Hopf action and Rankin–Cohen brackets on an Archimedean complex

Abstract

The Hopf algebra \mathcal H_1 of “codimension 1 foliations”, generated by operators X , Y and \delta_n , n\geq 1 , satisfying certain conditions, was introduced by Connes and Moscovici in [1]. In [2], it was shown that, for any congruence subgroup \Gamma of SL _2(\mathbb Z) , the action of \mathcal H_1 on the “modular Hecke algebra” \mathcal A(\Gamma) captures classical operators on modular forms. In this paper, we show that the action of \mathcal H_1 captures the monodromy and Frobenius actions on a certain module \mathbb B^*(\Gamma) that arises from the Archimedean complex of Consani [4]. The object \mathbb B^*(\Gamma) replaces the modular Hecke algebra \mathcal A(\Gamma) in our theory. We also introduce a “restricted” version \mathbb B^*_r(\Gamma) of the module \mathbb B^*(\Gamma) on which the operators \delta_n , n\geq 1 , of the Hopf algebra \mathcal H_1 act as zero. Thereafter, we construct Rankin–Cohen brackets of all orders on \mathbb B^*_r(\Gamma) .