On the Harish-Chandra Homomorphism

Using the Iwasawa decomposition <!-- MATH $\mathfrak{g} = \mathfrak{k} \oplus \mathfrak{a} \oplus \mathfrak{n}$ --> of a real semisimple Lie algebra <!-- MATH $\mathfrak{g}$ --> , Harish-Chandra has defined a now-classical homomorphism from the centralizer of <!-- MATH $\mathfrak{k}$ --> in the universal enveloping algebra of <!-- MATH $\mathfrak{g}$ --> into the enveloping algebra <!-- MATH $\mathcal{A}$ --> of <!-- MATH $\mathfrak{a}$ --> . He proved, using analysis, that its image is the space of Weyl group invariants in <!-- MATH $\mathcal{A}$ --> . Here the weaker fact that the image is contained in this space of invariants is proved ``purely algebraically". In fact, this proof is carried out in the general setting of semisimple symmetric Lie algebras over arbitrary fields of characteristic zero, so that Harish-Chandra's result is generalized. Related results are also obtained.