Math 224 - Representations of Reductive Lie

Groups

Taught by Dennis GaitsgoryNotes by Dongryul Kim

Spring 2018

This was a course taught by Dennis Gaitsgory. The class met on Tuesdaysand Thursdays from 11:30am to 1pm, and the main textbook was Springer’sLinear Algebraic Groups. There were a few assignments near the beginning ofthe course, and a final paper.

Contents

1 January 23, 2018 41.1 Linear algebraic groups . . . . . . . . . . . . . . . . . . . . . . . 41.2 Representations of linear algebraic groups . . . . . . . . . . . . . 6

2 January 25, 2018 72.1 Embedding of linear algebraic groups . . . . . . . . . . . . . . . . 72.2 The regular representation . . . . . . . . . . . . . . . . . . . . . . 8

3 February 1, 2018 103.1 Examples of groups and classification of representations . . . . . 103.2 Tannaka duality . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

4 February 6, 2018 144.1 Connected component . . . . . . . . . . . . . . . . . . . . . . . . 144.2 Image and surjectivity . . . . . . . . . . . . . . . . . . . . . . . . 15

5 February 8, 2018 175.1 Fun with Frobenius . . . . . . . . . . . . . . . . . . . . . . . . . . 175.2 More on orbits . . . . . . . . . . . . . . . . . . . . . . . . . . . . 175.3 Jordan decomposition . . . . . . . . . . . . . . . . . . . . . . . . 18

6 February 13, 2018 206.1 Representations of unipotent groups . . . . . . . . . . . . . . . . 206.2 Groups of dimension 1 . . . . . . . . . . . . . . . . . . . . . . . . 22

1 Last Update: August 27, 2018

7 February 15, 2018 237.1 Commutative groups with semi-simple points . . . . . . . . . . . 23

8 February 20, 2018 268.1 Continuous family of automorphisms . . . . . . . . . . . . . . . . 268.2 Automorphisms of projective space . . . . . . . . . . . . . . . . . 28

9 February 22, 2018 299.1 Ind-schemes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

10 February 27, 2018 3310.1 Comparing reduced parts of automorphism functors . . . . . . . 3310.2 Lie algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34

11 March 1, 2018 3611.1 Equivariant quasicoherent sheaf . . . . . . . . . . . . . . . . . . . 3611.2 Distributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37

12 March 6, 2018 4012.1 Lie algebra over characteristic p . . . . . . . . . . . . . . . . . . . 4012.2 Quotients . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42

13 March 8, 2018 4413.1 Parabolic subgroups and solvable groups . . . . . . . . . . . . . . 44

14 March 20, 2018 4814.1 Borel subgroup . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4814.2 Structure theory of connected solvable groups . . . . . . . . . . . 4914.3 Maximal tori . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50

15 March 27, 2018 5115.1 Reductive groups . . . . . . . . . . . . . . . . . . . . . . . . . . . 52

16 March 29, 2018 5416.1 Cartan subgroup . . . . . . . . . . . . . . . . . . . . . . . . . . . 55

17 April 3, 2018 5717.1 Nonsolvable groups with maximal torus of dimension 1 . . . . . . 57

18 April 5, 2018 6018.1 Groups of semi-simple rank 1 . . . . . . . . . . . . . . . . . . . . 6018.2 Roots . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61

19 April 10, 2018 6419.1 Examples of root systems . . . . . . . . . . . . . . . . . . . . . . 64

20 April 12, 2018 6620.1 Structure of reductive groups . . . . . . . . . . . . . . . . . . . . 66

2

21 April 17, 2018 6821.1 Structure of reductive groups II . . . . . . . . . . . . . . . . . . . 68

22 April 19, 2018 7222.1 Bruhat decomposition . . . . . . . . . . . . . . . . . . . . . . . . 7222.2 Simple roots . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73

23 April 24, 2018 7523.1 Classification of parabolics . . . . . . . . . . . . . . . . . . . . . . 7523.2 Length . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76

3

Math 224 Notes 4

1 January 23, 2018

This is a course on linear algebraic groups. We are going to try and followSpringer’s Linear algebraic groups. We will be working over a field k, but weare not going to restrict to varieties.

1.1 Linear algebraic groups

Definition 1.1. An algebraic group is a group object in the category Schft

of schemes of finite type.

Then for every scheme S, Hom(S,G) is a group, and is contravariant in S.

Definition 1.2. A linear algebraic group is a algebraic group that is affine.

Example 1.3. For V a vector space, we define GL(V ) in the following way:

Hom(S,GL(V )) = group of automorphisms of V ⊗OS as an OS-module.

This is something you have actually seen before. LetW be a finite-dimensionalvector space, and suppose we want to find W such that

Hom(S,W) = W ⊗OS .

Then W is actually Spec Sym(W ∗). This is because

Hom(S, Spec Sym(W ∗)) = HomComAlg(Sym(W ∗), OS) = HomVect(W∗, OS) = OS⊗W.

Let us generalize this and consider

Hom(S,M(V )) = monoid of all OS-linear endomorphisms of V ⊗ oS .

Then M(V ) is obtained from the vector space End(V ) in the previous construc-tion.

Example 1.4. For V = k, we have GL(k) = Gm. Here, we have

Hom(S,Gm) = OS .

Example 1.5. The group Ga is going to be Hom(S,Ga) = OS under addition.

A map between algebraic groups is a natural transformation of functors.Now let me try to explain what a determinant is. For each V , we can definedetV =

∧top(V ). We want to construct a map M(V ) → M(det(V )). To

construct this, we need to find a map between monoids. If α acts on V ⊗ OS ,its determinant det(α) acts on

∧topOS

(V ⊗ OS). This is the determinant map. IfV is of dimension 1, we have OS ∼= EndOS (V ). So we have the map

det : M(V )→M(det(V )) ∼= M(k) ∼= A1.

Math 224 Notes 5

So we have a map GL(V )→ Gm, and SL(V ) is the kernel of this map. This isa scheme because can take the fiber product.

Now fix a flag V = Vn ⊇ Vn−1 ⊇ · · · ⊇ V0 = 0. Now we want to define asubgroup P ⊆ GL(V ) by

Hom(S, P ) ⊆ Hom(S,GL(V ))

consisting of endomorphisms that send each Vi ⊗ OS to itself. This is going tobe the intersection of some MP and GL(V ) in M(V ). We want to show that Pis a closed subscheme of M(V ). To do this, it suffices to show that MP is closedin M(V ). Each Vi ⊗OS preserved by α means that

Vi ⊗Os → V ⊗Osα−→ V ⊗OS → (V/Vi)⊗OS

is zero. This is really Hom(Vi, V/Vi). So this MP fits in the fiber product

MP M(V )

∗∏iHom(Vi, V/Vi).

Now we have a map

1→ U → P →∏i

GL(Vi/Vi−1)→ 1.

Here, the map P →∏i GL(Vi/Vi−1) is given in the following way. For each

α : V ⊗ Os → V ⊗ OS , we have Vi ⊗ OS → Vi ⊗ OS so (Vi/Vi−1) ⊗ OS →(Vi/Vi−1)⊗OS . So we can consider its kernel.

Take a bilinear form V ⊗V → k. We can also define a H ⊆ GL(V ) given byHom(S,H) ⊆ Hom(S,GL(V )) consisting of α such that

(V ⊗OS)⊗OS (V ⊗OS) OS

(V ⊗OS)⊗OS (V ⊗OS) OS

α⊗α

commute.Now let OG be the algebra of regular functions on G. This OG is going to be

a commutative Hopf algebra, with maps ∆ : OG → OG⊗OG and γ : OG → OGand ε : OG → k.

There is also a notion of an algebraic group acting on a scheme. This is goingto be a morphism G×X → X. This is going to be an action of Hom(S,G) onHom(S,X) for each S that are compatible with maps on S.

Example 1.6. AnyG acts on itself on the left and on the right. Also GL(V ) actson V for a finite-dimensional V . This is tautological, because Hom(S,GL(V ))is the automorphisms of V ⊗OS and Hom(S,V) is V ⊗OS .

Math 224 Notes 6

The projective space is defined as

Hom(S,P(V )) = line bundles L on S with 0→ L → V⊗OS with locally free cokernel.

Example 1.7. GL(V ) acts on P(V ) because α is an automorphism of V ⊗OS .

More generally, consider n = nk ≥ nk−1 ≥ · · · ≥ n1 ≥ n0 = 0 where dimV =n. We can define the flag variety as the following: Hom(S,Fl(V )nk,...,n0) isthe collection of locally free sheaves 0 = M0 ⊆ M1 ⊆ · · · ⊆ Mk such thatrk(Mi) = ni and an identification Mk

∼= V ⊗OS such that Mi/Mi−1 are locallyfree.

1.2 Representations of linear algebraic groups

Definition 1.8. Let V be a vector space. A representation of G on V is, forevery S, an action of Hom(S,G) on V ⊗OS by automorphisms of OS-modules.

Lemma 1.9. To define on V a structure of G-representations is equivalent todefine on V a structure of an OG-comodule.

The idea is that V ⊗ OG is the functions on G with values in V . So acomodule structure V → V ⊗OG is something we want to send v to g 7→ gv.

Proof. Take S to be G, and the identity element on Hom(G,G). Then thereis a canonical automorphism α on V ⊗ OG. Then we can compose V → V ⊗OG

α−→ V ⊗ OG. For the other direction, we start with a comodule structureV → V ⊗ OG to a OG-linear map α : V ⊗ OG → V ⊗ OG. This turns outto be an automorphism. Then for any test scheme S with S → G, restrictV ⊗OG → V ⊗OG to S.

Corollary 1.10. If V is a G-representation, then for every finite-dimensionalW ⊆ V , there exists a W ⊆ V ′ such that V ′ is finite-dimensional and preservedby the G-action.

Proof. Given W , let V ′ be the minimal subspace in V such that im(W ) ⊆V ′ ⊗OG. We’ll continue next time.

Math 224 Notes 7

2 January 25, 2018

We’ll continue talking about representations. Let G be an algebraic group.A representation of G by V is an action of Hom(S,G) on V ⊗ OS . For g ∈Hom(S,G), the map V ⊗ OS → V ⊗ OS is just a map V → V ⊗ OS of vectorspaces. So this should be send an element v ∈ V to a function S → V . So wehad an equivalence Rep(G) ∼= OG−coMod.

2.1 Embedding of linear algebraic groups

Proposition 2.1. Let V ∈ Rep(G), then for any finite-dimensional W ⊆ Vthere exists a W ⊆ V ′ ⊆ V where V ′ is finite-dimensional and a subrepresenta-tion.

Proof. We have a coaction V → V ⊗OG and let V ′ be the minimal subspace in Vsuch that the co-action of W is contained in V ′⊗OG. So given a W ′ ⊆ V ⊗OG,what is the minimal V ′ such that W ′ ⊆ V ′ ⊗ OG? This can be computed bythe span of (id⊗ξ)(W ) for ξ ∈ U∗.

Now our claim is that V ′ is a subrepresentation. We need to show thatthe coaction of V ′ is contained in V ′ ⊗ OG. By definition, we need to showthat W → V ⊗ OG → V ⊗ OG ⊗ OG → V always lies in V ′. This is clearby interpreting the map V ⊗ OG → V ⊗ OG ⊗ OG as the comultiplication onOG.

Corollary 2.2. Any algebraic group G has a closed embedding to GL(V ) forsome finite-dimensional V .

Proof. We know that OG is a representation. Because G is finite type, thereexists a W ⊆ OG such that Sym(W ) → OG, and so we can replace W with aV ⊆ OG that is a representation. From this we get a map G → GL(V ). Wewould like to show that this is a closed embedding, and it suffices to show that

G → GL(V ) →M(V )

is a closed embedding, where GL(V ) →M(V ) is an open embedding.Now M(V ) = M(V ) is the variety associated to the vector space V ⊗ V ∗.

I’m trying to show that Sym((V ⊗V ∗))→ OG is surjective. But we started witha surjection Sym(V ) OG. But how is V → OG related to V ⊗ V ∗ → OG?There is a natural covector V → OG → k by evaluation on 1. It turns out thatthe original V → OG is

Vid⊗ ev1−−−−−→ V ⊗ V ∗ → OG.

So Sym(V ⊗ V ∗)→ OG is surjective.

Corollary 2.3. If G acts on X and X is affine, there exists a closed embeddingof X → V for V a finite-dimensional representation on which G acts.

Math 224 Notes 8

Proof. We have G×X → X which becomes OX → OX ⊗OG. Let W ⊆ OX bea subspace that generates it as an algebra. We can replace W by V that is asubrepresentation. We have a surjection Sym(V ) OX , or a closed embeddingX → V.

Proposition 2.4. HomRep(G)(V,OG) = V ∗.

Proof. The map in one direction is V → OGev1−−→ k. For the other direction, a

representation is given by V ⊗ V ∗ → OG. So any element of V ∗ gives a mapV → OG.

Actually, the map V ⊗ V ∗ → OG is a map of G×G-representations.

2.2 The regular representation

We would now like to construct the regular representations from the represen-tations. A map between representations ϕ : V → W will give a commutativediagram

V ⊗ V ∗ OG

V ⊗W ∗ W ⊗W ∗id⊗ϕ∗

ϕ⊗id

For a category C, we define the twisted arrow category TwArr(C) consist-ing of objects C0 → C1 and morphisms

C0 C1

C ′0 C ′1.

If we have two functors F,G : C → D, then we can look at the natural trans-formations from F to G. These are collections of αC ∈ Hom(F (C), G(C))satisfying some compatibility conditions. Now my claim is that

Nat(F,G) = lim←−TwArr(C)

H

where H isH(ϕ : C0 → C1) = HomD(F (C1), G(C0)).

This is because a limit is just a family satisfying some compatibility conditions.

Definition 2.5. Let F : C → Set and G : Cop → Set be the functors. We candefine a functor H : TwArr(C)→ Set defined by

H(α : C0 → C1) = F (C0)×G(C1).

Then we define the coend as

coend(F,G) = lim−→TwArr(C)

H.

Math 224 Notes 9

So let C = Rep(G)fd and D = Rep(G×G). Then we define

H(α : V →W ) = V ⊗W ∗.

Then the claim we want to show is that

lim−→TwArr(Rep(G))fd

H = OG.

Math 224 Notes 10

3 February 1, 2018

Last time I sold the product of thinking of representations as comodules.

3.1 Examples of groups and classification of representa-tions

Let’s consider the group Ga. This is A1 with addition, so it is given by OG = k[t]with

k[t]→ k[t]⊗ k[t]; t 7→ t⊗ 1 + 1⊗ t.

I would like to give a complete description of representations. These are goingto be

V → V ⊗ k[t],

and so we can dualize it tok[s]⊗ V → V.

How are we doing this? In general, if we have V1 → V2⊗W then we can dualizeit to W ∗ ⊗ V1 → V2. This will have the condition that any v1 ∈ V1 is killed bya sufficiently large quotient of W ∗. So this k[s]⊗ V → V should be a map suchthat any v ∈ V is killed by a sufficiently large power of s.

Let Γ be a finite group. This is an algebraic variety given by

OΓ =⊕γ∈Γ

kδγ ,

with comultiplication

∆(δγ) =∑

γ1γ2=γ

δγ1 ⊗ δγ2 .

So we can truly dualize V → V ⊗ OΓ, and get k[Γ] ⊗ V → V . There are nomore representations other than the normal ones.

Now let G = Gm. Here, OG = k[t, t−1] and ∆(t) = t ⊗ t. There arerepresentations Gm → Gm given by x 7→ xn, which we can think of as nthcharacters.

Proposition 3.1. Every representation of Gm is a direct sum V ∼= ⊕nV (n)

where V (n) is a direct sum of copies of the nth characters.

Proof. Look at the map V → V ⊗ k[t, t−1] and write it as

v 7→∑n

vn ⊗ tn.

Now let us apply todotodo

Math 224 Notes 11

Let us look at Λ ∼= Zn a lattice. We have

k[Λ] ∼= k[t1, t−11 , . . . , tn, t

−1n ],

and then the torus

T = Spec k[Λ] ∼= Gm × · · · ×Gm.

It can be checked that ∆(λ) = λ⊗ λ.

Proposition 3.2. Rep(T ) is equivalent to VectΛ.

You can do the exact same thing. Note that this discussion is contravariantin Λ.

Proposition 3.3. There is an isomorphism Hom(Λ1,Λ2) ∼= HomAlgGrp(T2, T1).

Proof. By definition, Hom(Λ1,Λ2) isomorphic to Hom(k[Λ1], k[Λ2]) where Homis as Hopf algebras. Given any λ1 ∈ Λ1, we want to show that phi(λ1) isprimitive. This is because primitive is equivalent to the statement that ∆(λ) =λ⊗ λ. So we get map Λ1 → Λ2.

3.2 Tannaka duality

Representations is a tensor category. That is, you can tensor representations.For a test scheme S, there is an action of Hom(S,G) on V ⊗OS . Then if I haveV1 and V2, we can take

V1 ⊗ V2 ⊗OS = (V1 ⊗OS)⊗OS (V2 ⊗OS).

At the level of comodules, we can describe this in the following way. If A is aHopf algebra, we can define the tensor product of V1 → V1⊗A and V2 → V2⊗Athen

V1 ⊗ V2 → V1 ⊗A⊗ V2 ⊗A→ V1 ⊗ V2 ⊗A⊗A→ V1 ⊗ V2 ⊗A.

