Let be the set of unipotent elements. This is a closed subset and hence an algebraic (affine) variety. Note that acts on by conjugation, the orbits of which are called unipotent orbits. We will see later that if is reductive then there are finitely many unipotent orbits. In the case of , it is due to Dynkin-Kostant. In the case of , it is proved by Richardson (classical groups) and Lusztig (exceptional groups). This is one of Lusztig’s motivation, that is to give a uniform proof of finiteness of unipotent orbits using Deligne-Lusztig theory.
In the case of , nilpotent matrices are those with determinant and trace zero. Thus can be identified with , sending to , which looks like a cone with singularity at origin. There are two unipotent orbits: the identity and the conjugacy class of . In over the complex numbers, the number of conjugacy classes is the number of partitions of by Jordan normal form.
Every elements of torus is semisimple. One proof over is use the criterion that an element is semisimple iff its order is coprime to and unipotent if its order is a power of . A characterization of torus is that is a torus iff is connected commutative algebraic group consisting of semi-simple elements. The idea is that we can choose a closed embedding of into the diagonal torus, and it remains to show that closed connected subgroups of the diagonal torus is of the form for some .
Let be a torus. The Weyl group is finite (proof is highly nontrivial).
Below are some facts from the theory of algebraic groups:
Irreducible iff connected iff geometrically irreducible iff geometrically connected;
geometrically reduced iff smooth (This is because of generic smoothness and homogeneity); But reduced need not imply geometrically reduced over nonperfect fields. The identity component is geometrically connected (since it contains a rational point, see here, Lemma 33.7.14 for a proof.)
Let be an algebraic group. There exists a maximal torus (with respect to inclusion).
The following is an important theorem: Any maximal torus in are all conjugate to each other. In the case of compact Lie groups, this follows from Lefschetz fixed-point formula due to Hermann Weyl, in the case of algebraic groups, this follows from Borel fixed-point formula.
Assume is defined over . A maximal torus of is a subgroup of the form where is a maximal torus stable under . For any , then is again a maximal torus.
The natural question is to classify maximal torus in up to -conjugacy. For the diagonal maximal torus is -stable. Let be a -stable torus, then we can write , then we can check that for to be -stable. Since , where is antidiagonal matrix with all ones. Thus is isomorphic to by sending to (the condition that is fixed by Frobenius means that ). If (or more generally ) then this is just but we get something new; but if , then . So we get two -conjugacy classes of , but we need to ensure that we can find such that . This is eassy, e.g. take , then .
From the above discussion we also see that it is important to consider the function and the level set for where is the Weyl group . This is precisely the Deligne-Lusztig variety .
Some preliminaries on reductive groups:
(Kolchin’s theorem) If is a unipotent group, then there exists such that the standard unipotent group. Equivalently, there exists a complete flag fixed by and .
(Lie-Kolchin’s theorem) Let be a connected solvable group. Then there exists such that the upper triangular Borel subgroup. Equivalently, there exists a complete flag fixed by . (connectedness is crucial, e.g. )
The first theorem is equivalent to the assertion that any representation of a unipotent group has a fixed vector. Similarly, the second theorem is equivalent to the assertion that any represention of a connected solvable group has a fixed line.
Another remark is that although Kolchin’s theorem holds true for any field but Lie-Kolchin does not, e.g. the standard representation has no fixed lines.
Kolchin’s theorem also implies that unipotent groups are nilpotent as abstract group. The following gives a structure theorem of connected solvable groups:
Let be a connected solvable group, then is a connected normal subgroup of and where is a maximal torus. Moreover, .
As a corollary, we have . It is actually remarkable that the quotient has an algebro-geometric structure.
Now we get to the notion of reductive groups. Let eb a connected algebraic group. The radical is the maximal connected, solvable, normal subgroup of . The unipotent part is called the unipotent radical. is called reductive if and is called semisimple if .
If , then is reductive iff any representation of is a direct sum of irreducibles. All the classical groups , and are reductive. The proof in the case of consists of showing is equal to where acts on via the standard representation. By Kolchin’s theorem we can find . but since is normal, is acted on by and so by transitivity of the action. For other classical groups use the same trick.
Reductive essentially means the group is an extension of torus by semisimple groups (and also the finite component group if we consider discconected reductive groups.)
Borel subgroups: Let be a conneccted algebraic group. A subgroup is Borel if is maximal among all connected solvable subgroups (no normality). is Borel (Exercise). By Lie-Kolchin theorem, any Borel subgroup is of the form for some . Also, (Exercise). There is a set bijection between to the set of Borel subgroups of by sending to . On the other hand, there is a geometric interpretation of as the set of complete flags in , and this turns out to have the structure of a projective variety.
