Let $k$ be an algebraically closed field of characteristic 0, and $V$ be a vector space on $k$ of dimension $>1$.

In this situation, is there a classification of connected semisimple groups (up to isogeny) that act on $V$ and the induced action on $\mathbb{P}(V)$ is transitive?

I thought that in this case

- The representation defined by the action of $G$ on $V$ is irreducible,
- Considering an orbit of a pure tensor, we deduce that $G$ is simple,
- The center of $G$ acts trivially on $\mathbb{P}(V)$, so we can only consider the adjoint case and $G(k)$ becomes a simple group by the theory of Tits' system,

but I got stuck here.

If you have any ideas or references, please let me know. Thanks!

Espaces homogènes complexes compacts, Comm. Math. Helvet. 37, 111-120 (1962). $\endgroup$Automorphismes et déformations des variétés de Borel, Invent. Math. 39 (1977), no. 2, 179-186. $\endgroup$1more comment