Topological computation of some Stokes phenomena on the affine line

Let \(\mathcal M\) be a holonomic algebraic \(\mathcal D\)-module on the affine line, regular everywhere including at infinity. Malgrange gave a complete description of the Fourier-Laplace transform \(\widehat{\mathcal M}\), including its Stokes multipliers at infinity, in terms of the quiver of \(\mathcal M\). Let \(F\) be the perverse sheaf of holomorphic solutions to \(\mathcal M\). By the irregular Riemann-Hilbert correspondence, \(\widehat{\mathcal M}\) is determined by the enhanced Fourier-Sato transform \(F^\curlywedge\) of \(F\). Our aim here is to recover Malgrange's result in a purely topological way, by computing \(F^\curlywedge\) using Borel-Moore cycles. In this paper, we also consider some irregular \(\mathcal M\)'s, like in the case of the Airy equation, where our cycles are related to steepest descent paths.