On the monodromies at infinity of Fourier transforms of holonomic D-modules

TL;DR

This paper investigates the monodromies at infinity of Fourier transforms of holonomic D-modules, providing formulas for eigenvalues and revealing a reciprocity law in one-dimensional cases, advancing understanding in irregular Riemann-Hilbert theory.

Contribution

It introduces explicit formulas for monodromy eigenvalues and establishes a reciprocity law linking monodromies of D-modules and their Fourier transforms in dimension one.

Findings

Formulas for monodromy eigenvalues derived using monodromy zeta functions.

Reciprocity law between monodromies at infinity of D-modules and their Fourier transforms in dimension one.

Enhanced solution complexes of Fourier transforms analyzed in the context of irregular Riemann-Hilbert correspondence.

Abstract

Based on the recent progress in the irregular Riemann-Hilbert correspondence, we study the monodromies at infinity of the holomorphic solutions of Fourier transforms of holonomic D-modules in some situations. Formulas for their eigenvalues are obtained by applying the theory of monodromy zeta functions to our previous results on the enhanced solution complexes of the Fourier transforms. In particular, in dimension one we thus find a reciprocity law between the monodromies at infinity of holonomic D-modules and their Fourier transforms.