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 .
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 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.
Reference: https://virtualmath1.stanford.edu/~conrad/252Page/handouts/alggroups.pdf