Title: Distinguished representations of the metaplectic cover of GL(n)
Description: One of the fundamental differences between automorphic representations of classical groups like GL(n) and their metaplectic covers is that in the latter case the space of Whittaker functionals usually has a dimension bigger than one. Gelbart and Piatetski-Shapiro called the metaplectic representations, which possess a unique Whittaker model, distinguished and classified them for the double cover of the group GL(2). Later Patterson and Piatetski-Shapiro used a converse theorem to list the distinguished representations for the degree three cover of GL(3). In their milestone paper on general metaplectic covers of GL(n) Kazhdan and Patterson construct examples of non-cuspidal distinguished representations, which come as residues of metaplectic Eisenstein series. These are generalizations of the classical Jacobi theta functions. Despite some impressive local results to date, cuspidal distinguished representations are not classified or even constructed outside rank 1 and 2. In her thesis Wang makes some progress toward the classification in rank 3.

In this dissertation we construct the distinguished representations for the degree four metaplectic cover of GL(4), applying a classical converse theorem like Patterson and Piatetski-Shapiro in the case of rank 2. We obtain the necessary local properties of the Rankin-Selberg convolutions at the ramified places and finish the proof of the construction of cuspidal distinguished representations in rank 3.