Vector invariants of the groups \(\text{GL}(n,\mathbb{C}[[t]])\) and \(\text{Sp}(2m,\mathbb{C}[[t]])\) (Q1320682)

Let a group \(G\) act on a finite dimensional space \(W\) over the complex field \(\mathbb{C}\). The main problem of invariant theory is to describe the ring of \(G\)-invariant polynomial functions on \(W\). Let an element \(w\) of the space \(W\) smoothly depend on a parameter \(t\). It is interesting to know polynomial functions of a finite number of derivatives of \(w\) at the point \(t = 0\) that are invariant under changes of coordinates from the group \(G\) also smoothly depending on \(t\). These functions are called local invariants. The coefficients of the Taylor series of a \(G\)- invariant are local invariants if we consider the \(G\)-invariant as a function of \(t\). The main result of this paper is that for \(\text{GL}(n,{\mathbb{C}}[[t]])\) or \(\text{Sp}(2m,\mathbb{C} [[t]])\) and for \(W\) described above, the coefficients of the Taylor series of classical basic variants generate the ring of local invariants. A similar description is obtained for relations between basic local invariants.