In 1952, Gelfand and Fomin noticed that classical modular forms were related to representations of SL2(R). As a result of this realization, Gelfand later defined GLr automorphic forms via representation theory. A metaplectic form is just an automorphic form defined on a cover of GLr, called a metaplectic group. In this talk, we will carefully construct the metaplectic covers of GL2(F) where F i...

We establish a link between certain Whittaker coefficients of the generalized metaplectic theta functions, first studied by Kazhdan and Patterson [15], and the coefficients of the stable Weyl group multiple Dirichlet series defined in [3]. The generalized theta functions are the residues of Eisenstein series on a metaplectic n-fold cover of the general linear group. For n sufficiently large, we...

has cardinality r ≥ 1. Let G be the F-rational points of a simple Chevalley group defined over F. In his thesis, Matsumoto [5] gave a beautiful construction for the metaplectic cover G̃ of G, a central extension of G by μr(F) whose existence is intimately connected with the deep properties of the r-th order Hilbert symbol (·, ·)F : F × × F → μr(F). Metaplectic groups figure prominently in the st...

The principal series representations of the n-fold metaplectic covers of the general linear group GLr(F) were described in the foundational paper “Metaplectic Forms,” by Kazhdan and Patterson (1984). In this paper, we study the local coefficient matrices for a certain class of principal series representations over GL2(F), where F is a nonarchimedean local field. The local coefficient matrices c...

where the image of A is in the center of G̃. If G and G̃ are topological groups, and if A is a discrete subgroup of the center of G̃, then we may think of G̃ as a cover of G, in the topological sense. So we will sometimes use the term covering group. Very often for us, A will be the group μn(F ) of n-th roots of unity, in a given field F . We will use this notation only if |μn(F )| = n. By a Metapl...

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 o...

We classify the irreducible, admissible, smooth, genuine mod p representations of the metaplectic double cover of SL2(F ), where F is a p-adic field and p 6= 2. We show, using a generalized Satake transform, that each such representation is isomorphic to a certain explicit quotient of a compact induction from a maximal compact subgroup by an action of a spherical Hecke operator, and we define a...

Let n be an integer and let F be a nonarchimedean local field whose characteristic is not a prime dividing n. Let μk be the group of k-th roots of unity in the algebraic closure of F ; we assume that μ2n ⊂ F . Let G be a split, simply-connected semisimple algebraic group over F . We assume that G is actually defined over the ring o of integers in F in such a way that K = G(o) is a special maxim...

We provide an explicit formula for the period integral of the unramified Eisenstein series on GL3(AQ) over the orthogonal subgroup associated with the identity matrix. The formula expresses the period integral as a finite sum of products of double Dirichlet series that are Fourier coefficients of Eisenstein series on the metaplectic double cover of GL3.

If F is a local field containing the group μn of n-th roots of unity, and if G is a split semisimple simply connected algebraic group, then Matsumoto [27] defined an n-fold covering group of G(F ), that is, a central extension of G(F ) by μn. Similarly if F is a global field with adele ring AF containing μn there is a cover G̃(AF ) of G(AF ) that splits over G(F ). The construction is built on i...