Let $G$ be a reductive group over a field $k$ of characteristic zero. The Jacobson-Morozov theorem gives a method of embedding any unipotent element into an $\mathfrak{sl}_2$ triple, which in turn induces a parabolic subgroup of $G$ given by of $\mathfrak p=\oplus_{n\ge0}\mathfrak g_n$, called the canonical parabolic.

It is known, e.g. from the discussion here, that not all parabolic subgroups of $G$ arise as canonical parabolics as we range over unipotent elements of $G$. But I am wondering if all Levi subgroups of $G$ do arise from the Jacobson-Morozov triples, namely from the collection of all such $\mathfrak g_0$?

  • $\begingroup$ I think no -- in the link, for $G = SL_3$, only the Borel and $G$ appear as canonical parabolics, so the only Levis that appear are the torus and $G$. $\endgroup$ Commented Mar 13, 2023 at 8:55


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