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Algebras of symbols and modular forms. (English) Zbl 0857.43015
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 $$\Pi$$, 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.

##### MSC:
 43A99 Abstract harmonic analysis 11F11 Holomorphic modular forms of integral weight 30F99 Riemann surfaces
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##### References:
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