Here, we’re using that A as an algebra is commutative, when we’re saying thatV1 ⊗ V2

∼= V2 ⊗ V1.

Corollary 3.4. (i) We have an isomorphism Λ ∼= Hom(T,Gm).

(ii) We have an isomorphism Λ∨ = Hom(Λ,Z) = Hom(Gm, T ).

Now here is Tannaka’s theorem. There is a forgetful functor

F : Rep(G)→ Vect

which is a tensor functor, which means that it sends tensor products to tensorproducts. Sometimes I will write F (V ) = V . Tannaka’s theorem says that wecan recover G by this functor.

Math 224 Notes 12

Given any element g ∈ G, for every V the element g acts on F (V ) asαV : F (V ) → F (V ). So Tannaka’s theorem is going to say that the auto-morphism group of this functor is going to be G. But it is going to be a tensorautomorphism. In particular, we have

F (V1 ⊗ V2) F (V1 ⊗ V2)

F (V1)⊗ F (V2) F (V1)⊗ F (V2).

∼=

αV1⊗V2

∼=αV1⊗αV2

I would like to recover S-points of G.So we can consider the functor

F fdS : Repfd(G)→ Coh(S); V 7→ V ⊗OS .

There is an obvious map

Hom(S,G)→ Auttensor(FS)

with the same imposed condition.

Theorem 3.5 (Tannaka). This map Hom(S,G)→ Auttensor(FS) is an isomor-phism.

Proof. We are going to construct the map in the other direction. Every tensorautomorphism of F fd

S canonically extends to a tensor automorphism of FS :Rep(G) → QCoh(S). Now we can consider maps A ⊗ A → A for A ∈ Rep(G).So it makes sense to talk about commutative algebras in Rep(G). If F is a tensorcategory, every Auttensor(FS) should be respect the algebra structure.

Now let us take A = OG. Then FS(A) = OS ⊗ OG, and we obtain anautomorphism of OS ⊗ OG as an OS-algebra. So we get an automorphismα : S × G → S × G over S. We need to check that this is given by leftmultiplication, and it suffices to show that the diagram

S ×G×G S ×G×G

S ×G S ×G

id×m

α⊗idG

id×m

α

commutes. To show this, we need to go back to the functor. This comes fromthe commutative diagram

F (OG ⊗OG) F (OG ⊗OG)

F (OG) F (OG)

αOG⊗OG

∆

αOG

∆

and taking spectra.

Math 224 Notes 13

What this also tells us is that if g1, g2 ∈ Auttensor(FS) acts in the same wayon OG, then it acts in the same way on each V . Here is the reason. Every Vcan be embedded in V ⊗OG, which is a bunch of copies of OG. Here is why.

We saw from last time that we have a map V ⊗ V ∗ → OG as G × G-representation. Then we can move V ∗ to the other side and get V → OG ⊗ Vand an OG-linear map

V ⊗OG → V ⊗OG.

Considering OG ⊗ V as functions on G with values in V , this is f 7→ f(g) =g−1f(g). Now we consider a G×G-action on the domain V ⊗OG, given by thefirst G acting on V ⊗OG and the second G acting on OG. Consider the G×G-action on the target V ⊗OG, given by the first G acting on the second OG andthe second G acting on V ⊗G. Then the map V ⊗ OG → V ⊗ OG intertwinesthe G × G-action. This is a wonderful property of the regular representation.For the first G, you lose the representation, and for the second G, you gain thisinformation. Now if we ignore the second G-action, we get an intertwining

V ⊗OG → V ⊗OG

that is an isomorphism of G-representations, where G acts on the left sidediagonally. This will prove that the two construction of the Tannaka dualityare inverses.

Math 224 Notes 14

4 February 6, 2018

Let G be an algebraic group.

4.1 Connected component

Theorem 4.1. There exists an irreducible component G0 ⊆ G such that

(a) it is a subgroup,

(b) it is a connected component,

(c) it is normal with finite index.

Proof. Suppose I have to components X,Y containing e. Then we can look atthe closure XY . Then we get X ⊆ XY and Y ⊆ XY and so X = XY = Y .Likewise, if e ∈ X then e ∈ X−1 and so X = X−1. This shows (a).

Now consider the decomposition∐g∈G

gG0 = G

as a scheme for g ∈ G(k) (if k is algebraically closed). The claim is that allg ∈ G0 are connected. This can be seen by checking g1G

0 ∩ g2G0 6= 0 implies

g1G0 = g2G

0.To see that it is normal, we look at the conjugation on G. It sends connected

components to connected components, and the identity to the identity. So itsends G0 to G0. It being finite index means that Hom(S,G0) is of finite indexin Hom(S,G), which is because G is quasi-compact.

The groups Ga, Gm, GLn, Bn, Un, SLn are all connected. SLn is because

GLn×GLn → SLn; (g1, g2) 7→ g1g2g−11 g−1

2

is surjective at the level of k-points. The source is connected, and so the targetshould be connected. But On is not connected for char k 6= 2 because

On ±1

GLn Gm.det

We also have that SOn and Sp2n are connected. To show this, we use thefact that these groups are generated by connected subgroups.

Proposition 4.2. Let Yi by irreducible with maps φi : Yi → G. Assume thate ∈ im(φi). Let H be the (closed) subgroup of G generated by them. Then H isirreducible.

Math 224 Notes 15

Proof. Let I = 1, . . . , n, and consider the multiplication (with inversions) map

Y1 × · · · × Yn × Y1 × · · · × Yn × · · · × Yn → G.

Let Zm be the closure of the image. Then Zm ⊆ Zm+1 ⊆ G are irreduciblevarieties, and because G has finite Krull dimension, it stabilizes. Then Zm·Zm ⊆Zm+m and also Z−1

m ⊂ Zm, so Zm is an irreducible subgroup.

4.2 Image and surjectivity

If Φ : G′ → G is a homomorphism, we can look at its image G′ → im(Φ). Thisis a subgroup, as you can see by the functor of points.

Proposition 4.3. The map G′ → im(Φ) is surjective on k-points.

Proof. In general, if we have a dominant morphism X → Y of reduced varieties,there exists some open dense Y 0 ⊆ Y such that Y 0(k) ⊆ im(X(k)). (Thisis some kind of Chevalley.) Let’s use the following lemma on U = Y 0 andV = im(G′(k)). Then we get

G(k) = U(k) · V ⊆ im(G′(k)) · im(G′(k)) ⊆ im(G′(k)).

This finishes the proof.

Lemma 4.4. If U ⊆ G is open and V ⊆ G(k) is dense, then U · V = G.

Proof. For g ∈ G(k), we have

g−1V ∩ U(k) 6= ∅

by definition of density. This means that g ∈ V · U(k).

Let G act on a variety X. Consider the action map G→ X and look at theclosure Gx.

Definition 4.5. An action of G on X is transitive if G(k) acts transitively onX(k).

This is not good if X is not reduced, but is reasonable if X is reduced.

Proposition 4.6. Let G be reduced. Then Gx contains a unique open G-stablesubvariety with a transitive action that contains the point x.

Proof. The map G→ Gx is dominant, and therefore there exists an open Y ⊆Gx such that Y (k) ⊆ im(G(k)). Define the desired open subvariety by Z =⋃g gY

. This is transitive and G-stable by construction. Suppose x /∈ Z(k), so

that x /∈ Gx−Z. But the right hand side is closed and G-stable, so Gx ⊆ Gx−Z,which is a contradiction.

Math 224 Notes 16

Proposition 4.7. Let X be reduced and G(k) act on X(k) act transitively.Then for all x ∈ X(k),

G→ X; g 7→ gx

is faithfully flat.

Proof. In general, if I have a dominant map ϕ : Z1 → Z2 with Z2 reduced,there exists a nonempty open Z2 ⊆ Z2 over which ϕ is faithfully flat. Applyingthis, you see that there exists a nonempty open X ⊆ X over which G → Xis faithfully flat. Then the same is true for gX for all g ∈ G(k), and thenX =

⋃g gX

.

Using a similar idea, you can show that if the kernel ker(G1 → G2) is finite,then the morphism G1 → G2 is finite. Quasi-finite does not imply finite, butyou can take a open on which it is finite and then translate it around.

Example 4.8. Let X be a point and G be any non-reduced algebra group.Then G→ X is not smooth. But if G is any non-reduced algebraic group thenthe kernel of Frob : G → G is never reduced. For instance, if G = Ga thenker(Frob) = Spec k[t]/tp.

Math 224 Notes 17

5 February 8, 2018

We were talking about the Frobenius X → X.

5.1 Fun with Frobenius

If X = An, the preimage of the Frobenius would be

k[t1, . . . , tn]/tq1 · · · tqn.

So if X is smooth of dimension n then Frob−1q (x) is a scheme of length qn. So

the inverse image is non-reduced unless the point is isolated.Consider X = Ga. If we look at ker(Frobp, then it looks like Spec(k[t]/tp).

Let us look at its representations. Its dual is O∗G = k[x]/xp, and so these areendomorphisms whose p-th power is zero.

Also consider X = Gm, and this is

ker(Frobp) = µp = Spec(k[t, t−1]/tp = 1).

Again, let us look at representations of G. These are modules for the dual O∗G.There is an element called ξ = t∂t ∈ O∗G that maps ti 7→ i. (The motivation istaking the Lie derivative t∂t.)

The claims is that ξp = ξ. This is becasue

ξp(ti) = 〈ξ ⊗ · · · ⊗ ξ, ti ⊗ · · · ⊗ ti〉 = ip = i

for any i. Then you can show that O∗G = k[ξ]/ξp − ξ. Now representationsRep(G) are vector spaces graded by Z/pZ. You can see this in the followingway: we have a short exact sequence

1→ G→ Gm → Gm → 1,

where Gm → Gm is given by taking the pth power. Representations of Gm aregiven by VectZ, so representations pull back by a factor-p scaling of indices.

5.2 More on orbits

We defined a transitive action of a reduced G on X to be an action such thatX is reduced and G(k) acts transitively on X(k).

Proposition 5.1. If G acts on X transitively, then for all x ∈ X, the actionmap G→ X is faithfully flat.

So we can deduce the strongest notion from the weakest notion.

Theorem 5.2. Let G reduced act on X. The for all x ∈ X(k), there exists aunique locally closed subset Y ⊆ X such that x ∈ Y and the action of G on Yis transitive. This Y is called the orbit of x in X.

Proposition 5.3. Let G act on X. There exist closed orbits.

Proof. Take an orbit of smallest possible dimension Y . Call it Y . If Y is not Y ,there is going to a point in Y outside Y , and then consider the orbit of anythingin Z.

Math 224 Notes 18

5.3 Jordan decomposition

Let V be a vector space and T be a endomorphism.

Theorem 5.4 (Jordan decomposition).

(1) There exists a decomposition T = Tn + Tss such that they commute.

(2) If T is an automorphism, there exists a decomposition T = Tu · Tss suchthat TuTss = TssTu. Moreover,

(a) there exist pu, pss ∈ k[t] such that Tu = pu(T ) and Tss = pss(T ),

(b) [S, T ] = 0 implies [S, Tu] = [S, Ts] = 0,

(c) if T = T ′u + T ′ss and [T ′u, T′ss] = 0 then T ′u = Tu and T ′ss = Tss,

(d) if S : V1 → V2 intertwines T1 and T2 so that S T1 = T2 S, thenST1,u = T2,uS and ST1,ss = T1,ssS,

(e) If T1 acts on V1 and T2 acts on V2 so that T1 ⊗ T2 acts on V1 ⊗ V2,then (T1 ⊗ T2)u = T1,u ⊗ T2,u and (T1 ⊗ T2)ss = T1,ss ⊗ T2,ss.

Proof. By Jordan decomposition, we know that V =⊕

λ V(λ) so that (T −

λ id)|V (λ) are nilpotent. Then we can set Tss|V (λ) = λ id. Similarly, we candefine Tu and Tss.

(a) Let πλ be the projector of V onto V (λ). By the Chinese remaindertheorem, there exists a pλ such that πλ = pλ(T ). Then we can set

∑λ λpλ(T ).

Then the others can be deduced.

The theorem is false if the field is perfect. Suppose char(k) = p and notperfect, so that there exists an element a ∈ k that does not admit a pth root.Consider the vector space

V = k′ = k[t]/(tp − a),

with T multiplication by t. If we tensor with the algebraic closure,

V ⊗k k = k[t]/(tp − a) = k[t]/(t− b)p.

In this case, Tn = t− b and Tss = b, and this cannot be done in k.Let G be reduced and g ∈ G(k).

Definition 5.5. g is semi-simple/unipotent if the following equivalent con-ditions hold:

(1) g acts on OG unipotently on the left (this is on finite-dimensional sub-spaces)

(2) g acts on OG unipotently on the right

(3) g acts unipotently on any finite-dimensional representation.

Lemma 5.6. If g ∈ G is unipotent/semi-simple and γ : G→ G′ is a homomor-phism, then ϕ(g) is unipotent/semi-simple.

Math 224 Notes 19

Proof. Given any G′-action on V , we can consider it as a G-action.

Theorem 5.7. For every g there exists a g = gu · gss with gssgu = gugss withgu unipotent and gss semi-simple. Furthermore,

(a) the decomposition survives homomorphisms G→ G′,

(b) the decomposition is unique.

Proof. This can be done using Tanaka, but we can do this in an elementaryway. Consider the decomposition of right multiplication r(g) = r(g)ur(g)ss inEnd(OG). You can do this by the locally finite action of G on OG. If we lookat OG, we have

OG ⊗OG → Og,

and the action on the left with rg ⊗ rg is the action on the right with rg. Thatis, we have

(r(g)⊗ r(g))u = r(g)u ⊗ r(g)u.

But these are also algebra homomorphisms. But to show that these actuallycome from right multiplication, it suffices to show that they commute with leftmultiplication. This can be checked.

Definition 5.8. A group is called unipotent if every element in it is unipotent.

Example 5.9. Ga is unipotent. Take an element a ∈ Ga, and let’s see how itacts on the regular representation k[t]. It acts as t 7→ t− a, and on the span of1, . . . , tn, it is an upper triangular matrix.

Theorem 5.10. Take G = GLn. A point g is semi-simple/unipotent in thisalgebraic group sense if and only if it is semi-simple/unipotent as an automor-phism of kn.

We’ll prove this later.

Lemma 5.11. If G is unipotent, and G′ is a subgroup of G, then G′ is unipotent.

Proof. For g ∈ G′ ⊆ G, we get OG OG′ and it preserves ideals by actions ofg.

Theorem 5.12. Let g ∈ G(k) be of finite order. Then it is finite order. (Aslong as the order is coprime to the characteristic of k.)

Proof. Assume Tm = id. Then tm − 1 has distinct roots.

For example, the subgroup Z/pZ ⊆ Ga is semi-simple if and only if charac-teristic of k is not p.

Math 224 Notes 20

6 February 13, 2018

Let G be an algebraic group.

Definition 6.1. A g ∈ G(k) is semi-simple/unipotent if its action on anyfinite-dimensional representation of G is semi-simple/unipotent. This is equiv-alent to its action on OG being semi-simple/unipotent.

We showed that any homomorphism φ : G1 → G2 sends semi-simple/unipotentelements to semi-simple/unipotent elements.

Example 6.2. Any element in Gm is semi-simple. Any element in Ga is unipo-tent.

Theorem 6.3. An element g ∈ GLn(k) is semi-simple/unipotent if and only ifits action on kn is semi-simple/unipotent.

Proof. Assume that g acts semi-simple/unipotently on kn = V . If g is semi-simple, we can write

V =⊕i

V i, g|V i = λi.

Then we can consider g ∈ T (k) where T → GLn is the torus∏

Gm. Nowassume that g is nilpotent in GLn(k). We want to define ϕ : Ga → GLn withg = ϕ(1). But defining ϕ is the same as defining a representation of Ga on kn,and this is the same as the datum of a unipotent operator on kn.

There was a fire alarm. But we moved to the next building.

6.1 Representations of unipotent groups

Proposition 6.4. (i) If G is a finite group and (|G|, p) = 1, then any elementof G(k) is semi-simple.

(ii) If |G| = pk, then any element of G(k) is unipotent.

Theorem 6.5 (Jordan decomposition). For every g ∈ G(k), there exists aunique decomposition g = gsgu where gs, gu are semi-simple and unipotent, andgsgu = gugs.

Theorem 6.6. Let G be unipotent and let V be an irreducible representation.Then V is trivial.

If G acts on V , then G(k) acts on V . If V1 is a subrepresentation in thefirst sense, then V1 is a subrepresentation in the second sense. But the otherdirection is true in the reduced case as well.

Lemma 6.7. If V1 ⊆ V is preserved by G(k), then it is a subrepresentation.

Proof. We know that G(k) is Zariski dense in G. So

G× V1 → G× V → G× V/V1

todotodo

Math 224 Notes 21

Let V be irreducible as a representation of G(k).

Theorem 6.8 (Burnside). Let Tα be a collection of endomorphisms of V suchthat there does not exist V1, 0 ( V1 ( V such that Tα|V1

⊆ V1. Then the spanof Tα and their compositions is End(V ).

So for instance, G(k) span End(V ).

Corollary 6.9. Let G be unipotent and let V be a finite-dimensional represen-tation. Then there exists a flag

0 = V0 ⊆ V1 ⊆ · · · ⊆ Vk = V

such that Vi are subrepresentations such that Vi/Vi−1 is trivial.

Proof. By induction.

