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 ${\delta }_n$, $n\ge 1$, 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 ${\mathbb{B}}^{\ast }(\Gamma )$ that arises from the Archimedean complex of Consani [4]. The object ${\mathbb{B}}^{\ast }(\Gamma )$ replaces the modular Hecke algebra $\mathcal{A}(\Gamma )$ in our theory. We also introduce a “restricted” version ${\mathbb{B}}_r^{\ast }(\Gamma )$ of the module ${\mathbb{B}}^{\ast }(\Gamma )$ on which the operators ${\delta }_n$, $n\ge 1$, of the Hopf algebra ${\mathcal{H}}_1$ act as zero. Thereafter, we construct Rankin–Cohen brackets of all orders on ${\mathbb{B}}_r^{\ast }(\Gamma )$.