I'm currently reading Representations Of Compact Lie Groups by T. Bröcker and T. Tom Dieck. In the section to Representations and Lie Algebras (p.112) they introduce the notion of (infinitesimal) weights of a (complex) $T$-module $V$, where $T$ is a maximal torus and $V$ a euclidian vector space. They define a weight of $V$ as follows:
A homomorphism $\vartheta\colon T\to U(1)$ is called a weight of $V$ if the corresponding weight space $$V(\vartheta) = \{v\in V\mid xv = \vartheta(x)\cdot v\ \text{for all}\ X\in \mathrm{L}T\}$$ is nonzero.
In Notation (9.7) they state ($L_X$ refers to the Lie-derivative at $X$):
An $\mathbb{R}$-linear form $\alpha\colon \mathrm{L}T\to \mathbb{R}$ is a real (infinitesimal) weight of the $T$-module $V$ if $2\pi i \alpha$ is an infinitesimal weight of $V$. The weight space of $\alpha$ is then $$V(\alpha) = \{v\in V\mid \mathrm{L}_Xv = 2\pi i \alpha(X)\cdot v\ \text{for all}\ X\in \mathrm{L}T\}.$$
For clarification, i provide the definition of an infinitesimal weight as in Definition 9.3 on p.112:
An $\mathbb{R}$-linear form $\Theta\colon \mathrm{L}T \to \operatorname{LU}(1) = i\mathbb{R} \subset \mathbb{C}$ is called an infinitesimal weight of the $T$-module $V$ if the corresponding weight space $$V(\Theta) = \{v\in V\mid \mathrm{L}_Xv = \Theta(X)\cdot v\ \text{for all}\ X\in \mathrm{L}T\}$$ is nonzero.
My question: What is meant by
if $2\pi i\alpha$ is an infinitesimal weight of $V$
?
According to the definitions, an infinitesimal weight of $V$ is an $\mathbb{R}$-linear form $\mathrm{L}T \to\operatorname{LU}(1)$. So what is meant by saying that $2\pi i\alpha$ is an infinitesimal weight of $V$?
Thanks for any help!