On the Harish-Chandra homomorphism

Abstract:

Using the Iwasawa decomposition $\mathfrak {g} = \mathfrak {k} \oplus \mathfrak {a} \oplus \mathfrak {n}$ of a real semisimple Lie algebra $\mathfrak {g}$, Harish-Chandra has defined a now-classical homomorphism from the centralizer of $\mathfrak {k}$ in the universal enveloping algebra of $\mathfrak {g}$ into the enveloping algebra $\mathcal {A}$ of $\mathfrak {a}$. He proved, using analysis, that its image is the space of Weyl group invariants in $\mathcal {A}$. 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.