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

Question

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 $\lambda\in\Lambda_I^+$.

Answer 1

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.)

Answer 2

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.