Harish-Chandra isomorphism

Remarks

The key fact behind the theorem is that $z \in Z(\mathfrak{g})$ acts on the Verma module of highest weight $\lambda$ by the scalar $\pi(z)(\lambda) = \gamma(z)(\lambda + \rho)$. Comparing Verma modules whose highest weights are linked by simple reflections shows that $\gamma(z)$ is $W$-invariant. Injectivity follows because a central element is determined by its action on all Verma modules. Surjectivity is harder; one proof uses the characters of the finite-dimensional modules and Chevalley's restriction theorem. Example for $\mathfrak{sl}_2(\C)$: identify $\mathfrak{h}^*$ with $\C$ by $\lambda \mapsto \lambda(h)$; then $W = \set{\pm 1}$ and $\rho = 1$. The Casimir element is $\Omega = ef + fe + \tfrac{1}{2}h^2 = 2fe + h + \tfrac{1}{2}h^2$, so $\pi(\Omega)(\lambda) = \lambda + \tfrac{1}{2}\lambda^2$. Then $\gamma(\Omega)(\mu) = (\mu - 1) + \tfrac{1}{2}(\mu - 1)^2 = \tfrac{1}{2}(\mu^2 - 1)$, which is invariant under $\mu \mapsto -\mu$, and $Z(\mathfrak{sl}_2) = \C[\Omega]$. Without the shift by $\rho$, the map $\pi$ is still an algebra isomorphism onto its image. But the image is invariant under the shifted action $w \cdot \lambda = w(\lambda + \rho) - \rho$, not under the linear action.

Going deeper

For a student

Each $z \in Z(\mathfrak{g})$ acts by a scalar on a highest weight module, and this defines a central character $\chi_\lambda$. The theorem shows that $\chi_\lambda = \chi_\mu$ exactly when $\mu + \rho = w(\lambda + \rho)$ for some $w \in W$. This linkage principle splits category O into summands indexed by central characters, and it is a main step in the algebraic proof of the Weyl character formula. By Chevalley's theorem, $Z(\mathfrak{g})$ is a polynomial ring in $\operatorname{rank} \mathfrak{g}$ generators; for $\mathfrak{sl}_n$ their degrees are $2, 3, \dots, n$.