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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 $U(\mathfrak{g})= U(\mathfrak{h})\oplus (\mathfrak{n}^{-}U(\mathfrak{g}) + U(\mathfrak{g})\mathfrak{n}^{+})$ 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.

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    $\begingroup$ en.wikipedia.org/wiki/Duflo_isomorphism $\endgroup$
    – Alexander Chervov
    Commented Nov 10, 2022 at 20:31
  • $\begingroup$ Particular example for the determinant : mathoverflow.net/questions/92348/… More generally there is bunch of activity to write explicit nice formulas for the center of U(gl_n), such that we can control HC images, related to math. physics. $\endgroup$
    – Alexander Chervov
    Commented Nov 10, 2022 at 20:45
  • $\begingroup$ Duflo isomorphism is general construction but it is not that explicit as these formulas $\endgroup$
    – Alexander Chervov
    Commented Nov 10, 2022 at 20:50
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    $\begingroup$ @VítTuček Well... About what part ? The interest to the field re-appeared again in early 90-ies motivated by constructions found by physicists - Yangian in particular. Many works have been done by Molev,Nazarov, Olshanski, Okounkov. See some surveys/books by Molev: arxiv.org/abs/math/0211288 - but it is more general - about Yangians and similar - the paradox insight is that it is more easy to look on classical Lie algebras via the Yangian/related point of view ! Thus exposition kind of not focused on gl_n, but with some experience one can downgrade to gl_n from Yangian without changes $\endgroup$
    – Alexander Chervov
    Commented Nov 11, 2022 at 20:06
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    $\begingroup$ Concerning the linear basis it itself - we should think about Schur functions - so we can ask what are their appropriate analogues in ZU(gl_n) - that can be found e.g. in arxiv.org/abs/q-alg/9602028 Andrei Okounkov , and his related papers at that time. Though somewhat digging literature similar results can be found in some unknown paper from 1980-ies. $\endgroup$
    – Alexander Chervov
    Commented Nov 11, 2022 at 20:11

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