Let T be a split torus over local or global function field. The theory of BrylinskiDeligne gives rise to the metaplectic central extensions of T by a finite cyclic group. The representation theory of these metaplectic tori has been developed to some extent in the works of M. Weissman, G. Savin, W. T. Gan, P. McNamara and others. In this paper we propose a geometrization of this theory in the fr...

The metaplectic representation describes a class of automorphisms of the Heisenberg group H = H(G), defined for a locally compact abelian groupG. ForG = R ,H is the usual Heisenberg group. For the case when G is the finite cyclic group Zn, only partial constructions are known. Here we present new results for this case and we obtain an explicit construction of the metaplectic operators on C. We ...

Let λ : G̃ → G be the non-trivial double covering of the symplectic group G = Sp(V, ω) of the symplectic vector space (V, ω) by the metaplectic group G̃ = Mp(V, ω). In this case, λ is also a representation of G̃ on the vector space V and thus, it gives rise to the representation of G̃ on the space of exterior forms ∧• V by taking wedge products. Let S be the minimal globalization of the Harish-Chan...

We classify all genuine unitary representations of the metaplectic group whose infinitesimal character is real and at least as regular as that of the oscillator representation. In a previous paper we exhibited a certain family of representations satisfying these conditions, obtained by cohomological induction from the tensor product of a one-dimensional representation and an oscillator represen...

Among all classes of pseudo-differential operators only the Weyl operators enjoy the property of symplectic covariance with respect to conjugation by elements of the metaplectic group. In this paper we show that there is, however, a weaker form of symplectic covariance for Shubin’s τ -dependent operators, in which the intertwiners no longer are metaplectic, but still are invertible non-unitary ...

In their paper Metaplectic Forms, D. A. Kazhdan and S. J. Patterson developed a generalization of automorphic forms that are defined on metaplectic groups. These groups are non-trivial covering groups of usual algebraic groups, and the forms defined on them are representations that respect the covering. As in the case for automorphic forms, these representations fall into a principle series, in...

We develop the theory of “Weyl group multiple Dirichlet series” for root systems of type C. For a root system of rank r and a positive integer n, these are Dirichlet series in r complex variables with analytic continuation and functional equations isomorphic to the associated Weyl group. They conjecturally arise as Whittaker coefficients of Eisenstein series on a metaplectic group with cover de...

This paper presents a new construction of the m-fold metaplectic cover of GLn over an algebraic number field k, where k contains a primitive m-th root of unity. A 2-cocycle on GLn(A) representing this extension is given and the splitting of the cocycle on GLn(k) is found explicitly. The cocycle is smooth at almost all places of k. As a consequence, a formula for the Kubota symbol on SLn is obta...

In this paper, we construct unipotent representations for the real orthogonal groups and the metaplectic groups in the sense of Vogan.