New mirror pairs of Calabi-Yau orbifolds

Abstract: We prove a representation-theoretic version of Borisov-Batyrev mirror symmetry, and use it to construct infinitely many new pairs of orbifolds with mirror Hodge diamonds, with respect to the usual Hodge structure on singular complex cohomology. We conjecture that the corresponding orbifold Hodge diamonds are also mirror. When $X$ is the Fermat quintic in $\P^4$, and $\tilde{X}^*$ is a $\Sym_5$-equivariant, toric resolution of its mirror $X^*$, we deduce that for any subgroup $\Gamma$ of the alternating group $A_5$, the $\Gamma$-Hilbert schemes $\Gamma$-$\Hilb(X)$ and $\Gamma$-$\Hilb(\tilde{X}^*)$ are smooth Calabi-Yau threefolds with (explicitly computed) mirror Hodge diamonds.