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Let $\mathfrak g$ be a (finite dimensional) simple (nonabelian) Lie algebra over $\mathbb C$. I need a complete list of irreducible (nontrivial) representations $V$ of $\mathfrak g$ such that the Weyl group of $\mathfrak g$ acts transitively on weights of $V$.

It is obvious that for any given $\mathfrak g$, one can list all such $V$. But I don't know whether it is possible to give a complete list. The examples I have are the natural actions of $\mathfrak{sl}_n$ on $\mathbb C^n$.

This question may looks unnatural to experts in Lie algebra. The reason I need it is to solve a problem in homogeneous dynamics and those $V$ are bad cases. I believe I can handle the example I give above, but in general if there are too many of them I am not sure whether I can handle them or not.

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    $\begingroup$ Google for cominuscule representations. $\endgroup$
    – Sasha
    Commented Feb 7, 2015 at 20:50
  • $\begingroup$ As Sasha points out, this is all standard material. You might also search the Math Overflow pages for questions involving the term "minuscule". $\endgroup$
    – Jim Humphreys
    Commented Feb 7, 2015 at 22:08
  • $\begingroup$ @ Sasha: Thanks, that's exactly what I want. $\endgroup$
    – ronggang
    Commented Feb 7, 2015 at 22:09
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    $\begingroup$ @ronggang: The usual term is "minuscule" (though "cominuscule" also occurs in some contexts). See the explicit entry in Wikipedia en.wikipedia.org/wiki/Minuscule_representation (but note that the concept comes originally from Bourbaki and others). There is also a recent monograph by Richard Green: Combinatorics of minuscule representations. Cambridge Tracts in Mathematics, 199. Cambridge University Press, Cambridge, 2013 $\endgroup$
    – Jim Humphreys
    Commented Feb 8, 2015 at 14:52
  • $\begingroup$ I'm definitely in favor of these being the minuscule representations of $G$, which are the cohomology groups of the cominuscule flag manifolds of the Langlands dual group $G^L$. The co- is for the Langlands duality. (The minuscule flag manifolds are also important; they're the ones that Hodge-degenerate to Stanley-Reisner schemes of order complexes of Bruhat order. I don't know any reason to look at the irreps associated to cominuscule fundamental weights.) $\endgroup$
    – Allen Knutson
    Commented Feb 9, 2015 at 14:52

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