D-modules from the b-function and Hamiltonian flow

Given a complete intersection, one can define Hamiltonian flow and their coinvariants,
generalizing Poisson (and Poisson-de Rham) homology, which in the case of symplectic
resolutions conjecturally recovers the cohomology of the resolution. On the other hand,
in the hypersurface case, the Bernstein-Sato polynomial (closely related to analytic
continuation of the gamma function) gives deep information about the singularities.
Both constructions involve D-modules closely related to nearby cycles. I will explain
how, in the case of quasihomogeneous hypersurfaces with isolated singularities, the
D-modules are isomorphic. As a consequence, we answer a folklore question (to which
M. Saito recently found a counterexample in the non-quasihomogeneous case): if $c$ is a
root of the b-function, is the D-module D $f^c/Df^{c+1}$ nonzero? We moreover compute
this D-module: for $c = -1$ its length is one more than the genus (conjecturally in the
non-quasihomogenous case), matching an analogous D-module in characteristic p.
This includes joint work with Bitoun and with Etingof.