Equivariant mirror symmetry for stringy invariants of dual reflexive hypersurfaces

Suppose that a finite group $G$ acts linearly on a lattice $M$ of rank $d$ via a homomorphism

$$\rho:G\rightarrow GL(M).$$

Let $P$ and $P^*$ be polar, $G$-invariant, reflexive polytopes, and let $X$ and $X^*$ be corresponding $G$-invariant, non-degenerate hypersurfaces. The equivariant mirror symmetry conjecture says that the equivariant stringy invariants $E_{\operatorname{st},G}(X;u,v)$ and $E_{\operatorname{st},G}(X^*;u,v)$ are rational functions satisfying

$$E_{\operatorname{st},G}(X;u,v)=(-u)^{d-1}\det(\rho)\cdot E_{\operatorname{st},G}(X^*;u^{-1},v).$$

This conjecturally extends Batyrev and Borisov's mirror symmetry formula to varieties with a group action. If $G$-equivariant, crepant resolutions exist, it implies the corresponding equality of Hodge representations in the representation ring $R(G)$; the source does not give a resolution of the equivariant conjecture in general.