Lang-Steinberg: If is a connected algebraic group defined over , define the Lang map by , then is surjective (connectedness is crucial). This is also true for abelian varieties. Some famous examples are and this reduced to Artin-Schreier exact sequence. In the case of it becomes Kummer exact sequence.
Corollary: Let be a variety acted by a connected group . Let be a -orbit. Assume and the action is defined over , and is stable under . Then . (Proof is that if then for some and use Lang-Steinberg to write then .) This is specific to finite fields, e.g. if then .
Corollary: Let inclusion of algebraic groups over . Then where is connected and could be disconnected. This is because the map is surjective by the previous corollary (because a -coset is an -orbit) and injectivity is easy. Connectedness of is crucial because if and then by the squaring map and the map can be identifies with it, which is not surjective from to .
Corollary: Reductive groups over finite fields are quasi-split (note that is -stable$ doesn’t mean is split; it just means is defined over but the Galois action on it could still be nontrivial). If is connected algebraic group over . There exists such that , . To find a -stable Borel we just let be the variety of Borel subgroups and this is a single -orbits stable under (since is still a Borel). The same argument applies to finding an -stable maximal torus.
Note that for a single orbit , the induced action on (which is nonempty by the previous corollary) is not necessarily transitive (e.g. acting on itself by ).
Theorem: -orbits on are in bijection with the -conjugacy classes of for any . The map is given by .
Corollary (DL, classification of -stable maximal torus): The -conjugacy classes of -stable maximal torus is in bijection with -conjugacy classes of (in the reductive case .) The map is given by mapping to . Moreover, there are isomorphisms , here is the image .
Example: If and the standard diagonal maximal torus. Then acts trivially on , so the conjugacy classes of -stable maximal torus are in bijection with (ordinary) conjugacy classes of .
Let be an algebraic group (possibly disconnected). There is a bijection between -conjugacy classes of to -conjugacy classes in if is a connected normal subgroup defined over . The nontrivial direction follows form using a different -struture (essentially corresdponding to embedding of into affine space and use the Frobenius on ) given by (by Lang-Steinberg theorem we can modify the embedding and hence it is a Frobenius structure), and apply Lang-Steinberg theorem to it. Thus if is reductive we get a bijection between -conjugacy classes in and -conjugacy classes in . Injectivity is easy and surjectivity again follows from Lang-Steinberg.
An algebraic group can have many different -rational structures. Over , is the set of fixed points under conjugations, but we can use a different conjugations, e.g. , then the set of fixed points are . They are two different real structures on .
Example: In , . In , for every partition , then . But to write down the explicit embedding of into requires solving . If we let be the longest element , then .
Let be a variety over with . Let be an automorphism such that for some . Then is a Frob map associated to a structure on .
Example: If denotes the inversion map, then is a different Frobenius with . We have a commutative diagram.
In particular we can let and (note that in general).
Define sending to . In this case and . We have is called the unitary group . If is the usual conjugation then this the usual unitary group. Unfortunately, still acts trivially on . The -stable maximal torus in the unitary group is still in bijection with partitions of .
Consider associated to the anti-diagonal all-one matrix. Take the maximal torus . Let be where is the permutation matrix corresponding to (so this is not an inner automorphism of ). It is an outer automorphism of the Dynkin diagram of swapping the branching nodes. Since and commutes, we have . We call the twisted special orthogonal group. The Weyl group of is where swap the -th and -th factor, where we only consider even number of swaps. It can be generated by where swaps with and swaps and . Alternatively it is generated by and -action just permute the last two generators.
(Steinberg) Let be the Chevalley group. Let be any outer-automoprhism of . Then is a Frobenius of and is a finite simple group called the twisted Chevalley group if .
A -stable maximal torus if . is split if it has a split maximal torus.
Exercise: is not split. The -stable maximal tori are and where and .
For non-simply-laced Dynkin diagram like $B_2, there are exceptional -structures when or and odd. They are known as Suzuki and Ree groups.
Goal: For each -conjugacy class of corresponding to the -stable maximal torus , construct representations of parametrized by (complex-valued) characters of .
The idea is to have a common geometric object for which both and acts and if their actions commute, then we can link characters of to representations of . For the standard maximal torus if we take then we get the principal series.
We have seen that we can always find a -stable maximal torus inside a -stable Borel. But there are -stable maximal torus that are not contained in any -stable Borel.
DL’s idea to construct such an object is to use the Lang’s map given by . Note that the fibers are -torsor (act by left multiplication). We now want to find a collection of fibers that are acted on by .
Pick a Borel containing , and let be the unipotent radical of . Let . Then acts by right mutiplication on . Actually also acts by right multiplication on . The quotient and are the so-called Deligne-Lusztig varieties.
Example: Assume , then , then and it is easy to see that (since is connected) and . In this and are zero-dimensionnal varieties so only is interesting.
Alternative description using Weyl group and twisted Frobenius structure: Pick -stable. Let and . For any , define . Note that . If we pick a lift of , define and .
Proposition: 1. acts on , on the left. 2. and act on on the right. 3. . 4. .
There are -equivariant isomorphism from to . The map is given by sending to where and similarly for the ’s. The advantage of the former is that it’s more canonical while the advantage of the latter is that we can do explicit equation by choosing the standard .
Example: For , pick , then consists of satisfies , which can be solved to yield and and , and hence . Taking -th root yield the Drinfeld curve. In this case , so is isomorphic to , and , which is the finite field analogue of the upper and lower half-plane . The -adic analogue is called the Drinfeld’s half-plane.
