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]D[s]-module D[s]hsD[s]hs admits a Spencer logarithmic resolution satisfies the symmetry property b(−s−2)=±b(s)b(−s−2)=±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 EE and of its dual E⁎E⁎ with respect to a free divisor of linear Jacobian type are related by the equality bE(s)=±bE⁎(−s−2)bE(s)=±bE⁎(−s−2). Our results are based on the behaviour of the modules D[s]hsD[s]hs and D[s]E[s]hsD[s]E[s]hs under duality.