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Let $\mathfrak{g}$ be a real semisimple Lie algebra. A subalgebra $\mathfrak{p}$ of $\mathfrak{g}$ is parabolic if its complexification is parabolic in $\mathfrak{g}_\mathbb{C}$, meaning it contains a Borel subalgebra of $\mathfrak{g}_\mathbb{C}$. A nilpotent element of $\mathfrak{g}$ is regular or principal if the dimension of its stabilizer is minimal with respect to the stabilizers of all nilpotent elements of $\mathfrak{g}$. In "Wolf, Remark on nilpotent orbits", before Corollary 2 it is claimed that:

If $e \in \mathfrak{g}$ is regular-nilpotent, then $e$ is contained in a unique minimal parabolic subalgebra of $\mathfrak{g}$.

Question: Is there an easy proof for this statement, or a reference?

(Bonus: Does this property uniquely characterize regular nilpotent elements, i.e. if a nilpotent element is contained in a unique minimal parabolic subalgebra, is it regular?)


No reference for this is provided, and I could also not find this claim in any of the sources of the article. In the complex setting, this is a standard statement which one can, for example, look up in Bourbaki -- remembering that "Borel = minimal parabolic". But I don't think the proofs are easy to adapt to the real setting, since the structure theory of real minimal parabolics seems a tiny bit more complicated than in the complex setting.

Lastly: If someone could recommend to me any further literature on the study of minimal parabolics in real semisimple Lie algebras, that would also be appreciated. At the moment, I'm mostly using Bourbaki's Chapter 7-9 and Knapp's "Lie Groups beyond an introduction".

Bourbaki, Nicolas, Elements of mathematics. Lie groups and Lie algebras. Chapters 7–9. Transl. from the French by Andrew Pressley, Berlin: Springer (ISBN 3-540-43405-4/hbk). xi, 434 p. (2005). ZBL1139.17002.

Knapp, Anthony W., Lie groups beyond an introduction, Progress in Mathematics (Boston, Mass.). 140. Boston: Birkhäuser. xv, 604 p. (1996). ZBL0862.22006.beyond an introduction".

Wolf, Joseph A., Remark on nilpotent orbits, Proc. Am. Math. Soc. 51, 213-216 (1975). ZBL0316.22013.

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I had a good go at this but I couldn't nail down a proof (either by searching the literature or making my own). What I believe to be true is that for any regular-nilpotent element $X\in\mathfrak{g}$ it is contained in a minimal parabolic subalgebra $\mathfrak{p}$ equal to $\mathrm{Ker}(\mathrm{ad}_X^k) $ for some $k$. Here $k-1$ is the height of the parabolic subalgebra which is a fixed number and corresponds to the length of the filtration the parabolic subalgebra induces on itself (or more accurately on its nilradical). This is certainly not true for all parabolic subalgebras and in general the kernel bigger than $\mathfrak{p}$, but it is true for "self-dual" parabolic subalgebras of height 1. By self-dual I mean they are conjugate to complementary parabolics. Minimal parabolics are far away from being height 1 (indeed they have maximal height) but they are automatically self-dual.

Assuming the above is true we can prove what you want. Let $\mathfrak{p} = \mathrm{Ker}(\mathrm{ad}_X^k)$ and let $\mathfrak{q}$ be another minimal parabolic subalgebra containing $X$. All minimal parabolic subalgebras are conjugate so $\mathfrak{q} = g\cdot\mathfrak{p}$ for some $g \in \mathrm{Inn}(\mathfrak{g})$. Thus $X=g\cdot Y$ for some $Y\in\mathfrak{p}$. Now $\mathfrak{p} = \mathrm{Ker}(\mathrm{ad}_X^k) = \mathrm{Ker}(\mathrm{ad}_{g\cdot Y}^k) = g\cdot\mathrm{Ker}(\mathrm{ad}_{Y}^k) $. It's not too hard to see $\mathfrak{p} \leq \mathrm{Ker}(\mathrm{ad}_{Y}^k)$, since $Y$ must be in the nilradical, but this implies $\mathfrak{p} = \mathrm{Ker}(\mathrm{ad}_{Y}^k)$. This gives us $ \mathfrak{p} = g \cdot \mathfrak{p} $ and thus $\mathfrak{p} = \mathfrak{q}$.

This is unlikely to be the idea Wolf has in mind but I think it should work if you can show that it is valid to identify the parabolic subalgebra in the way I described. Here are a few references to expand on what I've said:

David M. J. Calderbank and Passawan Noppakaew, Parabolic subalgebras, parabolic buildings and parabolic projection, 2016. arxiv:1607.00370v2.

[Just a nice thing to read if you're interested in parabolic subalgebras. Covers all the filtration ideas in exhaustive detail]

Francis E. Burstall, Neil M. Donaldson, Franz Pedit, and Ulrich Pinkall, Isothermic submanifolds of symmetric R-spaces, J. Reine Angew. Math. 660 (2011), 191-243.

[The preliminary chapter of this paper covers pretty much all the terminology I have used here and chapter 4 implicitly defines the equivalent of regular-nilpotent elements]

Marcos Salvai, Circles in self dual symmetric R-spaces, 2019. arXiv:1902.01467.

[Focuses on these elements which they call "prevalent"]

These papers don't specifically answer your question and they aren't focused on minimal parabolics but I thought they might be useful nonetheless.

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    $\begingroup$ Thank you very much for the response and the literature, I've not come across those papers! I've been orienting myself mostly around the 1970s literature. I believe that I've found a proof which is parallel to how one proceed with Borel subalgebras in the complex setting, which was probably what Wolf thought of, and I'll write it down in a bit. But I will think about your approach, I'd not seen that perspective before. $\endgroup$ Jul 28 '21 at 8:44
  • $\begingroup$ Awesome. Let me know what the actual proof is/a reference for it. I'm intrigued to see it. $\endgroup$
    – Callum
    Jul 30 '21 at 10:15

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