Hecke algebras with independent parameters

We study the Hecke algebra (Formula presented.) over an arbitrary field (Formula presented.) of a Coxeter system (W, S) with independent parameters (Formula presented.) for all generators. This algebra always has a spanning set indexed by the Coxeter group W, which is indeed a basis if and only if every pair of generators joined by an odd edge in the Coxeter diagram receives the same parameter. In general, the dimension of (Formula presented.) could be as small as 1. We construct a basis for (Formula presented.) when (W, S) is simply laced. We also characterize when (Formula presented.) 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.