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Automorphic pseudodifferential operators. (English) Zbl 1055.11514
Fokas, A. S. (ed.) et al., Algebraic aspects of integrable systems: in memory of Irene Dorfman. Boston, MA: Birkhäuser (ISBN 0-8176-3835-0/hbk). Prog. Nonlinear Differ. Equ. Appl. 26, 17-47 (1997).
The Rankin-Cohen brackets $$[f,g]_n^{(k,l)}$$ define bilinear maps $$M_{2k}\otimes M_{2l}\rightarrow M_{2k+2l+2n}$$ between spaces of modular forms of even weights for some discrete subgroup $$\Gamma$$ of $$\text{PSL}_2(\mathbb R)$$. The present article uses the lifting $$f\mapsto f(z) \partial^{-k/2}$$ from modular forms of even weight $$k$$ to $$\Gamma$$-invariant pseudodifferential operators ($$\Psi \text{DOs}$$) to show that suitable linear combinations of Rankin-Cohen brackets correspond to the natural multiplication in the ring of $$\Psi \text{DOs}$$. It is then shown that the multiplication on the space $$\mathcal M (\Gamma)$$ of sequences $$(f_k)$$ of (generalized) modular forms of weight $$2k$$ with only finitely many nonzero terms with negative index given this way is just a special case of a family $$\mu^\kappa, \kappa \in \mathbb C\cup \{\infty\}$$, of multiplications expressed in terms of Rankin-Cohen brackets. These multiplications are shown to correspond in the same way as above to the natural multiplication in the ring of conjugate automorphic pseudodifferential operators of weight $$\kappa$$. Generalizing further, one obtains a family of multiplication maps $$\mu^{\kappa_1,\kappa_2,\kappa_3}$$ on $$\mathcal M (\Gamma)$$ with groupoid-like associativity properties. The coefficients appearing in the expressions of the multiplications considered in terms of the Rankin-Cohen brackets are shown to satisfy lots of interesting identities as a consequence of symmetries of the situation. Some of these identities and symmetries are explained with the help of the noncommutative residue map introduced by Yu. I. Manin [Topics in noncommutative geometry, Princeton Univ. Press, Princeton, NJ (1991; Zbl 0724.17007)]. In the final sections analogues for supermodular forms and superpseudodifferential operators are studied and connections to the theory of completely integrable systems of nonlinear differential equations are sketched.
For the entire collection see [Zbl 0851.00060].

##### MSC:
 11F25 Hecke-Petersson operators, differential operators (one variable) 11F12 Automorphic forms, one variable