Let $G$ be a reductive algebraic group over an algebraically closed field.
Let $B \subseteq P_2 \subseteq P_1$ be a Borel subgroup and two parabolic subgroups. Then $P_1/P_2 \subseteq G/P_2$ is a closed subvariety.
Is $P_1/P_2$ a Schubert variety defined in $G/P_2$? If not, what are the intersection of $P_1/P_2$ with the Schubert cells, also defined in $G/P_2$?
More precisely, let $W_{P_i}$ be the subgroup of the Weyl group $W$ of $G$ defined with a maximal torus $T\subseteq B$, such that $W_{P_i} \simeq N_{P_i}(T)/T$, for $i=1,2$. Let $$W_{P_i}^{\text{min}} = \{w \in W | \ell(ww') = \ell(w)+\ell(w'), \text{ for all } w' \in W_{P_i}\},$$ for $i=1,2$. Let $w_0$ be the longest element in $W$. Then there is a unique element in $W_{P_i}^{\text{min}}$ with maximal length, which is denoted as $w_i$, satisfying $w_0 = w_i \cdot w_{P_i}$, where $w_{P_i}$ is the longest element in $W_{P_i}$, for $i=1,2$.
Since $W_{P_2}$ is a subgroup of $W_{P_1}$, there exists $w_1' \in W_{P_1}$, such that $w_{P_1} = w_1' \cdot w_{P_2}$. So $w_2 = w_1 \cdot w_1'$.
Let $X_i(\cdot)$ denote the Schubert variety defined in $G/P_i$ for $i=1,2$. Then $G/P_1 \simeq X_1(w_1)$ and $G/P_2 \simeq X_2(w_1 \cdot w_1')$. This makes me suspect that $P_1/P_2$ and $X_2(w_1')$ are related in some sense. Is it true?
Thank you very much.
P.S.
- In a Coxeter group, we say $x = y \cdot z$ if $x = yz$ and $\ell(x)=\ell(y)+\ell(z)$.
2.For the definition of all the terms, please check: Billey and Lakshmibai's Singular Loci of Schubert Varieties, Chapter 2.