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 $\mathbb B^\*\_r(\Gamma)$.