Rankin-Cohen Bracket for Vector-Valued Modular Forms

TL;DR

This paper investigates Rankin-Cohen brackets for vector-valued modular forms, linking them to Petersson's inner products and extending the analysis to Jacobi and skew-holomorphic Jacobi forms, with explicit descriptions of adjoint maps.

Contribution

It provides an explicit description of the adjoint map for Rankin-Cohen brackets and extends the framework to Jacobi and skew-holomorphic Jacobi forms.

Findings

Derived explicit adjoint maps for the bracket operators.

Established connections between vector-valued modular forms and Jacobi forms.

Extended the theory to include skew-holomorphic Jacobi forms.

Abstract

In this paper, we explore the relationship between Rankin-Cohen brackets for vector-valued modular forms and Petersson's inner products, deriving an explicit description of the adjoint map for the bracket operator. The study extends to the cases of Jacobi forms and skew-holomorphic Jacobi forms, establishing connections between their respective Rankin-Cohen brackets and those defined for vector-valued modular forms through an isomorphism. Adjoint maps for these extended bracket operators are also examined.