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## Enhanced equivariant Saito duality

In a previous paper, the authors defined an equivariant version of the so-called Saito duality between the monodromy zeta functions as a sort of Fourier transform between the Burnside rings of an abelian group and of its group of characters. Here, a so-called enhanced Burnside ring *B*ˆ(*G*) of a finite group *G* is defined. An element of it is represented by a finite *G*-set with a *G*-equivariant transformation and with characters of the isotropy subgroups associated to all points. One gives an enhanced version of the equivariant Saito duality. For a complex analytic *G*-manifold with a *G*-equivariant transformation of it one has an enhanced equivariant Euler characteristic with values in a completion of *B*ˆ(*G*). It is proved that the (reduced) enhanced equivariant Euler characteristics of the Milnor fibers of Berglund–Hübsch dual invertible polynomials are enhanced dual to each other up to sign. As a byproduct, this implies the result about the orbifold zeta functions of Berglund–Hübsch–Henningson dual pairs obtained earlier.