Haiman's Conjecture and Springer's Representations

Technical Summary: Haiman's Conjecture and Springer's Representations

Problem Statement
The paper addresses the generalization of a 1993 conjecture by Mark Haiman from symmetric groups ($S_n$) to arbitrary finite Weyl groups ($W$). Haiman's original conjecture posits that for any permutation $z \in S_n$ and any partition $\mu \vdash n$, the Laurent polynomial $\alpha_z^\mu(v)$, which encodes the decomposition of the Frobenius characteristic of a specific $S_n$-representation into monomial symmetric functions, has coefficients that are non-negative and form a unimodal sequence.

While Haiman's conjecture was proven for $S_n$ (and related to the Shareshian–Wachs conjecture on chromatic quasisymmetric functions), a meaningful generalization to other Weyl groups was obstructed by the lack of a natural analog for "monomial characters" ($mon_\mu$) outside of type A. Furthermore, initial attempts to generalize the conjecture using induced characters from parabolic subgroups failed, as these characters are insufficient to capture the necessary structure, and the resulting polynomials often exhibited negative coefficients or non-unimodal behavior.

Methodology
The author employs a geometric approach rooted in the representation theory of reductive groups, specifically utilizing the work of Lusztig and Abreu–Nigro. The methodology proceeds through the following steps:

1. Geometric Modeling: The paper replaces the symmetric group setting with a connected complex reductive group $G$ and its Weyl group $W$. The left-hand side of Haiman's defining equation is replaced by the graded $W$-character of the intersection cohomology ($IH^*$) of a Lusztig variety $\bar{Y}_{z,g}$. This variety is the Zariski closure of the set of Borel subgroups $B$ such that $(B, g^{-1}Bg)$ is in relative position $z$, where $g$ is a regular semisimple element.
2. Algebraic Formulation: Using the exotic Fourier transform $\{ \cdot, \cdot \}$ (a pairing on irreducible characters of $W$ originating from Lusztig's work on finite reductive groups), the author derives a purely algebraic formula for the graded character of $IH^*(\bar{Y}_{z,g})$. This formula expresses the character as a linear combination involving the Hecke algebra basis elements $c_z$.
3. Springer Correspondence: To generalize the right-hand side of Haiman's equation (the decomposition into "monomial" analogs), the paper utilizes the Springer correspondence. The irreducible characters $\psi \in \text{Irr}(W)$ are associated with Springer representations arising from unipotent elements $u \in G_{\text{uni}}$ and characters of their component groups. The author defines new Laurent polynomials $\alpha_z^{\psi, G}(v)$ by decomposing the Lusztig variety character into these Springer characters.
4. Induction and Cell Theory: The paper analyzes the properties of $\alpha_z^{\psi, G}$ using induction theorems for parabolic subgroups and the theory of two-sided cells in the Hecke algebra. This allows for the reduction of the problem to irreducible Weyl groups and the identification of vanishing conditions based on the cell order.
5. Computational Verification: Extensive computations using the GAP3 package CHEVIE are performed for irreducible Weyl groups of rank up to 6 (and partial data for rank 7) to test the positivity and unimodality of the coefficients of $\alpha_z^{\psi, G}$.

Key Contributions and Results

• Theorem 1.1 (Regular Semisimple Locus): The author proves that the graded $W$-character of the intersection cohomology of the Lusztig variety $\bar{Y}_{z,g}$ (for $g$ in the regular semisimple locus) is given by $v^{\ell(z)} \tau_{z,v}(w)$, where $\tau_{z,v}$ is defined via the exotic Fourier transform pairing. This generalizes the result of Abreu–Nigro from type A to all reductive groups.
• Theorem 1.3 (Unipotent Locus): A parallel result is established for the unipotent locus. For rationally smooth elements $z$, the equivariant compactly-supported cohomology of a related variety $\bar{Z}_z$ is related to $\tau_{z,v}$ via the Springer characters and the graded character of the symmetric algebra on the root lattice. This provides a new geometric model for unicellular LLT polynomials.
• Proposition 1.9 & 1.10: The paper establishes that if $z$ lies in a parabolic subgroup $W'$, the polynomials $\alpha_z^{\psi, G}$ are non-negative integer sums of the corresponding polynomials for $W'$. Furthermore, $\alpha_z^{\psi, G}$ vanishes if the two-sided cell of $z$ is strictly smaller than the cell associated with $\psi$ in the Kazhdan–Lusztig order.
• Conjecture B: Based on computational evidence, the author proposes a new conjecture generalizing Haiman's. It states that if an irreducible character $\psi$ is inflated from a quotient map $\phi: W \to \Omega$ where $\Omega$ is a product of symmetric groups, then the coefficients of $\alpha_z^{\psi, G}$ are unimodal in absolute value and have uniform sign. Moreover, if $\phi$ does not descend through the specific $G_2 \to A_2$ quotient, the coefficients are strictly positive.
• Counterexamples and Nuance: The paper demonstrates that the "strict positivity" property (all coefficients positive) fails for many characters even in low ranks (e.g., Type $B_7$, $C_7$) and for certain characters in Type $G_2$ and $D_4$. This suggests that the correct generalization of Haiman's conjecture involves "unimodality in absolute value" and "uniform sign" rather than strict positivity for all characters.

Significance
The paper claims to offer a definitive answer to a 1993 question posed by Haiman regarding the generalization of his conjecture to Weyl groups. By constructing a geometric model via Lusztig varieties and relating it to Springer representations, the author provides a framework that explains why the original conjecture fails in its strict form for general Weyl groups and proposes a refined version (Conjecture B) that holds for a specific class of characters (those inflated from products of symmetric groups).

The work bridges algebraic combinatorics (LLT polynomials, Hecke algebras) and algebraic geometry (intersection cohomology, Springer fibers). It clarifies the role of the exotic Fourier transform in relating Lusztig varieties to Springer theory and establishes that the positivity and unimodality properties are stable under Levi subgroup inclusions. The paper does not claim to prove Conjecture B for all cases but provides strong evidence for it in low ranks and identifies the precise structural conditions (inflation from symmetric groups) under which the generalized positivity holds.