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:
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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).
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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?
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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).
Best Answer
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 $x,y\in G/B$ in relative position $w$, and a coset $gU\in G/U$ 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 $gU\in G/U$ for which $F(gU)=w_{gB,F(gB)}(gU)$.
When does a point $(x,y)\in\mathbf{A}^2=G/U$ lie in $Y_w$? We need to calculate $w_{gB,F(gB)}(gU)$, where $g=(x,*,y,*)\in G$. We have $gB=g\cdot\infty=x/y$ 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 $0\leq i,j\leq n-1$.
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!