Corollary 6.10. Let ϕ : G→ GLn be a homomorphism. Then there exists anelement g ∈ GLn(k) such that conjugation by g gives

G→ U(n).

Proof. g is what takes the standard flag to the flag of the previous corollary.

Corollary 6.11. G has a filtration by normal group subschemes

1 = G0 ⊆ G1 ⊆ · · · ⊆ Gn = G

such that Gi/Gi−1 is a subgroup of Ga.

This is not very informative in characteristic p. Let G′ be the Zariski closureof the group generated by G×G→ G with (g1, g2) 7→ g1g2g

−11 g−1

2 .

Corollary 6.12. If G is unipotent, then G′ ( G.

Proof. This follows from the corresponding property of U(n).

Suppose unipotent G acts on an affine variety X. We have a notion of theorbit Gx ⊆ Gx ⊆ X.

Proposition 6.13. Let G be unipotent. Then Gx is closed.

Example 6.14. Consider Ga acting on P1. Then the orbit of 0 is A1, which isnot closed.

Proof. Let I ⊆ OGx be the ideal of Gx−Gx. Then I is a representation of G,and every representation has a nonzero 0 6= f ∈ IG. Then f |Gx is constant,and because Gx is dense in Gx, we have f |Gx is constant. But f |Gx−Gx = 0, sof = 0.

Math 224 Notes 22

6.2 Groups of dimension 1

Theorem 6.15. Let G be a connected algebraic group of dimension 1. Theneither G = Ga or G = Gm.

Proof. Because it is smooth, there exists a canonical curve G such that G is anopen subset. The claim I want to make is that the action of G on itself by lefttranslations extends to an action of G on G.

For now let’s just use it to prove the theorem. Consider S = G − G. Thisconsists of finitely many points and nonempty because G is not projective. G(k)acts on G because it is functorial in the naıve sense, and it preserves S becauseG is connected.

Now I claim that if a proper curve X admits an infinite group of auto-morphisms that preserve a nonempty subset S ⊆ X, then X = P1. The onlyother case possible is an elliptic curve, but this is not possible because the onlyautomorphisms are translations up to a finite group.

Now we show that G is commutative. We’ll get back to this later.

Theorem 6.16. Let G be a commutative algebraic group. Then G→ Gu ×Gswhere Gu is unipotent and all elements in Gs are semi-simple.

Math 224 Notes 23

7 February 15, 2018

Theorem 7.1. If G is unipotent and V is an irreducible representation, thenV ∼= k.

Corollary 7.2. If G is unipotent, every representation admits a flag such thatthe action of G on Vi/Vi−1 is trivial.

Corollary 7.3. If G is unipotent, it embeds into U(n).

Corollary 7.4. If G is unipotent, there exist 1 = G0 ⊆ G1 ⊆ · · · ⊆ Gn suchthat Gi is reduced and G′i ⊆ Gi−1.

Proof. Take Gi = (G ∩ U(n)i)red.

Theorem 7.5. If G is connected and reduced of dimension 1, then G ∼= Gm orG ∼= Ga.

Proof. We first prove that any G as above is commutative. Let ϕg = Adg forg ∈ G(k). We want to show that ϕg = id. We will do this later.

So G is commutative. We want to show that we can write G = Gu×Gs whereGu is unipotent and all elements of Gs are semi-simple. First note that the setof unipotent elements is Zariski-closed because on every representation V , it isT ∈ End(V ) : (T − id)dimV = 0. Also, if g1, g2 are unipotent and commute,then g1g2 is unipotent. So Gu is a closed subgroup of G whose k-points areexactly the unipotent elements.

Likewise, we have G → GL(V ) and because G commutes, we can diagonalizeall semi-simple elements of G(k) so that

V ∼=⊕i

Vi.

Then we can take Gs = G ∩ T where T is the torus that acts on each Vi. Thenevery semi-simple element of G will belong to Gs.

We naturally have a map Gs × Gu → G. We will construct a map ϕ :G→ Gs that at the level of k-points exacts the semi-simple part of the Jordandecomposition. To construct this, we note that with the diagonalization V =⊕

i Vi, we haveG→ Bn Tn

where Bn is the upper-triangular matrices and Bn → Tn is the projection. Atthe level of k-points, this map extracts the semi-simple part, and also it landsin Gs.

So G = Gs ×Gu, and so either Gu = G or Gs = G.

7.1 Commutative groups with semi-simple points

Theorem 7.6. Let G be reduced. The following are equivalent:

(1) G is commutative and all of points are semi-simple.

Math 224 Notes 24

(2) Every representation splits as a direct sum of 1-dimensional representa-tions.

(3) There exists an embedding G → Tn.

(4) Characters span OG.

(5) There exists a finite abelian group Λ such that (|Λtors|, char k) = 1 andOG ∼= k[Λ].

Definition 7.7. A character is a homomorphism G → Gm, which is thesame as k[t, t−1] → OG, which is the same as an element f ∈ O×G such that∆(f) = f ⊗ f .

Here is an application.

Corollary 7.8. If G is a commutative connected group of dimension 1 all ofwhose points are semi-simple, then G ∼= Gm.

Proof. Let us writeG = Spec k[Λ]. Then Λ = Λtors⊕Zn and soG = Spec k[Λtors]×(Gm)×n, and the reducedness, connectivity and dimension conditions forceG = Gm.

Now let us prove the theorem.

Proof. (1) ⇒ (2) is just simultaneous diagonalization. (2) ⇒ (3) follows fromthe embedding G → GL(V ). For (3) ⇒ (4), We have OTn OG, and OTn =k[t1, t

−11 , . . . , tn, t

−1n ]. So characters span OG. For (4) ⇒ (5) we define Λ to be

the group formed by the characters of G. Then we have a surjection

OG k[Λ]

of Hopf algebras. We need to show that the characters are linearly indepen-dent in OG. Suppose

∑i aifi = f1 with minimal cardinality. Then applying

comultiplication gives∑i

aifi ⊗ fi = f1 ⊗ f1 =∑i,j

aiajfi ⊗ fj .

Because fi are linearly independent, aiaj = 0 and a2i = ai. For (5) ⇒ (1), we

just useG = Spec k[Λtors]×Gm × · · · ×Gm.

Then it suffices to show that Spec k[Λtors] have semi-simple points, but this issomething we have seen. It is going to be a finite product of µn.

Now let us show that if G is a unipotent group of dimension 1 then G = Ga.We showed that

G = X → X ∼= P1.

Also, X −X = S 6= ∅ is finite and nonempty.

Math 224 Notes 25

Proposition 7.9. (1) PGL2∼= Aut(P1), i.e., Hom(S,PGL2) is isomorphic

to Aut(S × P1/S).

(2) The action of G on X by left multiplication extends to an action of G onX.

If we have the proposition, we obtain a map G → PGL(2). Then G →Perm(S), and because G is connected, G preserves elements of S. Choose oneelement S, and call it ∞. The subgroup of PGL2 that preserves ∞ is GmnGa:

1 Ga Ga nGa Gm 1

G

But the composite G → Gm is trivial, because G is unipotent and the onlyunipotent element of Gm is 1. (We will prove that G → PGL2 is a closedembedding.) So we get G → Ga, and because both are 1-dimensional, G→ Gais an isomorphism.

Now let us show that G → PGL2 is a closed embedding. We know thatϕ : G1 → G2 being a homomorphism with finite kernel implies that ϕ is finite.A finite map is a closed embedding if it is injective on S-points. So it is enough toshow that G→ PGL2 is injective at the level of S-points, and this is because anynontrivial automorphism S×X/S induces a nontrivial automorphism S×X/S.

To prove this proposition, we are going to use the following general fact. ForX a proper scheme and Y a quasi-projective scheme, we want to have

Hom(S,Map(X,Y )) = Hom(S ×X,Y ).

Theorem 7.10. Map(X,Y ) is representable.

Proof. If Y1 → Y2 is a closed embedding, Map(X,Y1)→ Map(X,Y2) is a closedembedding, and likewise for open embeddings. So it suffices to show this forY = P(V ).

We have S ×X → P(Y ) corresponding to a line bundle L on S ×X suchthat L → V ⊗OS×X with flat quotient. This means V ∗ ⊗OS×X L −1.

Recall that if X is proper and E is a coherent sheaf on X, then the functorthat sends S to the to the set of all quotient coherent sheaves

E ⊗OS F

on X×S that are S-flat is representable. This representing scheme is called theHilbert scheme of E . Now take E = V ∗ ⊗OS . In our functor, F has to be aline bundle on X × S, and this is an open condition. So it is representable.

We are actually interested in Aut(P1), not E = Map(P1,P1). Then we haveto look at the image of the inverse image of the identity of E × E → E.

Math 224 Notes 26

8 February 20, 2018

We are still trying to prove that if G is connected, reduced, 1-dimensional, thenG = Gm or G = Ga.

Theorem 8.1. Such G is commutative.

Proof. If there exists a g ∈ G(k) of infinite order, then G is commutative. Thisis because Adg = id coincides on gn. Then g is in the center, and also gk is inthe center, and this is also Zariski dense.

Now let us assume that there is no element of infinite order after any fieldextension. Now I claim that the order is bounded, i.e., gN = 1 for all g ∈ G(k).To show this, let us consider the map G→ G given by g 7→ gN , and look at thekernel GN . If we assume that the order is unbounded, all subschemes GN areproper. Let K be the field of rational functions on G. Then the generic point ηcan be considered as a closed point in the base change OG⊗kK. Now if Z ⊆ Gis any proper subvariety, then η ∈ G(K) is not in Z(K). So η /∈ GN (K) for allN , and this shows that η ∈ G(K) is of infinite order. (Here, commutativity canbe checked after a base change, because it’s a variety.)

Now take a embedding φ : G → GL(n). We know that φ(g)N = id. There-fore the characteristic polynomial of φ(g) divides (1 − tN )n. Therefore thereexist finitely many options for the characteristic polynomial ch(φ(g)). Now wecan consider the characteristic polynomial as

Gφ−→ GL(n)

ch−→ An.

But we know that the image lies in finitely many points, and G is connected.So it should all map to 1. Now matrices that have the characteristic polynomial(x− 1)n is unipotent by definition. So φ(G) consists of unipotent elements, andthus G is unipotent. This shows that G′ ( G, and also G′ is connected. SoG′ = e and G is commutative.

8.1 Continuous family of automorphisms

Then we showed that if G is a commutative algebra group then G = Gs × Guwhere Gs have all elements semi-simple and Gu is unipotent. Then we had thefollowing theorem.

Theorem 8.2. The following are equivalent:

(1) G is commutative and the elements are semi-simple.

(2) Every representation splits as a direct sum of 1-dimensional representa-tions.

(3) G ⊆ Gm × · · · ×Gm.

(4) OG is spanned by characters.

(5) G = Spec k[Λ] where Λ is a finitely generated abelian group whose torsionis coprime to p.

Math 224 Notes 27

Now it can be shown that Spec k[Λ1] → Spec k[Λ2] correspond to mapsΛ2 → Λ1. (We showed this for Λ torsion-free, and the torsion case can beproved in the same way.) This is very different from the unipotent case. Thereis a continuous family of automorphisms Ga → Ga, for instance, given by

Gm ×Ga → Ga.

This is actually the universal example.

Proposition 8.3. Let φ : S × G1 → G2 be a morphism where G1 and G2 arediagonalizable. If S is connected, then φ factors through S ×G1 → G1 → G2.

Proof. It suffices to show that G2 = Gm, because G2 embeds into some torus.Let us denote G1 = G. Then we have a map

S ×G→ Gm = Spec k[t, t−1]

where t maps to∑i fi ⊗ χi, where χi are characters on G. The condition that

this is a group homomorphism is(∑i

fi ⊗ χi)⊗OS

(∑i

fj ⊗ χj)

=∑k

fk ⊗ χk ⊗ χk.

Then ∑i,j

fifj ⊗ χi ⊗ χj =∑k

fk ⊗ χk ⊗ χk.

This shows that only fi is 1 and all other fi are 0, because S connected meansthat there are only two idempotents 0 and 1.

Now comes the fun part. We looked at G = P1. Then P1−G = S is a finiteset. We are assuming that G is unipotent and trying to show that G ∼= Ga.

Theorem 8.4. (1) There is an action PGL2 → Aut(P1).

(2) There exist a homomorphism G → Aut(P1, S) such that the action re-stricts to the left multiplication on G.

Here, if X and Y are algebraic varieties, then we can define a prestack

Hom(S,Map(X,Y )) = Hom(S ×X,Y ).

Theorem 8.5. If Y is quasi-projective and X is proper, then Map(X,Y ) isrepresentable.

Before proving Theorem 8.4, we finish the proof of G = Ga. We may assumethat ∞ ∈ S, and then we get

G→ Aut(P1,×) ∼= Gm nGa.

Because G is unipotent, G → Gm n Ga → Gm is trivial. So G maps to Ga. Ifwe can show that this is a closed embedding, we are done.

Math 224 Notes 28

Lemma 8.6. A homomorphism of algebra groups is a closed embedding if it isa categorical injection, i.e., injective S-points for all S.

Proof. We can factor G1 → φ(G2) → G2. Now it suffices to show that G1 →φ(G2) is a closed embedding. But it is a general fact that a finite morphism ofschemes that is a categorical embedding is a closed embedding. So it sufficesto show that G1 → φ(G1) is finite. But we know that it is quasi-finite becauseits kernel is a point, and then a quasi-finite morphism of algebraic group isfinite.

Now let us show that G → Aut(P1, S) is a categorical injection. This isactually a tautology because if something acts trivially on Aut(P1, S), then itacts trivially on G.

8.2 Automorphisms of projective space

We have this group PGLn, and this has the universal property

1 Gm GL(V ) PGL(V ) 1

H.

So we have a mapPGL(V )→ Aut(P(V ))

and want to show that this is an isomorphism.But first let us look at Maps(P(V1),P(V2)). The k-points of this should be

polynomial functions polyn(V1, V2) ∼= Symn(V ∗1 ) ⊗ V2, but with some nonvan-ishing condition.

Proposition 8.7. Maps(P(V1),P(V2)) ∼=∐n≥0

P(polyn(V1, V2)).

Math 224 Notes 29

9 February 22, 2018

We were trying to compute Maps(P(V1),P(V2)).

Proposition 9.1.∐n≥0

P(Sym(V ∗1 )⊗ V2) ∼= Maps(P(V1),P(V2)).

Proof. We use the functor of points. A map S × P(V1) → P(V2). Recall thata map Y → P(V ) is the same as a line bundle L on Y with an embeddingL → V ⊗ OY , with V ∗ ⊗ OY → L ⊗−1 surjective.

So we take Y = S×P(V1). Now it is a fact in algebraic geometry that everyline bundle on S × P(V ) is of the form

L ⊗−1S O(n)

for some L ⊗−1S on S. So the data is

V ∗2 ⊗ OS×P(V1) → L ⊗−1S O(n),

which is the same asπ∗(LS ⊗ V ∗2 )→ OS O(n),

where π : S × P(V )→ P(V ). Then this is the same as

LS ⊗ V ∗2 → OS ⊗ Symn(V ∗1 ),

which is just a LS → Symn(V ∗1 ) ⊗ V2. This is the same as a map S →P(Symn(V ∗1 ⊗ V2)). Now surjectivity will give us the condition that the im-age is nondegenerate.

Corollary 9.2. Isom(P(V ),P(V )) = P(Hom(V, V )) = PGL(V ).

So let us finish this proof that we have been working on for two weeks. ForS ⊆ P1 a finite set, we have

Aut(P1, S) = Aut(P1)×Map(S,P1) Map(S, S).

Theorem 9.3. The action of G on itself by left multiplication extends to a mapG→ Aut(P1, S).

9.1 Ind-schemes

Definition 9.4. An ind-scheme is a contravariant functor on the category ofaffine schemes that can be written as

colimn Yn

where Y1 → Y2 → Y3 → · · · is a closed embedding. (Or you can also allow afiltered indexing set.) Then Hom(S, Y ) =

⋃n Hom(S, Yn).

Math 224 Notes 30

Example 9.5. Consider A∞ that can be written A1 → A2 → · · · . This isdifferent from A∞,pro = Spec k[t1, . . .] = lim←−An.

There are two senses in which we can take colimit. First, we can take

Hom(S, “ colim ”Yi) = colim Hom(S, Yi).

The other thing we can do is to take the colimit colimYn in the category ofaffine schemes. If Yi = SpecAi, then we have

colimYi = Spec(lim←−i

Ai).

For instance, if Ai = Sym(V ∗i ), then

Hom(S, colimV i) = Hom(lim−→(Sym(V ∗1 )), OS)

and the set Hom(S, “ colim ”Vi) are only those that are continuous.

Example 9.6. Consider the sequence

Spec k[t]/t→ Spec k[t]/t2 → · · ·

and there is a notion of

Spf k[[t]] = “ colim ” Spec k[t]/tn,

which is an ind-scheme. There is also the scheme

Spec k[[t]] = colim Spec k[t]/ti = Spec k[[t]].

By definition,

Hom(S, Spec k[[t]]) = Hom(k[[t]], OS),

Hom(S,Spf k[[t]]) = colimn(k[t]/tn, OS) = (nilpotents in OS).

Definition 9.7. Let X be an affine scheme and Y ⊆ X a closed subscheme.We define

Hom(S,X∧Y ) = Hom(S,X)×Hom(Sred,X) Hom(Sred, Y ).

