PREFACE IX spinors". In a sense, this chapter is the most incomplete in the book. It is clear that the quantization scheme proposed here is not broad enough to include all interesting cases. A preliminary computation, outlined at the end of the chapter, indicates that the requirement that two polarizations be unitarily related—an essential ingredient in the scheme—is only slightly less restrictive than they be "Heisenberg related", i.e. that the pairing be geometrically equivalent to the classical Fourier transform. Thus more sophisticated procedures must be deve- loped, presumably along the lines of closer and closer approximations to the Feynmann path integral method. Even within the current framework many basic questions remain unanswered: What are the appropriate conditions guaranteeing the convergence of the integrals involved in the pairing? What is the correct formulation of the pairing in the case of complex polarizations, and its relation with such objects as the Bergmann kernel? Is there a discrete analogue of the notion of polarization, with an appropriate pairing so as to tie in with the theory of theta functions? Must one look at generalized sections, associated polariza- tions, or higher cohomologies in order to construct the quantized group represen- tations? To what extent are the recent examples, of Vergne and Rothschild-Wolf, where the group representation is not independent of the polarization, a consequence of geometrical pathology of the polarization in question? What is the relation between polarizations and pairings as introduced here geometrically to formally similar objects arising in the /?-adic theory, especially in Weil's fundamental papers? In short, it is quite clear that the subject matter of this chapter is only in its preliminary stages of development, and it is to be hoped that the present chapter will be completely out of date within a few years. Chapter VI. Geometric Aspects of Distributions This is perhaps the central chapter of the book from the mathematical point of view. In it we develop the theory of generalized functions (or sections of a vector bundle) in terms of their behavior under smooth maps. A density on a manifold X is an object which locally looks like a function, but transforms, under change of coordinates, in such a way that integration makes sense. If fx is a density and v is a smooth function of compact support on a manifold X9 we define n(v) by fx^ii, so that densities dejfine linear functionals on C™(X). By a generalized density we then mean any continuous linear functional on C™(X). Let X and Y be manifolds and /: X - Y a proper mapping. If /i is a generalized density on X the "push-forward" / is defined by the formula f^(v) = ii(f*v), VB C?(Y). It turns out that in certain instances one can define the "pull-back" /*/i of a generalized function on Y. (For example this is always the case if / is a fiber mapping.) The purpose of this chapter is to develop systematically the theory of
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