Lifting to GL(2) over a division quaternion algebra, and an explicit construction of cap representations

The aim of this paper is to carry out an explicit construction of CAP representations of GL(2) over a division quaternion algebra with discriminant two. We first construct cusp forms on such group explicitly by lifting from Maass cusp forms for the congruence subgroup Γ0(2). We show that this lifting is non-zero and Hecke-equivariant. This allows us to determine each local component of a cuspidal representation generated by such a lifting. We then know that our cuspidal representations provide examples of CAP representations, and in fact, counterexamples of the Generalized Ramanujan conjecture.