The charges are functions of the momentum and the central charge, invariant under the action of the Weyl group. In particular, is a polynomial of the momentum of degree , which under the Dynkin diagram automorphism behaves as . The conformal dimension is Let us parametrize the central charge in terms of a number such that If there is a positive root and two integers such that then the Verma module of momentum has a null vector at level . This null vector is itself a primary state of momentum or equivalently (by a Weyl reflection) . The number of independent null vectors is the number of positive roots such that (up to a Weyl reflection). The maximal number of null vectors is the number of positive roots . The corresponding momentums are of the type where are integral dominant weights, i.e. elements of , which are highest weights of irreducible finite-dimensional representations of . Let us call the corresponding fully degenerate representation of the W(N) algebra. The irreducible finite-dimensional representation of of highest weight has a finite set of weights , with . Its tensor product with a Verma module of weight is . The fusion product of the fully degenerate representation of W(N) with a Verma module of momentum is then Correlation functions Primary fields To a primary state of charge , the state-field correspondence associates a primary field , whose operator product expansions with the fields are On any field , the mode of the energy-momentum tensor acts as a derivative, . Ward identities On the Riemann sphere, if there is no field at infinity, we have . For , the identity may be inserted in any correlation function. Therefore, the field gives rise to global Ward identities. Local Ward identities are obtained by inserting , where is a meromorphic function such that . In a correlation function of primary fields, local Ward identities determine the action of with in terms of the action of with . For example, in the case of a three-point function on the sphere of W(3)-primary fields, local Ward identities determine all the descendant three-point functions as linear combinations of descendant three-point functions that involve only . Global Ward identities further reduce the problem to determining three-point functions of the type for . In the W(3) algebra, as in generic W-algebras, correlation functions of descendant fields can therefore not be deduced from