Let $\mathfrak{g}$ be a complex semisimple Lie algebra and let $\mathfrak{h}$ be a Cartan subalgebra of $\mathfrak{g}$. Fix a Borel subalgebra $\mathfrak{b}$ containing $\mathfrak{h}$ and fix a parabolic subalgebra $\mathfrak{p}$ containing $\mathfrak{b}$. Let $I \subseteq\Delta$ be the subset of simple roots corresponding to $\mathfrak{p}$. Denote by $\Phi_I$ the subsystem generated by $I$. i.e., $\Phi_I=\Phi\cap\sum_{\alpha\in I}\mathbb{Z}\alpha$. Let $\Phi^+_I=\Phi_I\cap\Phi^+$.

Let $\mathfrak{l} = \mathfrak{h}\oplus\sum_{\alpha\in \Phi_I}\mathfrak{g}_\alpha$ be the Levi subalgebra. Denote by $\mathfrak{u}$ the nilpotent radical of $\mathfrak{p}$ and let $\overline{\mathfrak{u}}$ be the dual space of $\mathfrak{u}$. Note that $\mathfrak{p}=\mathfrak{l}\oplus \mathfrak{u}$.

The category $\mathcal{O}$ is the category of all finitely generated, locally $\mathfrak{b}$-finite and $\mathfrak{h}$-semisimple $\mathfrak{g}$-modules, where $\mathfrak{g}$ is a complex semisimple Lie algebra with Cartan subalgebra $\mathfrak{h}$ and Borel subalgebra $\mathfrak{b}$ containing $\mathfrak{h}$.

The category $\mathcal{O}^\mathfrak{p}$ is the full subcategory of $U(\mathfrak{g})$-Mod such that every object $M$ in category $\mathcal{O}^\mathfrak{p}$ satisfies the following conditions.

1. $M$ is a finitely generated $U(\mathfrak{g})$-module.
2. $M$ is a direct sum of finite-dimensional simple $U(\mathfrak{l})$-modules.
3. $M$ is locally finite as a $U(\mathfrak{p})$-module.

The Verma module is of the form $M(\lambda) = U(\mathfrak{g})\otimes_{U(\mathfrak{b})} \mathbb{C}_\lambda$, where $\mathbb{C}_\lambda$ is a simple $\mathfrak{b}$-module with weight $\lambda$. Denote by $L(\lambda)$ the unique simple quotient of $M(\lambda)$.

The parabolic Verma module is defined to be $M_I(\lambda) = U(\mathfrak{g})\otimes_{U(\mathfrak{p})} F(\lambda)$, where $F(\lambda)$ is the simple finite-dimensional $\mathfrak{l}$-module with highest weight $\lambda$.

The set of $\Phi^+_I$-dominant integral weights in $\mathfrak{h}^*$ is defined to be $\Lambda^+_I = \{\lambda \in \mathfrak{h}^* : \langle\lambda,\alpha^\lor\rangle \in \mathbb{Z}^{\ge 0} \ \text{for all }\alpha \in \Phi^+_I\}$.

By Proposition 9.3 and Theorem 9.4 in Representations of Semisimple Lie Algebras in the BGG Category $\mathcal{O}$, we get $L(\lambda)\in\mathcal{O}^\mathfrak{p}\iff \lambda\in\Lambda_I^+$. Also $\lambda\in \Lambda_I^+ \implies M_I(\lambda)\in \mathcal{O}^\mathfrak{p}$.

My question: What is about $M(\lambda)$? Does $\lambda\in \Lambda_I^+$ implies $M(\lambda)\in \mathcal{O}^\mathfrak{p}$? If not, any counterexample?

Let $\mathfrak{g}$ be a complex semisimple Lie algebra and let $\mathfrak{h}$ be a Cartan subalgebra of $\mathfrak{g}$. Fix a Borel subalgebra $\mathfrak{b}$ containing $\mathfrak{h}$ and fix a parabolic subalgebra $\mathfrak{p}$ containing $\mathfrak{b}$. Let $I \subseteq\Delta$ be the subset of simple roots corresponding to $\mathfrak{p}$. Denote by $\Phi_I$ the subsystem generated by $I$. i.e., $\Phi_I=\Phi\cap\sum_{\alpha\in I}\mathbb{Z}\alpha$. Let $\Phi^+_I=\Phi_I\cap\Phi^+$.

