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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!

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    $\begingroup$ 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). $\endgroup$
    – abx
    Mar 29, 2022 at 5:54
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    $\begingroup$ @abx it should be $\mathrm{SL}(V)$ here rather than $\mathrm{PGL}(V)$, since $G$ is assumed to act on $V$. $\endgroup$
    – YCor
    Mar 29, 2022 at 7:56
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    $\begingroup$ @abx Where in this reference can you find such a statement? (link to the reference). $\endgroup$
    – YCor
    Mar 29, 2022 at 8:04
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    $\begingroup$ @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. $\endgroup$
    – abx
    Mar 29, 2022 at 16:24
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    $\begingroup$ ... and yes, it should be $\operatorname{SL}(V) $. $\endgroup$
    – abx
    Mar 29, 2022 at 16:26

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