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Let W be a finite Coxeter group. We define its Hecke-group algebra by gluingtogether appropriately its group algebra and its 0-Hecke algebra. We describein 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 symmetrypreserving operator algebra, as poset algebra, as commutant algebra, ...).  In type A, the Hecke-group algebra can be described as the algebra generatedsimultaneously by the elementary transpositions and the elementary sortingoperators acting on permutations. It turns out to be closely related to themonoid algebras of respectively nondecreasing functions and nondecreasingparking functions, the representation theory of which we describe as well.  This defines three towers of algebras, and we give explicitly theGrothendieck algebras and coalgebras given respectively by their inductionproducts and their restriction coproducts. This yields some new interpretationsof the classical bases of quasi-symmetric and noncommutative symmetricfunctions as well as some new bases.

The Hecke group algebra of a Coxeter group and its representation theory