A microlocal approach to the enhanced Fourier-Sato transform in dimension one

Abstract

Let $\mathcal{M}$ be a holonomic algebraic $\mathcal{D}$-module on the affine line. Its exponential factors are Puiseux germs describing the growth of holomorphic solutions to $\mathcal{M}$ at irregular points. The stationary phase formula states that the exponential factors of the Fourier transform of $\mathcal{M}$ are obtained by Legendre transform from the exponential factors of $\mathcal{M}$. We give a microlocal proof of this fact, by translating it in terms of enhanced ind-sheaves through the Riemann-Hilbert correspondence.

arXiv comment56 pages

References in corpus (12)

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• Microlocal condition for non-displaceablility
• Rigid irregular connections on P^1
• Riemann-Hilbert correspondence for holonomic D-modules
• Local Fourier transform and epsilon factors
• Local Fourier transforms and rigidity for D-modules
• Enhanced perversities
• Product formula for p-adic epsilon factors
• On the Laplace transform for tempered holomorphic functions
• Differential systems of pure Gaussian type
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Cited by in corpus (8)

• On a topological counterpart of regularization for holonomic D-modules
• D-modules of pure Gaussian type and enhanced ind-sheaves
• Perverse schobers and the Algebra of the Infrared
• On Irregularities of Fourier Transforms of Regular Holonomic D-Modules
• Enhanced nearby and vanishing cycles in dimension one and Fourier transform
• Categorification of Legendrian knots
• On Some Topological Properties of Fourier Transforms of Regular Holonomic D-Modules
• Kashiwara conjugation and the enhanced Riemann-Hilbert correspondence