Construction of Jacobi forms using adjoint of the Jacobi-Serre derivative (Q6612394)

In an unpublished work, Oberdieck defined a derivation, now called the ``Oberdieck derivation'', which maps any weak Jacobi form of weight \(k\) and index \(m\) to a weak Jacobi form of weight \(k+2\) and index \(m\). The authors show that the result holds for Jacobi forms. They then explicitly compute the adjoint for the Petersson inner product of one of the components of Oberdieck's derivation: the heat operator, followed by the adjoint of Oberdieck's derivation itself. From this, they deduce an explicit formula for the adjoint of the Jacobi-Serre derivative, defined for a form \(f\) of weight \(k\) and index \(m\), by:\N\[\N\mathcal{D}^{J}(f) = D_\tau(f) - \frac{k}{12} E_2 f - \frac{1}{1 - 4m} \left( D^2_z(f) - J_1 D_z(f) + m J_2 f - \frac{m}{6}E_2f \right),\N\]\Nwhere\N\[\ND_\tau= \frac{1}{2i\pi} \frac{\partial}{\partial \tau}, \quad D_z = \frac{1}{2i\pi} \frac{\partial}{\partial z}\N\]\Nare the derivatives with respect to the variable \(\tau\) of the Poincaré half-plane and the variable \(z\) of the complex plane;\N\(E_2\) is the Eisenstein series of weight 2, defined by\N\[\NE_2(\tau) = 1 - 24 \sum_{n=1}^{\infty} \left( \sum_{d|n} d \right) e^{2i\pi n \tau}\N\]\N\[\NJ_1(\tau, z) = \frac{1}{1-e^{-2i\pi z}}-\frac{1}{2} - \sum_{r,s \geq 1} \left( e^{2i\pi s z} - e^{-2i\pi sz} \right) e^{2i\pi rs\tau}\N\]\N\[\NJ_2(\tau, z) = \frac{1}{6} -2\sum_{r,s \geq 1} r \left( e^{2i\pi s z} +e^{2i\pi sz} \right) e^{2i\pi rs \tau}.\N\]\NThe heat operator is then defined by\N\[\N\mathcal{L}_{k,m}(f)=\left(4mD_\tau-D_z^2-\frac{m(2k-1)}{6}E_2\right)f\N\]\Nand Oberdieck's derivation is\N\[\N\mathcal{O}_{k,m}(f)=\mathcal{L}_{k,m}(f)-(4m-1)\partial^{J}(f).\N\]\NOne ingredient of the proof is the study of the action of the Jacobi group on the images of \((\tau,z)\mapsto e^{2i\pi(s\tau+rz)}\) through the previous applications.