An Explicit Lifting Construction of CAP Forms on O(1,5)

Abstract

In this paper, we explicitly construct nontempered cusp forms on the orthogonal group O(1,5) of signature [Formula: see text]. Given a definite quaternion algebra [Formula: see text] over [Formula: see text], the orthogonal group is attached to the indefinite quadratic space of rank 6 with the anisotropic part defined by the reduced norm of [Formula: see text]. Our construction can be viewed as a generalization of the previous work by the first two authors joint with Masanori Muto to the case of any definite quaternion algebras, for which we note that the work just mentioned takes up the case where the discriminant of [Formula: see text] is two. Unlike the previous work the method of the construction is to consider the theta lifting from Maass cusp forms to O(1,5), following the formulation by Borcherds. The cuspidal representations generated by our cusp forms are studied in detail. We determine all local components of the cuspidal representations and show that our cusp forms are CAP forms.