We study the action of metaplectic operators on Wiener amalgam spaces, giving upper bounds for their norms. As an application, we obtain new fixed-time estimates in these spaces for Schrödinger equations with general quadratic Hamiltonians and Strichartz estimates for the Schrödinger equation with potentials V (x) = ±|x|.

We derive some consequences of very recent results of Cordero and Nicola on the metaplectic representation, the Wiener amalgam spaces, (whose definition is due to Feichtinger), and their applications to the regularity of the solutions of Schrödinger equation with quadratic Weyl symbol. We do not however discuss the validity of Cordero and Nicola’s claims.

We establish a Kazhdan-Lusztig algorithm to compute characters of irreducible genuine representations of the (nonlinear) metaplectic group with half-integral infinitesimal character. We then prove a character multiplicity duality theorem for representations of Mp(2n,R) at fixed half-integral infinitesimal character. This allows us to extend some of Langlands’ ideas to Mp(2n,R).

We study the twisted Weyl symbol of metaplectic operators; this requires the definition of an index for symplectic paths related to the Conley–Zehnder index. We thereafter define a metaplectically covariant algebra of pseudo-differential operators acting on functions on symplectic space.

We generalize the methods of Moy-Prasad, in order to define and study the genuine depth zero representations of some nonlinear covers of reductive groups over p-adic local fields. In particular, we construct all depth zero supercuspidal representations of the metaplectic group Mp2n over a p-adic field of odd residue characteristic.