I have the following problem:
Let $\mathfrak{g} = \mathfrak{gl}(n)$ be the general linear Lie algebra over $\mathbb{C}$. Denote $\Pi = \{\alpha_i:=\epsilon_i-\epsilon_{i+1}\}_{i=1}^{n-1}$ by the simple system for the standard Cartan subalgebra $\mathfrak{h}: = \bigoplus_{1\leq i\leq n}\mathbb{C}E_{ii}$. Denote $\Phi$ by the corresponding root system. Let $\lambda = \sum_{i=1}^n \lambda_i \epsilon_i \in \mathfrak{h}^*$ and there exist $1\leq j_1<\ldots <j_t\leq n$ such that $\lambda_{i_j}-\lambda_{i_{k}} \in \mathbb{Z}$ for all $i_p \leq i_j, i_{k}\leq i_{p+1} $ and $1\leq p\leq t-1$. Set $\Lambda:=\{\alpha \in \Phi~ | ~ \frac{2<\lambda, \alpha >}{<\alpha,\alpha>} \in \mathbb{Z}\}$. It is natural to consider the corresponding Levi subalgebra $\mathfrak{l} : = \mathfrak{h} \oplus \bigoplus_{\alpha\in \Lambda}\mathfrak{g}_{\alpha}$, where $\mathfrak{g}_\alpha$ is the root space corresponding the root $\alpha$. Let $\mathfrak{u}$ be the nilradical corresponding to the Levi $\mathfrak{l}$.
$\bf My~ $ $\bf Question:$ Let $L$ be a finite-dimensional, $\mathfrak{h}$-weighted, and irreducible $\mathfrak{l}$-module (of highest weight $\lambda$). Let $u$ acts trivially on $L$, then $L$ is a $\mathfrak{l}+\mathfrak{u}$-module. Is it true that the parabolic induction $\textbf{Ind}_{\mathfrak{l}+\mathfrak{u}}^{\mathfrak{g}}L$ is also irreducible $\mathfrak{g}$-module ? Thanks very much!
The induced module will not be irreducible in general. For example, start with a $\mathfrak g$-dominant weight $\lambda$, viewed as a weight of $L$. Then there is a finite dimensional irreducible representation $V$ of $\mathfrak g$ with highest weight $\lambda$. If $W$ is the irreducible $L$-representation of highest weight $\lambda$, then there is a homomorphism $W\to V$ of $L$-modules. (Or otherwise put, $W$ can be realized as the $L$-submodule of $V$ generated by a highest weight vector.) By the universal property of induced representations, this gives rise to a $\mathfrak g$-homomorphism from the induced representation to $V$. The kernel of this is a $\mathfrak g$-invariant subspace in the induced representation.
Indeed, one of the important applications of induced modules is to realize finite-dimensional irreducible representations as their quotients. For this it is necessary, that they are not irreducible themselves.