The quaternionic Maass Spezialschar on split SO(8)

The classical Maass Spezialschar is a Hecke‐stable subspace of the space of holomorphic Siegel modular forms of genus two and level 1 cut out by certain linear relations among Fourier coefficients. We define an analogous quaternionic Maass Spezialschar, which consists of the quaternionic modular forms of level 1 on split whose Fourier coefficients satisfy certain linear relations. We characterize this space in terms of a theta lift from the space of holomorphic Siegel modular forms on , and in terms of periods. We also give a conjecture for the Dirichlet series of the standard ‐function of quaternionic modular eigenforms on and verify our conjecture on the quaternionic Maass Spezialschar.