Evaluations of Hecke algebra traces at Kazhdan-Lusztig basis elements

Abstract

For irreducible characters $\{{\chi }_q^{\lambda }|\lambda ⊢n\}$, induced sign characters $\{{\epsilon }_q^{\lambda }|\lambda ⊢n\}$, and induced trivial characters $\{{\eta }_q^{\lambda }|\lambda ⊢n\}$ of the Hecke algebra $H_n(q)$, and Kazhdan-Lusztig basis elements $C_w^{\prime }(q)$ with $w$ avoiding the patterns 3412 and 4231, we combinatorially interpret the polynomials ${\chi }_q^{\lambda }(q^{l(w)/2}{C}_w^{\prime }(q))$, ${\epsilon }_q^{\lambda }(q^{l(w)/2}{C}_w^{\prime }(q))$, and ${\eta }_q^{\lambda }(q^{l(w)/2}{C}_w^{\prime }(q))$. This gives a new algebraic interpretation of chromatic quasisymmetric functions of Shareshian and Wachs, and a new combinatorial interpretation of special cases of results of Haiman. We prove similar results for other $H_n(q)$-traces, and confirm a formula conjectured by Haiman.