Let $\mathfrak{l} = \mathfrak{h}\oplus\sum_{\alpha\in \Phi_I}\mathfrak{g}_\alpha$ be the Levi subalgebra. Denote by $\mathfrak{u}$ the nilpotent radical of $\mathfrak{p}$ and let $\overline{\mathfrak{u}}$ be the dual space of $\mathfrak{u}$. Note that $\mathfrak{p}=\mathfrak{l}\oplus \mathfrak{u}$.

The category $\mathcal{O}$ is the category of all finitely generated, locally $\mathfrak{b}$-finite and $\mathfrak{h}$-semisimple $\mathfrak{g}$-modules, where $\mathfrak{g}$ is a complex semisimple Lie algebra with Cartan subalgebra $\mathfrak{h}$ and Borel subalgebra $\mathfrak{b}$ containing $\mathfrak{h}$.

The category $\mathcal{O}^\mathfrak{p}$ is the full subcategory of $U(\mathfrak{g})$-Mod such that every object $M$ in category $\mathcal{O}^\mathfrak{p}$ satisfies the following conditions.

1. $M$ is a finitely generated $U(\mathfrak{g})$-module.
2. $M$ is a direct sum of finite-dimensional simple $U(\mathfrak{l})$-modules.
3. $M$ is locally finite as a $U(\mathfrak{p})$-module.

The Verma module is of the form $M(\lambda) = U(\mathfrak{g})\otimes_{U(\mathfrak{b})} \mathbb{C}_\lambda$, where $\mathbb{C}_\lambda$ is a simple $\mathfrak{b}$-module with weight $\lambda$. Denote by $L(\lambda)$ the unique simple quotient of $M(\lambda)$.

The parabolic Verma module is defined to be $M_I(\lambda) = U(\mathfrak{g})\otimes_{U(\mathfrak{p})} F(\lambda)$, where $F(\lambda)$ is the simple finite-dimensional $\mathfrak{l}$-module with highest weight $\lambda$.

The set of $\Phi^+_I$-dominant integral weights in $\mathfrak{h}^*$ is defined to be $\Lambda^+_I = \{\lambda \in \mathfrak{h}^* : \langle\lambda,\alpha^\lor\rangle \in \mathbb{Z}^{\ge 0} \ \text{for all }\alpha \in \Phi^+_I\}$.

By Proposition 9.3 and Theorem 9.4 in Representations of Semisimple Lie Algebras in the BGG Category $\mathcal{O}$, we get $L(\lambda)\in\mathcal{O}^\mathfrak{p}\iff \lambda\in\Lambda_I^+$. Also $\lambda\in \Lambda_I^+ \implies M_I(\lambda)\in \mathcal{O}^\mathfrak{p}$.

What about $M(\lambda)$? Does $\lambda\in \Lambda_I^+$ imply $M(\lambda)\in \mathcal{O}^\mathfrak{p}$? If not, any counterexample?

The category $\mathcal{O}$ is the category of all finitely generated, locally $\mathfrak{b}$-finite and $\mathfrak{h}$-semisimple $\mathfrak{g}$-modules, where $\mathfrak{g}$ is a complex semisimple Lie algebra with Cartan subalgebra $\mathfrak{h}$ and Borel subalgebra $\mathfrak{b}$ containing $\mathfrak{h}$.

The Verma module is of the form $M(\lambda) = U(\mathfrak{g})\otimes_{U(\mathfrak{b})} \mathbb{C}_\lambda$, where $\mathbb{C}_\lambda$ is a simple $\mathfrak{b}$-module with weight $\lambda$. Denote by $L(\lambda)$ the unique simple quotient of $M(\lambda)$.

The parabolic Verma module is defined to be $M_I(\lambda) = U(\mathfrak{g})\otimes_{U(\mathfrak{p})} F(\lambda)$, where $F(\lambda)$ is the simple finite-dimensional $\mathfrak{l}$-module with highest weight $\lambda$.

The set of $\Phi^+_I$-dominant integral weights in $\mathfrak{h}^*$ is defined to be $\Lambda^+_I = \{\lambda \in \mathfrak{h}^* : \langle\lambda,\alpha^\lor\rangle \in \mathbb{Z}^{\ge 0} \ \text{for all }\alpha \in \Phi^+_I\}$.

