  Let W be a finite Coxeter group. We define its Hecke-group algebra by gluing
together appropriately its group algebra and its 0-Hecke algebra. We describe
in detail this algebra (dimension, several bases, conjectural presentation,
combinatorial construction of simple and indecomposable projective modules,
Cartan map) and give several alternative equivalent definitions (as symmetry
preserving operator algebra, as poset algebra, as commutant algebra, ...).
  In type A, the Hecke-group algebra can be described as the algebra generated
simultaneously by the elementary transpositions and the elementary sorting
operators acting on permutations. It turns out to be closely related to the
monoid algebras of respectively nondecreasing functions and nondecreasing
parking functions, the representation theory of which we describe as well.
  This defines three towers of algebras, and we give explicitly the
Grothendieck algebras and coalgebras given respectively by their induction
products and their restriction coproducts. This yields some new interpretations
of the classical bases of quasi-symmetric and noncommutative symmetric
functions as well as some new bases.
