**Background**:
I am focusing on $G=GL_{2}(\overline{\mathbb{F_q}})$ here. If you wonder why I am interested in this, I am trying a problem relating to the Deligne-Lusztig varieties defined over local rings by Stasinksi, and this background theory is relevant there. The definition I am using is this:

- The first definition I have of Deligne-Lusztig varieties is this: Consider the Lang map $L(g) = g^{-1} F(g)$. Let $T$ be an $F$-stable maximal torus of $G$, and $B$ a Borel subgroup containing $T$ (not necessarily $F$-stable), and $U$ the unipotent radical of this Borel subgroup. Then the Deligne-Lusztig variety is defined as $X = L^{-1}(U)$

**Question**
Roughly: Using these definitions (I am not sure exactly to what extent these two definitions are "compatible"), how do I explicitly compute the Deligne-Lusztig variety for $GL_{2}(\mathbb{F}_{q})$ to be the Drinfeldt curve $xy^{q} - yx^{q} = 1$ in the non-split case? More precisely:

If I pick a torus that is not maximally split, $T$, and a Borel subgroup $B$ containing $T$, then apply the definition $1$ above - how do we relate this variety to the Drinfeldt curve? Do we explicitly get the Drinfeldt curve, and if not what do we get? If it is not the Drinfeldt curve that we get using this definition, how do we express this variety "nicely" (with a view towards counting $F_{q}$ points on it).

Roughly speaking (not explicitly!) how would I go about doing this for $GL_{3}(\mathbb{F_q})$? Is this computationally feasible? Are there any tricks that would help significantly with the computation? Even better, are there any references you know which do this?

I understand Deligne-Lusztig varieties, and these tori, correspond in some sense to Weyl group elements. Are there any specific Weyl group elements in $S_n$ for which the Deligne-Lusztig variety always has a "nice" / "tractable" description, or do they get out of hand very quickly?

**My Attempts at (1)**

(I'm not entirely sure how to write matrices in Latex here, so I did it crudely by writing it as $4$ numbers, Row 1 followed by Row 2).

Pick $\alpha, \beta \in F_{q^2}$, so that $(x- \alpha)(x- \beta)$ is irreducible over $F_{q}$. Then a unipotent subgroup of a Borel subgroup for a non-split torus by conjugating the ordinary unipotent subgroup of strictly upper triangular matrices, by the matrix $M$ with entries $(1, 1, \alpha, \beta)$. This is because we can obtain the matrix with entries $(0, 1, -\alpha \beta, \alpha + \beta)$ (lying inside $GL_{2}(\mathbb{F_q})$ as $M^{-1} X M$, where $X$ is the matrix with entries $( \alpha, 0, 0, \beta)$. Since when we conjugate the nilpotent matrix with entries $(0, 1, 0, 0)$ by $M$ we get a scalar multiple of the matrix with entries $ ( - \alpha, 1, - \alpha^2, \alpha)$, the end result of this calculation is that our non-split maximal torus consists of matrices of the form $(1-s \alpha, s, - s \alpha^2, 1 + s \alpha)$.

Now if $g = (a,b,c,d)$ (i.e. the matrix with those 4 entries), then $g^{-1} = \frac{1}{ad-bc}( d, -b, -c, a)$, $F(g) = (a^q, b^q, c^q, d^q)$, and $g^{-1} F(g) = \frac{1}{ad-bc} (da^q - bc^q, db^q - bd^q, -ca^q + ac^q, -cb^q + ad^q)$, so equating that $g^{-1} F(g)$ lies in the subgroup calculated in the last paragraph gives the following description of the Deligne-Lusztig variety (let $D = ad-bc$), by comparing entry by entry:

- $ \frac{1}{D} ( da^q - bc^q -cb^q + ad^q ) = 2 $
- $ -ca^q + ac^q = - \alpha^2 (db^q - bd^q) $

This might be easy but I cannot see how to finish off from here: why is this related to the Drinfeldt curve? All I can see from a first glimpse is that the terms $db^q - bd^q$ appears, but I don't know how to get rid of everything else. How can I simplify the defining equations of this variety (hopefully in a suitably simple version, so that counting $F_{q^{k}}$ points is straightforward, which is what I really need to do).