Derivatives of modular forms of negative weight (Q1001760)

Let \(\Gamma\) be a discrete subgroup of \(G= \text{Sp}(2n,\mathbb R)\) acting on the Siegel upper half-space \(\mathbb{H}_n\) of degree \(n\), and let \(\chi\) be a multiplier system on \(\Gamma\) giving rise to the automorphic factor \(j_k(\gamma, Z)\) \((\gamma\in\Gamma)\) of weight \(k\in{1\over 2}\mathbb Z\). Let \(M_k(\Gamma,\chi)\) denote the space of meromorphic functions \(f\) on \(\mathbb H\) such that \[ f(Z)= j_k(\gamma, Z)\,f(\gamma Z)\qquad (\gamma\in\Gamma,\,Z\in \mathbb H_n) \] and consider the differential operator \(\mathbb D_n= \text{det}(\partial_{ij})\), where \(\partial_{ij}\) is defined by \[ \partial_{ij}= {1\over 2}(1+ \delta_{ij}){\partial\over\partial Z_{ij}}. \] The motivation of the paper under review comes from Conjecture 1. If \(r\geq 0\) is an integer and \(f\in M_{-r+(n-1)/2}(\Gamma, \chi)\), then \(\mathbb D^{r+1}_n f\in M_{r+ 2+(n-1)/2}(\Gamma,\chi)\). The aim of this work is to reveal the Lie theory and representation theory behind this conjecture and to prove it when \(n\leq 2\). For \(n=1\) the assertion can be traced back to classical work by G. Bol and H. Maaß, and the authors also give another proof in the spirit of Lie theory and representation theory of \(\text{SL}_2(\mathbb R)\). Moreover, they give a review of the Maaß\ operators for the symplectic group, their origin in the Lie algebra and the representation theory of the metaplectic group, that is, the double cover of \(\text{Sp}(2n,\mathbb R)\). This leads to Conjecture 2 on the behaviour of certain holomorphic vectors under the action of certain Maaß\ operators, and it is shown in Theorem 2 that Conjecture 2 implies Conjecture 1. In the special case \(\text{Sp}(4,\mathbb R)\) the authors determine the ring of invariant differential operators on \(\mathbb H_2\) explicitly (by means of the centre of the universal enveloping algebra), and they show that Conjecture~2 (and hence Conjecture~1) is true if \(n= 2\).