Example 9.8. We claim that (A1)∧0 = Spf k[[t]]. This is because

Hom(S,A1)×Hom(Sred,A1)Hom(Sred,Spec k[t]/(t)) = f ∈ OS : f is in the nilradical.

Similarly, we can prove that

XˆY = “ colimn ” SpecA/In.

Proposition 9.9. Map(X,Y ) is an ind-affine ind-scheme if Y is affine. (Thisjust means that the Yi can be taken to be affine.)

Math 224 Notes 31

Proof. Let us first take Y = A1. Here, we claim

Map(X,A1) ∼= Γ(X,OX).

To show this, we compare functor of points. Then

Hom(S,Map(X,A1)) = Hom(S ×X,A1) = Γ(S ×X,OS×X) = OS ⊗ Γ(X,OX).

Then V is a ind-scheme for a vector space V . For Y = An, we can do the samething. Or we can use the fact that the product of ind-schemes is an ind-scheme.Also, it is clear that Map(X,Y1 × Y2) = Map(X,Y1)×Map(X,Y2).

Now let us look at arbitrary finite type Y . Then Y fits in the fiber product

Y An

∗ Am.

Now it suffices to show that the fiber product of ind-schemes is an ind-scheme.It can be checked that

(colimYi)×(colimZj) (colimWk) = colimi,k Yi ×Zj Wk,

where j is supposed to sufficiently large. This is because X mapping into themshould come as something into Yi and into Wk and into Zj for some large enoughindices. By filteredness, we can find a large enough Zj such that Yi → Zj andWk → Zj .

So what is Aut(A1)? We have

Hom(Spec(k),Aut(A1)) = Isom(A1,A1) = (Gm nGa)(k).

We have a map Gm n Ga → Aut(A1) defined by the action, but is this anisomorphism? A priori, Aut(A1) is only an ind-scheme.

Let us evaluate at R. We are looking at automorphisms R[t]→ R[t]. If R isa field, the only ones come from scaling and translation. But if R has nilpotents,like R = k[ε], then

R[t]→ R[t]; t 7→ t+ εt100

is an automorphism.

Proposition 9.10. Gm nGa ∼= (Aut(A1))red.

The reduced ind-scheme is just defined as taking “ colim ”(Yn)red. This isindependent of the choice of the Yn. Given X → “ colim ”Yi, the closure ofthe image makes sense because all the connecting maps are closed embeddings.Moreover,

Lemma 9.11. Let G → G1 be a homomorphism, where G is a group schemeand G1 is a group ind-scheme, and ϕ is injective on S-points, and ϕ is bijectiveon k-points. Then ϕ induces an isomorphism G→ (G1)red.

Math 224 Notes 32

Proof. So consider the closure Gϕ−→ G → G1. Then ϕ is injective on S-points,

so it is an isomorphism. Now I claim that if X → Y where X is reduced,and X(k) ∼= Y (k), then X ∼= Yred. We have X → Yi for some Yi, and thenX(k) ∼= Yi(k) implies X ∼= (Yi)red by Nullstellensatz.

Math 224 Notes 33

10 February 27, 2018

Last time we define this notion of an ind-scheme. Then we proved that if Y is(finite-type) affine, Maps(X,Y ) is an ind-scheme. Then for X affine, Aut(X)was a group ind-scheme. We had

Aut(A1) ⊇ Aut(A1)red∼= Gm nGa.

10.1 Comparing reduced parts of automorphism functors

We had a group G = X acting on X fixing S.

Theorem 10.1. We have a homomorphism G→ Aut(X,S).

We can look at Aut(X,Sset) and Aut(X,S) = Aut(X) ×Map(S,X) Aut(S).Also, given any functor, we can consider its reduced functor by a left Kanextension. That is, we are taking Fred(S) = colimS′ rem F (S′) for S → S′. ThenI have

Aut(X) Aut(X,Sset) Aut(X,S)

G Aut(X)red Aut(X,Sset)red Aut(X,S)red.

We want to map G→ Aut(X,S). We will do this by showing that the bottomtwo maps are isomorphisms.

But let me torture you a bit first. I claim that Aut(X,Sset)← Aut(X,S) isnot an isomorphism. If we look at a T -point of Aut(X,Sset), this is a diagram

T ×X X

T × S S′

φ

with the set-theoretic image of T × S being S. This means that the scheme-theoretic image is a nilpotent thickening S′ of S. But if we look at a T -point ofAut(X,Sset)red, we can just test at T reduced. Then T × S is reduced, and soT ×S → S′ factors through S. Therefore we the isomorphism Aut(X,Sset)red

∼=Aut(X,S)red.

Also, we need to show that Aut(X)red ← Aut(X,Sset)red is an isomorphism.

Lemma 10.2. Let G1 → G2 be a homomorphism where G1 a group scheme andG2 a group ind-scheme. If φ is injective on T -points and bijective on k-points,then φ is an isomorphism.

todotodoRecall that for X ⊇ Y , we have defined

Hom(T,X∧Y ) = Hom(T,X)×Hom(Tred,X) Hom(Tred, Y ).

Math 224 Notes 34

For S ⊇ H is a subgroup, then we can also define G∧H as a group ind-scheme.Take X = P1 and S =∞, we have

Aut(X,Sset) ⊇ Aut(X,S) = Gm nGa.

Proposition 10.3. Aut(X,Sset) = Aut(X)∧Aut(X,S)

.

10.2 Lie algebras

Definition 10.4. For G an algebraic group, we define its Lie algebra as g =TeG. The tangent space is defined as

TxX = Hom(Spec k[ε]/ε2, X)×Hom(Spec k,X) x.

This is going to be a vector space.

Theorem 10.5 (Lie). g has a structure of a Lie algebra.

I’ll use Tannaka’s theorem to give a different description.

Theorem 10.6. An element ξ ∈ g is the same thing as

• for every V ∈ Rep(G), an element Tξ ∈ End(V ) that is

• functorial in the sense that for φ : V1 → V2 we have φ Tξ = Tx φ and

• Tξ acts on V1 ⊗ V2 as Tξ ⊗ id + id⊗Tξ.

Proof. By Tanaka, an element of g is the same as, for every V we have auto-morphisms Tε on V ⊗ k[ε]/ε2 which is the identity modulo ε. Then if we writeTξ = id +εTξ, then Tξ|V1⊕V2

= Tξ|V1⊗ Tξ|V2

gives the last condition.

Now let us prove Lie’s theorem. For ξ1 and ξ2, we can define [ξ1, ξ2] as

T[ξ1,ξ2] = [Tξ1 , Tξ2 ].

Corollary 10.7. Every element ξ ∈ g admits a unique Jordan decompositionξ = ξn + ξss such that ξn acts nilpotently on every representation and ξss actssemi-simply on every representation and [ξn, ξss] = 0.

Proof. You do Jordan decomposition on every Tξ.

Corollary 10.8. TeG is in bijection with endomorphisms of OG that are

• compatible with the left left action of G on itself by right translations,

• and with the map OG ⊗OG → OG.

So these are vector fields that are invariant under right translations.

Theorem 10.9. If k has characteristic 0, then every algebraic group is reduced.

Math 224 Notes 35

So in characteristic 0, every algebraic group is smooth. This is because everyreduced scheme has a nonempty open on which the scheme is smooth, and thenwe can translate.

Proof. Note that OX,x is regular if and only if OX,x is regular. So regular isequivalent to Sym(m/m2)

⊕im

i/mi+1 being an isomorphism. We are going

to construct the map OG,e → Og,0 given by

Sym(m/m2) ⊕i

mi/mi+1 →⊕i

mi/mi+1.

The composition is the identity, and this shows that all the maps are isomor-phism.

So we want to construct his map. First we will define an exponential mapg∧0 → G∧e and then induce Spec Og,0 → Spec OG,e. Now we need to define

Hom(SpecR, g∧0 )→ Hom(SpecR,G∧0 ).

The left hand side is ξ ∈ R⊗g such that ξ = 0 in the reduced R of R. The righthand side is compatible family of automorphisms of R⊗ V that are identity onR⊗V . The left hand side are endomorphims of R⊗V that vanishes on R⊗V .Then we can define

Tξ = exp(Tξ) = 1 + Tξ +T 2ξ

2+ · · · ,

because this is actually finite.

In characteristic 0, we can define

OX,x = lim←−OX,x/mi.

Then we can define distributions

DistX,x = colim(OX,x/mi)∗.

We will later prove that in characteristic 0, DistG,e ∼= U(g). For characteristicpositive, U(g) will looks something more like the wrong symmetric power.

Math 224 Notes 36

11 March 1, 2018

Let G be an algebraic group. We defined

g = TeG = Hom(Spec k[ε]/ε2, G)×Hom(Spec k,G) e.

Theorem 11.1. An element ξ ∈ g is equivalent to the functor taking V ∈Rep(G) to TV (ξ) ∈ End(V ) that is compatible with the tensor structure

TV1⊗V2(ξ) = TV1

(ξ)⊗ id + id⊗TV2(ξ).

There is the regular representation OG ⊗ OG → OG, and then we get anendomorphism Lielξ of OG. This commutes with right translations, and so Lielξis a derivation of OG. We can think of this as a right-invariant vector field.

Now let me do this same construction with a different perspective. For eachx ∈ X, we can take TxX = mx/m

2x, and TxX = (mx/m

2x)∗. We can also say

ΩX,x = T ∗xX. But we can also define the sheaf

TX = H omX(ΩX ,OX).

These are the tangent vectors, informally. Now we can take

TX,x → TxX.

This is a special case of doing (F∨)x → (Fx)∨. This is not in general anisomorphism, but is an isomorphism if F is locally free.

Proposition 11.2. There exists an canonical isomorphism ΩG ∼= OG ⊗ g∗ andTG ' OG ⊗ g such that sections of ΩG (repsectively TG) that is right-invariantcorresponds to g∗ → OG ⊗ g∗ and g → OG ⊗ g.

But we want to refine this.

11.1 Equivariant quasicoherent sheaf

Let G act on Y , and F ∈ QCoh(Y ).

Definition 11.3. A structure of G-equivariance on F is, for all S considerF OS ∈ QCoh(Y × S) and make Hom(S,G) act on it, compatibly with pull-backs with respect to S′ → S.

This has all functorial properties built in.

• If G acts on Y1 and on Y2, and if Y1 → Y2 is an equivariant morphism,then pullbacks and pushforwards of equivariant sheafs are equivariant.

• If Y is a point, then QCoh(pt)G = Rep(G).

Most textbooks will define this differently. A structure of an equivarianceon F is an isomorphism

ϕ : pr∗(F )→ act∗(F ).

Here, pr, act : G × Y → Y are the projection and the action. This ϕ shouldsatisfy some associativity condition.

Math 224 Notes 37

Lemma 11.4. Let G act on Y . Then ΩY has a canonical structure of a G-equivariance.

Proof. For any g ∈ Hom(S,G), this defines Y ×S → Y ×S. Then it immediatelygives an automorphism of ΩY OS = ΩY×S/S .

Let Y = G, with an action by G by left translations.

Lemma 11.5. There exists an equivalence of categories QCoh(G)G ' Vect.

Proof. You take the vector space V and take this to V ⊗ OG. This is sendingVect→ QCoh(pt)G → QCoh(G)G. The functor in the opposite direction is goingto be F 7→ Fe.

Now this immediately shows that ΩG ∼= g∗ ⊗OG.

11.2 Distributions

For a point x ∈ X of a scheme, we defines the distributions as

Dist(X)x = topological dual (OX,x)∗.

Her is another definition. Take Xi ⊆ X be points so that Xi,red = x bydefining Xi = Spec(Ox/mix). Then we have the ind-scheme

X∧x = colimiXi.

There, we haveDist(X)x = colimi(OXi)

∗.

This is a vector space, and it is going to be a cocommutative coalgebra. If youknow about differential operators, there is a sheaf D(X) such that D(X)x =Dist(X)x.

For example, let us take X = V and x = 0. Then we have

OX = Sym(V ∗), OXi = Sym≤i(V ∗).

Now consider(Symn(V ))∗ = Sym

n(V ∗).

This normally, Symn(V ) is a quotient of V ⊗n, and so Symn(V ) is going to be

a subspace of V ⊗n. If characteristic 0, we are going to get the invariants, but

for other characteristic, we have to be a bit careful. Sym2(V ) is going to be the

vector space spanned by v ⊗ v. Then we have

Symn(V ) = (Sym2(V )⊗ V ⊗(n−2)) ∩ (V ⊗ Sym2(V )⊗ V ⊗(n−3)) ∩ · · · .

Then we can writeDist(V )0 = Sym(V ).

Math 224 Notes 38

Note that if f : X → Y is a map sending x ∈ X to y ∈ Y , then we have amap in the other direction of rings, and so on duals we get

Dist(X)x → Dist(Y )y.

Also, we can check that

Dist(X1 ×X2)x1×x2 = Dist(X1)x1 ⊗Dist(X2)x2 .

This shows that we have a map

Dist(G)e ⊗Dist(G)e → Dist(G)e

coming from m : G × G → G. This gives an additional algebra structure, soDist(G)e has a structure of a cocommutative Hopf algebra on Dist(G)e.

There is something called the universal enveloping algebra U(g) for aLie algebra g. This is an algebra U(g) such that a map U(g)→ A of algebras isthe same as a map g→ A of Lie algebras (with the Lie algebra structure givenby the commutator.) We can construct this explicitly as

U(g) = T (g)/(ξ1 ⊗ ξ2 − ξ2 ⊗ ξ1 − [ξ1, ξ2]).

There is a natural filtration given on the tensor algebra, which descends to U(g).

Theorem 11.6 (Poincare–Birkhoff–Witt). The map Sym(g) → grU(g) is anisomorphism.

Note that there is a natural map g → Dist(G)e. This is because there isgenerally a map Tx → Dist(X)x given by O/m2 → k. You can check that thismap commutes with the bracket. So we get a map

U(g)→ Dist(G)e.

Consider the case G = V . Then we have

U(V ) Dist(V )0

Sym(V ) Sym(V )

T (V )

Now this map is defined as

Tn(V )→ Symn(V ); v1 ⊗ · · · ⊗ vn →

∑σ∈Σn

vσ(1) ⊗ · · · ⊗ vσ(n).

This is not necessarily an isomorphism if characteristic is not zero. In particular,for V = k we get

k[t]→ k[t]; tn → n!tn.

Math 224 Notes 39

Theorem 11.7. The map U(g)→ Dist(G)e is an isomorphism in characteristic0.

This will also prove the Poincare–Birkhoff–Witt theorem.

Proof. It is enough to show that grU(g)→ gr Dist(G)e is an isomorphism. Butwe have

grU(g) gr Dist(G)e

Sym(g) Sym(TeG)∼=

and so we should have and isomorphism.

Note that we have an exponential map g∧0 → G∧e , and this induces a map

Dist(g)0 → Dist(G)e

of only cocommutative coalgebras. We can interpret this as

Dist(g)0 Dist(G)e

Sym(g) U(g)

T (g)

exp

where T (g)→ U(g) is dividing by n!. This is what Baker–Campbell–Hausdorffreally is.

Math 224 Notes 40

12 March 6, 2018

For a Lie algebra g, we have the universal enveloping algebra U(g). We alsohave for each point x ∈ X the distributions Dist(X)x, and Dist(G)e becomes acocommutative Hopf algebra by TxX → Dist(X)x. Then we had a map

U(g) Dist(G)e

g TeG.

Theorem 12.1. In characteristic 0, the map U(g) → Dist(G)e is an isomor-phism.

Corollary 12.2. OG,e = (U(g))∗ and G∧e = Spf(U(g))∗.

Today I want to say a few words about what happens in characteristic p.

12.1 Lie algebra over characteristic p

There is a map g → g written as x → x[p] defined in the following way. Any ξon X is a differential operator of order 1. Then ξn is also a differential operatorof order 1 because

ξp(fg) = ξp−1(ξ(f, g)) = ξp(f)g + p(· · · ) + fξp(g).

Let’s look at some examples.

Example 12.3. Take G = Ga. Then g = span( ∂∂t ). Then we have( ∂∂t

)[p]

(t) = 0

and so ( ∂∂t )[p] = 0.

Example 12.4. Take G = Gm. Then we have g = span(t ∂∂t ) and(t∂

∂t

)[p]

(t) = t

and so (t ∂∂t )[p] = t ∂∂t .

Now we can define the restricted universal enveloping algebra as

U(g)restr = U(g)/(ξ[p] − ξp).

Corollary 12.5. The map U(g)→ Diff(G)e factors through U(g)restr.

Math 224 Notes 41

Recall that we had Sym(g) ∼= grU(g). If we define Sym(V )restr = Sym(V )/(vp)then we will have

Sym(g)restr grU(g)restr.

If you choose a basis V = spane1, . . . , en, then this is going to look like

Sym(V )rest =⊗i

k[t]/tp.

Recall that there is a Frobenius map Frob : G → G, and its kernel if nilpo-tent, and the same is true for ker(Frobn). Because the Frobenius kills all tangentvectors, Spec(O/m2) ⊆ ker(Frob) and then

Spec(O/m···) ⊆ ker(Frobn).

This shows thatG∧e = colimn ker(Frobn).

Hereker(Frob : G∧e → G∧e ) = ker(Frob : G→ G)

but ker(Frob) are non-isomorphic for Ga for Gm. This shows that there is noway to express this purely in terms of the Lie algebra.

We can consider (Oker(Frob))∗ ⊆ Dist(G)e.

Lemma 12.6. The map U(g)→ Dist(G)e factor through (Oker(Frob))∗.

