Harish-Chandra isomorphism

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 \( K \).