Maass space for lifting to GL(2) over a division quaternion algebra

Abstract

Muto, Narita and Pitale construct counterexamples to the Generalized Ramanujan Conjecture for GL(2,B) over the division quaternion algebra B with discriminant two via a lift from SL(2). In this paper, we try to exactly characterize the image of this lift. The previous methods of Maass, Kohnen or Kojima do not apply here, hence we approach this problem via a combination of classical and representation theory techniques to identify the image. Crucially, we use the Jacquet Langlands correspondence described by Badulescu and Renard to characterize the representations.