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?



About subcategory of parabolic Category $\mathcal{O}^\mathfrak{p}$
$\mathcal{O}\_\chi^{\mathfrak{p}}$ 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 $\mu$.


