On p.24 of the book **"Representations of Semisimple Lie Algebras in the BGG Category $\mathcal{O}$"**.


**I quote:**
"For example, the fixed point $-\rho$ under the dot action lies in a linkage
class by itself (and no other weight does). The usual notion of regular
weight for $\lambda \in\mathfrak{h}^\*$, requiring that the isotropy group of $\lambda$ in W be trivial, or
$|W\lambda| = |W|$, is replaced in the setting of linkage classes by a shifted notion of regular weight (which might also be called dot-regular): the **weight**
$\lambda \in\mathfrak{h}^\*$ is regular if $|W \cdot\lambda| = |W|$, or in other words, $\langle \lambda + \rho, \alpha^\lor\rangle \neq 0$ for
all $\alpha \in \Phi$. **Weights** which are not regular may be called singular."


**My questions:**


1. What is the definition of "**weight**" in bold? Is it just mean an element in $\mathfrak{h}^\*$. Where can I find its definition? I just know the definition of a weight of a module.
2. Two members of my PhD committee insist that the terminology "weight" means an element in $\mathfrak{h}^\*$ with some integral conditions, is that true?
3. If 2 is true, can I still define $\lambda\in\mathfrak{h}^\*$ is a regular element if $|W\cdot\lambda|=|W|$?



About weights in $\mathfrak{h}^*$
1. Any element of $\mathfrak{h}^\*$ is called weight. I don't know why in this case the author used definite article. As a non-native speaker I would use indefinite one, i.e. "a weight is called regular if ..."
2. That sounds wrong. One can speak about weights which are integral (or dominant) with respect to some (positive) root system. Perhaps the members of your committee are talking about weights of some (class of) module(s)? In that case it might be the case that all such weights have some integrality property.
3. Sure. Integrality and regularity are independent definitions.


