Harish-Chandra isomorphism ($Z(U(\mathfrak{g}))\cong S(\mathfrak{h}{)}^W$)

For a finite-dimensional complex semisimple Lie algebra with Cartan subalgebra $\mathfrak{h}$, the shifted Harish-Chandra map identifies the centre of the universal enveloping algebra with the Weyl group-invariant polynomial functions on ${\mathfrak{h}}^{\ast }$. To normalize the shift, let $M^{\chi }$ have highest weight $\chi -\rho$, where $\rho$ is the Weyl vector. A central element $z$ acts by a scalar $j_z(\chi )$, and $z\mapsto j_z$ is the displayed algebra isomorphism. The Poincaré-Birkhoff-Witt theorem gives polynomial dependence on $\chi$ by projecting $z$ onto its pure Cartan component and evaluating at $\chi -\rho$. The associated central characters agree exactly when their shifted parameters lie in the same Weyl orbit. Without incorporating the shift into the parameter, the invariance is expressed using the dot action $w\cdot \lambda =w(\lambda +\rho )-\rho$.