Abstract
We apply differential operators to modular forms on orthogonal groups
O
(
2
,
ℓ
)
{\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 ϕ are theta lifts of partial development coefficients of ϕ. For certain lattices of signature
(
2
,
2
)
{(2,2)}
and
(
2
,
3
)
{(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.
Low-dimensional representations of finite orthogonal groups
