The Harish-Chandra isomorphism is a fundamental result in the representation theory of Lie groups and Lie algebras, particularly in the context of semisimple Lie groups. It relates the spaces of invariant differential operators on a symmetric space to the space of functions on the Lie algebra of the group. More specifically, consider a semisimple Lie group \( G \) and a maximal compact subgroup \( K \).
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For a finite-dimensional complex semisimple Lie algebra with Cartan subalgebra , the shifted Harish-Chandra map identifies the centre of the universal enveloping algebra with the Weyl group-invariant polynomial functions on . To normalize the shift, let have highest weight , where is the Weyl vector. A central element acts by a scalar , and is the displayed algebra isomorphism. The Poincaré-Birkhoff-Witt theorem gives polynomial dependence on by projecting onto its pure Cartan component and evaluating at . 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 .