Extend the method to exceptional complex reflection groups

Construct suitable $W$-stable multiarrangements and prove the coinvariant phenomenon for exceptional complex reflection groups beyond the non-exceptional groups $G(m,p,n)$.

Background

The paper establishes generalized hit results for the non-exceptional complex reflection groups G(m,p,n)G(m,p,n) by using free multiarrangements, invariant logarithmic derivations, and top-degree concentration of twisted logarithmic de Rham cohomology.

The authors identify extension to exceptional complex reflection groups as a subsequent problem. The final section reports preliminary evidence for many exceptional groups but leaves the construction and verification of suitable multiarrangements unresolved, particularly for groups such as G12G_{12}, G13G_{13}, and G22G_{22}.

References

Two natural questions for subsequent work concern possible generalizations to exceptional complex reflection groups and possible approaches to questions of for $\ell>1$ in the same way, making use of suitable multiarrangements of subspaces of codimension $\ell$, as well as the associated logarithmic vector fields and the arising Lie--Rinehart homology. It is not yet clear to us whether these questions, especially the second one, have promise, but they are certainly worth examining.

— Twisted logarithmic de Rham complexes and the rational hit conjecture  (2610.26028 - Dotsenko, 19 Oct 2026) in Section 1.3, paragraph beginning “Two natural questions for subsequent work”; Section 8, Section 8 title

sixteen of the nineteen irreducible rank two cases, that is, all except for $G_{12}$, $G_{13}$, and $G_{22}$ (we do not know if that is related to the fact that these are precisely the rank two cases for which natural Saito structures do not exist Sec.~9.4.1)

— Twisted logarithmic de Rham complexes and the rational hit conjecture  (2610.26028 - Dotsenko, 19 Oct 2026) in Section 8, paragraph discussing exceptional rank-two groups