Let $\mathfrak g$ be a real semisimple Lie algebra (without compact factors) with Iwasawa decomposition $\mathfrak g=\mathfrak k\oplus \mathfrak a\oplus \mathfrak u$. Let $\mathfrak p$ be a parabolic subalgebra of $\mathfrak g$ containing $\mathfrak a$ and $\mathfrak u$. Let $\mathfrak u_{\mathfrak p}$ be the unipotent radical of $\mathfrak p$. Let $\Phi(\mathfrak u_{\mathfrak p})$ be the relative roots of $\Phi(\mathfrak g, \mathfrak a)$ appeared in the root space decomposition of $\mathfrak u_{\mathfrak p}$. Let $\beta\in \mathfrak a^*$ be an algebraically integral weight dominated by $\Phi(\mathfrak u_{\mathfrak p})$, i.e $\langle \beta, \alpha \rangle\ge 0 $ for every $\alpha\in \Phi(\mathfrak u_{\mathfrak p})$ with respect to the usual inner product induced by the Killing form.
Is it true that there is an irreducible real representation $(\rho, V)$ of $\mathfrak g$ such that $\beta$ is a weight of $V$ and there is a nonzero vector $v\in V_{\beta}$ annihilated by $\mathfrak u_{\mathfrak p}$? I.e. $\rho(u)v=0$ for every $u \in \mathfrak u_{\mathfrak p}$.
In the case where $\mathfrak g$ is split and $\mathfrak p$ is the minimal parabolic subalgebra we get an affirmative answer from highest weight theorem. In this sense what I am asking can be considered as a generalization of highest weight theorem.
It would also be interesting to have an answer in the case where $\mathfrak g $ is split. Maybe experts in Lie algebra can answer this question immediately
