Let $G$ be a connected, reductive algebraic group over $\mathbb{C}$. Fix a maximal torus $T$ and Borel subgroup $B$. Let $L$ be a generalized Levi ($L = Z_G(s)^\circ$, for some semisimple element $s \in T$). Then $L$ is also a reductive group, and $B \cap L$ is a Borel subgroup in $L$. Therefore there is a closed embedding of flag varieties

$$ i \colon Fl_L = L/B \cap L \hookrightarrow Fl_G = G/B. $$

I am wondering if this embedding respects the Schubert stratification in the following sense: There is an inclusion of weyl groups $j\colon W_L \hookrightarrow W_G$. For any $w \in W_L$, there is a schubert cell $C_w \subset Fl_L$, and similarly for $G$. Then is $$ i(C_w) = C_{j(w)}? $$

If $L$ is an actual Levi subgroup, I believe then the answer is yes. Therefore I am wondering about the slight generalization to pseudo-levis.

Thank you.

$\newcommand\Fl{\mathrm{Fl}}$Let $G$ be a connected, reductive algebraic group over $\mathbb{C}$. Fix a maximal torus $T$ and Borel subgroup $B$. Let $L$ be a generalized Levi ($L = Z_G(s)^\circ$, for some semisimple element $s \in T$). Then $L$ is also a reductive group, and $B \cap L$ is a Borel subgroup in $L$. Therefore there is a closed embedding of flag varieties

$$ i \colon \Fl_L = L/B \cap L \hookrightarrow \Fl_G = G/B. $$

I am wondering if this embedding respects the Schubert stratification in the following sense: There is an inclusion of Weyl groups $j\colon W_L \hookrightarrow W_G$. For any $w \in W_L$, there is a Schubert cell $C_w \subset \Fl_L$, and similarly for $G$. Then is $$ i(C_w) = C_{j(w)}? $$

If $L$ is an actual Levi subgroup, I believe then the answer is yes. Therefore I am wondering about the slight generalization to generalized Levis.