BCFG Drinfeld–Sokolov hierarchies and FJRW-theory

According to the ADE Witten conjecture, which is proved by Fan, Jarvis and Ruan, the total descendant potential of the FJRW invariants of an ADE singularity is a tau function of the corresponding mirror ADE Drinfeld-Sokolov hierarchy. In the present paper, we show that there is a finite group $\Gamma$ acting on a certain ADE singularity which induces an action on the corresponding FJRW-theory, and the $\Gamma$-invariant sector also satisfies the axioms of a cohomological field theory except the gluing loop axiom. On the other hand, we show that there is also a $\Gamma$-action on the mirror Drinfeld-Sokolov hierarchy, and the $\Gamma$-invariant flows yield the BCFG Drinfeld-Sokolov hierarchy. We prove that the total descendant potential of the $\Gamma$-invariant sector of a FJRW-theory is a tau function of the corresponding BCFG Drinfeld-Sokolov hierarchy.