Evaluations of Hecke algebra traces at Kazhdan-Lusztig basis elements

For irreducible characters $\{{\chi }_q^{\lambda }\mathrm{\mid }\lambda ⊢n\}$, induced sign characters $\{{ϵ}_q^{\lambda }\mathrm{\mid }\lambda ⊢n\}$, and induced trivial characters $\{{\eta }_q^{\lambda }\mathrm{\mid }\lambda ⊢n\}$ of the Hecke algebra $H_n(q)$, and Kazhdan-Lusztig basis elements $C_w^{\mathrm{\prime }}(q)$ with $w$ avoiding the patterns 3412 and 4231, we combinatorially interpret the polynomials ${\chi }_q^{\lambda }(q^{l(w)\mathrm{/}2}{C}_w^{\mathrm{\prime }}(q))$, $\epsilon_q^\lambda(q^{l(w)/2} C’_w(q))$,and$\smash{\eta_q^\lambda(q^{l(w)/2} C’_w(q))}$. 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.