Lie Groups Beyond an Introduction
《Lie Groups Beyond an Introduction》 · 第 290–291 页 · Chapter V，Section 5.4「Complete Reducibility」
Let $\mathfrak g$ be a complex Lie algebra, and let $U(\mathfrak g)$ be its universal enveloping algebra. As a consequence of the generalization of Schur's Lemma given in Proposition 5.19 below, the center $\mathcal Z(\mathfrak g)$ of $U(\mathfrak g)$ acts by scalars in any irreducible unital left $U(\mathfrak g)$ module, even an infinite-dimensional one. The resulting homomorphism $\chi\colon\mathcal Z(\mathfrak g)\to\mathbb C$ is the first serious algebraic invariant of an irreducible representation of $\mathfrak g$ and is called the infinitesimal character.