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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 $|W\cdot\lambda|=|W|$?

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    $\begingroup$ I would say it depends on context whether or not you want to consider non-integral weights as weights. Just like it depends on context whether or not you will consider infinite-dimensional representations. Often someone is interested exclusively in finite-dimensional representations: it's good to point this out at the beginning of your article/thesis/whatever, but you don't want to write "finite-dimensional" every time so you can note that from then on all representations will be finite-dimensional. Same story with integral weights. $\endgroup$
    – Sam Hopkins
    Commented Jun 9, 2019 at 12:22

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  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.
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  • $\begingroup$ Do you have any reference about your first claim? I agree with you that any element of $\mathfrak{h}^*$ is called weight, but I want to make sure my PhD thesis contain no mistake. $\endgroup$
    – James Cheung
    Commented Jun 9, 2019 at 8:58
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    $\begingroup$ As Sam Hopkins says, conceptually and historically the primitive notion is that of weight of (some vector in some module), so it all depends on what modules enter. “Weights” in (1972, §13.1) were members of the lattice $\Lambda$, now this book must “broaden the idea of “weight”” and qualify members of $\Lambda$ as “integral” (§0.7). $\endgroup$
    – Francois Ziegler
    Commented Jun 9, 2019 at 13:11
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In Humphreys, weights are defined in section 0.7 "Representations". He notes that in the finite case all weights are integral, but that this is not true for infinite dimensional representations.

Note that by definition, if $V_\lambda \neq 0$, then $\lambda$ is a weight of $V$. Since for any $\lambda\in\mathfrak{h}^*$ there are modules V with $V_\lambda\neq 0$ (for example the Verma module of highest weight $\lambda$), there are no integrality conditions on $\lambda$ for it to be a weight.

Regarding question 3: If $\lambda$ is regular then $\lambda(\alpha) \in\mathbb{Z}$ for all $\alpha\in\Phi$. This should be easy to see in the case of $\mathfrak{sl}_2$, which immediately extends to higher ranks.

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