rt.representation theory – When does the null-cone consist entirely of eigenvectors? Answer

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rt.representation theory – When does the null-cone consist entirely of eigenvectors?

let $V$ be a finite-dimensional representation of a complex reductive Lie algebra $\mathfrak g$.

For our purposes, we may define the null cone like this: $v\in V$ belongs to the null cone if and only if there is a semisimple $h\in\mathfrak g$ and a decomposition $v=v_1+…+v_k$ look for that $hv_1=\lambda_1v_1$…, $hv_k=\lambda_kv_k$ for some strictly positive $\lambda_1,…,\lambda_k$.

By the Jacobson-Morozov theorem, for the adjoint representation one does not need the decomposition: $v$ is nilpotent if and only if it is the eigenvector with nonzero eigenvalue for some semisimple element. By Vinberg’s theory of theta groups this is true more generally for the action of $\mathfrak g=\mathfrak a^{(0)}$ on $V=\mathfrak a^{(1)}$ for a cyclically graded Lie algebra $\mathfrak a=\bigoplus_{j\in \mathbb Z/n\mathbb Z}\mathfrak a^{(j)}$.

There are however very easy examples showing that in general the decomposition is necessary. For example, the form $f=x^5+x^3y^2$ is in the null-cone of the 6-dimensional irrep of $\mathfrak{sl}_2$ but $gf=\lambda f$ implies $g=0$ for any $g\in \mathfrak{sl}_2$.

I do know one example of a representation which does not come from a theta group but still has the property that a vector is in the nullcone if and only if it is the eigenvector with nonzero eigenvalue for a semisimple element. This representation is however so ugly and obscure that I do not want to reproduce it here.

Is there known the characterization of some natural nice class of representations, not contained in the one coming from theta-groups, which has the same property? That is, such that any element of the null-cone is the eigenvector for a semisimple element with nonzero eigenvalue?

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