Let $q$ be a power of a prime, $\mathbb{F}_q$ the field with $q$ elements, $\mathbb{F}$ an algebraic closure. I am trying to check whether $\mathbb{P}^1 \setminus \mathbb{P}^1 (\mathbb{F}_q)$ appears as some Deligne-Lusztig variety $X(w)$ for $\operatorname{GL}_3$. For dimension reasons, I know that the length of $w$ must be 1, so I looked into the variety $X((1 2))$. Borel subgroups in it correspond to complete flags $$ \{0\} \subsetneq V_1 \subsetneq V_2 \subsetneq \mathbb{F}^3$$ with ($F$ being the Frobenius) $$ F(V_1) \ne V_1, \quad F(V_2) = V_2.$$

But while this is helpful for concrete computations, I don't see how it helps me with the set in question? That is, how do I obtain an identification with some projective space?

A somewhat more basic question would be: how can I map the flag variety into projective space? For $\operatorname{GL}_2$ and a fixed Borel subgroup $B$, we have the isomorphism $$ \{\text{Borel-subgroups}\} \to \mathbb{P}^1, \quad B' = gBg^{-1} \mapsto [g e_1].$$ How does this generalize to $\operatorname{GL}_n$?


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