Proof. It suffices to show that generators are mapped into (Oker(Frob))∗. But g

is mapped to (O/m2)∗ and this is contained in (Oker(Frob))∗.

We thus obtain a map

U(g)rest → (Oker(Frob))∗.

Theorem 12.7. This is an isomorphism.

Example 12.8. Last time we had Symn(V ) = (Symn(V ∗))∗ and then

Sym(V )rest ⊆ Sym(V ).

Then we get

Sym(V )rest =⊗i

(k[s]<p).

You can see that the map Sym(V )→ Dist(V )e factors as

Sym(V )→ Sym(V )rest

∼=−→ Sym(V )rest → Dist(V )e

where the isomorphism is explicitly described as k[t]/tp → k[s]<p with tn 7→n!sn.

Math 224 Notes 42

Proof. We prove this on gr. We have

grU(g)rest gr(Oker(Frob))∗

Sym(g)rest Sym(g)rest∼=

∼=

where the right vertical map is seen to be an isomorphism by choosing etalecoordinates.

12.2 Quotients

Given H ⊆ G, we want to define the quotient. That is, we want to find G/Hsuch that

HomG(G/H, Y ) = Y (k)H .

Definition 12.9. A G-homogeneous space is a scheme X equipped with anaction of G such that for any x ∈ X(k), the map G → X given by g 7→ gx isfaithfully flat.

Proposition 12.10. Suppose X is a G-homogeneous space and h = StabG(x).Then X has the universal property.

Proof. We want to show that HomG(X,Y )→ Y (k)H is an isomorphism. I wantto construct the map in the other direction. There are two maps G×H → G,first projection and multiply. Then the two maps G×H → G→ Y agree, andso by faithfully flat descent, we get

G×H

G Y

X

the map X → Y .

Theorem 12.11. Let G be reduced and H ⊆ G. Then there exists a G-schemeX and x ∈ X such that StabG(x) = H.

We need to construct this from somewhere.

Theorem 12.12. There exists a representation V and a subrepresentation V ′ ⊆V such that the stabilizer StabG(V ′) = H. Here, by definition,

Hom(S, StabG(V ′)) ⊆ Hom(S,G)

consists of automorphisms of V ⊗OS that maps V ′ ⊗OS to itself.

Math 224 Notes 43

Theorem 12.13. We can take V ′ to be 1-dimensional.

If we assume this, here is how you prove it. For 0 ≤ k ≤ dimV , thereis the Grassmannian Grk(V ) with Hom(S,Grk(V )) given by the locally freesubsheaves of OS ⊗ V of rank k. Take Y = Grk(V ). Then G→ GL(V ) acts onY , and V ′ is a k-point of Y = Grk(V ). The stabilizer of this is going to be H.So we get the first theorem from the second.

Here is how you prove the third theorem from the second. We have a naturaltransformation

Grk(V )→ P(∧k(V )),

given by sending E ⊆ OS ⊗ V to∧k(E ) ⊆ OS ⊗

∧k(V ), called the Plucker

embedding.

Proof of second theorem. Let H correspond to the ideal I ⊆ OG. Then thereexist a subrepresentation V ⊆ OG such that V ∩ I generates I. Then we firstnote StabG(I) = H. Also, StabG(V ∩I) = StabG(I). So we take V ′ = V ∩I.

Corollary 12.14. There exists a homogeneous space such that StabG(x) = H.

Proof. The homogeneous space is the orbit Gx.

Corollary 12.15. If G is reduced, then G/H is reduced.

Also, note that G/H is quasi-projective because we took it from the Grass-mannian. We are assuming that G is reduced.

Proposition 12.16. The map G→ G/H is smooth if and only if H is reduced.

Proof. Because G/H is reduced and G → G/H is faithfully flat, we can checksmoothness of fibers. But then H is smooth if it is reduced.

Proposition 12.17. Let H ⊆ G be normal. Then G/H has a unique groupstructure such that G→ G/H is a group homomorphism and G/H is affine.

Proof. We can use faithfully flat descent to construct G/H × G/H → G/H.To prove that G/H is affine, we first take G/G′ ∩ H. This is a connectedcomponent, so we may assume that G is connected. Take V a G-representationsuch that StabG(kv) = H. H acts on v by a character χ. But characters arerigid, so the action of G and H be conjugation preserves the character. Let

V1 = v1 ∈ V1 : h(v1) = χ(h)v1.

This a subrepresentation, and so we can assume that H|V acts via χ. This givesa map

G/H → PGL(V ).

This is categorically injective, so G/H is isomorphic to a closed subschemeof PGL(V ). Now it suffices to show that PGL(V ) is affine. This sat insideP(End(V )) as an open subset, with nonzero determinant.

Math 224 Notes 44

13 March 8, 2018

Last time we introduced quotients. For H ⊆ G, we characterized G/H with asthe universal property.

Definition 13.1. Let H acts on X. We say that X → Y is a faithfully flatquotient of X by H if

(1) π is H-invariant,

(2) π is faithfully flat,

(3) H ×X ∼= X ×Y X.

Lemma 13.2. Hom(Y,Z) = Hom(X,Z)H .

Proof. The proof is basically faithfully flat descent. The two maps X ×Y X ⇒X → Z agree if X → Z is H-invariant, so we get Y → Z.

Observe that if X is a G-homogeneous space, and H = StabG(x), then themap G → Y is a faithfully flat quotient of G with respect to the action ofH by right translations. We showed last time that for any H there exists ahomogeneous space with this property. It sufficed to construct a G-space Y andy ∈ Y such that StabG(y) = H.

Theorem 13.3. Let a 1-dimensional connected group scheme G act on a com-plete variety. Then it has a fixed point.

Proof. Take an arbitrary point. Then we get a map G → X, and then we cancompactify G → G. Because X is proper, we can extend G → X to G → Xusing the valuative criterion, and this is a map of G-varieties. Then the pointsof G \G are fixed.

13.1 Parabolic subgroups and solvable groups

Definition 13.4. A subgroup P ⊆ G is parabolic if G/P is complete.

Lemma 13.5. If P is parabolic in G and P1 is parabolic in P , then P1 isparabolic in G.

Note that base being proper and the fiber being proper does not implyproper.

Proof. Note that being proper is local in the faithfully flat topology. So we cantake

G/P ′ G× P/P ′ P/P ′

G/P G pt

todotodo

Math 224 Notes 45

Lemma 13.6. If P ⊆ G is parabolic and P ⊆ Q then Q is parabolic.

Proof. We have a surjective G/P → G/Q. Note that if X is complete andπ : X → Y is surjective, then Y is complete. To show this, we need to showthat Y × Z → Z is closed. But we have X × Z → Y × Z surjective, so we cantake image of V ⊆ Y × Z by looking at the inverse image in X × Z and thenlooking at the image in Z.

Definition 13.7. A group G is solvable if the following equivalent conditionshold:

(1) If we define G0 = G and Gi = [Gi−1, Gi−1] then Gn = 1 for n 0.

(2) There exists 1 = Gn ⊆ · · · ⊆ G1 ⊆ G0 = G of normal reduced subgroupssuch that Gi/Gi−1 are abelian.

(3) (2) without assumption on reducedness.

It is clear that (1) implies (2) implies (3). For (3) implies (1), we note thatGi ⊆ Gi.

Lemma 13.8. The following conditions are equivalent:

(1) G is solvable.

(2) Any representation contains a nonzero G-fixed line.

(3) Any representation V has a G-stable flag.

(4) G → Bn where Bn is the upper-triangular matrices.

Proof. (2) implies (3) implies (4) implies (1) is clear. We have to show (1)implies (2). We induct on the dimension of the group. Then we know thatV G1 6= 0. Then G/G1 acts on this, and the quotient is abelian so we findsomething that is invariant.

Lemma 13.9. Let G be solvable and connected. Then it has a filtration by nor-mal subgroups such that the subquotients are either Gm and Ga. G is unipotentif and only if all these subquotients are Ga.

Note that all unipotent groups are solvable. Let me postpone the proof.

Theorem 13.10 (Borel). If G is solvable and connected, acting on X complete,then it has a fixed point.

Proof. For G1 ⊆ G, we have XG = (XG1)G/G1 .

Lemma 13.11. P ⊆ G is parabolic if and only if P 0 ⊆ G0 is parabolic.

Proof. We note that (G/P )0 = G0/G0 ∩ P . So G/P is proper if and only if(G/P )0 is proper if and only if G0/G0 ∩P is proper. But P 0 ⊂ G0 ∩P is finiteindex.

Theorem 13.12. (1) If G has no proper parabolics, then G is solvable.

Math 224 Notes 46

(2) If G is connected and solvable, then it has no proper parabolics.

Proof. For (2), suppose that G has a proper parabolic. Then G acts on G/P ,which is proper, but it does not have a fixed point. This contradicts Borel’stheorem. For (1), let V be a representation. We want to show that there existsa G-stable line. Consider the action of G on P(V ) and take a closed orbit.Either this is a single point and we get a fixed point, or this is a not a pointand we get a proper parabolic.

Let us now prove that lemma. First reduce to the case when G is abelian.Then we have G = Gu × Gss. We know that Gss is a product of a bunch ofGms. So we can assume that G is unipotent. By induction, it suffices to finda subgroup isomorphic to Ga. Embed G into Um. We know that there exists afiltraion

0 = G0 ⊆ G1 ⊆ · · · ⊆ Gn = Um

such that dim(Gi/Gi−1) = 1. Now take i such that dim(G ∩Gi) > 0. Then weget G ∩Gi/G ∩Gi−1 → Gi/Gi−1 so dim(G ∩Gi) = 1. So ((G ∩Gi)red)0 = Ga.

Proposition 13.13. In characteristic 0, a commutative unipotent group is V .

Proof. Recall that we had Gu as a Zariski closed subgroup, because we wereable to phrase this as a condition on characteristic polynomials. In the sameway, we similarly define gn. Now the claim is that

gnexp−−→ Gu

is an isomorphism. Note that the S-points on the right are the automorphismsof V ⊗ OS that are unipotent. On the other hand, gn are endomorphisms onV ⊗OS that are nilpotent. So we get this from

EndnilR (V ⊗R)

exp−−→ AutuniR (V ⊗R).

Note that this is a homomorphism because exp(A + B) = exp(A) exp(B) bycommutativity.

Note that in characteristic p, we have

Ga(k) = k = (Z/pZ)⊕···.

So we can take any finite subgroup and the quotient is going to be Ga becauseit is 1-dimensional reduced.

For G a group over characteristic p, we can define

L : G→ G; L(g) = Frob(g)g−1.

Theorem 13.14 (Lang). L is always etale. If G is connected, then it is sur-jective. Also ker(L) = G(Fp).

Math 224 Notes 47

Proof. For etale, we need to show that the differential is an isomorphism. Firstreduce to the case g = e. This is because given an infinitesimal g0, we can take

Frob(gg0)g−10 g−1 = Frob(g)(Frob(g0)g−1

0 )g−1.

Now note that the map is the composition of

Gid×Frob−−−−−−→ G×G inv×id−−−−→ G×G mult−−−→ G.

But its differential at e is then g→ g⊕ g→ g⊕ g→ g given by −1, because thederivative of the Frobenius is 0.

Now let us compute its kernel. Consider g ∈ G(k) such that Frob(g) = g.Note that a point in X(Fq) satisfies Frob(x) = x if and only if x ∈ X(Fp). todotodo

For the second part, consider theG action given byG with g∗g1 = Frob(g)g1g−1.

ButL′(g) = g−1

1 Frob(g)g1g−1

is etale because we can use the same argument on the tangent spaces. So allorbits are opens, and because G has one connected component, there is onlyone orbit.

Math 224 Notes 48

14 March 20, 2018

We were talking about solvable groups. A group G is solvable if there exist

1 = G0 ⊆ G1 ⊆ · · · ⊆ Gn = G

such that Gi/Gi−1 are commutative. You can impose the condition that Gi arereduced, and even let Gi = [Gi−1, Gi−1].

Theorem 14.1. The following are equivalent:

(1) G is solvable.

(2) Every representation has an invariant line.

(3) Every representation has a G-stable flag.

(4) Every G can be realized as a subgroup of Bn.

Theorem 14.2. Let G be a solvable connected group. Then there exists 1 =G0 ⊆ · · · ⊆ Gn = G such that Gi/Gi−1 is either Ga or Gm. Moreover, G isunipotent if and only if all subquotients are Ga.

Corollary 14.3. If G is unipotent, there exist 1 = G0 ⊆ · · · ⊆ Gn = G witheach of the normal subgroups such that the adjoint action of G on Gi/Gi−1 istrivial.

Definition 14.4. P ⊆ G is parabolic if G/P is complete.

Lemma 14.5. (a) If P is parabolic and P ⊆ Q then Q is parabolic.

(b) If P is parabolic in G and P ′ is parabolic in P then P ′ is parabolic in G.

(c) If ϕ : G G′ is a surjective morphism, and P is parabolic, then ϕ(P ) isparabolic.

Theorem 14.6. Let G be connected. Then G contains a proper parabolic if andonly if G is not solvable.

To prove this, we used the Borel fixed point theorem.

Theorem 14.7. If G is connected solvable, and acts on X a complete variety,then it has a fixed point.

14.1 Borel subgroup

Definition 14.8. A subgroup B ⊆ G is Borel if it is connected, solvable, andmaximal with these properties.

Theorem 14.9. (a) D is a parabolic if and only if it contains a Borel.

(b) A Borel is parabolic.

(c) Any given Borel can be conjugated into any given parabolic.

(d) Any two Borels are conjugate.

Math 224 Notes 49

Proof. (c) We have B acting on G/P , and this has a fixed point. If we picksomething in this fixed point and then conjugate by this, B goes inside P .

(b) If G is solvable, then this is trivial. If G is not solvable, it has a properparabolic P , and we may assume that B ⊆ P ⊆ G. Clearly, B is a Borel ofP . By induction on the dimension, we have that B is parabolic of P , and thenparabolic is transitive.

(a) If it contains a Borel, it contains a parabolic and is parabolic. If it isparabolic, some Borel can be conjugated into it. (d) is a consequence of (b) and(c).

Corollary 14.10. Z(G) ⊆ B.

14.2 Structure theory of connected solvable groups

Unless said otherwise, we are going to assume that G is solvable and connected.

Proposition 14.11. (a) There exists a

1→ U → G→ T → 1

where U is connected and unipotent, and T is a torus. Here, g ∈ G isunipotent if and only if g ∈ U .

(b) There exists a non-canonical splitting, so that G = T o U .

Proof. (a) We first have

1→ Un → Bn → Tn → 1.

Also, we know that G embeds into Bn. So we can take U = G ∩ Un. These areprecisely going to be the unipotent elements of G. Then G/(G ∩ Un) embedsinto Tn, and hence is a torus.

For (b), we have G T . Assume first that k 6= Fq. Then find an elementλ ∈ T such that λ generates a dense subset of T . Find a semi-stable elementg ∈ GG that projects to λ. Define G′ = 〈g〉. Then G′ surjects to T . Here, G′

is a semi-simple commutative group, because under G → GLn, G′ should lieinside some torus Tn. But the kernel of G′ Gm should be trivial because itunipotent as well (as it lies in U). So G′ ∼= Gm and we get a section.

If k 6= Fq, we use the fact that the fiber product

Y Mapgroup(T,G)

id Mapgroup(T, T ).

We want to show that Y (k) is nonempty, but we know that Y (k′) is nonemptyfor other algebraically closed k′. So the ind-scheme Y should be nonempty as ascheme, and then Y (k) is nonempty by Nullstellensatz.

Math 224 Notes 50

Proposition 14.12. Mapgroup(T,G) is a scheme.

Note that we have proved before that Mapgroup(G,T ) is a discrete scheme.

Proof. Because we can embed Mapgroup(T,G)→ Mapgroup(T,GLn), we reduceto the case of GLn. Note that the k-points of this is going to be the ways ofGm acting on V , which is the ways to write V =

⊕i Vi with Gm acting on Vi

by the ni-th power. So we will have∐dimV=

∑i dimVi

GL(V )/∏i

GL(Vi)

This is because the S-points are locally free sheaves E of rank dimV over S,equipped with an action of Gm, which is equal to to splittings E =

⊕i Ei.

14.3 Maximal tori

Definition 14.13. A maximal torus in G is T ′ ⊆ G that projects isomorphi-cally to T .

We just showed that maximal tori exists for connected solvable group G.

Theorem 14.14. Let G be a connected solvable.

(a) All maximal tori are conjugate.

(b) If s1, s2 are two semisimple elements that project to the same elements inT , then they are conjugate.

(c) Centralizers of semisimple elements are connected.

Proof. (b) We first reduce to when U = Ga. Let us induct on the dimension ofGa. Let us pick Ga → U and take the quotient of G by Ga and let G′ = G/Gaand U ′ = G′/Ga. Then 1 → U ′ → G′ → T → 1. Now any two maximal toriare conjugate in G′. So given any two elements in G, we can make them haveequal image in G′. That is, we may now assume that G is the inverse image ofT in the map G→ G′. That is, we can assume that G = T nGa.

But this is now completely explicit. Every action of Ga by automorphismsof Ga is the n-th power of the standard characters. Consider G = Gm nGa forsimplicity. Take two elements (s, a) and (s, 0). Assume that Gm acts on Ga byn-th powers. If n = 0, we are done. If n 6= 0, we need to solve the equationbsb−1 = bs, which is b− λnb = λn. This is solvable if λn 6= 1, which is the casebecause if as = sa then we get a contradiction to that (s, a) is semisimple.

