# Flipping operators and locally harmonic Maass forms

## TL;DR

This paper explores flipping operators and their effects on locally harmonic Maass forms, extending known properties to new types of mathematical objects.

## Contribution

The paper proves a new property of flipping operators applied to hyperbolic-type Poincaré series.

## Key findings

- Flipping operators exchange parts of harmonic Maass forms.
- Locally harmonic Maass forms are lifts of hyperbolic-type Poincaré series.
- Flipping negates the index of these series.

## 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–Poincaré 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 Poincaré 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 Poincaré series.