  A classical result of Artin states that the ideal generated by symmetric
polynomials in $n$ variables is of codimension $n!$. The author, F. Bergeron
and N. Bergeron have recently obtained a surprising analogous in the case of
quasi-symmetric polynomials. In this case, the ideal is of codimension given by
$C_n$, the $n$-th Catalan number. Quasi-symmetric polynomials are the
invariants of a certain action of the symmetric group $S_n$ defined by F.
Hivert. The aim of this work is to generalize these results to the wreath
product $S_n\wr \Z_m$, also known as the generalized symmetric group $G\nm$. We
first define a quasi-symmetrizing action of $G\nm$ on $\C[x_1,...,x_n]$, then
obtain a description of the invariants and the codimension of the associated
ideal, which is $m^n C_n$.
