Take $G=\operatorname{GL}\_3$, defined over the algebraic closure of a finite field $\mathbb{F}\_q$ and let $X$ be the set of Borel subgroups of $G$.
The Frobenius morphism $F:G\to G$ induces a map $F:X\to X$, as $G/B\cong X$ for any chosen Borel subset $B$.
The Weyl group $W$ of $G$ is isomorphic to the symmetric group $S\_3$, and we have a bijection between $W$ and the set of $G$-orbits on $X\times X$.
Fix a Borel subset $B^+\in X$.
For $w\in W$, define $\mathcal{O}(w)$ to be the orbit of $(B^+,\dot wB^+\dot w^{-1})$ in $X\times X$.
We say that $B\_1,B\_2\in X$ are in relative position $w$ if $(B\_1,B\_2)\in\mathcal{O}(w)$.


For any $w\in W$, we can define the Deligne–Lusztig variety associated to it as $X(w)=\{B\in X:B\text{ and }F(B)\text{ are in relative position }w\}$.


**My Question**


I want to compute $X((1\ 2))$.


I know that $X$ can be identified with the full flag variety $\mathcal{F}\_3$, but how I can interpret the condition of $B$ and $F(B)$ being in relative position $w$ in terms of flags?


Any help is appreciated.



An example of a Deligne–Lusztig variety for a general linear group
Let $\mathcal{F}\colon V\_1\subseteq\dotsb\subseteq V\_n$ and $\mathcal{F}'\colon U\_1\subseteq\dotsb\subseteq U\_n$ be two complete flags. Then $\mathcal{F}$ and $\mathcal{F}'$ are said to be in relative position $w$, where $w\in S\_n$ is a permutation, if
$$\dim V\_i\cap U\_j=\#(\{ 1,\dotsc,i \}\cap \{ w(1),\dots,w(j) \})$$
for any $i,j\in\{1,\dotsc,n\}$.


In particular, the variety $X((1\ 2))$ consists of the flags $V\_1\subseteq V\_2$ in which $V\_2$ is an $F$-stable plane and $V\_1$ is an $F$-unstable line. So it is
$$\operatorname{Gr}(1,3)^F\times \mathbb{P}^1\setminus\mathbb{P}^1(\mathbb{F}\_q),$$
a disjoint union of copies of the Drinfeld curve.


Computations like this are very useful for making explicit connections with combinatorics. For a reference, you may want to take a look at Steinberg's paper *[An occurrence of the Robinson–Schensted correspondence](https://doi.org/10.1016/0021-8693(88)90177-9)*.


