb{Z}): a \equiv d \equiv 1 \mod N, \; N \mid c\right\}$ and $\chi$ runs over all Dirichlet characters modulo $N$. Any hints or proof directions?

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This requires some basic representation theory (or you can reproduce the representation-theoretic theorems in a special case).

Note that $\Gamma_1(N)$ is a normal subgroup of $\Gamma_0(N)$. So take any $\gamma \in \Gamma_0(N)$. Then because $\gamma \Gamma_1(N)=\Gamma_1(N)\gamma$ one can show that for $f \in M_k(\Gamma_1(N))$, we have $f|_k[\gamma] \in M_k(\Gamma_1(N))$. In this way, we obtain an action of $\Gamma_0(N)$ on $M_k(\Gamma_1(N))$. For this action, $\Gamma_1(N)$ acts trivially, so we obtain an action of the quotient group $\Gamma_0(N)/\Gamma_1(N)$. Now this quotient is actually isomorphic to $(\Bbb Z/N\Bbb Z)^\times$.

So we have an action $(\Bbb Z/N\Bbb Z)^\times$ on $M_k(\Gamma_1(N))$. Because the Petersson slash operator is $\Bbb C$-linear, we actually have a representation of the group $(\Bbb Z/N\Bbb Z)^\times$ on $M_k(\Gamma_1(N))$. By elementary representation theory, this representation decomposes into $\rho$-isotypic components, where $\rho$ runs over all irreducible representations of $(\Bbb Z/N\Bbb Z)^\times$. But irreducible representations of $(\Bbb Z/N\Bbb Z)^\times$ are exactly the Dirichlet characters and the isotypic component is just $M_k(N,\chi)$.


Bonus: all of the above works more generally in the case where $\Gamma$ and $G$ are two congruence subgroups such that $\Gamma$ is a normal subgroup of $G$. In this setting, let $g \in \Delta$, then we have $g \Gamma = \Gamma g$ by normality, so that we have $f|_k[g] \in M_k(\Gamma)$ for all $f \in M_k(\Gamma)$. So we obtain a linear action of $G$ on $M_k(\Gamma)$ for which $\Gamma$ acts trvially, thus we have a representation of the finite group $G/\Gamma$ on the vector space $M_k(\Gamma)$.

By representation theory, this decomposes as a direct sum of $\rho$-isotypic components, where $\rho$ runs over all irreducible representations of $G/\Gamma$. These $\rho$-isotypic components can be considered "modular forms of generalized Nebentypus". For example, $\Gamma(N)$ is normal in $\Gamma(1)$ and we have $\Gamma(1)/\Gamma(N) \cong \mathrm{SL}_2(\Bbb Z/N\Bbb Z)$, so $M_k(\Gamma(N))$ decomposes accordings to irreducible representations of $\mathrm{SL}_2(\Bbb Z/N\Bbb Z)$.

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  • $\begingroup$ Yes, though the projection map has an easy expression $P_\chi f = \frac1{\phi(N)}\sum_{d\bmod N} \overline{\chi(d)} \langle d\rangle f$ $\endgroup$
    – reuns
    May 19, 2022 at 14:36
  • $\begingroup$ @reuns yes, that formula is easy and useful and one can cook up an ad hoc proof based on it that avoids the representation theoretic language. I still think that the representation theory is "the correct way" to look at it, because it clarifies what's going on conceptually. $\endgroup$ May 19, 2022 at 14:50
  • $\begingroup$ @reuns also of course such a formula is not specific to the example at hand: it's just a special case of the projection formula onto the isotypic component. $\endgroup$ May 19, 2022 at 14:58
  • $\begingroup$ What does $f\vert_{k}[\gamma]$ stand for? Can you specify what the action you assert is, and why it needs the normality of $\Gamma_1(N)$ in $\Gamma_0(N)$? $\endgroup$
    – ferhenk
    May 19, 2022 at 16:12
  • $\begingroup$ @ferhenk If $\gamma=\begin{pmatrix} a & b \\ c & d\end{pmatrix}$, then $f|_k[\gamma]$ is defined via $f|_k[\gamma](z)=(c+dz)^{-k}f(\frac{az+b}{cz+d})$. The condition that a holomorphic function $f:\Bbb H \to \Bbb C$ that is also holomorphic at infinity is in $M_k(\Gamma)$ can be written as $f=f|_k[\gamma]$ for all $\gamma \in \Gamma$ $\endgroup$ May 19, 2022 at 16:50

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