# About finite direct sum of full subcategory of category $\mathcal{O}^\mathfrak{p}$

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?

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 $$t$$.

• 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? Dec 17, 2018 at 11:49
• @JamesCheung What is $M_i$? Dec 17, 2018 at 15:07
• 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 $i\neq j$. Dec 17, 2018 at 15:15
• 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 $M = \bigoplus M^\chi$. Dec 17, 2018 at 15:59