Quantum quasi-symmetric functions and Hecke algebras

. The algebra of quasi-symmetric functions is known to describe the characters of the Hecke algebra Hn (v) of type An\Gamma1 at v = 0. We present a quantization of this algebra, defined in terms of filtrations of induced representations of the 0Hecke algebra. We show that this q-deformed algebra admits a simple realization in terms of quantum polynomials. For generic values of q, the algebra of quantum quasisymmetric functions is isomorphic to the one of noncommutative symmetric functions. This gives rise to a one parameter family of Hilbert space structures on the algebra of noncommutative symmetric functions, as well as to new interesting bases. 1. Introduction It is well known that characters of the symmetric group S n are encoded by symmetric functions, and this correspondence is the cornerstone of many computational methods in representation theory (cf. [15, 24]). The same correspondence works as well for the Hecke algebra H n (v) when v is neither 0 nor a root of unity (see [1])..