vin PREFACE Chapter IV. Symplectic Geometry We have already made sporadic use of symplectic methods. Our intention here is to provide a much more systematic treatment than in the earlier chapters. We begin by giving Weinstein's proof of Darboux's theorem (and the Kostant- Weinstein generalization which says that near a Lagrangian submanifold A a given symplectic manifold looks locally like T* A.) In §2 we discuss linear symplectic geometry, the Langrangian Grassmannian and its universal cover- ing space, and define the Maslov index following the ideas of Leray and Souriau. In §3 we discuss Hormander's cross ratio construction of the Maslov class. This material is based on §3.3 of Hormander's Acta paper on Fourier integral operators, with some simplifications suggested to us by Kostant. In §§4 and 5 we develop the main facts we will need in Chapters VI and VII about Lagrangian manifolds. In preparation for Chapters VI and VII we have put considerable stress on their "functorial" properties. In §6 we discuss periodic orbits of Hamiltonian systems and make an application to the Kepler problem (following Moser). In §7 we discuss symplectic manifolds on which a group G acts transitively. Finally, in §8 we discuss relations between symplectic geometry and the calculus of variations. Chapter V. Geometric Quantization In this chapter we discuss the current state of knowledge concerning the geometry of quantization as introduced (independently) by Kostant and Souriau. The main function of this chapter, as far as the rest of the book is concerned, is the introduction of metalinear structures, half-forms, and metaplectic structures. These notions will be of crucial importance for us in Chapters VI and VII where we develop the symbol calculus in the metalinear category. The first two sections deal with the concept, due to Kostant and Souriau, of prequantization. This has the effect of selecting, among symplectic manifolds, those satisfying certain integrality conditions. (In the case of relativistic particles this has the effect of constraining the "spin" to take on half integral values. In the case of the hydrogen atom this constrains the negative energy levels to their appropriate discrete values.) In §3 we introduce the notion of polarization, a concept introduced independently by Kostant and Souriau in the real case, while Auslander and Kostant introduced the notion of a complex polarization. The rest of the chapter represents joint research by Blattner, Kostant, and Sternberg, dealing with half-forms and a pairing between sections associated to different polarizations, together with some physical examples due to Simms. The concept of a metaplectic structure is introduced together with the pairing, mentioned above which can be viewed as a generalization of the Fourier transform. The metaplectic representation of Segal-Shale-Weil is constructed by geometric methods using the pairing, and is applied to the construction of "symplectic

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