Quaternionic Automorphic Forms on Sp(1,1)
We define a quaternionic automorphic form on a quaternionic symmetric domain as a smooth function, with values in an irreducible representation of SU(2), satisfying suitable automorphy and growth rate conditions. A regularity condition defined by a linear differential equation is then the analogue of holomorphicity. We construct examples on the quaternionic ball, corresponding to the group Sp(1,1). We prove that the quotient of the quaternions by a lattice can be embedded into quaternionic projective space using the analogue of the Weierstrass P-function and its derivatives.