= Harish-Chandra isomorphism
{c}
{title2=$Z(U(\mathfrak g))\cong S(\mathfrak h)^W$}
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 $\mathfrak h^*$. 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 $w\cdot\lambda=w(\lambda+\rho)-\rho$.
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