On vector-valued Poincar\'e series of weight 2

TL;DR

This paper constructs vector-valued Poincaré series of weight 2 for Fuchsian groups with arbitrary rank representations, providing explicit bases in genus zero cases and exploring implications for parabolic bundles.

Contribution

It introduces a method to construct vector-valued Poincaré series of weight 2 for arbitrary rank representations and explicitly describes bases in genus zero cases.

Findings

Explicit bases for vector-valued Poincaré series in genus zero.

Connections established between Poincaré series and parabolic bundles.

Framework applicable to arbitrary rank unitary representations.

Abstract

Given a pair $(\Gamma,\rho)$ of a Fuchsian group of the first kind, and a unitary representation $\rho$ of $\Gamma$ of arbitrary rank, the problem of construction of vector-valued Poincar\'e series of weight 2 is considered. Implications in the theory of parabolic bundles are discussed. When the genus of the group is zero, it is shown how an explicit basis for the space of these functions can be constructed.