Let $\mathfrak g$ be a semisimple Lie algebra over $\mathbb C$, $\rho : \mathfrak g \to \operatorname{End}(V)$ a finite-dimensional irreducible representation and $x \in \mathfrak g$ regular with centralizer $Z$.

Does $\rho(Z)$ consist of polynomials in $\rho(x)$?

Examples where I know this is true:

• $x$ is semisimple: then $Z$ is a Cartan subalgebra, $\rho(x)$ acts by distinct scalars on the weight spaces $V_\lambda$ and the claim follows from Lagrange interpolation.
• $\mathfrak g = \mathfrak{sl}_n$ and $\rho = $ standard rep., because $x$ regular implies its minimal polynomial has degree $n$ so the (trace $0$) polynomials in $x$ give all of $Z$.
• $\mathfrak g = \mathfrak{sl}_2$. For semisimple elements we already know it, and a nilpotent acts by a Jordan matrix in a basis of weight vectors for a suitable Cartan, so $\rho(x) \in \mathfrak{sl}(V)$ is regular and then its centralizer consists of polynomials, as above.

Let $\mathfrak g$ be a semisimple Lie algebra over $\mathbb C$, $\rho : \mathfrak g \to \operatorname{End}(V)$ a finite-dimensional irreducible representation and $x \in \mathfrak g$ regular with centralizer $Z$.

Does $\rho(Z)$ consist of polynomials in $\rho(x)$?

Examples where I know this is true:

• $x$ is semisimple: then $Z$ is a Cartan subalgebra, $\rho(x)$ acts by distinct scalars on the weight spaces $V_\lambda$ and the claim follows from Lagrange interpolation. (This argument doesn't work: the scalars need not be distinct.)
• $\mathfrak g = \mathfrak{sl}_n$ and $\rho = $ standard rep., because $x$ regular implies its minimal polynomial has degree $n$ so the (trace $0$) polynomials in $x$ give all of $Z$.