For example, consider . Note that is a Borel subgroup of , by writing it as . The former is and the latter is . The latter can be written as product of four unipotent one-dimensional subgroup
In terms of flags, it looks like . This is an example of isotropic flags. Thus can be identified with the set of isotropic flags.
To go from a Borel of to an isotropic flag, by Lie-Kolchin theorem there is a line fixed by . The orthogonal complement is three-dimensional and contains . But then acts on , and there is a (non-degenerate) symplectic form that is again preserved by . Using Lie-Kolchin theorem has another fixed line, call the pullback . The general form of isotropic flags are .
For a split torus, the character group is defined to be and the cocharacter group is . Both are free -modules. In the nonsplit case we define to be , which is equipped with a discrete action of (via action on both domain and codomain, see this answer for an example. This turns out to be an equivaluence between category of -tori and category of discrete -modules.
Sketch why an affine algebraic group admits a closed immersion into some : An action of an affine algebraic group on a vector space (possibly infinite-dimensional) can be defined as a natural transformation , so it is the same as a functorial action of acting on for each -algebra . The idea is that acts on the coordinate ring , but is infinite-dimensional. However, it turns out that the action is via a -algebra automorphism, so we are done if we can show that the finite number of generators is contained in a finite-dimensional -stable subspace. This turns out to reduced to the associativity axiom of a group action, for details see this handout, section 12.2.
Key lemma in showing Jordan decomposition for arbitrary algebraic groups: If is given as a closed subgroup of , how do we single out the elements of ? For classical groups they are defined as groups of matrices that preserve a certain bilinear or sesquilinear form. Thus we would like to similarly realize an arbitrary algebraic group as stablizer of a certain action. It turns out this is indeed possible (see here Theorem 14.1.1; the idea is that stablizes the kernel ). Using this we can reduced showing the existence of Jordan decomposition of an arbitrary affine algebraic group to showing a closed immersion preserves semisimple and unipotent part, for details see here.
Proof of Kolchin’s theorem: The idea is to reduced to the case of algebraically closed fields, and use Wedderburn’s lemma:
If and is such that is an irreducible -representation, then is generated as a -algebra by .
Using this lemma, we could show any irreducible representation of a unipotent group is trivial. The idea is to write , where is nilpotent, and unipotence of implies that for any . Then by Wedderburn’s lemma for any , and hence by the nondegeneracy of the trace pairing.
The importance of Borel subgroups is that they cover the group. More precisely, the subgroups for cover . In particular, every element of lies in a Borel subgroup ( algebraically closed). From this we can also derive that every semisimple element lies in a torus. (There is real content to this, since might not lie in the identity component of the Zariski closure of , e.g. may have finite order, as for all unipotent elements when .) For the proof see this handout (The idea is to use an algebro-geometric version of covering lemma, similar to the case of compact Lie groups, see this question as well, especially the proof using mapping degree).
We have the following important property of orbits of action by algebraic group:
is an open in .
This essentially follows from Chevalley’s theorem and the homogeneity of orbits. A corollary is that if is connected (so irreducible), then is a disjoint union of orbits of strictly lower dimension. In particular, orbits of minimal dimension are closed.
A nice characterization of semisimple elements is that they are precisely the closed orbits in the adjoint action of (i.e. conjugacy classes).
Another corollary is that the image of a homomorphism of algebraic groups is closed, because all orbits are of the same dimension.
Borel fixed point theorem: Let be a connected solvable group acting on a projective variety , then has a fixed point. The idea is to induct on . Since is also connected solvable and has lower dimension, it has a fixed point . Let , which is closed (by separatedness) and nonempty (since it contains ), so it is projective. We have an action of on since is normal subgroup of . Let be a closed orbit, so (see this question for orbit-stablizer theorem in algebraic group), and this is both affine ( is normal subgroup fo since it contains )
From this we deduce that 1. is projective. 2. All Borel subgroups are conjugate. For proof of 1, let be a Borel subgroup of maximal dimension. Choose a representation and a line such that is the stablizer of . Thus acts on , and let be a flag fixed by by the Lie-Kolchin’s theorem and by the choice of we have . Then embeds into by the orbit stablizer theorem. It remains to show that the orbit is closed, but in fact it is a -orbit of minimal dimension (since we choose to have maximal dimensions among all Borels). For proof of 2, we consider an arbitrary Borel acting on (which is already projective) by left multiplication, then by the Borel fixed-point theorem let be a fixed point of . That means . By definition of Borel this means that .
