The quaternionic Maass Spezialschar on split $\mathrm{SO}(8)$

Abstract

The classical Maass Spezialschar is a Hecke-stable subspace of the level one holomorphic Siegel modular forms of genus two cut out by certain linear relations between their Fourier coefficients. We define an analogous quaternionic Maass Spezialschar, which consists of level one quaternionic modular forms on split $\mathrm{SO}(8)$ whose Fourier coefficients satisfy certain linear relations. We characterize this space in terms of the theta lift from holomorphic Siegel modular forms on $\mathrm{Sp}(4)$ and in terms of periods. We also give a conjecture for the Dirichlet series of the standard $L$-function of quaternionic modular eigenforms on $\mathrm{SO}(8)$ and verify our conjecture on the quaternionic Maass Spezialschar.