Invariant theory for coincidental complex reflection groups

V.F. Molchanov considered the Hilbert series for the space of invariant skew-symmetric tensors and dual tensors with polynomial coefficients under the action of a real reflection group, and he speculated that it had a certain product formula involving the exponents of the group. We show that Molchanov's speculation is false in general but holds for allcoincidentalcomplex reflection groups when appropriately modified using exponents and co-exponents. These are the irreducible well-generated (i.e., duality) reflection groups with exponents forming an arithmetic progression and include many real reflection groups and all non-real Shephard groups, e.g., the Shephard-Todd infinite familyG(d, 1, n). We highlight consequences for theq-Narayana andq-Kirkman polynomials, giving simple product formulas for both, and give aq-analogue of the identity transforming theh-vector to thef-vector for the coincidental finite type cluster/Cambrian complexes of Fomin-Zelevinsky and Reading. We include the determination of the Hilbert series for the non-coincidental irreducible complex reflection groups as well.