In the case of unitary group and , . acts on since . This case is done by Tate-Thompson, and the induced action on the contains interesting unipotent cuspidal representation. In the real case it is related to spherical harmonics.
To state the main theorem of Deligne-Lusztig it is better to use cohomology with compact support because it makes Lefschetz trace formula works, which holds for non-smooth varieties as well. The homology counterpart is the Borel-Moore homology, which are formed by replacing chains with locally finite chains (Borel−Moore homology is a covariant functor with respect to proper maps, e.g. is a counterexample. Similarly for cohomology with compact support).
We define the virtual representation and its trace .
Main properties of :
for any containing .
Any irreducible representation of appears with nonzero coefficient in some .
For most choice of , is of irreducible representation.
iff
where is the set of intertwiners that sends to .
, in particular it is independent of . There is an explicit way to determine the sign.
Fact: Let be a variety with , then if , . If is a permutation of , then induces the permutation representation of .
Example: consider where is -stable. Then , so . Then is the usual parabolic induction.
For the Drinfeld curve, we have iff is regular, i.e. doesn’t factor through the norm map, and .
(Lefschetz fixed point formula) If is the Frobenius map (in particular it is proepr so it induces maps on compactly supported cohomology), then .
Example: if , then the only nonzero cohomology is and , so .
Let be a finite-order automorphism. Then
.
and it is independent of .
(Kunneth formula) If and , then .
If then .
If is the Jordan decomposition where has order coprime to and has order power of , then . In particular, if , then .
If is a surjective map with fibers isomorphic to for some fixed , and and commute w.r.t. , then .
(Homotopy invariance) Let be a connected algebraic group, then and .
Fact: is affine. So . The following lemma reduces the computation of character of to that of Lefschetz number:
Lemma: (because is a projection from to )
Proof of independence of for : calculate (note that )
Define the Green function by (it turns out to have values in ). It is also the same for other character . In particular, the dimension is independent of , and equal to the Euler characteristic . Note that this depends on the Frobenius structure on the variety.
Example: For , we have and . We have and .
Kazhdan-Springer formula for : Assume , let . Let be the Killing form. Fix a regular semisimple element whose centralizer is a torus.
Let be a nilpotent element. Let and . Then , the -primary part of . When , and , so .
Let of type . Springer showed that and is the famous Springer fiber and acts on via the Springer representation.
Example: If , , acts on by trivial and by sign representation. Then (since acts trivially on ) and (because acts by on ).
We have the following character formula for :
where are the Jordan decomposition of . Note that and is a connected reductive group stable under (since is)
Corollary: If is not -conjugate to , then ( and and is not -conjugate to ).
Another corollary is when restricted to (because and for ).
A third corollary is if is regular semisimple (i.e. is a torus) Then for (This is because implies ). For other the trace is zero by the first corollary.
Proof of character formula: Use the expression for trace in terms of Lefschetz number. We have the Jordan decomposition and use property 5.
Fact:
The subgroup is a Borel subgroup of .
If is a Jordan decomposition, then .
(Steinberg) If the derived subgroup is simply connected (i.e. ) then is connected.
E.g. If and , then is connected. If and , then and so it is disconnected (and admits a 2-fold cover).
A semi-simple element is regular if is a maximal torus. The set of regular semi-simple elements is dense.
Levi subgroup: Subgroup of the form for a torus is called Levi subgroup.
Fact: Levi subgroups are connected reductive groups (unlike the case of semisimple elements). Proper Levi subgroup has positive dimensional center.
Note: may not be a Levi subgroup (in the case of it is). A counterexamle is , and , then and it is not a Levi since its center is the Klein-four group (this also generalize to ).
Consider and and and is not a Levi subgroup by the same reason. The Langlands dual of is itself, which is not a subgroup of the Langlands dual of , which is . It is an example of endoscopic subgroup of .
In a sense that the character of a regular semisimple element should determine the character. Though we are in the context of finite groups of Lie type we should consider regular unipotent elements as well.
For a non-semisimple element, we can still define what it means for it to be regular (dimension of centralizer is minimal). Fact: If is regular, then for maximal torus . Example: is regular since its centralizer is .
Fact (c.f. Steinberg’s regular elements of semi-simple algebraic group):
is open. If is semisimple .
and are nonempty.
For any , the centralizer is abelian and .
For example: If , then is everything but the identity. The regular semisimple elements are conjugate to , and regular unipotent elements are conjugate to and centralizer are the standard maximal torus and (in particular disconnected).
Exercise: For , TFAE:
is regular
minimal polynomial of has degree .
is a cyclic module over where acts by .
Moreover, the centralizer has a concrete interpretation: given by mapping to where is the minimal polynomial of .
Proof of character formula: The key geometric lemma is
;
where the isomorphism is given by and this isomorphism is equivariant for the action of (by mapping to ).
By (i) and (ii), we have
By doing a change of variable, this is equal to Finally, we note that .
Proof of (i): means that and , which implies that implying which implies so . By Lang’s theorem, there exists such that . Let , then , so and , hence .
Proof of inner product formula: The key is the orthogonality of Green functions:
In particular, if is not -conjugate to then the sum is zero. This is first proved (in the case of ) by J. A. Green. Using this we compute by grouping the together, this is equal to
Note that there is a bijection from to by . Plug this into the sum, we have which is equal to (letting )