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Let $\mathfrak g$ be a semisimple Lie algebra and $V$ - its representation, both finite-dimensional over $\mathbb C$. By the well-known results, one can choose in $\mathfrak g$ a Cartan subalgebra and decompose the vector space $V$ as a direct sum of its weight spaces. Elements of the weight spaces are called weight vectors. This decomposition depends on the choice of a Cartan subalgebra. Define a set $\Xi$ to consist of all $x \in V$ such that $x$ is a weight vector for some choice of a Cartan subalgebra. What can be said about the set $\Xi$? I'm asking about topological properties (both with respect to the usual and the Zariski topology), characterizations in terms of explicit equations, if possible finer geometric characteristics (e.g. dimension).

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    $\begingroup$ For $V =$ the adjoint representation, it seems to me that by Jacobson-Morozov, $\Xi$ is the set of all nilpotent elements of $\mathfrak{g}$; which, further, seems to be Zariski-closed and irreducible, and as variety has dimension $=$ total number of roots in the root system of $\mathfrak{g}$. $\endgroup$ – Torsten Schoeneberg Jun 6 at 16:40
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    $\begingroup$ Thank you, this is very interesting. I'm somewhat worried that for other representations such nice characterizations may be unavailable, since the algebraic structure is not so rich in general. At the very least it is clear that the set I'm after is the union of finitely many group orbits, but unfortunately these are typically neither closed nor simple to understand. $\endgroup$ – Blazej Jun 7 at 10:36
  • $\begingroup$ By the way, in my above comment I assumed that "weight" means "non-zero weight". $\endgroup$ – Torsten Schoeneberg Aug 6 at 5:43
  • $\begingroup$ Do you happen to know what is the answer for the case of zero weights, or in other words what is the union of all Cartan subalgebras of $\mathfrak g$? One observation is that this is a Zariski dense subset of $\mathfrak g$, since it contains the nonempty, Zariski-open set of all regular elements. This is related to another question of mine, math.stackexchange.com/questions/3314046/… $\endgroup$ – Blazej Aug 6 at 8:43

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