Hi,

I've a question regarding the Petersson operator.

We have the following definition and the lemma.

Definition

Let $k, m \in \mathbb{Z}$ and $\phi: \mathbb{H} \times \mathbb{C} \rightarrow \mathbb{C}$.

For

$\begin{pmatrix} a & b \\ c & d \\ \end{pmatrix} \in \Gamma_1$ und $\begin{pmatrix} \lambda & \mu \end{pmatrix} \in \mathbb{Z}^{2}$

we set

i) $\left( \phi|_{k,m} \begin{pmatrix} a & b \\ c & d \\ \end{pmatrix} \right) (\tau,z) := (c\tau+d)^{-k} e^{2\pi i m \frac{-cz^2}{c\tau+d}} \phi(\frac{a\tau+b}{c\tau+d},\frac{z}{c\tau+d})$ `and

ii) $( \phi|_m \begin{pmatrix} \lambda & \mu \\ \end{pmatrix}) (\tau,z) := e^{2\pi i m (\lambda^2\tau + 2 \lambda z)} \phi(\tau, z + \lambda\tau+\mu) $

Lemma

Let $k, m \in \mathbb{Z}$ and $\phi: \mathbb{H} \times \mathbb{C} \rightarrow \mathbb{C}$.

$M, M_1, M_2 \in \Gamma_1;\; X, X_1, X_2 \in \mathbb{Z}^2$ satisfy

i) $(\phi|_{k,m} M_1)|_{k,m} M_2 = \phi|_{k,m} (M_1 M_2)$

ii) $(\phi|_m X_1)|_m X_2 = \phi|_m (X_1 + X_2)$

iii) $( \phi|_{k,m} M)|_m XM = (\phi|_m X)|_{k,m}M $

I've to prove that

$ \phi|(M,X)|(M',X') = \phi|(MM',XM'+X')$

Unfortunately, after spending a lot of time, I am not clear how I can prove this. The main problem is that I don't know which calculation rule I can use. The lemma only deals with one variable ($M$ or $X$) but not with 2 variables $(M,X)$.

Do you have any hints how I could proceed.

Thanks and regards, Jan