Abstract
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 SO ( 8 ) 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 Sp ( 4 ) , and in terms of periods. We also give a conjecture for the Dirichlet series of the standard L -function of quaternionic modular eigenforms on SO ( 8 ) and verify our conjecture on the quaternionic Maass Spezialschar.