Evaluations of Hecke algebra traces at Kazhdan-Lusztig basis elements

Abstract: For irreducible characters ${ \chi_q^\lambda \,|\, \lambda \vdash n }$, induced sign characters ${ \epsilon_q^\lambda \,|\, \lambda \vdash n }$, and induced trivial characters ${ \eta_q^\lambda \,|\, \lambda \vdash n }$ 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 $\chi_q^\lambda(q^{l(w)/2}C'_w(q))$, $\epsilon_q^\lambda(q^{l(w)/2} C'_w(q))$, 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.