[Paper Review] A $(-q)$-analogue of weight multiplicities

G. Lusztig, Zhiwei Yun|arXiv (Cornell University)|Mar 2, 2012

Advanced Algebra and Geometry9 references3 citations

TL;DR

This paper proves that certain polynomials $P^{ ho}_{d_{ ho},d_{ ho}}(q)$ associated with twisted involutions in an affine Weyl group provide a $(-q)$-analogue of weight multiplicities for the Langlands dual group $\check{G}$, establishing a deep connection between Hecke algebra combinatorics and representation theory. The key result confirms a conjecture by Lusztig, showing $P^{ ho}_{d_{ ho},d_{ ho}}(q) = P_{d_{ ho},d_{ ho}}(-q)$, and further links these polynomials to the signature of hermitian forms on irreducible representations of $\check{G}$.

ABSTRACT

We prove a conjecture in \cite{L} stating that certain polynomials $P^σ_{y,w}(q)$ introduced in \cite{LV1} for twisted involutions in an affine Weyl group give $(-q)$-analogues of weight multiplicities of the Langlands dual group $\check{G}$. We also prove that the signature of a naturally defined hermitian form on each irreducible representation of $\check{G}$ can be expressed in terms of these polynomials $P^σ_{y,w}(q)$.

Motivation & Objective

• To prove a conjecture by Lusztig (2011) that $P^{ ho}_{d_{ ho},d_{ ho}}(q)$ is a $(-q)$-analogue of weight multiplicities in the irreducible representations of the Langlands dual group $\check{G}$.
• To establish a geometric and algebraic connection between the refined $P^{ ho}$-polynomials and intersection cohomology of affine Schubert varieties.
• To show that the signature of a natural hermitian form on irreducible representations of $\check{G}$ is expressible in terms of $P^{ ho}_{d_{ ho},d_{ ho}}(q)$.
• To generalize the result to other involutions $\diamond$ on the affine Weyl group, extending the $(-q)$-analogue construction.

Proposed method

• Use the geometric Satake equivalence to interpret $P^{ ho}_{d_{ ho},d_{ ho}}(q)$ as the Poincaré polynomial of the global intersection cohomology of $G[[t]]$-orbits in $G((t))/\mathbf{P}$.
• Define a modified $\mathcal{A}$-linear map $\zeta: M \to \mathbb{Q}(q)$ via $a_w \mapsto q^{\ell(w)} \left(\frac{q-1}{q+1}\right)^{\phi(w)}$, which generalizes the standard $\chi$-map for $Z_w(q)$.
• Leverage the action of the involution $*$ on the affine Weyl group to define $*$-twisted involutions $I_*$ and construct the $P^\sigma$-polynomials as refined analogues of Kazhdan-Lusztig polynomials.
• Use the action of $\widetilde{g} \in \check{T}^{\text{sc}}$ lifting $(-1)^\rho$ to compute traces on intersection cohomology sheaves, showing that $\mathcal{H}^{2j}_{\mu}\Psi_\lambda$ acts by $(-1)^j$.
• Apply the weight decomposition of cohomology groups and the relation $j = \langle \rho, \nu + \lambda - 2\mu \rangle$ to compute the trace of the involution on each weight space.
• Establish the identity $P^{\sigma,\diamond}_{d_1,d_2}(q) = \sum_j \text{tr}(\mathcal{H}^{2j}_{\mu}\Psi_\lambda, \mathcal{H}^{2j}_{\mu}\mathbf{C}_\lambda) q^j$, leading to the $(-q)$-analogue result.

Experimental results

Research questions

• RQ1 Does the $P^\sigma$-polynomial $P^\sigma_{d_\mu,d_\lambda}(q)$ provide a $(-q)$-analogue of the weight multiplicity $m(\mu, \lambda)$ in the irreducible representation $V_\lambda$ of the Langlands dual group $\check{G}$?
• RQ2 Can the signature of the natural hermitian form on $V_\lambda$ be expressed in terms of the $P^\sigma$-polynomials?
• RQ3 How do the $P^\sigma$-polynomials relate to the global intersection cohomology of $G[[t]]$-orbits in the affine flag variety $G((t))/\mathbf{P}$?
• RQ4 Can the $(-q)$-analogue construction be generalized to other involutions $\diamond$ on the affine Weyl group?

Key findings

• The main result confirms that $P^\sigma_{d_\mu,d_\lambda}(q) = P_{d_\mu,d_\lambda}(-q)$, proving that the $P^\sigma$-polynomials are indeed $(-q)$-analogues of weight multiplicities in $V_\lambda$.
• The signature of the hermitian form on each irreducible representation $V_\lambda$ of $\check{G}$ is given by the alternating sum of coefficients of $P^\sigma_{d_\mu,d_\lambda}(q)$, reflecting the action of the involution on cohomology.
• The $Z^\sigma_w(q)$-polynomials, defined via the $\zeta$-map, generalize the standard $Z_w(q)$-polynomials and encode global intersection cohomology data under the $*$-involution.
• The trace of the involution $\mathcal{H}^{2j}_{\mu}\Psi_\lambda$ on $\mathcal{H}^{2j}_{\mu}\mathbf{C}_\lambda$ is shown to be $(-1)^j$, which is essential to proving the $(-q)$-analogue identity.
• The result extends to other involutions $\diamond$ on the affine Weyl group, with the same $(-q)$-analogue property holding under the generalized $P^{\sigma,\diamond}$-polynomials.
• The proof relies on the geometric Satake equivalence and the action of $\widetilde{g} \in \check{T}^{\text{sc}}$ lifting $(-1)^\rho$, which induces the correct sign behavior on cohomology sheaves.