A Non-formal Formula for the Rankin-Cohen Deformation Quantization

Abstract

Don Zagier’s superposition of Rankin-Cohen brackets on the Lie group $SL_2(\mathbb {R})$ defines an associative formal deformation of the algebra of modular forms on the hyperbolic plane [9]. This formal deformation has been used in [6] to establish strong connections between the theory of modular forms and that of regular foliations of co-dimension one. Alain Connes and Henri Moscovici also proved that Rankin-Cohen’s deformation gives rise to a formal universal deformation formula (UDF) for actions of the group $ax+b$. In a joint earlier work [5], the first author, Xiang Tang and Yijun Yao proved that this UDF is realized as a truncated Moyal star-product. In the present work, we use a method to explicitly produce an equivariant intertwiner between the above mentioned truncated Moyal star-product (i.e. Rankin-Cohen deformation) and a non-formal star-product on $ax+b$ defined by the first author in an earlier work [2]. The specific form of the intertwiner then yields an oscillatory integral formula for Zagier’s Rankin-Cohen UDF, answering a question raised by Alain Connes.