Hecke algebras with independent parameters

We study the Hecke algebra $\H(\bq)$ over an arbitrary field $\FF$ of a Coxeter system $(W,S)$ with independent parameters $\bq=(q_s\in\FF:s\in S)$ for all generators. This algebra is always linearly spanned by elements indexed by the Coxeter group $W$. This spanning set is indeed a basis if and only if every pair of generators joined by an odd edge in the Coxeter diagram receive the same parameter. In general, the dimension of $\H(\bq)$ could be as small as $1$. We construct a basis for $\H(\bq)$ when $(W,S)$ is simply laced. We also characterize when $\H(\bq)$ is commutative, which happens only if the Coxeter diagram of $(W,S)$ is simply laced and bipartite. In particular, for type A we obtain a tower of semisimple commutative algebras whose dimensions are the Fibonacci numbers. We show that the representation theory of these algebras has some features in analogy/connection with the representation theory of the symmetric groups and the 0-Hecke algebras.