Math 224 Notes 51

15 March 27, 2018

We talked about connected solvable groups. We should that any solvable groupfits in a short exact sequence

1→ U → G→ T → 1

with a splitting T → G, and moreover U has a filtration with subquotients Ga.

Theorem 15.1. Let g1 and g2 be two semisimple elements that project to thesame element in T . Then they are conjugate.

Proof. By induction, we reduce to the case when U : Ga. Now G is T n Ga,and so T Acts on Ga. This is given by some character χ. Say that our twopoints are (t, 0) and (t, a). If χ(t) = 1, then (t, a) = (t, 0)(1, a) and so theycan’t both be semisimple unless a 6= 0. If χ(t) 6= 1, then we want to solve(1, b)(t, 0)(1,−b) = (t, a). Then this is the same as solving χ(b) = a + b, whichis a = χ(t)b− b. Because χ(t) 6= 1, we can solve it.

Recall that a maximal torus in G is a T ′ ⊆ G that projects isomorphicallyto T .

Corollary 15.2. Any torus can be conjugated into any given maximal torus.So any two maximal tori are conjugate.

Proof. Consider g : Adg(S) ⊆ T. We want to say that this has a point, so wehave to show that it has a point in some field extension. todotodo

Theorem 15.3. Let t ∈ T ⊆ G be semisimple. Then ZG(t) is connected.

Proof. By induction, we can take out a Ga-factor from U . By Let this by0→ U ′ → G′ → T → 1. Then we assume ZG′(t) is connected. So we now have

1→ Ga → G→ G′ → 1

where t ∈ T ⊆ G′, and moreover that the image of t in G′ is central. Now Tacts on Ga, by some character χ.

First suppose that χ(t) 6= 1. Then, consider the map ZG(t) → ZG′(t), andwe want to show that this is an isomorphism. This map is injective becasue thekernel should be in Ga but Ga acts non-trivially. (If characteristic is 0, thenwe can conclude this by looking at Lie algebras and showing that (g)t (g′)t.)Given any g′ ∈ G′, we want to lift it to g that is in the centralizer. Let us firstlift to g ∈ G′ to g ∈ G. Then we can write

Adt(g) = ag

for some a ∈ Ga. Then we find a b such that Adt(bg) = bg, by solving the sameequation we did.

Now consider the case when χ(t) = 1. Here, we have G = T n Ga. Let uswrite G = T n U in general, where U is a unipotent group. Then U has a lift

Math 224 Notes 52

such that the adjoint action of t on the subquotients is trivial. This means thatwe have a map

T → Autflag(U)

of flag-preserving automorphisms. Here, you can show that (Autflag(U))red isan algebraic group, and that the kernel

1→ Aut,flag(U)→ Autflag(U)→∏

Gm

of the map given by taking the subquotients has (Aut,flag)red unipotent. So tmaps to something that is semisimple, but it lies in a unipotent group so it isthe identity. This means that t acts as the identity on U . So ZG(t) = G.

Corollary 15.4. Let G be solvable, and consider a maximal torus T . Thenormalizer is NG(T ) = ZG(T ) and is connected.

Proof. After field extension, there is going to be a point that generates the torus.So ZG(T ) = ZG(t) for some t, and is connected. To show that NG(T ) = ZG(T ),we look at ntn−1t−1 for any normalizing n. Then ntn−1t−1 ∈ T so it is semi-simple. But if we project to the torus, this should be 1, so the commutatorshould lie in U . This means that is is unipotent, and so 1.

We also talked about Borel subgroups. For G a connected group, B ⊆ G isBorel if it is solvable and maximal with this property.

Theorem 15.5. (1) If B is a Borel subgroup, then G/B is complete.

(2) Any connected solvable can be conjugated into any parabolic.

Corollary 15.6. Any Borel can be conjugated into any parabolic. Any solvablecan be conjugated into any Borel.

Corollary 15.7. Any two Borels are conjugate.

Lemma 15.8. Z(G) ⊆ Z(B) ⊆ Z(G).

Proof. Take any g ∈ Z(B). Then we have gxg−1x−1 : G → G. This factorsthough G/B, and G/B is proper and G is affine.

Lemma 15.9. If B is normal, then B = G.

Proof. Consider G/B. This is proper, and affine.

15.1 Reductive groups

Definition 15.10. The unipotent radical Ru(G) is the maximal normalunipotent connected subgroup. We say that G is reductive if Ru(G) = 1.

Lemma 15.11. Ru(G) = (⋂B BU ) where B runs over all Borels and BU is

the unipotent part.

Math 224 Notes 53

Proof. One direction is by definition, because the right hand side is a normalunipotent connected. In the other direction, Ru is in some BU and then conju-gate them around.

Consider B = T nBU .

Proposition 15.12. Suppose that the adjoint action of T on BU is trivial.Then G = B.

Proof. If B = T × BU , then there exists some Ga ⊆ Z(BU ) ⊆ Z(B). ThenGa ⊆ Z(G) and then we can quotient out G′ = G/Ga. Then we can proceed byinduction.

Corollary 15.13. B = T implies G = B = T .

Corollary 15.14. B ⊆ (NG(B)) is an equality.

Proof. Let H = (NG(B)). Because B ⊆ H, we have B = H. todotodo

Theorem 15.15. Every element of G is contained in a Borel.

Proof. We define

G = (G×B)/(g, b1) ∼ (gb−1, bb1b−1).

Then we can consider

G G×G/B

G.

π

This is going to be (g, b1) being sent to (gb1b−1, g mod B). We’ll do this next

time.

Math 224 Notes 54

16 March 29, 2018

Let G be a connected group.

Lemma 16.1. Maximal tori are conjugate.

Theorem 16.2. ZG(T ) is connected. (Later, if G is reductive, then T ⊆ ZG(T )is an equality.)

Theorem 16.3. Every element of G belongs to a Borel, and every semisimpleelement of G belongs to a maximal torus.

Theorem 16.4. B → NG(B) is an equality.

Before proving, I will play around with these theorems.

Corollary 16.5. If P is a parabolic, then P is connected and P → NG(P ) isan equality.

Proof. We will show that P → NG(P ) is an equality. Suppose x ∈ NG(P ).Then for B ⊆ P , we have Adx(B) ⊆ P . Because both are Borels in P , thereexists a y ∈ P such that Adx(B) = Ady(B). Then if we write x′ = y−1x, weget Adx′(B) = B. So x′ ∈ B ⊆ P by the third theorem.

Corollary 16.6. If B ⊆ P 1 and B ⊆ P 2 are conjugate parabolics, then P 1 =P 2.

Corollary 16.7. Let T be a torus, and let S be the Borels containing T . ThenNG(T ) acts on S.

(i) This action is transitive.

(ii) ZG(T ) acts trivially.

(iii) NG(T )/ZG(T ) acts simply.

Proof. (1) Suppose T ⊆ B1, B2. Then we can find g ∈ G such that Adg(B1) =

B2. But inside the solvable group, any two maximal tori are conjugate. So wecan assume Adg(T ) = T .

(2) If T ⊆ B, it suffices to show that ZG(T ) ⊆ B. We will prove uncondi-tionally that ZG(T ) ⊆ B. Let C = ZG(T ). Last time we proved that if amaximal torus is central in the Borel, then BC = C. So C is connected solvable,so T ⊆ C ⊆ B′ for some other Borel B′. Now we can play our usual game.

(3) For T ⊆ B, assume that x ∈ NG(T ) and x ∈ NG(B). By the thirdtheorem, we have x ∈ NB(T ). But then recall that NB(T ) = ZG(T ). (This wasproved by ntn−1t−1 being both semisimple and unipotent.)

Math 224 Notes 55

16.1 Cartan subgroup

Definition 16.8. The definition of the Cartan is C = ZG(T ).

Proposition 16.9. There exists a dense open subset in G such that all of itselements can be conjugated into C.

Corollary 16.10. Every element of G belongs to a Borel.

Proof. The theorem implies that there exists a dense open of elements thatbelong to a Borel. But we introduced this Grothendieck alteration G ⊆ G/B×Gsuch that the fiber over G/B is the conjugate of the Borel B. The image is denseopen, and because it is closed, the image is the whole thing.

Theorem 16.11. For S ⊆ G a torus, ZG(S) is connected.

Proof. Let g ∈ ZG(S), and let B be a Borel such that g ∈ B. Then let X ⊆ G/Bconsisting of those x such that Adx(B) 3 g. This is nonempty, and S acts onX. This means that the action has a fixed point. Let x be the fixed point, andB′ = Adx(B). Then g ∈ B′, so we have S ⊆ (NG(B′)) implies S ⊆ B′. Theng ∈ ZB′(S). But we have proved that inside a solvable group, centralizers oftori are connected. So g ∈ ZB′(S) ⊆ ZG(S).

Proof. Consider similarly the

G/C ×G ⊇ ˜Gπ−→ G.

We want to show that π is dominant. It is enough to show that there existg ∈ G such that π has a finite nonempty fiber over G, by semi-continuity.

We use our trick of enlarging the field, and take t ∈ T such that the closureof 〈t〉 is T . We want to see that there are finitely many (modulo C) elementsg such that t ∈ Adg(C). Here, t ∈ Adg(C) implies Adg′(t) ⊆ C, which impliesAdg′(T ) = T . So g ∈ NG(T ), and we want to see that NG(T )/ZG(T ) is finite.This is because the action on a torus is given by a discrete character, and so(NG(T )) ⊆ ZG(T ).

Now we show that the normalizer of a Borel is itself.

Proposition 16.12. Let S is a torus and B ⊇ S, then ZG(S) ∩ B is a Borelin ZG(S).

Using this we can prove the theorem.

Proof. Let x ∈ NG(B). Then we can assume that Adx(T ) = T , because we canmove T back. Consider the following strange map

ψ : T → T ; t 7→ Adx(t)t−1.

Let S = (kerψ). First consider the case S 6= 1. We know that x ∈ ZG(S) andalso it normalizes B ∩ ZG(S), which is a Borel inside ZG(S). If ZG(S) ( G,

Math 224 Notes 56

then by the induction hypothesis on ZG(S), we get x ∈ B ∩ ZG(S) ⊆ B. IfZG(S) = G, then we can replace G by G/S, and we are again done by induction.

Now assume that S = 1. Then ψ : T → T is surjective. Let H = NG(B).Then there exists a representation V and v ∈ V such that

H = g ∈ G : gv ⊆ spanv.

(This came up in our construction of the quotient.) Then we get a map

G→ V ; g 7→ gv.

The claim is that this factors through G → G/B → V . To show this, we needto show that B acts trivially on v. A priori, the action of NG(B) on V is givenby a character χ : NG(B)→ Gm. We want χ|B to be trivial. But we know thatB = Bu o T , and we need to see that γ|T is trivial. But T ⊆ [NG(H), NG(H)]because ψ was surjective.

So we are done, up to that proposition.

Math 224 Notes 57

17 April 3, 2018

Today we are going to do more examples. But last time, we needed something.

Proposition 17.1. Let G be connected, and S ⊆ B ⊆ G where S is a torus.Then ZG(S) ∩B ⊆ ZG(B) is a Borel subgroup.

Proof. We want to show that ZG(S)/ZG(S) ∩B → G/B is closed. This meansthat Z(G) × B has close image in G. Let Y be the image, so that we want toshow Y = Y .

Consider the map

Y × S → B; (y, s) 7→ Ady−1(s).

But we can map further to Y ×S → B → B/BU = T . This is a family of mapsof tori, parametrized by Y . So Ad−1

y (s) = s modulo BU , for y ∈ Y . todotodo

By the centralizer of S, I could have meant two things. I could have meantthe scheme-theoretic centralizer, or I could have meant the reduced subschemeof the k-points.

Proposition 17.2. Scheme-theoretic ZG(S) is reduced.

Proof. It is enough to prove that for s ∈ G a semi-simple element, its centralizerZG(s) is reduced. Let s be a semi-simple automorphism of a smooth affinealgebraic variety Y . (This means that it is semi-simple on the finite stables ofthe ring of functions.) Now we can take Y s as

Hom(Z, Y s) = (Hom(Z, Y ))s.

If Y = SpecA, then Y s = SpecA/(f − s(f)). The claim is that Y s is smooth.Smoothness is equivalent to every n-jet extending to an n+ 1-jet.

Spec k[t]/tn Y s

Spec k[t]/tn+1 Yϕ0

Here, we have ϕ : A → k[t]/tn+1, and we want to find ϕ = ϕ0 + tnl so thatϕ is s-invariant. The claim is that if ϕ′ = ϕ0 + tnl′ where l′ is the s-invariantprojection of l, then ϕ′ is an algebra homomorphism.

17.1 Nonsolvable groups with maximal torus of dimension1

Let G be connected and T be a maximal torus. Assume dimT = 1. (This iscalled a group of rank 1.) If G is solvable, it is a semi-direct product and I can’tsay much about this.

Math 224 Notes 58

Theorem 17.3. If G is not solvable, then

1→ K → G→ PGL2 → 1,

where K is unipotent, so that it is the unipotent radical.

Proof. Consider G/B. We’ll prove that dim(G/B) = 1 as a variety. If we havethis, B acts on G/B which is proper, and fixes a point. So G/B should beP1. Because the automorphisms is PGL2, we get a map G → PGL2. ThenG→ PGL2 is surjective because it acts transitively on the flag variety. BecauseK is a connected algebra group, B(K) is unipotent. Then B(K) = K.

So dim(G/B) = 1 is what we want to prove. We have W = NG(T )/ZG(T )acts on T . So |W | ≤ 2 because T = Gm. Because W acts simply transitively onthe set of Borels containing T , there are at most 2 Borels. Take a representationv ∈ V such that B = g ∈ G : gv ∈ kv. I can restrict to subrepresentationscontaining v, so we can assume that G-translates of v span V .

Choose a basis e1, . . . , em such that Gm acts on ei by ni-th power of thestandard character. Assume that n1 ≥ n2 ≥ · · · ≥ nm. We know that T acts onG/B, and we are interested in the T -fixed points. But G/B sites inside P(V ),and so let’s study the Gm-fixed points in P(V ). A line

∑i aiei can be a fixed

point if and only if ai, aj 6= 0 implies ni = nj .But we also wan to look at attractors. Take an arbitrary element v′ =

∑aiei

and look at the map Gm → P(V ) given by λ 7→ λv′. By the valuative criterion,this extends to P1 → P(V ). The image of 0 and ∞ will be something likechucking away all ai except for the minimal or maximal ni.

Without restriction of generality, we can assume that G → GL(V ). Then Iclaim that n1 > nm. If it wasn’t, T acts on V by scalars and T ⊆ Z(G), then Gwould be solvable. Now I will be playing with the fact that there are only twoBorels. There exists a translate g1v such that

g1v = a1e1 +∑n≥2

aiei

with a1 6= 0. Here, we will have

limλ→0

(λgv) = line in the space of ei with eigenvalue n1.

Because G/B is the space of Borel, and this is a fixed point in projective spacewith the T -action, it is a Borel. Now by the similar logic, we take g2 such thatg2v = amem +

∑n≤m−1 anen and then limλ→∞ has eigenvalue nm. This is

another Borel.Now take V ′ = spane1, . . . , em−1. Then P(V ′) ⊆ P(V ). We have P(V ′) ∩

G/B ⊆ G/B, and we will prove that P(V ′) ∩ (G/B) = B1. If we show this,hitting a projective variety by a hyperplane only reduces the dimension by atmost 1, so dim(G/B) = 1. STo show this, take

∑i aiei ∈ G/B with am = 0.

Now limλ→∞ λ∑aiei must be B1 because it cannot be B2. Then it actually

should be B1.

Math 224 Notes 59

Theorem 17.4. Let G be connected reductive with dim(T ) = 1. Then G =PGL2 or G = SL2.

Proof. Let G′ = PGL2. There is a map G → G′ and this should induce anisomorphism G/B ∼= G′/B′ because both are P1. There is

1 K G G′ 1

1 K B B′ 1.

This gives B = T n BU → T n B′U = B′. The claim is that BU → B′U is anisomorphism. If we believe this, we have

K = ker(T → T ′) = ker(Gm → Gm; t 7→ tn).

The claim is that either n = 1 or n = 2. We have

1→ µn → G→ G′ → 1.

Even if characteristic is nonzer, the representation theory of µn is governed bycharacters. So G being connected implies that µn ∈ Z(G). Take n a nontrivialelement in W . Then Adn(t) = t−1. Because µn lives in the center, this is trivial.Therefore inversion is trivial, and so n = 1 or n = 2.

For n = 1, this is easy. For n = 2, we want to show that G is SL2. Take

G′′′ = (SL2×G′G)red SL2 = G′′

G G′.

If we can show that the left vertical is an isomorphism, we have B → B′′ → B′,and then T → T ′′ → T ′ because it is an isomorphism on the unipotent. ThenT → T ′ is covering of degree 2 and T ′′ is degree 2, so T → T ′′ is an isomorphism.Then B → B′′ is an isomorphism, so G → G′′ is an isomorphism. To see thatG′′′ → G is an isomorphism, we can again take Borels and tori, and see thatT ′′′ → T is an isomorphism. This again can be seen by looking at degree. todotodo

Math 224 Notes 60

18 April 5, 2018

Last time we did the following.

Theorem 18.1. Let G be a connected nonsolvable and T ∼= Gm. Then thereexists a nonsolvable map G → PGL2 that is surjective and induces an isomor-phism G/B P1.

