Algebras of symbols and modular forms (Q1922216)

Given a holomorphic function \(f\) on the Poincaré upper half-plane \(\Pi\) and a nonnegative integer \(k\); set, as usual, \[ (f|_k)(z)=(cz+d)^{-k} f\left({az+b\over cz+d}\right), \] so that the validity of the equation \(f|_k \gamma=f\) whenever \(\gamma\) belongs to some arithmetic subgroup of \(G=SL(2, \mathbb{R})\) characterizes weakly modular forms of weight \(k\) with respect to the given group. Given three nonnegative integers \(k_1\), \(k_2\) and \(j\), and any two holomorphic functions \(f\) and \(g\) on \(\Pi\), \textit{H. Cohen} introduced in [Math. Ann. 217, 271-285 (1975; Zbl 0311.10030)] the function \[ F_j(f,g)=\sum^j_{l=0} (-1)^l {k_1+j-1\choose l} {k_2+j-1\choose j-l} f^{(j-l)} g^{(l)} \] with \(g^{(l)}=({\partial \over \partial z})^\ell g\), and proved, for all \(\gamma \in G\), the identity \[ F_j(f|_{k_1}\gamma,g|_{k_2}\gamma)=F_j(f,g)|_{k_1+k_2+2j} \gamma. \] We here show that the sequence \(\{F_j(f,g)\}\) appears in the composition formula relative to a certain symbolic calculus of operators. The phase space \(\Pi_i\) to be used is the one-sheeted hyperboloid, on which \(G\) acts under the coadjoint action. The calculus, studied in some earlier work of the authors, permits to associate with every function \(f\) in \(L^2(\Pi_i)\) a Hilbert-Schmidt operator \(Op(f)\) on the Hilbert space of any representation taken from the principal series \(\pi_{i\lambda}\) of \(G\). Decomposing the space \(L^2(\Pi_i)\) under the quasi-regular action of \(G\) brings to light a family \(\{E^\pm_n\}_{n\geq 0}\) of discrete summands. The term \(E^+_n\) can be identified, under some intertwining map \(T_n\), with some weighted \(L^2\)-space of holomorphic functions on \(\Pi\), the natural Hilbert space for the representation of \(G\), taken from the discrete series, sometimes denoted \({\mathcal D}^+_{2n+2}\). It now turns out that, given \(f\in E^+_m\) and \(g\in E^+_n\), the composition of symbols \(f\# g\) can be written \(j\in E^+_{m+n+j+1}\), convergent in \(L^2(\Pi_i)\). Finally, setting \((k_1,k_2)=(2m+2,2n+2)\), one can make the various terms in the series explicit as \[ T_{m+n+j+1} h_j=\Phi(m,n,\lambda) F_j(T_mf,T_ng), \] where the numerical coefficient \(\Phi(m,n,\lambda)\) can be computed. In this setting, one can view Cohen's formula as quoted above as a consequence of the covariance of the symbolic calculus under consideration.