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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?

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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 $t$.

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  • $\begingroup$ First, do you means for each $i$, $M_i$ decompose into modules from $\mathcal{O}_{\chi_{i_1}}$, $\cdots$, $\mathcal{O}_{\chi_{i_{k(i)}}}$, where $k(i)$ is a number depend on $i$. While for each $i$, $k(i)$ is finite, the set $X=\bigcup_{i\in\mathbb{N}}\bigcup_{j=1}^{k(i)}\{\chi_{i_{j}}\}$ can be a infinitely set? $\endgroup$ – James Cheung Dec 17 '18 at 11:49
  • $\begingroup$ @JamesCheung What is $M_i$? $\endgroup$ – Vít Tuček Dec 17 '18 at 15:07
  • $\begingroup$ I just want to index the any countable subcollection of modules in $\mathcal{O}$ with $\mathbb{N}$ in order to clarify whether the direct sum for the category $\mathcal{O}=\bigoplus_{\chi}\mathcal{O}_{\chi}$ is an infinitely direct sum while for each module $M\in \mathcal{O}$ will be decompose into finitely many $\mathcal{O}_{\chi}$, say $\mathcal{O}_{\chi_1}$,$\cdots$,$\mathcal{O}_{\chi_n}$. while for some other $N\in \mathcal{O}$ will be decompose into finitely many $\mathcal{O}_{\chi}$, say $\mathcal{O}_{\chi_{n+1}}$,$\cdots$,$\mathcal{O}_{\chi_m}$, where $\chi_i\neq \chi_j$ for $i\neq j$. $\endgroup$ – James Cheung Dec 17 '18 at 15:15
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    $\begingroup$ Aha. Yes, I think you got it now. To repeat: The set of possible (generalized) infinitesimal characters is infinite and corresponds to $\mathfrak{h}^* / W$ but since each module $M$ in $\mathcal{O}$ is finitely generated and since on each cyclic module the center of $\mathfrak{U(g)}$ acts by some character, you get only finite direct sum $M = \bigoplus M^\chi$. $\endgroup$ – Vít Tuček Dec 17 '18 at 15:59

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