  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.
