# About subcategory of parabolic Category $\mathcal{O}^\mathfrak{p}$

It follows from Exercise 1.13 in Humphreys' Category $$\mathcal{O}$$ book that $$M\in\mathcal{O}^\mathfrak{p}_{\chi_\lambda}$$ has a direct sum decomposition $$M=\oplus M_i$$ such that all weights of each $$M_i$$ are contained in a single coset of root lattice $$\Lambda_r$$ in $$\mathfrak{h}^*$$. Then, the category $$\mathcal{O}^\mathfrak{p}_{\chi_\lambda}$$ decomposes as a direct sum of full subcategories, which can be indexed by the nonempty intersection of the orbit $$W\cdot\lambda$$ with the cosets $$\mathfrak{h}^*/\Lambda_r$$. From now on, we use the anti-dominant weight $$\mu$$ in the intersection to parameterize the corresponding subcategory of $$\mathcal{O}^\mathfrak{p}_{\chi_\lambda}$$. We denote this subcategories by $$\mathcal{O}^\mathfrak{p}_{\mu}$$. In particular, if $$I=\emptyset$$, then the parabolic subalgebra $$\mathfrak{p}$$ collapses to a Borel subalgebra $$\mathfrak{b}$$ of $$\mathfrak{g}$$ and $$\mathcal{O}^\mathfrak{p}_{\mu}$$ turns into the ordinary block $$\mathcal{O}_{\mu}$$ of the BGG category $$\mathcal{O}$$.

I would like to know whether $$\mathcal{O}^\mathfrak{p}_{\mu}=\mathcal{O}^\mathfrak{p}_{\chi_\mu}$$ for $$\mu$$ being regular integral?

$$\mathcal{O}_\chi^{\mathfrak{p}}$$ is a full subcategory of $$\mathcal{O}_\chi$$, so the answer is yes. That is:
• $$\mathcal{O}_\mu = \mathcal{O}_{\chi_\mu}$$, for all $$\mu$$
• $$\mathcal{O}_\mu^{\mathfrak{p}}$$ is the full subcategory of $$\mathcal{O}_\mu$$ of locally $$\mathfrak{l}$$-finite modules.
• $$\mathcal{O}_{\chi_\mu}^{\mathfrak{p}}$$ is the full subcategory of $$\mathcal{O}_{\chi_\mu}$$ of locally $$\mathfrak{l}$$-finite modules.
Hence $$\mathcal{O}_\mu^{\mathfrak{p}} = \mathcal{O}_{\chi_\mu}^{\mathfrak{p}}$$ for all $$\mu$$.
• I think the right argument requires the fact that $\mu$ is integral. – James Cheung Sep 27 '18 at 13:31
• Actually, reading the question again I don't think I understand it. Using your notation, the $\mu$ in $\mathcal{O}_\mu^{\mathfrak{p}}$ is required to be anti-dominant, you define it as such. How is $\mathcal{O}_\mu^{\mathfrak{p}}$ different from $\mathcal{O}_{\chi_\mu}^{\mathfrak{p}}$? – Johan Kåhrström Sep 28 '18 at 7:48
• $\mathcal{O}_{\chi_\mu}^\mathfrak{p}$ can be decomposed into direct sum of full.subcategories, which can be indexed by the nonempty intersection of the orbit $W\cdot\mu$ with the cosets $\mathfrak{h}^*/\Lambda_r$. While $\mathcal{O}_{\mu}^\mathfrak{p}$ is just one direct summand of direct sum of full.subcategories. I would like to know whether the direct sum of full.subcategories is just trivial direct sum when $\mu$ is integral. – James Cheung Sep 28 '18 at 19:21