Fourier expansions of vector-valued automorphic functions with non-unitary twists

TL;DR

This paper derives Fourier expansions for vector-valued automorphic functions with non-unitary twists, detailing their coefficients and growth, extending classical automorphic form theory to more general twist scenarios.

Contribution

It introduces a comprehensive Fourier expansion framework for vector-valued automorphic functions with arbitrary endomorphism twists, including non-invertible and non-unitary cases.

Findings

Explicit formulas for Fourier coefficients depending on eigenvalues and Jordan blocks.

Analysis of the growth properties of Fourier coefficients.

Application to vector-valued twisted automorphic forms of Fuchsian groups.

Abstract

We provide Fourier expansions of vector-valued eigenfunctions of the hyperbolic Laplacian that are twist-periodic in a horocycle direction. The twist may be given by any endomorphism of a finite-dimensional vector space; no assumptions on invertibility or unitarity are made. Examples of such eigenfunctions include vector-valued twisted automorphic forms of Fuchsian groups. We further provide a detailed description of the Fourier coefficients and explicitly identify each of their constituents, which intimately depend on the eigenvalues of the twisting endomorphism and the size of its Jordan blocks. In addition, we determine the growth properties of the Fourier coefficients.