This paper extends a 1993 idea of Mark Haiman from the symmetric group to a much broader class of symmetry groups called Weyl groups. The author gives a concrete geometric formula for the graded character (the bookkeeping of symmetries in each cohomological degree) of the intersection cohomology of certain varieties constructed from a Weyl-group element z. These varieties, called Lusztig varieties, can be singular, so intersection cohomology is needed to capture their topology correctly.

Building on earlier work of Lusztig and of Abreu–Nigro, the paper proves an explicit identity (Theorem 1.1) that expresses the graded action of the Weyl group on the intersection cohomology of a closed Lusztig variety over the regular semisimple part of the group. The identity packages the result in terms of a linear combination τ_{z,v} that comes from Lusztig’s “exotic Fourier transform,” a pairing on characters from his work on representations of reductive groups. In the special case of type A (the symmetric group case studied by Haiman), the formula reduces to known results of Abreu–Nigro.

The author also gives a related description over the unipotent locus (roughly, the part of the group near the identity) and explains how this produces a new geometric model for certain Lascoux–Leclerc–Thibon (LLT) polynomials of a simple kind, called unicellular LLT polynomials. At a high level, the method uses monodromy (how cohomology transforms when parameters move) and convolution actions coming from a Steinberg-style variety, together with the Springer correspondence, which relates geometry of unipotent orbits to representations of the Weyl group.

One of the paper’s main combinatorial outputs is a family of Laurent polynomials α^z_{ψ,G}, indexed by irreducible characters ψ, that record how the geometric formula breaks into the ungraded pieces that appear in Springer theory. From computations in small examples the author conjectures that when ψ comes from type A in a specific way, the nonzero coefficients of these α-polynomials are positive and unimodal (they increase then decrease). This is proposed as a natural generalization of Haiman’s original positivity conjecture for symmetric groups. The paper also proves structural facts: the matrix of these α-polynomials is partially triangular, and the guessed positivity and unimodality are stable when passing to certain subgroup inclusions (Levi subgroups).