Action de Hopf sur les opérateurs de Hecke modulaires tordus

Abstract

Let $\Gamma \subseteq S{L}_2(\mathbb{Z})$ be a principal congruence subgroup. For each $\sigma \in S{L}_2(\mathbb{Z})$, we introduce the collection ${\mathcal{A}}_{\sigma }(\Gamma )$ of modular Hecke operators twisted by $\sigma$. Then, ${\mathcal{A}}_{\sigma }(\Gamma )$ is a right $\mathcal{A}(\Gamma )$-module, where $\mathcal{A}(\Gamma )$ is the modular Hecke algebra introduced by Connes and Moscovici. Using the action of a Hopf algebra ${\mathfrak{h}}_0$ on ${\mathcal{A}}_{\sigma }(\Gamma )$, we define reduced Rankin–Cohen brackets on ${\mathcal{A}}_{\sigma }(\Gamma )$. Moreover ${\mathcal{A}}_{\sigma }(\Gamma )$ carries an action of ${\mathcal{H}}_1$, where ${\mathcal{H}}_1$ is the Hopf algebra of foliations of codimension 1. Finally, we consider operators between the levels ${\mathcal{A}}_{\sigma }(\Gamma )$, $\sigma \in S{L}_2(\mathbb{Z})$. We show that the action of these operators can be expressed in terms of a Hopf algebra ${\mathfrak{h}}_{\mathbb{Z}}$.