The only nontrivial case is when dimG = 2.

Lemma 18.2. If dim(G) = 2 then G is solvable.

Proof. Consider B ⊆ G. Either G = B and G is solvable. If dim(B) ≤ 1, thenB is commutative and is contained in the center of G. So we can quotient by itand G is solvable as well.

Theorem 18.3. If G is reductive and dimT = 1, then G is either SL2 or PGL2.

We looked at G → PGL2 and looked at T → Gm. Then the Borels aregoing to be B = T n BU → B′ = T ′ n B′U . The claim is that BU → B′U isan isomorphism. We know that dim(BU ) = 1 and so BU = Ga. So we have amorphism ϕ : Ga → Ga. This shows that

φ =∑

ai Frobi .

But this has to be Gm-equivariant, with some Gm → Gm given by x 7→ xn.This will show that there cannot be higher Frobenius.

18.1 Groups of semi-simple rank 1

Today we will tighten our control. Let G be reductive. The rank of G is definedas dimT and the semi-simple rank is the rank of G/Z(G).

Theorem 18.4. Let G be reductive of semi-simple rank 1.

(1) Either [G,G] is SL2 or PGL2.

(2) If [G,G] = PGL2, then G = PGL2×T0. If [G,G] = SL2, then G =SL2×T0 or G = GL2×T 1

0 .

Theorem 18.5. Let G be reductive.

(1) [G,G] ∩ Z(G) is finite.

(2) [G,G]× Z(G)/([G,G] ∩ Z(G))→ G is an isomorphism.

Using the short exact sequences

1→ [G,G]→ G→ Z(G)/[G,G] ∩ Z(G)′ → 1,

1→ Z(G)′ → G→ [G,G]/[G,G] ∩ Z(G)′ → 1,

we get the following.

Math 224 Notes 61

Corollary 18.6. (1) [G,G] is reductive.

(2) G = [G,G] if and only if Z(G) = 1 if and only if Z(G) is finite.

(3) Z([G,G]) ⊆ Z(G).

(4) Z(G)′ → G→ G/[G,G] is surjective with finite kernel. (It is an isogeny.)

Proof. What we really have to prove is (4). To prove that the kernel is finite,we need to map G → T so that Z(G)′ → G → T has finite kernel. To dothis, find a faithful representation G on V such that G → GL(V ). Insidethere, there is a torus Z(G) and it acts on V by a finite number of characters.Therefore we can write V as V =

⊕i Vi. Then G respects this so that so that

Z(G)′ → G→∏i GL(Vi). Then we can take the determinant

Z(G)′ → G→∏i

GL(Vi)→∏i

Gm.

This has finite kernel.Now we show that it is surjective. Define the adjoint quotient Gad = G/Z(G)

and takeZ(G)→ G→ G/[G,G]→ Gad/[Gad, Gad].

The claim is that [Gad, Gad] = Gad implies that the map is surjective. If we havethis fact, then Z(G) surjects onto G/[G,G]. But because G/[G,G] is connected,Z(G)′ already surjects. We will prove that [Gad, Gad] = Gad later.

Now let us prove the classification theorem of semi-simple rank 1 groups.

Proof. (1) We know that G = [G,G]× Z(G)′/[G,G] ∩ Z(G)′.(2) First if [G,G] = PGL2, then we have [G,G]∩Z(G)′ = 1 so we are taking

the product with Z(G)′. If [G,G] = SL2, then either G/Z(G)′ = SL2 or PGL2.In the first case, we also have [G,G] ∩ Z(G)′ = 1 and we get G = SL2×Z(G)′.If G/Z(G)′ = PGL2, then [G,G] ∩ Z(G)′ = µ2. Then we have

(SL2×T )/µ2 = (SL2×Gm/µ2)× T

because µ2 → T looks like µ2 → Gm×T with the µ2 action on Gm part.

18.2 Roots

We don’t have the ABC of the structure theory of reductive groups, but we willdevelop them with roots. Let G be a connected group and let T be a maximaltorus.

Definition 18.7. A fake root ζ is a nonzero eigenvalue of the adjoint actionof T on g.

Math 224 Notes 62

Let α be a fake root. We write

ZG((ker(α))) = Gα.

Suppose G is reductive. If we appeal to your picture of reductive groups, thiswill look like

Lie(Gα) = gα ⊕ t⊕ g−α.

In general, your group is the semi-direct product

G = Ru(G) oGred.

There are several scenarios for the fake root.

1. α occurs as a root on Gred. In this case, we get

Gα = Uα oGα,red,

where Uα is the part of Ru(G) centralized by ker(α).

2. α occurs as a root on Gred, but its rational multiple appears in Ru(G). Inthis case, we again have Gα = Uα oGα,red.

3. α occurs on Ru(G), but its rational multiple α′ occurs in Gred. ThenGα = Uα oGα′,red.

4. No rational multiple of α occurs in Gred. Then Gα = Uα × T so that Gαis solvable.

Definition 18.8. A true root is a fake root that occurs on the reductive partof Gα.

Lemma 18.9. For α a true root, either Gα is solvable or Gα/Ru(Gα) is ofsemi-simple rank 1.

Proof. Consider Gα/Ru(Gα) ⊇ T such that ker(α) is contained in the center.Consider Gα/Ru(Gα) ker(α). todotodo

We can consider R ⊆ X∗(T ) the set of true roots.

Definition 18.10. Let Λ be a lattice and Λ∨ be the dual roots. For α, weconsider sα(λ) = λ− 〈λ, α〉α. A finite root system is a pair of sets

R ⊆ Λ, R∨ ⊆ Λ∨

and a correspondence R↔ R∨ such that

• 〈α, α〉 = 2,

• sα preserve R ⊆ Λ,

• the subgroup that they generate Aut(Λ) is finite.

Math 224 Notes 63

The claim is that the set of true roots form a root system. I need to producecoroots for you. Recall that

1→ Ru(Gα)→ Gα → Gα/Ru(Gα)→ 1

where Gα/Ru(Gα) is a reductive group of semi-simple rank 1. Whichever case,there is a canonical map

SL2 → Gα/Ru(Gα).

Then there is a corresponding map Gm → T of tori. This we define to beα ∈ Λ∨ = X∗(T ). We can check 〈α, α〉 = 2 inside SL2.

We can now consider the Weyl group

W = NG(T )/ZG(T ).

The action of W on X∗(T ) preserves the set of true roots. Also, sα ⊆W (Gα) ⊆W by pushing ( 0 1

−1 0 ) in SL2.

Proposition 18.11. W is generated by the reflections sα.

Proof. We do this by induction on the dimension of G. Let w ∈ W . BecauseT is commutative, consider the map ϕ(t) = w(t)t−1 from T to T . Then eitherdim(ker(ϕ)) ≥ 1 or ker(ϕ) is finite.

If dim(ker) ≥ 1, let S = ker(ϕ). Take G′ = ZG(S) so that w ∈ G′. IfG′ < G, then we are done by induction. If G′ = G, then S is central in G andso we can take G′′ = G/S. Then we are also done by induction.

If ker(ϕ) is finite, it is an isogeny ϕ : T → T . Consider ΛQ = Λ ⊗Z Q. ϕdefines an isomorphism, and is defined by v 7→ w(v) − v. Let α ∈ R and letv ∈ ΛQ be such that w(v)− v = α. Now we claim that 〈α, v〉 = −1. If we knowthat,

sαw(v) = sα(v + α) = v − 〈αv〉α− α = v.

Then we can set w′ = sαW and w′ has eigenvalue 1. Then we can peel this offand do induction again.

Now let’s prove that 〈α, v〉 = −1. I have an invariant form, and I can averageover W . Then we have

〈α, x〉 =2(α, x)

(α, α)

and (v + α, v + α) = (w(v), w(v)) = (v, v) implies that 〈α, v〉 = −1.

Math 224 Notes 64

19 April 10, 2018

Miraculously, we are developing the theory of roots without the structure theoryof reductive groups. Let G be connected and T ⊆ G be a maximal torus. Fakeroots are just the eigenvalues of T on g. For γ a character, we can take

Gγ = ZG(ker(γ)).

If you look at the reductive of Gγ , we can write

1→ ker(γ) → Gγ,red → Gγ,red/ ker(γ) → 1.

By the classification from before, there are two scenarios: either Gγ,red is Gmor it is PGL2 or SL2.

We say that γ is a true root if both of the following are satisfied:

(i) the second scenario occurs,

Look at the adjoint action of Gγ/ ker(γ) on gγ , which is a 1-dimensionaltorus because G/ ker(γ). Then we get a short exact

0→ Lie(Ru(Gγ))→ gγ → Lie(Gγ,red)→ 0.

(ii) γ appears as a character of T on Lie(Gγ,red).

We consider the Weyl group W = NG(T )/ZG(T ). For α a true root, we getan element sα ∈ W (Gα) ⊆ w. Let X∗(T ) be the lattice of characters of T andX∗(T ) be the lattice of cocharacters of T . For all α, there exists a α ∈ X∗(T )such that

sα(λ) = λ− 〈α, λ〉α, sα(λ) = λ− 〈α, λ〉α.for λ ∈ X∗(T ) and λ ∈ X∗(T ).

19.1 Examples of root systems

We defined root systems last class.

Theorem 19.1. sα generate W .

Example 19.2. Consider Λ = Z and Λ = Z. Then we can have roots ±1 andcoroots ±2. Or we can have roots ±2 and coroots ±1. The first correspondsto PGL2 and the second corresponds to SL2. There are no others, because weneed 〈α, α〉 = 2.

Example 19.3. Consider (Ga×Ga)oGm with the action given by (5, 7) timesthe standard character. Then 5, 7 are fake roots in X∗(T ) ∼= Z, because in bothcases, Gγ is the entire group.

Example 19.4. Consider (Ga × Ga) o SL2. Then X∗(T ) = Z and the actionis given by (1,−1). Then the eigenvalues are these two and ±2. In all the cases(except for α = 0), we have Gα = G and so the true roots are only those comingfrom SL2, which are ±2. For α = 0, we have Gα = ZG(Gm) so this is a fakeroot. You can do this for other representations, like (−2, 0, 2) o SL2.

Math 224 Notes 65

Example 19.5. Consider A2, which is α1, α2, α1 + α2,−α1,−α2,−α1 − α2.Consider the dual basis ωi such that 〈ωi, αj〉 = δij . Then we will have α1 =2ω1 − ω2 and α2 = −ω1 + 2ω2. This is PGL3 and the dual is the SL3.

Example 19.6. Consider B2. Here, Λ = Zω1 ⊕ Zω2 and the roots are ±ωiand ±ω1 ± ω2. Then for α = ω1 and β = −ω1 − ω2 and we have α = 2ω1

and β = ω2. In this case, Λ = Spanα, β and Spanα, β ⊆ Λ has index2. The corresponding group and its dual are SO5 and Sp4

∼= Spin5. (Index 2corresponds to this being a double cover.)

Example 19.7. We consider G2. For this, both the group and its dual aregenerated by the roots.

Definition 19.8. A set of positive roots is a choice of R+ ⊆ R such thatthere exists a ξ ∈ Λ or ξ ∈ ΛQ such that 〈α, ξ〉 6= 0 for every α ∈ R andR+ = α : 〈α, ξ〉 > 0.

The Weyl group acts on the choices of positive roots. We will see that itdoes so simply transitively.

Theorem 19.9. Let G be a connected group and T ⊆ B ⊆ G. Consider theaction of true roots that is contained in B. Then it is a system of positive roots.

Proof. There exists a representation V and a vector v ∈ V such that g : gv ⊆kv = B. Then the action is given by some character tv = λ(t)v for λ ∈ X∗(T ).We will show that α ∈ B if and only if 〈α, λ〉 > 0.

Math 224 Notes 66

20 April 12, 2018

We talked about root systems. We said that R+ ⊆ R is a set of positive rootsif there exists a λ ∈ Λ∨ such that α ∈ R+ if and only if 〈λ, α〉 > 0.

Theorem 20.1. Choose T ⊆ B ⊆ G. Note that we have shown that Gα ∩B isa Borel in Gα. Then for any root α, either α or −α is in the Borel. The claimis that α : α ∈ B is a system of positive roots.

Proof. Let V be a representation and v such that gv ⊆ kv if and only if g ∈ B.Then tv = λ(t)v for some λ ∈ Λ = X∗(T ). We will show that α ∈ B implies〈α, λ〉 > 0.

We can replace G by Gα. Then we have the uniponent radical Ru(Gα) ⊆ Gαand this acts trivially on v. So we can replace V by V Ru(Gα). Then we canreplace Gα with Gα/Ru(Gα). This will come from SL2 → Gα and my α will bein SL2.

Now we assume that G = SL2, and V is generated by gv for g ∈ G. Letn1 ≥ n2 ≥ · · · ≥ nm be the eigenvalues. Because

∑i ni = 0, we have ni > 0.

Then we can look at G/B ⊆ P(V ). We can look at G/B → P(V ) given byg 7→ gv, and v should be a Gm-attractor. This shows that v has eigenvalue n1,which is positive.

Here is another proof. For ξ : V → k, an equivariant line bundle on apoint is the same as a G-equivariant line bundle L on G/B. Now the conditionthat gv generates V is equivalent to the condition that the G-equivariant V →Γ(G/B,L ) is injective. todotodo

20.1 Structure of reductive groups

Theorem 20.2. Fix T ⊆ G.

(1)⋂B⊇T Ru(B) = Ru(G).

(2) Lie(Ru(G)) is the span of the fake roots.

For (1), it suffices to show that the left hand side is normal.

Corollary 20.3. Let S be a torus. If G is reductive then ZG(S) is reductive.

Proof. Take S ⊆ T . For B′ Borel subgroup of ZG(S), we have⋂B′⊇T

Ru(B′) =⋂B⊇T

(Ru(B) ∩ ZG(S)) ⊆⋂B⊇T

Ru(B) = 1,

where B′ runs over Borel subgroups of ZG(S).

Corollary 20.4. For G reductive, ZG(T ) = T .

Proof. We know that ZG(T ) is reductive. So ZG(T ) = T ×Ru(ZG(T )) = T .

Corollary 20.5. If α is a root, then

Math 224 Notes 67

1. dim gα = 1,

2. cα is root if and only if c = ±1.

Proof. Take α to be a root and take Gα = ZG(ker(α)). We know that gα liesin the Lie algebra of Gα but we know that this is reductive. So Gα/ ker(α) iseither SL2 or PGL2. It suffices to check in this case.

Let’s no prove the theorem.

Proof. We’ll first prove (2). The inclusion Lie(Ru(G)) ⊆ span is obvious. Takethe eigenspace gγ and consider the corresponding Gγ . Then gγ is in the Liealgebra of Ru(G) and is contained in Gγ . In the other direction, if γ is a fakeroot then gγ ⊆ Lie(Ru(Gγ)) and is contained in Ru or every Borel of Gγ .

For (1), we need to show that⋂T⊆B Ru(B) is normal in G. The claim is

that G is generated by Gγ . This is because g =⊕

γ gγ and gγ ⊆ Lie(Gγ). So it

suffices to show that⋂T⊆B Ru(B) is preserved under conjugation by each Gγ .

If Gγ is solvable, then Gγ∩B = Gγ for some B. Then Gγ ⊆⋂B ⊆

⋂T⊆B B.

Then it is uninteresting. Suppose Gγ is not reductive. Then

1→ Ru(Gα)→ Gα → Gα,red → 1

andGα,red is generated by its two Borel subgroups that contains a given maximaltorus. Then for T ⊆ Bα ⊆ Gα we want to show that Bα preserves

⋂T⊆B Ru(B).

We want to create a unipotent subgroupRα such thatRα ⊇ R =⋂T⊆B Ru(B)

that contains Ru(Gα) and has dim(Rα)− dim(R) = 1. If such Rα exists, thenR is necessarily normal in it by the following lemma. We define

R =⋂T⊆B

Ru(B), Rα =⋂

T⊆B,α∈B

Ru(B).

Then we need to check dim(Rα)−dim(R) = 1. But Lie(Rα)/Lie(R) is spannedby those true roots β for which the following happen: if α ∈ B then β ∈ B.

In this case, we show that this actually implies α = β. The Weyl group Wacts on the set of positive root systems. A combinatorial fact is that W actssimply transitivity. So every system of positive roots is of the form α ∈ B forsome Borel B. The next proposition solves this.

Lemma 20.6. If H is connected and unipotent, and H ′ ⊆ H is such thatdim(H)− dim(H ′) = 1, then H ′ is normal.

Lemma 20.7. If H is a connected unipotent with H ′ ⊆ H, if dim(H ′) <dim(H) then dim(NH(H ′)) > dim(H ′).

Proof. We have some Ga ⊆ ZG(H). If H ′ does not contain Ga, we can ad it.Else, quotient out.

Math 224 Notes 68

21 April 17, 2018

We proved the following structural theorem. Let G be connected.

Theorem 21.1. (⋂T⊆B Ru(B)) = Ru(G). Also, we have

Lie(Ru(G)) =⊕

α6=0 fake

gα ⊕ Lie(Ru(ZG(T ))).

Here, ZG(T ) = T ×Ru(ZG(T )).

Corollary 21.2. If S ⊆ G is a torus and G is reductive, then ZG(S) is reduc-tive. If G is reductive, then ZG(T ) = T .

From now on, assume G is reductive.

Corollary 21.3. Z(G) ⊆ T .

