Higher pullbacks of modular forms on orthogonal groups

TL;DR

This paper develops higher pullback operators for modular forms on orthogonal groups, generalizing previous methods and connecting to theta lifts and differential operators, with applications to special cycles and Hilbert-Siegel forms.

Contribution

It introduces higher pullback operators for orthogonal modular forms, extending the quasi-pullback concept and linking to Cohen and Ibukiyama's differential operators.

Findings

Higher pullbacks preserve theta lift subspaces.

Higher pullbacks of lifts relate to partial coefficients of Jacobi forms.

For certain lattices, higher pullbacks match known differential operators.

Abstract

We apply differential operators to modular forms on orthogonal groups $\mathrm{O}(2, \ell)$ to construct infinite families of modular forms on special cycles. These operators generalize the quasi-pullback. The subspaces of theta lifts are preserved; in particular, the higher pullbacks of the lift of a (lattice-index) Jacobi form $\phi$ are theta lifts of partial development coefficients of $\phi$. For certain lattices of signature (2, 2) and (2, 3), for which there are interpretations as Hilbert-Siegel modular forms, we observe that the higher pullbacks coincide with differential operators introduced by Cohen and Ibukiyama.