Haiman's Conjecture and Springer's Representations

Abstract

For any connected complex reductive group $G$ and element $z$ of its Weyl group $W$, we use work of Lusztig and Abreu-Nigro to compute the graded $W$-character of the intersection cohomology of any closed Lusztig variety for $z$ over the regular semisimple locus of $G$. We relate the resulting formula to unipotent Lusztig varieties, giving a new geometric model for unicellular LLT polynomials. We then consider Laurent polynomials $\alpha_{\psi, G}^z$ indexed by irreducible characters $\psi$, encoding how our formula decomposes into ungraded characters arising from the Springer theory of $G$. From evidence in low rank, we conjecture that if $\psi$ is inflated from type $A$ in a particular way, then the nonzero coefficients of $\alpha_{\psi, G}^z$ are positive and unimodal. This offers an answer to a 1993 question of Haiman about generalizing a conjecture he posed for symmetric groups. We also prove that the matrix formed by the $\alpha_{\psi, G}^z$ is partially triangular, and that their positivity and unimodality properties are stable under inclusions of Levi subgroups.