# Symmetrization map for universal enveloppings. What are Harish-Chandra images of symmetrization (poisson-center of S(gl_n) ) ?

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...

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Ok, I think that I now understand your question (it wasn't clear from the original post). Let $$\sigma: S(\mathfrak{g}) \to U(\mathfrak{g})$$ be the standard symmetrization map, $$HC: S(\mathfrak{h})^W \to Z(\mathfrak{g})$$ be the Harish-Chandra homomorphism and $$HC_{qc}: S(\mathfrak{h})^W \to S(\mathfrak{g})^{\mathfrak{g}}$$ be its quasi-classical version. Your question, then, concerns an explicit form of the composite map $$HC^{-1}\circ\sigma\circ HC_{qc}: S(\mathfrak{h})^W \to S(\mathfrak{h})^W.$$ I am busy this week (exams), but if it's what you want, I'll think about it later. – Victor Protsak Dec 12 '11 at 21:47