# Explicit computation of Deligne-Lusztig varieties for $GL_n$

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 $$\operatorname{GL}_n$$?