TITLE: How to understand the Harish-Chandra isomorphism?
QUESTION [22 upvotes]: The Harish-Chandra isomorphism describe the center $Z(\mathfrak{g})$ of $U(\mathfrak{g})$ as invariants of $\text{Sym}^*\mathfrak{h}$ under the action of the Weyl group. (One need to twist the action or make a change of coordinate on the affine space $\mathfrak{h}^*$ for this isomorphism work.)
My question is, how does one understand this isomorphism, and what's the geometric context of it? Why should one expect something like this might be true?

REPLY [5 votes]: Depending what one means by "intuition", the computation of the behavior of the center $\mathfrak z$ of $U\mathfrak g$ on unramified principal series repns not only suggests the form of the Harish-Chandra isomorphism, but also suggests the $W$-invariance, since the intertwinings (generically isomorphisms) among principal series are exactly given by Weyl group elements (with the shift by the half-sum $\rho$ of positive roots as a result of a change-of-measure at the group level).