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 : D(g)^g -> D(h)^W, that reduces to the restriction map: C[g]^g -> C[h]^W on zero order differential operators, and such that \Phi Laplacian_g = Laplacian_h "

Question:

Consider some higher order Laplacians - what are their image under this map ? Are there "nice explicit" formulas ? I am interested mostly in gl_n case.

Naive answer (probably incorrect): (ACTUALLY THIS CORRECT see answer) one may naively expect that e.g. cubic Laplacian on 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.

===

Further questions: 1) what will happen for "deformed Harish-Chandra" homomorphism ? (i.e. what about the algebra of quantum Calogero-Moser hamiltonians ? )

2) Consider the center of U(g) \in D(g)^g, 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(gl_n), might be some nice formula for its image under this map ... ?

=====

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

Setup: Consider Lie algebra g = gl_n, denote D(g)^g - invariant with respect to conjugation differential operators on it, denote C[g]^g - invariant functions. h \in gl_n - subspace of diagonal matrices, C[h]^W - symmetric function on h.

It is clear that D(g)^g acts on C[g]^g. It is well-known that C[g]^g isomorphic C[h]^W.

So we get map from \tilde Psi: D(g)^g -> D(h)^W.

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

Question: What happens with higher order Laplacians ?