Evaluations of Hecke algebra traces at Kazhdan-Lusztig basis elements

For irreducible characters $\{ χ_q^λ\,|\, λ\vdash n \}$, induced sign characters $\{ ε_q^λ\,|\, λ\vdash n \}$, and induced trivial characters $\{ η_q^λ\,|\, λ\vdash n \}$ of the Hecke algebra $H_n(q)$, and Kazhdan-Lusztig basis elements $C'_w(q)$ with $w$ avoiding the patterns 3412 and 4231, we combinatorially interpret the polynomials $χ_q^λ(q^{l(w)/2}C'_w(q))$, $ε_q^λ(q^{l(w)/2} C'_w(q))$, and $\smash{η_q^λ(q^{l(w)/2} C'_w(q))}$. 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.

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