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

TL;DR

This paper characterizes the image of a lift from SL(2) to GL(2) over a division quaternion algebra, using classical and representation theory techniques, including the Jacquet Langlands correspondence.

Contribution

It provides a precise description of the Maass space for lifting to GL(2) over a division quaternion algebra, extending previous methods.

Findings

Identifies the image of the lift from SL(2) to GL(2) over B.

Utilizes Jacquet Langlands correspondence for characterization.

Overcomes limitations of previous methods by combining classical and representation theory techniques.

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.