Corollary 21.4. If Γ ⊆ G is finite normal unipotent then Γ = 1.

Proof. G acts on Γ by conjugation, so Γ is central. So Γ is in a torus, whichmeans that Γ is semi-simple.

Let’s look at root systems again. Note that if R1 and R2 are two rootsystems, we can make a stupid root system R = R1 qR2 and Λ = Λ1 ⊕ Λ2. Sothere actually is another root system or rank 2, namely A1 × A1. These fourA1 ×A1, A2, B2, G2 are the all the rank 2 root systems.

Here is how you see it. Take α, β a basis. Then we can take 〈α, β〉 ≤ 0. Inthis case sβ sα acts on Λ and preserves orientation. So the matrix(

−1 −〈α, β〉〈β, α〉 〈α, β〉〈β, α〉 − 1

)has trace strictly less than 2. This shows that 〈α, β〉〈β, α〉 ≤ 3.

21.1 Structure of reductive groups II

For G a reductive group, its Lie algebra is

g =⊕α∈R+

gα ⊕ t⊕⊕α∈R+

g−α.

Then Gα = ZG(ker(α)0 and

Lie(Gα) = gα ⊕ t⊕ g−α.

Its commutator subgroup will have

Lie([Gα, Gα]) = gα ⊕ Span(α)⊕ g−α.

Math 224 Notes 69

Lemma 21.5. t ∈ Z(G) if and only if α(t) = 1 for all α ∈ R+.

Proof. Because Gα generate G, it is enough to show this for Gα. But we knowwhat these look like.

Corollary 21.6. If Z(G)0 = 1 if and only if [G,G] = G.

Proof. It is enough to show that Lie([G,G]) is all of g. But gα ⊆ Lie([Gα, Gα]).So it suffices to show that Z(G) = 1 implies that t is spanned by α. Thisfollows from Lie(Z(G)) = (spanα)⊥.

Definition 21.7. G is semi-simple if it is reductive and Z(G) = 1.

Theorem 21.8. If G is reductive, then

Z(G) → G→ G/[G,G]

is a isogeny, i.e., it is a surjective map of tori with finite kernel.

Corollary 21.9. Z(G) ∩ [G,G] is finite. Also,

[G,G] ∩ Z(G)

[G,G] ∩ Z(G)→ G

is an isomorphism.

Corollary 21.10. Z([G,G]) ⊆ Z(G). Z([G,G]) finite implies that [G,G] issemi-simple.

We proved that Z(G) = t ∈ T : α(t) = 1 for all ζ ∈ R.

Corollary 21.11. X∗(Z(G)) = X∗(Z(G)′) as a subset of X∗(T ) is the setλ ∈ X∗(T ) : 〈λ, α〉 = 0.

Lemma 21.12. The map T → G → [G,G] is a surjection and X∗(G/[G,G])as a subset of X∗(T ) is λ : 〈λ, α〉 = 0.

Corollary 21.13. The following are equivalent:

(1) Z(G) = 1(2) [G,G] = G

(3) SpanQα = ΛQ

(4) SpanQα = ΛQ.

Definition 21.14. G is said to be of adjoint type if Z(G) = 1.

Lemma 21.15. G is of adjoint type if SpanZα = X∗(T ).

For G reductive, G/Z(G) is actually adjoint type.

Definition 21.16. G is said to be simply-connected if Spanα = X∗(T ).

Math 224 Notes 70

Lemma 21.17. For every G reductive there exists

Gsc → [G,G] → G

which is the dual picture of G G/Z(G) G/Z(G).

Definition 21.18. For G1, G2 reductive, we say that the map G1 → G2 is anisogeny if it is surjective and has finite kernel on the torus.

Theorem 21.19. Let G be semi-simple and G1 ⊆ G be a connected normalsubgroup. Then there exists a unique G2 ⊆ G such that G1 × G2 → G is anisogeny.

Proof. Take a Cartan T and T1 = (T ∩ C1) ⊆ T . We can take G2 =[ZG(T1), ZG(T1)]. This is the right think to look at, because ZG1×G2(T1) =T1 ×G2.

Let’s now look at Borels. Take B ⊆ G and look at the unipotent part U .Then we have

Lie(U)⊕α∈R+

gα.

For each α, we actually have SL2 → Gα → G, and we can look at the image ofGa and write Uα.

Proposition 21.20. The multiplication map∏R+

Uα → U

is an isomorphism of varieties equipped with a T -action.

Proof. Assume that the characters of T on Lie(U) appear with multiplicity 1.Then there exists a Ga ⊆ Z(U) such that the action of T on Ga is given by α0.Then we can look at ∏

α Uα U

∏α 6=α0

Uα U/Uα0.

Now we can use the following fact.

Suppose we have

X Y

SpecA,

πx

φ

πy

where everything is equipped with a Gm-action, and the grading on A is positive.(This means that A0 = k and An = 0 for n < 0.) If πx, πy are smooth, φ

Math 224 Notes 71

defines an isomorphism on the fiber π−1x (0) → π−1

y (0), and the action on Y iscontractible, then φ is an isomorphism.

Let G be reductive. To each B, we have B/Ru(B). Note that if g′ and g′′

both conjugates B1 to B2, they define the same map B1/Ru(B1)→ B2/Ru(B2).This is because the normalizer of B is itself and then the action of b ∈ B onB/Ru(B) is trivial. So all B/Ru(B) are canonically isomorphic, and this iscalled the abstract Cartan. Likewise, you can define the abstract Weylgroup.

Theorem 21.21 (Bruhat decomposition). Let X = G/B be the flag variety.Then the orbits of the diagonal action of G on X ×X are in bijection with W .

Given an element w ∈W , we are going to take (B,Bw) ⊆ X ×X.

Math 224 Notes 72

22 April 19, 2018

We had this notion of an abstract Cartan that is canonically attached to G. IfG is reductive, we look at a Borel B and take B/Ru(B). If g1, g2 conjugates B1

to B2, they give the same isomorphism B1/Ru(B1)→ B2/Ru(B2). Similarly wecan define the abstract Weyl group. For T , we look at NG(T )/T . If g conjugates(T1 ⊆ B1)→ (T2 ⊆ B2), we will get map Adg : NG(T1)/T1 → NG(T2)/T2. Butthen any normalizer of (T1 ⊆ B1) is in B1, so it is T1 itself. So we can definethe abstract Weyl group like this.

22.1 Bruhat decomposition

Let X be the flag variety, the variety of Borels.

Theorem 22.1. G-orbits of X ×X are in bijection with W .

Proof. We need to construct the maps. Given T ⊆ B and w ∈ NG(T )/T , weconsider the orbit of (B,Adw(B)) ∈ X ×X. This is independent of the choiceof T ⊆ B, because we are going to conjugate anyways. The construction in theother way is more remarkable. Consider B1, B2 two Borels. Then we get

(B1 ∩B2)/Ru(B1 ∩B2) B2/Ru(B2)

B1/Ru(B1) T

∼=∼=

that doesn’t necessarily commute. Then by the following lemma, B1 ∩B2 con-tains some maximal torus, so all the maps are isomorphisms. The noncommu-tativity is going to be given by some automorphism of T , and this is going tobe the w we want.

Lemma 22.2. The intersection of any two Borels contains a maximal torus.

Now choose T ⊆ B.

Corollary 22.3. The B-orbits on G/B are in bijection with W = NG(T )/T .

Consider Xw = Bw inside G/B. Then this looks like

Xw = B/B ∩Adw(B).

These are called Schubert cells. This is a homogeneous space, but we can saymore things. Recall thatB = TnU and we proved last time that U =

∏α∈R+ Uα

where Uα = Ga. Then we can also write Xw = U/U ∩Adw(U).

Lemma 22.4. (a) U ∩ Adw(U) ⊆ U can be described as∏α,w(α)∈R+ Uα ⊆∏

α∈R+ Uα.

(b) U/U ∩Adw(U) ∼=∏α∈R+,w(α)∈R− Uα.

Math 224 Notes 73

This lemma is highly believable. At least it is evident at the level of Liealgebras.

Corollary 22.5. B ×B-orbits on G (with left and right multiplication) are inbijection with W :

w 7→ BwB = Gw.

These are called Bruhat cells.

We can also write this as Gw = (∏α∈R+,w(α)∈R− Uα)wB. Let w0 ∈ W be

the element such that Xw0 ⊆W is open. Then w0 satisfies

dim(X) = dim(G)− dim(B) = dim(g)− dim(b) = |R+|.

So dim(Xw0) = dim(X) means that w0 satisfies w0(R+) = R−. This is calledthe longest element of the Weyl group. In this case,

Gw0 = Uw0B, Xw0 = Uw0.

22.2 Simple roots

Suppose we have a root system R and pick a system of positive roots R+ ⊆ R.

Definition 22.6. α ∈ R+ is simple if it cannot be written as a sum of positiveroots with nonnegative integral coefficients α 6= α1 = α2.

Let I ≡ R+ be the set of simple roots.

Lemma 22.7. (1) αi form a basis for SpanQ(α) = (Λ[G,G])Q.

(2) si generates W .

In the semi-simple case, we can look at

Λad = Spanα ⊆ Λ.

We already had Λsc = Spanα and so we have

Λad ⊆ Λ ⊆ Λsc.

So there are only finitely many choices for Λ.

Lemma 22.8. Λad can be uniquely recovered by the Cartan matrix (I, 〈αi, αj〉).

Proof. We have R+ ⊆ Λad = Spanα. Then we can just reflect si(αj) = αj −〈αj , αi〉αi. Then you can look at the set generated by all these reflections.

Theorem 22.9. Let B be a Borel. Parabolics that contain B are in bijectionwith subsets of I.

Math 224 Notes 74

Let J ⊆ I. We will find a parabolic PJ with

Lie(PJ) = Lie(B)⊕⊕β

g−β

where β lies in the positive integral span of αj for j ∈ J . This is going to begiven by

PJ = Ru(PJ) oMJ , MJ = ZG((⋂j∈J ker(αj))

).

Then Lie(MJ) =⊕

β gβ ⊕ t⊕⊕

β g−β and Lie(Ru(PJ)) =⊕

α6=β gα.Let’s now show that these are all the parabolics. If Gm acts on Z, then we

can think of Z+ = MapsGm(A1, Z), which is the z ∈ Z such that limt→0 tz exists.There is a natural map Z+ ⊆ Z. We can also think of Z0 = MapsGa(pt, Z).Then there is a natural map Z+ → Z0 by taking the attractor, and a Z0 → Z+

by looking at the constant map. If Z = SpecA, with grading, we have Z+ =SpecA+ where A+ is A quotiented out by all the negative degree things. ThenA0 is Aquotiented out by all nonzero degree things.

Lemma 22.10. (a) If Z is smooth then Z+, Z0 are smooth.

(b) If z ∈ Z0, then Tz(Z0) is the direct sum of TzZ on which Gm acts triv-

ially. Tz(Z+) is the direct sum of TzZ on which Gm acts with nonnegative

eigenvalues.

So here is the construction. For J ⊆ I, we look at λ ∈ X+(t) such that〈λ, αj〉 = 0 for j ∈ J and 〈λ, αi〉 > 0 for j /∈ J . Then λ : Gm → T . Now wedefine PJ = G+ and MJ = G0.

Math 224 Notes 75

23 April 24, 2018

I want to classify parabolic subgroups. Fix a Borel T ⊆ B and we want toclassify the parabolics containing B.

23.1 Classification of parabolics

Theorem 23.1. There is a bijection between J ⊆ I and PJ a parabolic con-taining B, such that

Lie(PJ) = Lie(B)⊕⊕

α∈Spanαj :j∈J

gα,

and

PJ = MJ nRu(PJ), MJ = ZG((⋂j∈J ker(αj))

0),∏α∈R+,α/∈Spanαj

Gα = Ru(PJ) ⊆ Ru(B) =∏α∈R+

Gα.

Our method was to look at the Gm action on Z and look at Z0 → Z+ → Z.For example, on Z = V >0 × V 0 × V <0 an affine space, we will have Z+ =V >0 × V 0 and Z0 = V 0.

Lemma 23.2. If Z is smooth, then Z+ and Z0 are smooth. If z ∈ Z0, thenTzZ

0 = (TzZ)0 and TzZ+ = (TzZ)>0.

Now consider a coroot λ : Gm → T such that 〈λ, αj〉 = 0 if j ∈ J and〈λ, αj〉 > 0 if j /∈ J . Then we define PJ = G+. This PJ is going to contain B,because we have

PJB = B+ ⊆ B

and TeB+ = TeB by design. Now consider MJ = G0. We have maps PJ →MJ

and MJ → PJ . Then we have MJ = ZG(λ(Gm)). The claim is that

ker(PJ →MJ) = Ru(PJ).

This can be shown again by comparing Lie algebras.Let us now do this other direction. Let B ⊆ P be a parabolic.

Lemma 23.3. On G/P , there is a very ample G-equivariant line bundle L onG/P .

Proof. We look at a representation V of G and look at G/P → P(V ).

Let V = Γ(G/P,L ). Then we have a P -invariant functional V → L1 anddualizing gives (L1)∗ → V ∗. The stabilizer of a line/coline is exactly P .

Let us be given ` ⊆ V and StabG(`) = P ⊇ B. Let B act on ` via somecharacter λ and consider

J = j : 〈λ, αj〉 = 0.

Math 224 Notes 76

I then claim that P = PJ . Let us first do the inclusion PJ ⊆ P . We know thatPJ = Ru(PJ)oMJ where MJ is generated by Gα for α in the span of J . Here,Gα acts on V with Bα acing on v via a character. Because

1→ [Gα, Gα]→ Gα → Gα/[Gα, Gα]→ 1,

[Gα, Gα] receives a surjection from SL2, and Gα/[Gα, Gα] receives an actionfrom a torus, it suffices to show that SL2 acts trivially on v. This can be done.

Now because PJ ⊆ P , it is enough to show that Lie(PJ) = Lie(P ). Forα ∈ R+ such that g−α ⊆ Lie(P ), we want to show that g−α ⊆ Lie(PJ). But Gα

stabilizes kv and the same proof on Gα works.We talked about the Bruhat decomposition. We said that orbits in are in

bijection with W . Now we can be interested in G-orbits of (G/P1) × (G/P2).These each correspond to J1 ⊆ I and J2 ⊆ I.

Proposition 23.4. The Weyl group WJ = W (MJ) = 〈sj : j ∈ J〉 is naturallya subgroup of W . Then the G-orbits are in bijection with WJ1 \W/WJ2 .

23.2 Length

Let w ∈W . We define its length as

l(w) = #α ∈ R+, w(α) ∈ R−.

There is a different way to define length. For any w ∈W , we can write

w = si1 · · · sin .

Definition 23.5. We say that this is a reduced decomposition if l(w) = n.

We will talk about the interplay between length in the Bruhat decomposition.Suppose w = w1w2 be such that l(w1) + l(w2) = l(w). Consider (X ×X)w thecorresponding G-orbit on X ×X. Then there is a map

(X ×X)w1×X (X ×X)w2

→ X ×X.

Theorem 23.6. This is an isomorphism onto (X ×X)w.

There is something called the Bruhat order. Then for the Schubert cells,you can prove that

Xw ⊆ B · · ·w ⊆ X, Xw =⋃w′≤w

Xw′ .

Suppose w = si1 · · · sin . According to the previous theorem, we have an isomor-phism

(X ×X)si1 ×X (X ×X)si2 ×X · · · ×X (X ×X)sin → (X ×X)sin .

Math 224 Notes 77

Then you can take closures on both sides. Then we get a resolution of singu-larities (X ×X)w. Here, (X ×X)si is going to be some P1 fibered over the flagvariety.

We know how to get a root data from a reductive group. We want to see towhat extent this classifies reductive groups. Consider the data of T ⊆ B ⊆ Gand a pinning, i.e., a choice of a nonzero vector in gαi for i ∈ I. This categoryis called C1. There is another category C2 of just R ⊆ Λ and R∨ ≡ Λ∨. Thereis a functor

C1 → C2.

Theorem 23.7. This is an equivalence of categories.

The pinning is needed because the group has more automorphisms than theroot system. T/Z(G) will act nontrivially on G, and this can be identified with∏i∈I Gm given by αi.Because we don’t want to see pinnings, which seem to be artificial, we can

define the following category. Objects are reductive groups, and morphismMor(G1, G2) are isomorphisms up to inner automorphisms. You can also definea category with G with some enhancement.

Index

abstract Cartan, 71abstract Weyl group, 71adjoint type, 69algebraic group, 4

Borel subgroup, 48Bruhat cell, 73Bruhat decomposition, 71

Cartan subgroup, 55character, 24coend, 8

distribution, 35

equivariant sheaf, 36

faithfully flat quotient, 44flag variety, 6

Grassmannian, 43

Hilbert scheme, 25homogeneous space, 42

ind-scheme, 29isogeny, 70

Jordan decomposition, 18

length, 76Lie algebra, 34linear algebraic group, 4

maximal torus, 50

orbit, 17

parabolic subgroup, 44Poincare–Birkhoff–Witt theorem,

38positive roots, 65

rank, 60reductive group, 52representation, 6root, 61, 62root system, 62

Schubert cell, 72semi-simple, 18, 69semi-simple rank, 60simple root, 73simply-connected, 69solvable group, 45

tangent space, 34Tannaka duality, 12torus, 11transitive, 15twisted arrow category, 8

unipotent, 18, 19unipotent radical, 52universal enveloping algebra, 38

universal, 40

Weyl group, 63

78