Lalg`ebre des invariants dun groupe de Coxeter agissant sur un mutiple de sa representation standard

Let G be a Coxeter group of type A_n, B_n, D_n or I_2(N), or a complex reflection group of type G(de,e,n). Let V be its standard representation and let k be an integer greater than 2. Then G acts on S(V)^{otimes k}. We show that the algebra of invariants (S(V)^{otimes k})^G is a free (S(V)^G)^{otimes k}-module of rank |G|^{k-1}, and that S(V)^{otimes k} is not a free (S(V)^{otimes k})^G-module.