I am stuck on a seeming elementary Lie-theoretic question arising from a study of components of affine Springer fibers. Will be very grateful if somebody would like to share some insight, or interested in writing their answer as an appendix!
$\S$1. Lie-theoretic question without motivation
Begin with a root system $\Phi$, we consider an associated semisimple Lie group $G$ over $\mathbb{C}$ (adjoint or not, doesn't matter). Consider the Borel subgroup $B\subset G$ and respective Lie algebras $\mathfrak{b}\subset\mathfrak{g}$. The quotient $\mathfrak{g}/\mathfrak{b}$ can be realized (just like any vector space) as an abelian complex Lie group. Consider the semidirect product $M:=B\ltimes\mathfrak{g}/\mathfrak{b}$. We also have its Lie algebra $\mathfrak{m}=\mathfrak{b}\ltimes\mathfrak{g}/\mathfrak{b}$.
Now, for any root $\alpha\in\Phi$ there is a $1$-dimensional "root subspace" $\mathfrak{m}_{\alpha}$ of $\mathfrak{m}$, defined to be the usual one in $\mathfrak{b}$ if $\alpha\in\Phi^+$ is positive, and is the image of the usual one in $\mathfrak{g}/\mathfrak{b}$ if $\alpha\in\Phi^-$. For any $w\in W$ in the Weyl group, it is easy to see that $\mathfrak{m}_w:=\sum_{\alpha\in w.\Phi^+}\mathfrak{m}_a\subset \mathfrak{m}$ is a subalgebra. Consider $$ X_w:=\{\operatorname{Ad}(g)x\;|\;g\in M,\; x\in \mathfrak{m}_w \} $$ and let $\overline{X_w}$ be the closure of $X_w$ in $\mathfrak{m}$, an irreducible closed subvariety of $\mathfrak{m}$. The statement we hope for is:
Conjecture. For any $w\not=w'$, we always have $\overline{X_w}\not=\overline{X_{w'}}$.
I don't know if the conjecture is true, except that no counterexample is found after some effort.
$\S$2 Motivation
I am working on extending my own result in Components of affine Springer fibers, which roughly says that under a specific but reasonably general condition, the components of an affine Springer fiber mod centralizer are predicted to be in bijection with the Weyl group. (The cited paper proves the weaker numerical result and some other numerical evidences).
I have a method to construct the desired bijection in the following way: inside the loop group $LG$ (with $LG(\mathbb{C})=G(\mathbb{C}(\!(t)\!))$ we have the Iwahori subgroup $I$. It has a natural normal subgroup $I_1$ (the Moy-Prasad subgroup at depth $1$, with $\operatorname{Lie}I_1=t\operatorname{Lie}I$) such that $I/I_1\cong M$ constructed above. Recall that the affine Springer fiber is given by $$ \DeclareMathOperator\Fl{Fl} \Fl_{\gamma}=\{g\in LG/I\;|\;\operatorname{Ad}(g^{-1})\gamma \in\operatorname{Lie} I\}. $$ Consider also $$ \Fl^1_{\gamma}=\{g\in LG/I_1\;|\;\operatorname{Ad}(g^{-1})\gamma \in\operatorname{Lie} I\}. $$ The components of $\Fl^1_{\gamma}$ and those of $\Fl_{\gamma}$ are evidently in bijection. Now we have a map $\mu:\Fl^1_{\gamma}\rightarrow \operatorname{Lie}I/{\operatorname{Lie}I_1}=\mathfrak{m}$ sending $g$ to $\overline{\operatorname{Ad}(g^{-1})\gamma}$. I can prove that if $t^{-1}\gamma\in L\mathfrak{g}$ is bounded (compact) and regular semisimple, then there exists a bijection between components of $\Fl^1_{\gamma}$ mod centralizer to the set $W$, given by the property that the component indexed by $w$ is mapped to $\overline{X_w}$ dominantly by $\mu$. What stops that from being a complete result is just the aforementioned conjecture; if we want to make this bijection canonical, we need these $\overline{X_w}$ to be distinct.
