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In mathematical representation theory, a Harish-Chandra homomorphism is a homomorphism from a subalgebra of the universal enveloping algebra of a semisimple Lie algebra to the universal enveloping algebra of a subalgebra. A particularly important special case is the Harish-Chandra isomorphism identifying the center of the universal enveloping algebra with the invariant polynomials on a Cartan subalgebra.

In the case of the K-invariant elements of the universal enveloping algebra for a maximal compact subgroup K, the Harish-Chandra homomorphism was studied by Harish-Chandra (1958).
References

Harish-Chandra (1958), "Spherical functions on a semisimple Lie group. I", American Journal of Mathematics 80: 241–310, ISSN 0002-9327, JSTOR 2372786, MR 0094407
Howe, Roger E. (2000), "Harish-Chandra homomorphisms", in Doran, Robert S.; Varadarajan., V. S., The mathematical legacy of Harish-Chandra (Baltimore, MD, 1998), Proc. Sympos. Pure Math., 68, Providence, R.I.: American Mathematical Society, pp. 321–332, ISBN 978-0-8218-1197-9, MR 1767901

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