A Class of Polynomial Modules over Map Lie Algebras

For any finitely generated unital commutative associative algebra \(\mathcal {R}\) $\mathcal{R}$ over \(\mathbb {C}\) $\mathbb{C}$ and any complex finite-dimensional simple Lie algebra \(\mathfrak {g}\) $\mathfrak{g}$ with a fixed Cartan subalgebra \(\mathfrak {h}\) $\mathfrak{h}$ , we classify all \(\mathfrak {g}\otimes \mathcal {R}\) $\mathfrak{g}\otimes \mathcal{R}$ -modules on \(U(\mathfrak {h})\) $U(\mathfrak{h})$ such that \(\mathfrak {h}\) $\mathfrak{h}$ as a subalgebra of \(\mathfrak {g}\otimes \mathcal {R}\) $\mathfrak{g}\otimes \mathcal{R}$ , acts on \(U(\mathfrak {h})\) $U(\mathfrak{h})$ by the multiplication. We construct these modules explicitly and study their module structures.