The Harish-Chandra homomorphism is a mathematical concept in the field of representation theory, particularly concerning Lie groups and Lie algebras. It provides a way to relate the representations of a semisimple Lie group to those of its corresponding Lie algebra. This homomorphism is crucial for understanding how these algebraic structures interact and can be represented. In essence, the Harish-Chandra homomorphism maps elements from the center of the universal enveloping algebra of a Lie algebra to the center of the group algebra of the corresponding Lie group. This mapping helps in studying the characters of representations and plays a significant role in the broader context of harmonic analysis on Lie groups.