We study Whittaker functions on nonlinear coverings of simple algebraic groups over a non-archimedean local field. We produce a recipe for expressing such a Whittaker function as a weighted sum over a crystal graph, and show that in type A, these expressions agree with known formulae for the p-part of Multiple Dirichlet Series.

Let F be a global field with ring o of integers, and let μn denote the group of n-th roots of unity in F . Let Γ(f) be the principal congruence subgroup of SL3(o) consisting of elements that are congruent to the identity modulo f. The Kubota symbol κ : Γ(f) −→ μn is a character constructed by Bass, Milnor and Serre [1]. We will give a direct construction, obtaining on the way new formulas for t...

The set of paraxial optical systems is the manifold of the group of symplectic matrices. The structure of this group is nontrivial: It is not simply connected and is not of an exponential type. Our analysis clarifies the origin of the metaplectic phase and the inherent limitations for optical map fractionalization. We describe, for the first time to our knowledge, an image girator and a cross g...

We construct quantum evolution operators on the space of states, that realize the metaplectic representation of the modular group SL(2, Z 2 n). This representation acts in a natural way on the coordinates of the non-commutative 2-torus, T 2 2 n and thus is relevant for noncommutative field theories as well as theories of quantum space-time. The larger class of operators, thus defined, may be us...

We discuss two topics that are usually considered to be exclusively ”quantum”: the Schrödinger equation, and the uncertainty principle. We show (or rather recall) that the Schrödinger equation can be derived from Hamilton’s equations using the metaplectic representation. We also show that the uncertainty principle, stated in the form of the Robertson-Schrödinger-Heisenberg inequalities can be f...

A theory of frames that extend Gabor analysis by including chirping is discussed. The chirping parameter in thesètime-frequency localization frames' depends on time and/or frequency shift parameters that can be adapted to analyze and detect chirps in noisy signals. Radar/sonar applications are outlined. The frame theory is motivated by a generalized notion of square-integrable group representat...

We construct unitary evolution operators on a phase space with power of two discretization. These operators realize the metaplectic representation of the modular group SL(2,Z2n). It acts in a natural way on the coordinates of the non-commutative 2-torus, T2n and thus is relevant for noncommutative field theories as well as theories of quantum space-time. The class of operators may also be usefu...

Abstract In analogy with the Barbasch–Vogan duality for real reductive linear groups, we introduce a notion useful representation theory of metaplectic groups. This is map on set nilpotent orbits in complex symplectic Lie algebra, whose range consists so-called special orbits. We relate this primitive ideals and extend unipotent representations to also interpret terms double cells Weyl group re...