Non-commutative analogue of the Berglund-Hübsch-Henningson duality and symmetries of orbifold invariants of singularities

The first regular construction of (conjecturally) mirror symmetric orbifolds belongs to Berglund, Hübsch and Henningson. The Berglund-Hübsch-Henningson- (BHH- for short) duality is a duality on the set of pairs (f,G) consisting of an invertible polynomial group and a subgroup G of diagonal symmetries of f. Symmetries of (orbifold) invariants of BHH-dual pairs are related to mirror symmetry. There were prooved symmetries for the orbifold Euler characteristic, orbifold monodromy zeta-function, and orbifold E-function. One has a method to extend the BBH-duality to the set of pairs (f,G^, where G^ is the semidirect product of a group G of diagonal symmetries of f and a group S of permutations of the coordinates preserving f. The construction is based on ideas of A.Takahashi and therefore is called the Berglund-Hübsch-Henningson-Takahashi- (BHHT-) duality. Invariants of BHHT-dual pairs have symmetries similar to mirror ones only under some restrictions on the group S: the so-called parity condition (PC). Under the PC-condition it is possible to prove some symmetries of the orbifold invariants of BHHT-dual pairs.