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](https://en.wikipedia.org/wiki/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.



Explicit formulas for certain elements in $Z(U(\mathfrak{gl_n}))$
I believe the elements of $\mathcal{Z}(\mathfrak{g})$ you are looking for are (suitably renormalised) the [Capelli elements](https://en.wikipedia.org/wiki/Capelli%27s_identity); these come from the (renormalised) Capelli determinant
\[C(u) = \det\left(E\_{jk} + \left(u - \frac{n - 2j + 1}{2}\right) \delta\_{jk}\right),\]
where by the determinant of a matrix $A = (A\_{jk})$ we mean the column determinant
\[\det A = \sum\_{\sigma \in S\_n} \mathrm{sgn}(\sigma) A\_{\sigma(1),1} \cdots A\_{\sigma(n),n}. \]
For all $u$, $C(u)$ lies in $\mathcal{Z}(\mathfrak{g})$.


One can show that given an irreducible representation $\pi\_{\lambda}$ of $\mathrm{GL}\_n(\mathbb{C})$ of highest weight $\lambda = (\lambda\_1,\ldots,\lambda\_n)$, the Harish-Chandra isomorphism $\chi\_{\pi\_{\lambda}}$ acts on $\mathcal{Z}(\mathfrak{g})$ by sending $E\_{jj}$ to $\lambda\_j + \frac{n - 2j + 1}{2}$ and $E\_{jk}$ to $0$ whenever $j \neq k$. So from the definition of $C(u)$, $\pi\_{\lambda}(C(u))$ acts by the scalar
\[\prod\_{\ell = 1}^{n} (u + \lambda\_{\ell}).\]
Expanding $C(u)$ as a polynomial in $u$, we see that the coefficients of each power of $u$ are elements of $\mathcal{Z}(\mathfrak{g})$ that act by the scalar $e\_m(\lambda\_1, \lambda\_2, \ldots, \lambda\_n)$.


