Invariant polynomials on truncated multicurrent algebras

We construct invariant polynomials on truncated multicurrent algebras, which are Lie algebras of the form g⊗FF[t1,…,tℓ]/I, where g is a finite-dimensional Lie algebra over a field F of characteristic zero, and I is a finite-codimensional ideal of F[t1,…,tℓ] generated by monomials. In particular, when g is semisimple and F is algebraically closed, we construct a set of algebraically independent generators for the algebra of invariant polynomials. In addition, we describe a transversal slice to the space of regular orbits in g⊗FF[t1,…,tℓ]/I. As an application of our main result, we show that the center of the universal enveloping algebra of g⊗FF[t1,…,tℓ]/I acts trivially on all irreducible finite-dimensional representations provided I has codimension at least two.