Definition 3

Let $g$ be a Lie algebra and $M$ a $g$-module. An element $m \in M$ is called invariant (with respect to the $g$-module structure on $M$ or with respect to the corresponding representation of $g$) if $x_M \cdot m = 0$ for all $x \in g$.

Let $G$ be a connected real Lie group, $g$ its Lie algebra, $\theta$ an analytic representation of $G$ on a finite-dimensional real vector space $E$ and $\rho$ the corresponding representation of $g$ on $E$. Let $m \in E$. The element $m$ is invariant with respect to $\rho$ if and only if $\theta(g) \cdot m = m$ for all $g \in G$. This justifies the use of the word "invariant".