Enhanced equivariant Saito duality

TL;DR

This paper extends the concept of equivariant Saito duality by introducing an enhanced Burnside ring and demonstrates that the enhanced equivariant Euler characteristics of Milnor fibers for dual polynomials coincide up to sign, linking to orbifold zeta functions.

Contribution

It defines an enhanced Burnside ring and an enhanced equivariant Saito duality, providing a new framework for analyzing equivariant Euler characteristics and duality in complex analytic G-manifolds.

Findings

Enhanced equivariant Euler characteristics of Milnor fibers coincide up to sign for dual polynomials.

The new framework links orbifold zeta functions of dual pairs.

The enhanced Burnside ring captures additional symmetry information.

Abstract

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 $\widehat{B}(G)$ 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 $\widehat{B}(G)$. It is proved that the (reduced) enhanced equivariant Euler characteristics of the Milnor fibres of Berglund-H\"ubsch dual invertible polynomials coincide up to sign and show that this implies the result about orbifold zeta functions of Berglund-H\"ubsch-Henningson dual pairs obtained earlier.