Differential operators on Hermitian modular forms on $\mathrm{U}(n, n)$

TL;DR

This paper develops explicit differential operators for Hermitian modular forms on U(n,n), extending existing methods from Siegel modular forms, and applies them to derive a pullback formula for Eisenstein series.

Contribution

It introduces new explicit differential operators for Hermitian modular forms and constructs bases for related pluriharmonic polynomials, enabling concrete applications.

Findings

Constructed explicit differential operators for Hermitian modular forms.

Developed bases for two-variable spherical pluriharmonic polynomials.

Derived an exact pullback formula for Hermitian Eisenstein series.

Abstract

We construct explicit differential operators on hermitian modular forms, extending methods developed for Siegel modular forms. These differential operators are closely related to the two-variable spherical pluriharmonic polynomials. We construct explicit bases for the space of such polynomials and use them to build concrete operators. As an application, we derive an exact pullback formula for hermitian Eisenstein series.