Harish–Chandra isomorphism - Imported Knowlpedia Content

Statement

Let $\mathfrak g$ be a complex semisimple Lie algebra, let $\mathfrak h$ be a Cartan subalgebra, and choose positive roots with half-sum $\rho$ . The Poincaré–Birkhoff–Witt decomposition defines a projection from $Z(U(\mathfrak g))$ to $U(\mathfrak h)\cong S(\mathfrak h)$ . Composing it with the translation $H\mapsto H-\rho(H)$ gives the Harish–Chandra homomorphism

\gamma:Z(U(\mathfrak g))\longrightarrow S(\mathfrak h).

The Harish–Chandra isomorphism theorem states that $\gamma$ is an algebra isomorphism from $Z(U(\mathfrak g))$ onto $S(\mathfrak h)^{W}$ , where $W$ is the Weyl group.

Construction

For the triangular decomposition $\mathfrak g=\mathfrak n^-\oplus\mathfrak h\oplus\mathfrak n^+$ , the PBW theorem gives a direct-sum complement to $U(\mathfrak h)$ containing $\mathfrak n^-U(\mathfrak g)+U(\mathfrak g)\mathfrak n^+$ . Projecting a central element along this complement gives its unshifted Harish–Chandra projection. The $\rho$ -translation converts invariance for the shifted Weyl action into ordinary $W$ -invariance Dixmier, §7.4.

Consequences for central characters

Algebra homomorphisms $Z(U(\mathfrak g))\to\mathbb C$ are thereby parameterized by $W$ -orbits in $\mathfrak h^*$ . If a highest-weight module has highest weight $\lambda$ , then its central character is obtained by evaluating $\gamma(z)$ at $\lambda+\rho$ . Thus highest weights in the same shifted Weyl orbit have the same infinitesimal character.

Conventions

Some authors call the unshifted PBW projection the Harish–Chandra homomorphism and state that its image is invariant for the dot action $w\mathbin{\cdot}\lambda=w(\lambda+\rho)-\rho$ . Others incorporate the shift, as above, and obtain ordinary Weyl invariants. The two formulations are equivalent, but their evaluation formulas differ by $\rho$ .

References

Jacques Dixmier, Enveloping Algebras, Graduate Studies in Mathematics 11, American Mathematical Society, 1996. DOI record. Relevant: §7.4 on the Harish–Chandra homomorphism.
Anthony W. Knapp, Lie Groups Beyond an Introduction, 2nd ed., Birkhäuser, 2002. DOI record. Relevant: Chapter V, §5 on the center of the enveloping algebra.