If $(V,\Phi)$ is a root system of rank $n$, one knows that its Weyl group $W$ acts simply and transitively on Weyl chambers. But in general, if $d\lt n$, the action of $W$ on faces of dimension $d$ is no more transitive.

In a root system of type A, for any face $F$, one still has i $\Rightarrow$ ii, with :

i. $0$ is the only vector of $V_F$ invariant under $W_F$, where $V_F$ is the subspace spanned by $F$ and $W_F$ is the stabilizer of $F$ in $W$.

ii. Any face $F'$ which spans the same subspace as $F$ is in the $W$-orbite of $F$.

I wonder if i $\Rightarrow$ ii is valid for any root system.

From split reductive groups point of view, that is equivalent to iii $\Rightarrow$ iv, where $M$ is a Levi subgroup of a parabolic subgroup of a split reductive group $G$ (I will abusively say $M$ is a Levi subgroup of $G$) :

iii. $G$ has no proper Levi subgroup that contains the normalizer $N_G(M)$.

iv. Two parabolic subgroups having same Levi $M$ are conjugate under $G$.

The implication is true for $GL_n$. Is it also true for any split reductive group ?


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