# Product of $V_N$ operator(index changing) and its adjoint on Jacobi forms

In Kohnen & Skoruppa's 1989 inventiones paper, page 549, the operator $$V_N: J_{k,1}^\text{cusp} \longrightarrow J_{k,N}^\text{cusp}$$ is defined by the action $$\sum_{\substack{D<0,r \in \mathbb{Z}\\D\equiv r^2(4)}} c(D,r)\, e\left( \frac{r^2-D}{4}\tau+rz\right) \longmapsto \sum_{\substack{D<0,r \in \mathbb{Z}\\D\equiv r^2(4N)}} \left(\sum_{\substack{d|(r,N)\\D\equiv r^2(4Nd)}} d^{k-1} c\left(\frac{D}{d^2},\frac{r}{d}\right) \right) \,e\left( \frac{r^2-D}{4N}\tau+rz\right)$$

Its adjoint (wrt Petersson inner product) $$V_N^*: J_{k,N}^\text{cusp} \longrightarrow J_{k,1}^\text{cusp}$$ is then given by $$\sum_{\substack{D<0,r \in \mathbb{Z}\\D\equiv r^2(4N)}} c(D,r)\, e\left( \frac{r^2-D}{4N}\tau+rz\right) \longmapsto \sum_{\substack{D<0,r \in \mathbb{Z}\\D\equiv r^2(4)}} \left(\sum_{d|N} d^{k-2} \sum_{\substack{s(2d)\\s^2\equiv D(4d)}} c\left(\frac{N^2}{d^2}D,\frac{N}{d}s\right) \right) \,e\left( \frac{r^2-D}{4}\tau+rz\right)$$

Then, in the Propostion (ii) [page 549-550] the action of $$V_N^*V_N$$ is given by $$V_N^*V_N = \sum_{t|N}\Psi(t)t^{k-2}T\left( \frac{N}{t}\right),$$ where $$\Psi(t)=t\prod_{t|p}\left(1+\frac{1}{p}\right)$$ and $$T(n)$$ denotes the Hecke operator on $$J_{k,1}^\text{cusp}.$$

I am not able to follow how $$V_N^*V_N$$ is derived. I am not sure about action of $$T(n)$$ also. I first applied $$V_N$$ and then taking its coefficients to be $$c(D',d')$$ applied $$V_N^*$$ on it, but cannot proceed further. Any help?