Gauss Sum Combinatorics and Metaplectic Eisenstein Series

1.1. Steve Gelbart, Whittaker models and the metaplectic group. The metaplectic double cover of Sp(2r) and the Weil representation were introduced by Weil [37] in order to formulate results of Siegel on theta functions in the adelic setting. This was followed by two initially independent developments. First, Shimura [32], [33] gave two extremely important constructions involving modular forms of half-integral weight. Both Shimura integrals involved Rankin-Selberg convolutions with a theta function which, in the modern view, lives on the metaplectic double cover of SL2 (or GL2). Second, the mainstream modern context of automorphic representations of adele groups emerged with Jacquet and Langlands [23] and Godement and Jacquet [22]. (Gelbart’s book [11] was important in making this modern point of view accessible to a generation of workers in the field.) It became clear that the metaplectic group and Shimura’s constructions needed to be reworked in the modern language. Gelbart [12] was perhaps the first to talk about automorphic forms of half-integral weight, particularly theta functions, in completely modern terms. The Shimura constructions were carried out on the adele group by Gelbart and Piatetski-Shapiro [17], [18] for the Shimura correspondence, and by Gelbart and Jacquet [14], [15] for the symmetric square. Shimura’s important constructions were thus extended, and as a particular important point the lifting from GL2 to GL3 was established. Jacquet and Langlands [23] emphasized the uniqueness of Whittaker models for representations of GL2. Shalika [31] and Piatetski-Shapiro [29] showed that uniqueness holds over nonarchimedean local fields. (See also Gelfand and Kazhdan [21] and Bernstein and Zelevinsky [1].) For metaplectic covers of these groups, the Whittaker models may or may not be unique. Gelbart and Piatetski-Shapiro considered the representations of the double cover of SL2 that have unique Whittaker models and found them to be associated with theta functions. On the other hand Gelbart, Howe and Piatetski-Shapiro [13] showed that representations of the double cover of GL2 have Whittaker models (in a slightly modified sense) that are unique. The failure of uniqueness of Whittaker models might seem a defect, but Waldspurger [36] showed that precisely this