Let $\mathfrak{g}$ be a complex reductive Lie algebra (however, we are mainly interested in the case $\mathfrak{g}=\mathfrak{gl}_n$) and let $\mathfrak{h}$ be a Cartan subalgebra of $\mathfrak{g}$.

It is known that the Harish-Chandra map $\gamma\colon Z(U(\mathfrak{g}))\to S(\mathfrak{h})^{\widetilde{W}}$ is an isomorphism. Here $\widetilde{W}$ is the Weyl group "twisted" by $\rho\in\mathfrak{h}^*$ (half-sum of all positive roots) and $\gamma$ is basically the projection of $U(\mathfrak{g})=U(\mathfrak{g})\mathfrak{n}^{+}\oplus U(\mathfrak{h})\oplus\mathfrak{n}^{-}U(\mathfrak{g})$ onto $U(\mathfrak{h})\simeq S(\mathfrak{h})$.

Question. Is there any known explicit description of the inverse map $\gamma^{-1}$? In other words, how one would construct for a given $\widetilde{W}$-invariant polynomial in $p\in S(\mathfrak{h})$ the corresponding element $\gamma^{-1}(p)$ in the center of the universal enveloping algebra?

Any comments or references would be appreciated.