Let $\Gamma$ be a subgroup of $\operatorname{SL}_2(\textbf{Z})$. I want to understand how the Fricke involution $w_N : f \longrightarrow (\tau \longrightarrow N^{-k/2}\tau^{-k}f(-1/N\tau))$ acts on the space $\mathcal{M}_k(\Gamma)$.
Since $$\begin{pmatrix} 0 & -1/N \\ 1 & 0 \end{pmatrix} \begin{pmatrix} a & b \\ c & d \end{pmatrix}=\begin{pmatrix} d & -c/N \\ -bN & a \end{pmatrix}\begin{pmatrix} 0 & -1/N \\ 1 & 0 \end{pmatrix}$$ it seems that if we introduce the subgroup $$\Gamma_N:=\left\{ \begin{pmatrix} a & b \\ c & d \end{pmatrix} \in \Gamma \quad , \quad \begin{pmatrix} a & b \\ c & d \end{pmatrix} \equiv \begin{pmatrix} * & * \\ 0 & * \end{pmatrix} \pmod N \right\}$$ then $w_Nf$ is invariant under the usual transformation of weight $k$ on $\Gamma_N$ (if $f$ belongs to $\mathcal{M}_k(\Gamma)$). But under which conditions can we conclude that $w_Nf$ belongs to $\mathcal{M}_k(\Gamma_N)$ ? Is that true for all $N$ ?
Thanks !
There is no reason why $w_N$ should act on $M_k(\Gamma)$ for a general subgroup $\Gamma$. The action will make sense if, and only if, we have $$\Gamma = w_N^{-1} \Gamma w_N,$$ which is true if $\Gamma_1(N) \subseteq \Gamma \subseteq \Gamma_0(N)$ and false for most other subgroups. (E.g. it's clearly false for $\Gamma = \Gamma_1(MN)$ for $M > 1$).