Microlocalization and Stationary Phase

Abstract

In [6], G. Laumon defined a set of functors, called by him local Fourier transformations, which allow to analyze the local structure of the l-adic Fourier transform of a constructible l-adic sheaf G on the affine line in terms of the local behaviour of G at infinity and at the points where it is not lisse. These local Fourier transforms play a major role in his cohomological interpretation of the local constants and in his proof of the product formula (see loc. cit. and [5]). In this article, we are concerned with differential systems defined over a field K of characteristic zero. We define a set of functors which allow to prove a stationary phase formula, expressing the formal germ at infinity of the D-module theoretic Fourier transform of a holonomic K[t]h.ti-module M in terms of the formal germs defined by M at its singular points and at infinity. These functors might therefore be regarded as formal analogues of Laumon's local Fourier transformations ... We will use some results from D-module theory for which we refer e.g. to [14], [15]. Our proof of the formal stationary phase formula follows the leitfaden of the one given by C. Sabbah in [15] for modules with regular singularities. I thank C. Sabbah, B. Malgrange, G. Christol and W. Messing for their useful remarks. I thank also G. Lyubeznik and S. Sperber for their invitation to the University of Minnesota, during which part of this work was done.