Quasiautomorphic forms are isomorphic to vector-valued automorphic forms

TL;DR

This paper establishes an isomorphism between quasiautomorphic forms over Hecke triangle groups and vector-valued automorphic forms, extending to quasimodular forms for the modular group.

Contribution

It introduces a bijective correspondence between quasiautomorphic forms and vector-valued automorphic forms for Hecke triangle groups, including the modular group case.

Findings

Proved functional equations for Hecke vector-forms.

Established the bijection between quasiautomorphic and vector-valued automorphic forms.

Extended results to quasimodular forms for the modular group.

Abstract

We utilize the structure of quasiautomorphic forms over a Hecke triangle group to define a mapping from a quasiautomorphic form to a vector-valued automorphic form (vvaf). This kind of vvaf we call a Hecke vector-form. First we supply a proof of the functional equations that hold for Hecke vector-forms modulo the group generators. Then, utilizing the multiplier system for these Hecke vector-forms, we prove the opposite direction and complete the bijection. Since the modular group is a special instance of the Hecke triangle groups, our results hold for quasimodular forms.