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Let $G$ be a simple complex algebraic group acting on its Lie algebra $\mathfrak{g}$ via the adjoint representation.

What is the largest integer $d$ such that every subspace $U \subseteq \mathfrak{g}$ with $\dim(U) \leq d$ has trivial intersection with some Cartan subalgebra?

In other words, I am interested in the value of $$ \max \{ d \mid \forall U \leq \mathfrak{g} \colon \; \dim(U) \leq d \Rightarrow U \cap \mathfrak{t} = 0 \text{ for some Cartan subalgebra } \mathfrak{t} \}. $$

I am particularly interested in the case where $G = \mathrm{SL}_n(\mathbf{C})$ acting by conjugation on $\mathfrak{g} = \mathfrak{sl}_n(\mathbf{C})$.

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    $\begingroup$ My first guess would be $d$ to be the maximal possible candidate, i.e., equals the codimension of $T$ in $\mathfrak{g}$. This is obvious for $\mathfrak{sl}_2$ ($d=2$) and I have a rough sketch in mind for $\mathfrak{sl}_3$ ($d=6$?), but with some details to check. $\endgroup$
    – YCor
    Commented Nov 7 at 0:50
  • $\begingroup$ Where does the intersection $U\cap T$ live if, say, $G$ is of type $G_2$? $\endgroup$
    – Paul Levy
    Commented Nov 7 at 13:09
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    $\begingroup$ On reflection I guess what you mean is that $U\cap {\mathfrak t} =0$ for any Cartan subalgebra ${\mathfrak t}$ of ${\mathfrak g}$. $\endgroup$
    – Paul Levy
    Commented Nov 7 at 14:09
  • $\begingroup$ @YCor: Yes, I think I also have an idea on how to do $\mathfrak{sl}_3$ with $d = 6$, but I do not know how to generalize this to $\mathfrak{sl}_n$. I suspect the asnwer should be $n^2 - n$ in general. $\endgroup$
    – darko
    Commented Nov 7 at 14:43
  • $\begingroup$ @PaulLevy: Right, thanks for the correction, I have edited the question. $\endgroup$
    – darko
    Commented Nov 7 at 14:44

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