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Let $\mathfrak{g}$ be a simple complex Lie algebra and let $\Delta$ be a system of positive roots relative a choice of Cartan subalgebra and $\mathfrak{b}$ the corresponding Borel subalgebra. Let $B<G$ the corresponding Borel subgroup in the connected Lie group $G$ with Lie algebra $\mathfrak{g}.$ Then, there is a theorem of Kostant which states the following:

Let $V=\{u + \sum_{\alpha \in \Delta} v_{\alpha} \ | u \in \mathfrak{b}, 0\neq v_{\alpha}\in \mathfrak{g}_{-\alpha}\}$ where $\mathfrak{g}_{-\alpha}$ is the root space corresponding to the negative simple root $-\alpha.$

Then, there exists an affine subspace $L\subset V$ such that the mapping $\phi: B\times L\rightarrow V$ defined by $\phi(b, v)=Ad(b)(v)$ is an isomorphism of algebraic varieties.

Is there a generalization of this theorem to parabolic subgroups/subalgebras other than the Borel? If there is not a generalization, is it because no such result can be true? I feel like there's probably a discussion of this somewhere in the vast literature on Lie algebras, but I haven't been able to turn anything up with google searches. Thank you very much for any help you can offer.

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    $\begingroup$ I am not sure what kind of generalization you are looking for, but the content of the theorem is that the adjoint $B$-action on $V$ admits a linear section $L$. If I remember correctly, this is termed "Kostant section" in the literature --- perhaps, that would be useful in searches. $\endgroup$
    – Victor Protsak
    Commented Oct 27, 2017 at 3:42
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    $\begingroup$ Thank you for the reminder. I have in mind something specific for a generalization, but per mathoverflow standards I think I need to let this question stand as is. If it helps at all, I think $V$ in this generalization should be the $\mathfrak{p}$-translate of the unique open $P$-orbit in the $-1$ part of the grading corresponding to the parabolic $P.$ $\endgroup$
    – Andy Sanders
    Commented Oct 27, 2017 at 18:10
  • $\begingroup$ @Andy: It might help to add a specific reference to the earliest source where Kostant's theorem occurs. (Many though not all of his papers are by now freely accessible online.) In recent decades MathSciNet has included citation information, so it might be possible to track further developments (if any) in the parabolic case. $\endgroup$
    – Jim Humphreys
    Commented Oct 28, 2017 at 20:00
  • $\begingroup$ P.S. It does make sense to me to seek a generalization to all parabolics, though the formulation may involve some subtlety. What you've suggested in your short comment seems reasonable. $\endgroup$
    – Jim Humphreys
    Commented Oct 28, 2017 at 20:05
  • $\begingroup$ Just for clarification, in my comment above, I should have said unique open $L$-orbit where $L$ is the appropriate Levi of the parabolic $P.$ Thank you for your suggestion Jim, I'll try to track this original reference down. $\endgroup$
    – Andy Sanders
    Commented Oct 29, 2017 at 14:23

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