For any Lie algebra symmetrization maps Poisson center of S(g) to center of U(g). Consider g=gl_n, the Poisson center of S(gl_n) is isomorphic to algebra of symmetric polynomials in eigenvalues. Consider some element c \in S(gl_n), which corresponds to some symmetric polynomial P(l1,...ln).

Question: is there something known about value of symmetrization(c) in irrep of highest weight (w1,...wn), i.e. what is the Harish-Chandra image of symmetrization(c) ?

We should expect that this value is P(w1,...,wn)+correction, but it is quite difficult to calculate this "correction".

For sl(2) the answer is known - see Kirillov's paper "Merits and demerits of orbit method" section 3.5 (http://www.ams.org/bull/1999-36-04/S0273-0979-99-00849-6/S0273-0979-99-00849-6.pdf). Answer and proof are rather non-trivial and obtained with the help of the Duflo map...