Theta lifting from elliptic cusp forms to automorphic forms on Sp(1, q)

Tsuneo Arakawa formulated a theta lifting from elliptic cusp forms to automorphic forms on Sp(1,q) in his unpublished note, which was inspired by "Kudla lifting", i.e. a theta lifting from elliptic modular forms to holomorphic automorphic forms on SU(1,q). We prove that the images of Arakawa's theta lifting belong to the space of bounded automorphic forms generating quaternionic discrete series, which are non-holomorphic forms. In the appendix we provide the construction of Eisenstein series and Poincaré series generating such discrete series.