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Transformations canoniques, dualité projective, théorie de Lefschetz, transformations de Fourier et sommes trigonométriques. (Canonical transformations, projective duality, Lefschetz theory, Fourier transformations and trigonometric sums). (French) Zbl 0624.32009
Géométrie et analyse microlocales, Astérisque 140-141, 3-134 (1986).
[For the entire collection see Zbl 0588.00023.]
In the first part of the paper a topological version of canonical transformations is presented. By studying the special transformation (sheaves on \(P^ n)\to (sheaves\) on \((P^ n)^ v)\) one obtains in particular a new proof (independent of the Picard-Lefschetz formulae) of the irreducibility of the monodromy action on the middle cohomology of a hyperplane section of a smooth variety. One also studies a topological analogue of the Radon transform, and one obtains for instance analogues of results due to I. M. Gel’fand, S. G. Gindikin and M. I. Graev [cf. Funct. Anal. Appl. 13, 288-290 (1980); translation from Funkts. Anal. Prilozh. 13, No.4, 64-66 (1979; Zbl 0423.58001)].
The second part of the paper is devoted to the theory of Fourier transform as developped by Malgrange over \({\mathbb{C}}\) and by Deligne in characteristic p. One computes the generic rank of the Fourier transform of a \({\mathbb{C}}[\underline x,\partial /\partial \underline x]\)-module with regular singularities. The work of Katz-Laumon on trigonometric sums is then presented with an improvement in their uniform estimation formula where the generic rank of a certain Fourier transform is shown to play a role. Finally the irreducible representations of Weyl groups are considered from the viewpoint of Fourier transforms (as was implicitely done by T. A. Springer in Invent. Math. 36, 173-207 (1976; Zbl 0374.20054)).
Reviewer: A.Buium

32C37 Duality theorems for analytic spaces
42B10 Fourier and Fourier-Stieltjes transforms and other transforms of Fourier type
32C38 Sheaves of differential operators and their modules, \(D\)-modules