A team of mathematicians has shown that a construction called the Hessenberg functor, defined on the geometric Hecke category, exactly reproduces the graded W–representation that appears in the cohomology of regular semisimple Hessenberg varieties—at least for basis elements indexed by smooth Weyl group elements.

The Hessenberg functor is built by first microlocalizing along the diagonal inside the product of the flag variety with itself and then pushing forward along the Springer map. This produces a complex living on the coadjoint nilpotent cone, from which the Hessenberg character of the Iwahori–Hecke algebra is extracted. When this construction is applied to a Kazhdan‑Lusztig basis element whose index is a smooth element of the Weyl group, the resulting graded representation coincides with the cohomology of the corresponding regular semisimple Hessenberg variety.

A deeper analysis shows that the Hessenberg functor is essentially the same as Lusztig’s character functor followed by microlocalization at the identity, up to a shift and a Tate twist. Moreover, the functor satisfies a trace property: applying it to a convolution product of two Hecke category objects produces the same result whether the order of the factors is reversed.

The authors also clarify the scope and limits of the construction. While the non‑principal part of the functor vanishes for smooth elements and for the general linear group, it can be nonzero for other groups or for singular Weyl group elements. This means that the full behavior of the functor depends on both the group and the choice of element.

These results give a uniform framework for computing the graded W–characters of regular semisimple Hessenberg varieties in all Lie types up to rank ten, covering all classical and exceptional series. However, the method relies on the smoothness of the Weyl element and fails for singular cases, indicating an area for future refinement.",