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I am reading this paper: Representation type of the blocks of category $\mathcal{O}_S$

On p. 199, it said that enter image description here

While on p. 183 (Section 9.2) of Representations of Semisimple Lie Algebras in the BGG Category $\mathcal{O}^\mathfrak{p}$. Humphreys said that

enter image description here

Let $\Lambda_I^+:=\{\lambda\in\mathfrak{h}^*: \langle\lambda,\alpha^\lor\rangle\in\mathbb{Z}^{\ge 0} \ \forall \alpha\in I\}$ and $X_I^+:=\{\lambda\in\mathfrak{h}^*: \langle\lambda,\alpha^\lor\rangle\in\mathbb{Z}^{\ge 0} \ \forall \alpha\in\Phi_I^+\}$.

Obviously, they are talking about the same set. It is obviously that $X_I^+\subset \Lambda_I^+$.

But how to prove the other containment?

I think it is not trivial to see if $\alpha,\beta\in I$ such that $\alpha+\beta\in\Phi_I$, then $\dfrac{2\langle\lambda,\alpha+\beta\rangle}{\langle\alpha,\alpha\rangle+2\langle\alpha,\beta\rangle+\langle\beta,\beta\rangle}\in\mathbb{Z}^{\ge 0}$ for $\lambda\in\Lambda_I^+$.

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  • $\begingroup$ Note that the problem still seems to be open, of classifying all "blocks" of an arbitrary subcategory $\mathcal{O}^\mathfrak{p}$, as indicated in the abstract of the cited paper by Boe and Nakano. [comment expanded somewhat] $\endgroup$ – Jim Humphreys Aug 10 '19 at 21:30
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Just think about the dual parabolic root system $\Phi^\vee_I := \{\alpha^\vee\colon \alpha \in \Phi_I\}$. You can choose $\{\alpha^\vee\colon \alpha\in I\}$ to be a set of simple roots for $\Phi^\vee_I$, in which case we will have $(\Phi_I^\vee)^+ = \{\alpha^\vee\colon \alpha \in \Phi^+_I\}$ (these are well-known facts, should be in any book on root systems). Hence for any $\beta^\vee \in \Phi^+_I$ we have $\beta^\vee = \sum_{\alpha \in I} c_\alpha \alpha^\vee$ for some $c_\alpha \in \mathbb{Z}_{\geq 0}$, which proves the inclusion $\Lambda^+_I \subseteq X^+_I$ you want.

(Note that if we write $\beta = \sum_{\alpha \in I} c'_\alpha \alpha$, we will not in general have $c_\alpha=c'_\alpha$, which is probably your source of confusion, and something that can trip people up when first working with root systems.)

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The parabolic root system doesn't really play a role here. This question is basically:

Why is the integrality wrt simple roots the same as integrality wrt all roots?

This can be explained either of:

  1. Direct case by case calculation in the "$\epsilon$-basis".
  2. Induction on the height using properties of root systems.
  3. Showing that any coroot is integral linear combination of simple coroots. See this answer math.stackexchange which does it in one-line calculation.

edit: The third option is of course also implied by the fact that Sam Hopkins uses in his answer: If $\Phi$ is a root system, then $\Phi^\vee$ is also a root system.

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    $\begingroup$ Right, I should've noted that the parabolic aspects of the question are irrelevant. Also, just so there is no potential for confusion: "root system" here and everywhere for this question means "crystallographic root system." $\endgroup$ – Sam Hopkins Aug 9 '19 at 22:53
  • $\begingroup$ @Vit: Maybe it's easier to get a unified argument by appealing to the well-known fact that reflections relative to roots in 𝐼 generate $W_I$? [misprint corrected] $\endgroup$ – Jim Humphreys Aug 10 '19 at 21:21

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