2 Fix spelling and display of mathematical formulae

The usual definition of the parabolic category $\mathcal{O}^{\mathfrak{p}}$ is the following. We consider Lie algebra $\mathfrak{g}$ of the rank $r$ with the root system $\Delta$ and the set of positive roots $\Delta^+$.

We then select some subset $I$ in the set of simple roots $\alpha_1,\dots, \alpha_r$ and construct the corresponding standard parabolic subalgebra $\mathfrak{p}I$\mathfrak p_I$, Levi subalgebra$\mathfrak{l}{I}=\mathfrak{h}\oplus\sum_{\alpha\in \mathfrak l_{I}=\mathfrak h\oplus\sum_{\alpha\in \Delta_I} mathfrak{g}_{\alpha}$\mathfrak g_{\alpha}$ and the nilradical $\mathfrak{u}I=\oplus{\alpha\in \mathfrak u_I=\oplus_{\alpha\in \Delta\setminus \Delta_I}\mathfrak{g}_{\alpha}$. Then the category $\mathcal{O}^{\mathfrak{p}}$ is the subcategory of finitely-generated $U(\mathfrak{g})$-modules, which are locally $\mathfrak{u}_I$-finite and are a sums of finite-dimensional simple $U(\mathfrak{l}_I)$-modules. Then the genralized The generalized Verma modules are defined and the analogue of Bernstein-Gelfand-Gelfand resolution is proved.

The question is if it is possible to drop the requirement of standartness of parabolic subalgebra? What parts of the theory can be developed for the arbitrary parabolic subalgebra constructed from the arbitrary subset of positive roots $\Delta^+$?

Are there any papers discussing this generalisation?

1

# Is it possible to construct category $\mathcal{O}^{\mathfrak{p}}$ with non-standard parabolic subalgebra

The usual definition of the parabolic category $\mathcal{O}^{\mathfrak{p}}$ is the following. We consider Lie algebra $\mathfrak{g}$ of the rank $r$ with the root system $\Delta$ and the set of positive roots $\Delta^+$.

We then select some subset $I$ in the set of simple roots $\alpha_1,\dots, \alpha_r$ and construct the corresponding standard parabolic subalgebra $\mathfrak{p}I$, Levi subalgebra $\mathfrak{l}{I}=\mathfrak{h}\oplus\sum_{\alpha\in \Delta_I} mathfrak{g}_{\alpha}$ and the nilradical $\mathfrak{u}I=\oplus{\alpha\in \Delta\setminus \Delta_I}\mathfrak{g}_{\alpha}$. Then the category $\mathcal{O}^{\mathfrak{p}}$ is the subcategory of finitely-generated $U(\mathfrak{g})$-modules, which are locally $\mathfrak{u}_I$-finite and are a sums of finite-dimensional simple $U(\mathfrak{l}_I)$-modules. Then the genralized Verma modules are defined and analogue of Bernstein-Gelfand-Gelfand resolution is proved.

The question is if it is possible to drop the requirement of standartness of parabolic subalgebra? What parts of the theory can be developed for the arbitrary parabolic subalgebra constructed from the arbitrary subset of positive roots $\Delta^+$? Are there papers discussing this generalisation?