  We consider the partition of a finite Coxeter group $W$ into left cells with
respect to a weight function $L$. In the equal parameter case, Lusztig has
shown that the representations carried by the left cells are precisely the
so-called constructible ones. We show that this holds for general $L$, if the
conjectural properties (P1)--(P15) in Lusztig's book on Hecke algebras with
unequal parameters hold for $W,L$. Our proofs use the idea (Gyoja, Rouquier)
that left cell representations are projective in the sense of modular
representation theory. This also gives partly new proofs for Lusztig's result
in the equal parameter case.
