1
$\begingroup$

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.

$\endgroup$
5
  • 4
    $\begingroup$ Google for cominuscule representations. $\endgroup$
    – Sasha
    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". $\endgroup$ Feb 7, 2015 at 22:08
  • $\begingroup$ @ Sasha: Thanks, that's exactly what I want. $\endgroup$
    – ronggang
    Feb 7, 2015 at 22:09
  • 1
    $\begingroup$ @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 $\endgroup$ 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.) $\endgroup$ Feb 9, 2015 at 14:52

0

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.

Browse other questions tagged or ask your own question.