$M_k(\Gamma_1(N)) = \bigoplus_{\chi \mod N} M_k(N, \chi)$

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?

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 $$\mathrm{SL}_2(\Bbb Z/N\Bbb Z)$$.

• Yes, though the projection map has an easy expression $P_\chi f = \frac1{\phi(N)}\sum_{d\bmod N} \overline{\chi(d)} \langle d\rangle f$ May 19, 2022 at 14:36
• @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. May 19, 2022 at 14:50
• @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. May 19, 2022 at 14:58
• 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 $\Gamma_0(N)$? May 19, 2022 at 16:12
• @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 $\gamma \in \Gamma$ May 19, 2022 at 16:50