# Classification of semisimple algebraic groups which act transitively on a projective space

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

1. The representation defined by the action of $$G$$ on $$V$$ is irreducible,
2. Considering an orbit of a pure tensor, we deduce that $$G$$ is simple,
3. 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!

• The only possibilities are $G=\operatorname{PGL}(V)$ and $G=\operatorname{Sp}(V)$ for some skew-symmetric, non-degenerate bilinear form on $V$ (in case $\dim V$ is even). This was worked out by J. Tits: Espaces homogènes complexes compacts, Comm. Math. Helvet. 37, 111-120 (1962).
– abx
Mar 29, 2022 at 5:54
• @abx it should be $\mathrm{SL}(V)$ here rather than $\mathrm{PGL}(V)$, since $G$ is assumed to act on $V$.
– YCor
Mar 29, 2022 at 7:56
• @abx Where in this reference can you find such a statement? (link to the reference).
– YCor
Mar 29, 2022 at 8:04
• @YCor: p. 118, footnote 6. For a more detailed proof, valid over any field, I should have given: M. Demazure, Automorphismes et déformations des variétés de Borel, Invent. Math. 39 (1977), no. 2, 179-186.
– abx
Mar 29, 2022 at 16:24
• ... and yes, it should be $\operatorname{SL}(V)$.
– abx
Mar 29, 2022 at 16:26