In their great paper "Symplectic reflection algs. and Harish-Chandra hom." http://arxiv.org/abs/math/0011114, Etingof and Ginzburg write (page 9):

"In 1964, Harish-Chandra [HC] defined an algebra homomorphism $\Phi: D(\mathfrak{g})^{\mathfrak{g}} \rightarrow D(\mathfrak{h})^W$ that reduces to the restriction map: $\mathbb{C}[\mathfrak{g}]^{\mathfrak{g}} \rightarrow \mathbb{C}[\mathfrak{h}]^W$ on zero order differential operators, and such that $\Phi \Delta_{\mathfrak{g}} = \Delta_{\mathfrak{h}}$", where $\Delta_{\mathfrak{g}}$ is the second order Laplacian on $\mathfrak{g}$.

Question:

Consider some higher order Laplacians - what are their image under this map? Are there "nice explicit" formulas? I am interested mostly in $\mathfrak{gl}_n$ case.

Naive answer (probably incorrect): (ACTUALLY THIS IS CORRECT, see answer) one may naively expect that e.g. cubic Laplacian on $\mathfrak{g}$ will go to $\partial_1 ^3 + \partial_2 ^3 + ...$ in the same way as it is stated above for quadratic Laplacian. But probably this is not correct.

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Further questions:

What will happen for "deformed Harish-Chandra" homomorphism? (i.e. what about the algebra of quantum Calogero-Moser Hamiltonians?)

Consider the center of $U(\mathfrak{g})$ in $D(\mathfrak{g})^{\mathfrak{g}}$, and the same question for it. In particular we can take Talalaev's "oper" $\det(d/dz - E_{ij}/z)$ - some special generating "function" for generators of the center of $U(\mathfrak{gl}_n)$, might be some nice formula for its image under this map ...?

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Let me translate the question in simple-minded terms, it order to try to be self-contained.

Setup: Consider Lie algebra $\mathfrak{g} = \mathfrak{gl}_n$, denote $D(\mathfrak{g})^{\mathfrak{g}}$ - invariant with respect to conjugation differential operators on it, denote $\mathbb{C}[\mathfrak{g}]^{\mathfrak{g}}$ - invariant functions. $\mathfrak{h} \subseteq \mathfrak{gl}_n$ - subspace of diagonal matrices, $\mathbb{C}[\mathfrak{h}]^W$ - symmetric function on $\mathfrak{h}$.

It is clear that $D(\mathfrak{g})^{\mathfrak{g}}$ acts on $\mathbb{C}[\mathfrak{g}]^{\mathfrak{g}}$. It is well-known that $\mathbb{C}[\mathfrak{g}]^{\mathfrak{g}}$ isomorphic $\mathbb{C}[\mathfrak{h}]^W$.

So we get map from $\tilde \Psi: D(\mathfrak{g})^{\mathfrak{g}} \rightarrow D(\mathfrak{h})^W$.

Obviously zero-order differential operators (e.g. $\mathbb{C}[\mathfrak{g}]^{\mathfrak{g}}$ ) goes to $\mathbb{C}[\mathfrak{h}]^W$. Obviously second order Laplacian on $\mathfrak{g}$ will NOT go to second order Laplacian on $\mathfrak{h}$. However some educated guess is that we should conjugate it by the $\sqrt(\text{"volume of the orbit"})$ (which is Vandermonde of $h_i$ in $\mathfrak{gl}_n$ case) i.e. consider $\delta \circ \Psi \circ \delta^{-1}$. Then some calculations see (page 45 loc. cit.) claim that second order Laplacian on $\mathfrak{g}$ will go to second order Laplacian on $\mathfrak{h}$.

Question: What happens with higher order Laplacians?