Exercise: . This implies that .
As a corollary, we can show that all maximal torus are conjugate (if is algebraically closed). For details see this handout and 24.1 of the reference.
The Cartan subgroups are connected component of centralizer of maximal torus . Facts: 1. All Cartan subgroups are conjugate to each other. 2. . 3. is nilpotent. This implies is connected and solvable, so we can find a Borel containing . Bruhat decomposition: The natural map is a bijection. Note that is normal in , so is a finite group, called the Weyl group of . Corollary: and also by noting that . The Bruhat order on Weyl group is given by if . The closure are called the Schubert varieties. The advantage of using is that it is independent of the choice of Borel (it is -orbit rather than -orbit).
Example: If and be a complete flag and a pair of flags . If we look at the subspace , at some point its dimension jumps from to , denote it by . Repeat it with and get . We get a permutation , and .
What is the closure relation: For , the dense open orbit is the one for which the full flags are and . There is a Bruhat order that one can read off the closure relation.
For , acting on which is on the first entries in the diagonal and on the last entries. The factors are just swapping with . If and . If we use the above recipe and get and where is the permutation obtained by taking of and . But since we have isotropic flags, we gave , implying .
Exercise: Try to work out the Bruhat order for ().
For reductive and the quotient identifies the Bruhat decomposition (since is contained in every Borel ) so lots of combinatroics reduce to the reductive case.
A few remark: is often called (open) Schubert variety rather than . This is because we can define a twisted version as follows: Let be a surjective group homomorphism (so it induces maps on because it sends Borel to Borel), we can consider the graph embedding from to . Then we define to be the inverse image of . The two most important examples are (conjugation by ) which gives rise to Lusztig variety and (if ) which gives rise to Deligne-Lusztig variety.
A character is called a root if there is a unipotent subgroup (called a root subgroup) such that for and . It turns out if is reductive, then and it is uniquely determined. See corollary 2.1 of the second reference. This relies on the classification of connected reductive split rank 1 group, see the last theorem of the first reference.
Construction of coroot: Let be a root and the corresponding root subgroup. Let (a codimension-one subtorus of ) and . Note that contains and , and . The quotient is a connected reductive split rank 1 group, so it is either or . There is a unique cocharacter such that is a reflection of and maps the set of all roots to itself; moreover, we have (i.e. .
Example: For , the character is a root with root subgroup the standard unipotent subgroup. The coroot is given by
Root datum: See this handout for reference. The definition is given in Wikipedia. Denote , we call the associated root system to the root datum. Note that could be strictly smaller than .
Theorem (Chevalley): To a reductive group the quadraple is a reduced root datum. Moreover, iff . Finally, let be a reduced root datum, then there exists a connected reductive group such that .
Remark: for connected reductive . The Weyl group is isomorphic to . If we fix a Borel containing , then we can define the positive roots. Let be the set of simple roots. Then the set of reflections for already generates the Weyl group, and it turns out that it is a Coxeter group.
Example: For , then . For , it is (we have and , and
For , we have . Note that is a codimension one subspace of (dimension ). For , if , then we have with the corresponding root subgroup given previously The root system is type .
Construction of finite Chevalley group: Recall if the base field is complex, the Lie algebra of decomposes as where is a Cartan subalgebra. For any such that , let be the greatest integer such that . There exists a basis satisfying
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The upshot is that we can define since is nilpotent. The integral property of Chevalley basis gurantees that where all coefficients are in . Thus is a matrix with entries are polynomials in with coefficients in . Therefore we can define to be the subgroup generated by for and .
Chevalley proved that if then is an adjoint group of type . If , then is a finite simple group if . The group is defined over since the Frobenius morphism takes to itself ().
There is an alternate characterization of length of an element , which is the number of positive roots such that is a negative root. Its geometric meaning is that it is the dimension of the open Schubert variety . More precisely we have .
Example: For , we have and . Moreover, Finally . In this case all the closed Schubert varieties are smooth. But for , and are singular. If we fix a complete flag then . There is a map from to whose image consists of . Kazhdan-Lusztig noticed in 1983 that for , the Jordan-Holder series in principal series of have multiplicity 1. But for this fails.
Fact: For , the closed Schubert variety is smooth iff avoid pattern like and .
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 .
Reference:
https://virtualmath1.stanford.edu/~conrad/252Page/handouts/alggroups.pdf
https://virtualmath1.stanford.edu/~conrad/249BW16Page/handouts/249B_2016.pdf