Following Lusztig, we investigate constructible characters, leading coefficients and left cells for a finite Coxeter group
W
W
in the case of unequal parameters. We obtain explicit results for
W
W
of type
F
4
F_4
,
B
n
B_n
and
I
2
(
m
)
I_2(m)
(
m
m
even) which support Lusztig’s conjecture that known results about left cells in the equal parameter case should remain valid in the case of unequal parameters.