Weyl invariant Jacobi forms: a new approach

TL;DR

This paper introduces a new criterion for the algebraic structure of Weyl invariant weak Jacobi forms, providing automorphic proofs and new insights into their polynomial nature for various root systems.

Contribution

It establishes a criterion for when the algebra of weak Jacobi forms is free and applies automorphic methods to prove polynomiality for all root systems except E8.

Findings

Proves the algebra of Weyl invariant weak Jacobi forms is polynomial for most root systems.

Provides an automorphic proof of Wirthmüller's theorem.

Derives a new structure result for E8 Jacobi forms despite non-freeness.

Abstract

The weak Jacobi forms of integral weight and integral index associated to an even positive definite lattice form a bigraded algebra. In this paper we prove a criterion for this type of algebra being free. As an application, we give an automorphic proof of K. Wirthm\"{u}ller's theorem which asserts that the bigraded algebra of weak Jacobi forms invariant under the Weyl group is a polynomial algebra for any irreducible root system not of type $E_8$. This approach is also applicable to $E_8$. Even if the algebra of $E_8$ Jacobi forms is known to be non-free, we still derive a new structure result.