As to Coxeter group with weight function, most people are concerned with positive weight functions, including Lusztig, writing Hecke algebras with unequal parameters. This book don't deny the negative weight functions or zero weight functions, but discuss really little about them.
I think this is because only positive weight functions arise "naturally". But somehow negative weight functions are also of interest. So would anyone please introduce me some books or papers on Coxeter group with negative weight function. I am most concerned with the cell decompositions of them? Thanks to everyone.