Let $\lambda$ be a partition with $\leq n$ rows and let $L_{\lambda}$ be the corresponding irreducible representation of ${\rm GL}_n(\mathbb{C})$. Let $e_m(X_1,\dots,X_n)$ be the $m$th elementary symmetric polynomial. Is there a known formula for the element $c_m \in Z(U(\mathfrak{gl}_n))$ which acts on $L_{\lambda}$ by the scalar $e_m(\lambda_1,\dots,\lambda_n)$?

It is easy to check that $c_1 = E_{11} + \dots + E_{nn}$, so I suspect that $c_m$ should be related to the coefficients of the characteristic polynomial. Also, I am aware of the Harish-Chandra isomorphism, but have not been able to use it to produce a formula.

I would be particularly interested in references where this is worked out. I couldn't find it in any of the standard representation theory text books.