Evaluations of Hecke algebra traces at Kazhdan-Lusztig basis elements

For irreducible characters {chi(lambda)(q) vertical bar lambda proves n}, induced sign characters {epsilon(lambda)(q) vertical bar lambda proves n}, and induced trivial characters {eta(lambda)(q) vertical bar lambda proves 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(lambda)(q)(q(l(w)/2)C(w)'(q)), epsilon(lambda)(q)(q(l(w)/2)C(w)' (q)), and eta(lambda)(q) (q(l(w)/2)C(w)' (q)). This provides 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.