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## Enhanced equivariant Saito duality

Journal of Algebra and its Applications. 2018. Vol. 17. No. 10. P. 1-13.

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 Bˆ(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 Bˆ(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.