We calculate the character of the Weil representation using previous results which express the Weyl symbol of metaplectic operators in terms of the symplectic Cayley transform and the Conley–Zehnder index.

Smooth irreducible representations of tori over local fields have been parameterized by Langlands, using class field theory and Galois cohomology. This paper extends this parameterization to central extensions of such tori, which arise naturally in the setting of nonlinear covers of reductive groups.

The theory of Whittaker functions is of crucial importance in the classical study of automorphic forms on adele groups. Motivated by the appearance of Whittaker functions for covers of reductive groups in the theory of multiple Dirichlet series, we provide a study of Whittaker functions on metaplectic covers of reductive groups over local fields. Thesis Supervisor: Benjamin Brubaker Title: Assi...

We study the principal series representations of central extensions of a split reductive algebraic group by a cyclic group of order n. We compute the Plancherel measure of the representation using Eisenstein series and a comparison method. In addition, we construct genuine central characters of the metaplectic torus in the simply-laced case.

A lifting map from cuspidal automorphic representations of the Jacobi group G J to cuspidal auto-morphic representations of the group PGL(2) is constructed. The lifting also has a local deenition, and the local and global versions are compatible. The local lifts can be described in terms of equivalence classes of local representations. The main idea in the construction is to exploit the close r...

We study symplectic manifolds (M, ω) equipped with a symplectic torsion-free affine (also called Fedosov) connection ∇ and admitting a metaplectic structure. Let S be the so called symplectic spinor bundle and let R be the curvature tensor field of the symplectic spinor covariant derivative ∇ associated to the Fedosov connection ∇. It is known that the space of symplectic spinor valued exterior...

We prove that a certain vector valued double Dirichlet series satisfying appropriate functional equations is a Mellin transform of a vector valued metaplectic Eisenstein series. We establish an analogous result for scalar double Dirichlet series of the type studied by Siegel.