Flipping operators and locally harmonic Maass forms

TL;DR

This paper explores the flipping operator in the context of harmonic Maass forms, revealing its effect on Poincaré series and its relation to locally harmonic Maass forms and classical lifts.

Contribution

It demonstrates that the flipping operator acts similarly on locally harmonic Maass forms derived from hyperbolic Poincaré series, extending known properties from classical harmonic Maass forms.

Findings

Flipping operator negates the index of Poincaré series.

The operator exchanges parts of harmonic Maass forms.

Similar properties hold for locally harmonic Maass forms.

Abstract

In the theory of integral weight harmonic Maass forms of manageable growth, two key differential operators, the Bol operator and the shadow operator, play a fundamental role. Harmonic Maass forms of manageable growth canonically split into two parts, and each operator controls one of these parts. A third operator, called the flipping operator, exchanges the role of these two parts. Maass--Poincar\'e series (of parabolic type) form a convenient basis of negative weight harmonic Maass forms of manageable growth, and flipping has the effect of negating an index. Recently, there has been much interest in locally harmonic Maass forms defined by the first author, Kane, and Kohnen. These are lifts of Poincar\'e series of hyperbolic type, and are intimately related to the Shimura and Shintani lifts. In this note, we prove that a similar property holds for the flipping operator applied to these Poincar\'e series.