Constructing vector-valued automorphic forms on unitary groups

TL;DR

This paper presents a new method to generate vector-valued automorphic forms on unitary groups from scalar-valued forms, using differential operators, and provides an explicit example demonstrating the approach.

Contribution

It introduces a novel technique inspired by Siegel modular forms to construct vector-valued automorphic forms on unitary groups, overcoming unique challenges.

Findings

Constructed an explicit example of vector-valued automorphic form.

Developed a differential operator-based method for form construction.

Extended ideas from Siegel modular forms to unitary groups.

Abstract

We introduce a method for producing vector-valued automorphic forms on unitary groups from scalar-valued ones. As an application, we construct an explicit example. Our strategy employs certain differential operators. It is inspired by work of Cl\'ery and van der Geer in the setting of Siegel modular forms, but it also requires overcoming challenges that do not arise in the Siegel setting.