As is shown in ***Representations of Semisimple Lie Algebras in the BGG Category $\mathcal{O}$***, every nonzero module $M \in \mathcal{O}^\mathfrak{p}$ has a finite filtration with nonzero quotients, each of which is a highest weight module in $\mathcal{O}^\mathfrak{p}$. 
Thus the action of $Z(g)$ on $M$ is finite. 


Let
$
M^\chi = \{v \in M \ | \ (z - \chi(z))^nv = 0 \ \text{for some $n \in \mathbb{Z}\_{>0}$ depending on $z$}\},
$
then $z - \chi(z)$ acts locally nilpotently on $M^\chi$ for all $z \in Z(\mathfrak{g})$ and $M^\chi$ is a $U(\mathfrak{g})$-submodule of $M$.


Denote by $\mathcal{O}^\mathfrak{p}\_\chi$ the full subcategory of $\mathcal{O}^\mathfrak{p}$ whose objects are of the form $M^\chi$, then we have the following direct sum decomposition
$
\mathcal{O}^\mathfrak{p}=\bigoplus\_{\chi}\mathcal{O}^\mathfrak{p}\_\chi,
$
where $\chi = \chi\_\lambda$ for some $\lambda \in\mathfrak{h}^\*$.


Because $M$ is generated by finitely many weight
vectors, it must therefore be the direct sum of finitely many nonzero submodules $M^\chi$.


We call $\lambda,\mu\in\mathfrak{h}^\*$ are linked if $\lambda=w\cdot\mu:=w(\mu+\rho)-\rho$ for some $w\in W$ where $\rho$ is the half sum of positive roots and $W$ is the Weyl group of $\Phi$.


The set of $\Phi^+\_I$-dominant integral weights in $\mathfrak{h}^\*$ is
$
\Lambda^+\_I = \{\lambda \in \mathfrak{h}^\* : \langle \lambda, \alpha^\lor\rangle \in \mathbb{Z}^{\ge 0} \ \text{for all }\alpha \in \Phi^+\_I\},
$ there is a fact that $L(\lambda)\in\mathcal{O}^\mathfrak{p}$ iff $\lambda\in\Lambda^+\_I$.


Then here the question, let $\mathfrak{g}=\mathfrak{sl}(2,\mathbb{C})$, then denote $\Phi$ the root system and $\Delta=\{\alpha,\beta\}$ the set of simple roots in $\Phi$. Consider $t\alpha+(1-t)(\alpha+\beta)\in \Lambda\_I^+$ with $0\le t\le 1$. Then there should have infinitely many non-linked weights (since $t$ infinite and $t\alpha+(1-t)(\alpha+\beta)$ are weights lie between $\alpha$ and $\alpha+\beta$) such that $L(t\alpha+(1-t)(\alpha+\beta))\in\mathcal{O}^\mathfrak{p}\_{\chi\_{t\alpha+(1-t)(\alpha+\beta)}}$, 


which then should gives infinitely many $\mathcal{O}^\mathfrak{p}\_{\chi}$.


Can anyone give me some examples/ explain why my intuition fails to convince me that there is indeed finitely many $\Phi^+\_I$-dominant integral weights up to linkage? 



About finite direct sum of full subcategory of category $\mathcal{O}^\mathfrak{p}$
There is infinitely many linkage classes each containing some $\Phi^+\_I$-dominant elements. But since any module from $\mathcal{O}$ is finitely generated it will decompose only into finitely many modules from $\mathcal{O}\_\chi$.


I hope I understood your question correctly. There are some problems, e.g. $\mathfrak{sl}\_2$ has just one simple root and $t\alpha + (1-t)(\alpha + \beta)$ would be integral (i.e. element of $\Lambda\_I$) only for finitely many $t$.


