rt.representation theory - Explicit computations of small Deligne-Lusztig varieties (e.g. Drinfeld curve)

I found Teruyoshi Yoshida's exposition of the subject very helpful:

http://www.dpmms.cam.ac.uk/~ty245/Yoshida_2003_introDL.pdf

As JT commented, the curve you wrote down is really the Deligne-Lusztig variety for SL_2, not GL_2. Ben is also right about the curve being $mathbf{P}^1 - mathbf{P}^1(mathbf{F}_q)$, only he is using a different definition of DL variety from you I would presume. The way Ben has it, the DL variety is a subvariety of G/B, but the curve you want is a subvariety of G/U, where U is the unipotent radical. One formulation is a cover of the other with galois group equal to the rational points on a twist of the torus T. We'll take the G/U point of view here.

So let's start with $G=text{SL}_2$ over the field $mathbf{F}_q$. We'll let $B$ be the usual Borel and $U$ its unipotent radical. We can then identify $G/B$ with $mathbf{P}^1$ and $G/U$ with $mathbf{A}^2$. The latter identification sends $(a,b,c,d)$ to $(a,c)$.

Let $w=(0,1,1,0)$ be the nontrivial Weyl element. We let $X_w$ be the subvariety of $G/B$ consisting of elements $x$ for which $x$ and $F(x)$ are in relative position $w$, where $F$ is the Frobenius map. This is $mathbf{P}^1 - mathbf{P}^1(mathbf{F}_q)$ as Ben says.

For cosets $x,yin G/B$ in relative position $w$, and a coset $gUin G/U$ for which $gB=x$, we are going to define a new coset $w_{x,y}(gU)in G/U$ as follows. First find a $g'in G$ for which $g'B=x$ and $g'wB = y$. We may further take $g'$ so that $g'U=gU$. (This can be done because of the Bruhat decomposition of $G/B times G/B$--wait a moment to see how this plays out for $text{SL}_2$.) Then define $w_{x,y}(gU) = g'wU$. (Pardon the abuse of notation of the symbol $w$.) The Deligne-Lusztig variety $Y_w$ is defined as the set of $gUin G/U$ for which $F(gU)=w_{gB,F(gB)}(gU)$.

When does a point $(x,y)inmathbf{A}^2=G/U$ lie in $Y_w$? We need to calculate $w_{gB,F(gB)}(gU)$, where $g=(x,*,y,*)in G$. We have $gB=gcdotinfty=x/y$ and $F(gB)=(x/y)^q$. So we must now find $g'in G$ with $g'U=gU$ and $g'wB=F(g)wB$. The first condition means that $g'=(x,*,y,*)$ and the second means that $g'cdot 0=(x/y)^q$. Thus $g'=(x,ux^q,y,uy^q)$, where $u$ must satisfy $u(xy^q-x^qy)=1$. We find that $w_{gB,F(gB)}(gU)=g'wU=(ux^q,uy^q)$. The condition that $(x,y)in Y_w$ is exactly that $(x^q,y^q)=(ux^q,uy^q)$, which implies that $u^{-1}=x^qy-xy^q=1$. So that's the equation for the Deligne-Lusztig variety.

The equation for the DL variety for the longest cyclic permutation in the Weyl group of $text{SL}_n$ is $det(x_i^{q^j})=1$, where $0leq i,jleq n-1$.

I believe Lusztig calculated the zeta functions of his varieties in a very general setting, but I was never able to trudge through it all. There must be a simple answer for the behavior of the zeta functions for the $text{GL}_n$ varieties--if you ever write it up I'd certainly love to read it! I can start you off: for $text{SL}_2$ over $mathbf{F}_q$, the DL curve has a compactly supported $H^1$ of dimension $q(q-1)$, and the $q^2$-power Frobenius acts as the constant $-q$. (The behavior of the $q$-power Frobenius might be a little subtle--I suspect it has to do with Gauss sums.)

Good luck!