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We introduce a new definition of a generalized logarithmic module ofmultiarrangements by uniting those of the logarithmic derivation and thedifferential modules. This module is realized as a logarithmic derivationmodule of an arrangement of hyperplanes with a multiplicity consisting of bothpositive and negative integers. We consider several properties of this moduleincluding Saito's criterion and reflexivity. As applications, we prove a shiftisomorphism and duality of some Coxeter multiarrangements by using theprimitive derivation.

A generalized logarithmic module and duality of Coxeter multiarrangements