A duality approach to the symmetry of Bernstein-Sato polynomials of free divisors

In this paper we prove that the Bernstein-Sato polynomial of any free divisor for which the $D[s]$-module $D[s]h^s$ admits a Spencer logarithmic resolution satisfies the symmetry property $b(-s-2)=\pm b(s)$. This applies in particular to locally quasi-homogeneous free divisors (for instance, to free hyperplane arrangements), or more generally, to free divisors of linear Jacobian type. We also prove that the Bernstein-Sato polynomial of an integrable logarithmic connection $E$ and of its dual $E^{\ast }$ with respect to a free divisor of linear Jacobian type are related by the equality $b_{E}(s)=\pm b_{E^*}(-s-2)$\mathrm{.}Ourresultsarebasedonthebehaviourofthemodules$D[s] h^s$and$D[s] E[s]h^s $ under duality.