By Proposition 9.3 and Theorem 9.4 in Representations of Semisimple Lie Algebras in the BGG Category $\mathcal{O}$, we get $L(\lambda)\in\mathcal{O}^\mathfrak{p}\iff \lambda\in\Lambda_I^+$. Also $\lambda\in \Lambda_I^+ \implies M_I(\lambda)\in \mathcal{O}^\mathfrak{p}$.

My question: What is about $M(\lambda)$? Does $\lambda\in \Lambda_I^+$ implies $M(\lambda)\in \mathcal{O}^\mathfrak{p}$? If not, any counterexample?

Let $\mathfrak{g}$ be a complex semisimple Lie algebra and let $\mathfrak{h}$ be a Cartan subalgebra of $\mathfrak{g}$. Fix a Borel subalgebra $\mathfrak{b}$ containing $\mathfrak{h}$ and fix a parabolic subalgebra $\mathfrak{p}$ containing $\mathfrak{b}$. Let $I \subseteq\Delta$ be the subset of simple roots corresponding to $\mathfrak{p}$. Denote by $\Phi_I$ the subsystem generated by $I$. i.e., $\Phi_I=\Phi\cap\sum_{\alpha\in I}\mathbb{Z}\alpha$. Let $\Phi^+_I=\Phi_I\cap\Phi^+$.

Let $\mathfrak{l} = \mathfrak{h}\oplus\sum_{\alpha\in \Phi_I}\mathfrak{g}_\alpha$ be the Levi subalgebra. Denote by $\mathfrak{u}$ the nilpotent radical of $\mathfrak{p}$ and let $\overline{\mathfrak{u}}$ be the dual space of $\mathfrak{u}$. Note that $\mathfrak{p}=\mathfrak{l}\oplus \mathfrak{u}$.

The category $\mathcal{O}$ is the category of all finitely generated, locally $\mathfrak{b}$-finite and $\mathfrak{h}$-semisimple $\mathfrak{g}$-modules, where $\mathfrak{g}$ is a complex semisimple Lie algebra with Cartan subalgebra $\mathfrak{h}$ and Borel subalgebra $\mathfrak{b}$ containing $\mathfrak{h}$.

The category $\mathcal{O}^\mathfrak{p}$ is the full subcategory of $U(\mathfrak{g})$-Mod such that every object $M$ in category $\mathcal{O}^\mathfrak{p}$ satisfies the following conditions.

1. $M$ is a finitely generated $U(\mathfrak{g})$-module.
2. $M$ is a direct sum of finite-dimensional simple $U(\mathfrak{l})$-modules.
3. $M$ is locally finite as a $U(\mathfrak{p})$-module.

The Verma module is of the form $M(\lambda) = U(\mathfrak{g})\otimes_{U(\mathfrak{b})} \mathbb{C}_\lambda$, where $\mathbb{C}_\lambda$ is a simple $\mathfrak{b}$-module with weight $\lambda$. Denote by $L(\lambda)$ the unique simple quotient of $M(\lambda)$.

The parabolic Verma module is defined to be $M_I(\lambda) = U(\mathfrak{g})\otimes_{U(\mathfrak{p})} F(\lambda)$, where $F(\lambda)$ is the simple finite-dimensional $\mathfrak{l}$-module with highest weight $\lambda$.

The set of $\Phi^+_I$-dominant integral weights in $\mathfrak{h}^*$ is defined to be $\Lambda^+_I = \{\lambda \in \mathfrak{h}^* : \langle\lambda,\alpha^\lor\rangle \in \mathbb{Z}^{\ge 0} \ \text{for all }\alpha \in \Phi^+_I\}$.

By Proposition 9.3 and Theorem 9.4 in Representations of Semisimple Lie Algebras in the BGG Category $\mathcal{O}$, we get $L(\lambda)\in\mathcal{O}^\mathfrak{p}\iff \lambda\in\Lambda_I^+$. Also $\lambda\in \Lambda_I^+ \implies M_I(\lambda)\in \mathcal{O}^\mathfrak{p}$.

My question: What is about $M(\lambda)$? Does $\lambda\in \Lambda_I^+$ implies $M(\lambda)\in \mathcal{O}^\mathfrak{p}$? If not, any counterexample?