Inclusion of flag varieties and Schubert decomposition

Question

$\newcommand\Fl{\mathrm{Fl}}$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.

Answer 1

As you suspect, the equality of inflated Schubert cells does hold for actual Levis $L$. Let $N$ be the unipotent radical of the parabolic subgroup of $G$ containing $B$ that has $L$ as a Levi componnet. Then $w^{-1}N w$ is contained in $B$ for all $w \in W_L$, and $(B \cap L)w(w^{-1}N w)$ equals $B w$.

However, the equality need not hold for generalised Levis $L$. In fact, I think for every non-Levi $L$ it will fail for certain $w \in W_L$, and my example below will probably suggest how you would check this.

$\DeclareMathOperator\SL{SL}\DeclareMathOperator\Sp{Sp}$Consider $L = \SL_2 \times \SL_2$ in $\Sp_4$ and $w = (w_0, w_0)$, where $w_0$ is the long element in the Weyl group of $\SL_2$. Let $\alpha_1$ and $\alpha_2$ be the positive roots of the two $\SL_2$'s, numbered so that $\beta \mathrel{:=} \tfrac1 2(\alpha_1 - \alpha_2)$ is positive. We have that $U_w \times B \to U w B$ is an isomorphism of varieties, where $U_w = U \cap w U^- w^{-1}$, and analogously for $U_{L, w} \times B \to U_L w B$, with hopefully obvious notation. Then $U_\beta w B$ is obviously contained in $B w B$, but does not contain any element of $U_{L, w}w B = U_L w B$.