On the centralizer of \(K\) in \(U(\mathfrak g)\) (Q2370305)

Let \({\mathfrak g}={\mathfrak k}+{\mathfrak p}\) be a complexified Cartan decomposition of a complex semisimple Lie algebra \(\mathfrak g\) and let \(K\) be the subgroup of the adjoint group of \(\mathfrak g\) corresponding to \(\mathfrak k\). The irreducible Harish-Chandra modules \(H\) of \(U({\mathfrak g})\) are determined by the finite dimensional action of the centralizer \(U({\mathfrak g})^K\) on a fixed primary \(\mathfrak k\) component of \(H\). The purpose of the paper under review is to give bounds which allow to find efficiently the generators of \(U({\mathfrak g})^K\). Replacing the problem with the commutative problem for the generators of \(S({\mathfrak g})^K\), the author shows that the field of fractions of \(S({\mathfrak g})^K\) can be generated by \(K\)-invariant polynomials of degree \(\leq n=\dim{\mathfrak g})\). The paper also determines the variety \(\mathrm{Nil}_K\) of unstable points with respect to the action of \(K\) on \(\mathfrak g\) and shows that it is defined by the invariants of degree \(\leq 2n\). Together with a result in [\textit{H. Derksen}, Proc. Am. Math. Soc. 129, No. 4, 955--963 (2001; Zbl 0969.13003)] this gives the bound \({2n\choose 2}\text{ dim}({\mathfrak p})\) for the degree of the generators of \(S({\mathfrak g})^K\).