Let $G_\mathbb{C}$ be a complex simple Lie group and let $V_\lambda$ be its finite dimensional irreducible representation with highest weight $\lambda$. Define $\mathcal{H}\_{\mathbb{C}} \subset V_\lambda$ to be the set of all possible highest weight vectors for all possible choices of Borel subgroups in $G_\mathbb{C}$.

It is well known that the projectivization $\mathbb{P}\mathcal{H}\_\mathbb{C} \subset \mathbb{P}V$ is homogeneous space $G\_\mathbb{C}/P$ where $P$ is a parabolic subgroup easily determined by $\lambda$. As was reminded to me by Sasha and Alex, the action of $G_\mathbb{C}$ is even transitive on $\mathcal{H}\_\mathbb{C}$. It is also known that the space $G\_\mathbb{C} / P$ is in fact a projective algebraic variety cut out by quadrics.

Now consider a real form $G_\mathbb{R}$ and its real irreducible representation $V\_\mathbb{R}$ such that its complexification is $V_\lambda$.

**Question 1:** Does $G\_\mathbb{R}$ act transitively on the set of real points of $\mathcal{H}\_\mathbb{C}$ and of $\mathbb{P}\mathcal{H}\_\mathbb{C}$?

Consider a real form $G_\mathbb{R}$ with irreducible complex representation $V$ such that $V \simeq V_\lambda$ as a $G\_\mathbb{C}$ representations and let $v_\lambda$ be a highest weight vector.

**Question 2:** What are the $G\_\mathbb{R}$ orbits of $v_\lambda$ and $[v_\lambda]\in\mathbb{P}V_\lambda$?

*Edit:* Due to a highly nontrivial amount of confusion on my part the question was edited three times and so some of the answers and comments may seem out of place. I deeply apologize for that.

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