Harmonic Maa{\ss}-Jacobi forms of degree 1 with higher rank indices

TL;DR

This paper introduces and studies real analytic weak Jacobi forms of degree 1 with higher rank indices, analyzing their properties and associated operators, and establishing equivalences between harmonicity and semi-holomorphicity in higher ranks.

Contribution

It defines higher rank real analytic weak Jacobi forms, computes the Casimir operator for their symmetry group, and shows the equivalence of H-harmonicity and semi-holomorphicity for ranks above 1.

Findings

Casimir operator of order 4 for ranks > 1

H-harmonicity equals semi-holomorphicity in higher ranks

Explicit structure of real analytic weak Jacobi forms

Abstract

We define and investigate real analytic weak Jacobi forms of degree 1 and arbitrary rank. En route we calculate the Casimir operator associated to the maximal central extension of the real Jacobi group, which for rank exceeding 1 is of order 4. In ranks exceeding 1, the notions of H-harmonicity and semi-holomorphicity are the same.