# Confusion about $\lambda\in\mathfrak{h}^*$ such that $L(\lambda)\in\mathcal{O}^\mathfrak{p}$

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

On p. 199, it said that

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

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 $$\lambda\in\Lambda_I^+$$.

• 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] – Jim Humphreys Aug 10 '19 at 21:30

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.)

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.

• 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." – Sam Hopkins Aug 9 '19 at 22:53
• @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] – Jim Humphreys Aug 10 '19 at 21:21