The structure of the symmetric invariants of the Lie algebra \(W_ n({\mathbf m})\) (Q1358420)

Let \(L\) be a finite-dimensional Lie algebra, \(P\) and \(P^*\) contragredient \(L\)-modules contained in the universal enveloping algebra \(U(L)\) with respect to the adjoint action, \(P=\langle u_i\rangle\), \(P^*=\langle u_i^*\rangle\), where \(\{u_i\}\), \(\{u_i^*\}\) is a pair of dual bases. Then \(z=\sum u_iu_i^*\) is in the center \(Z(L)\) of \(U(L)\), and \(z\) is called a generalized Casimir element. If the ground field is of zero characteristic, then every element of \(Z(L)\) is a generalized Casimir element. The above construction remains valid if \(U(L)\) is replaced by \(S(L)\), the symmetric algebra over \(L\), and \(Z(L)\) by \(S(L)^L\). An element obtained by the modified procedure will be called a generalized symmetric Casimir element. In the article under review, the author shows that, for \(L=W(m,{\mathfrak n})\) (of characteristic \(p\)), any nontrivial element (i.e., not a \(p\)-th power of an element of \(S(L)\)) in \(S(L)^L\) is a generalized Casimir element.