The theorem on restriction of invariants, and nilpotent elements in $W_n$

The ring of invariant polynomial functions on the general algebra of Cartan type $W_n$ is described explicitly. It is assumed that the ground field is algebraically closed and its characteristic is greater than 2. This result is used to prove that the variety of nilpotent elements in $W_n$ is an irreducible complete intersection and contains an open orbit whose complement consists of singular points. Moreover, a criterion for orbits in $W_n$ to be closed is obtained, and it is proved that the action of the commutator subgroup of the automorphism group in $W_n$ is stable.