Subgroups of Linear Algebraic Groups

Subgroups of Linear Algebraic Groups

Contents

Introduction 1Acknowledgements 41. Basic definitions and examples 51.1. Introduction to Linear Algebraic Groups 51.2. Connectedness 82. Background in algebraic geometry 103. Tori, Unipotent and Connected Solvable Groups 123.1. Unipotent Groups 123.2. Tori 153.3. Connected Solvable Groups 174. Borel Subgroups 184.1. Actions of algebraic groups 194.2. The Borel fixed point theorem and consequences 225. Connected reductive groups 275.1. Reductive and semisimple groups 285.2. Lie algebras and root systems 295.3. Bruhat decomposition 366. Parabolic subgroups 386.1. Standard parabolic subgroups and the Levi decomposition 396.2. The Borel-Tits Theorem 447. G-complete reducibility 46References 50

Introduction

Let G be a variety over an algebraically closed field k. We call G an algebraicgroup if it is equipped with a group structure such that the multiplication mapG×G→ G and the inversion map G→ G are morphisms of varieties. Furthermore,when G is an affine variety, it is called a linear algebraic group. For example, the

set Mn(k) of n× n matrices over k can be easily identified with kn2

so is an affinevariety. Now, since the determinant of a matrix is a polynomial in its entries, we seethat SLn is also an affine variety, and it’s easy to show that it is a linear algebraicgroup. One can also show that GLn is a linear algebraic group.

Algebraic groups have applications to several areas of pure mathematics. Forinstance, they are notably central to the Langlands program in Number Theory.They can also be a good way to construct an important class of finite groups, calledfinite groups of Lie type. These groups arise as fixed point sets of certain types

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of endomorphisms of some linear algebraic group over k = Fq, where q is a primepower. For example, most of the finite simple groups are finite groups of Lie type.We will unfortunately not have time to discuss these applications, and the readeris referred to [MT, Part III] for a detailed introduction to finite groups of Lie type.

The combination of the group structure with the variety structure on G forcesit to have some nice properties. For instance, a linear algebraic group is irreducibleas a variety if and only if it is connected (see Proposition 1.8). Moreover, anylinear algebraic group can be embedded as a closed subgroup of GLn for some n(see Corollary 4.8). Thus linear algebraic groups can be viewed as certain groupsof matrices. However, the embedding into GLn is not canonical, and in generalwe have no control over what it is. Therefore, our results will usually be statedand proved in full generality, without assuming that our groups have already beenembedded into some GLn.

An important ingredient in studying linear algebraic groups is the notion of aBorel subgroup, which is a maximal closed connected solvable subgroup. For exam-ple, the group of invertible upper triangular matrices is a Borel subgroup of GLn.This naturally leads to the study of a larger class of subgroups, called parabolicsubgroups. These are closed subgroups which contain a Borel subgroup of G. Whenthe group G is connected and reductive, which means it has no non-trivial properclosed connected unipotent normal subgroups, the structure of its parabolic sub-groups is well understood. For instance, they can be expressed as a disjoint unionof double cosets of the Borel subgroup they contain (see Theorem 6.6). Parabolicsubgroups of connected reductive groups can then be used to generalise some fa-miliar concepts from representation theory. In particular, Serre [Se] introduced thenotion of a G-completely reducible (G-cr for short) subgroup (see Definition 7.1),which generalises the notion of a group acting completely reducibly on a vectorspace V . Indeed, in the special case G = GL(V ) (V a finite dimensional k-vectorspace), a closed subgroup H of G is G-cr if and only if V is a semisimple H-module.

A recent theorem of Bate, Martin and Rohrle [BMR] asserts that G-completereducibility is equivalent to the notion of strong reductivity, due to Richardson [R].They then used this to show that a closed normal subgroup of a G-cr subgroup isitself G-cr (see [BMR, Theorem 3.10]). In the case G = GL(V ), this reduces to awell known result in Clifford theory: if V is a semisimple H-module and N � H,then V is a semisimple N -module. The purpose of this essay is to give an accountof the general theory of linear algebraic groups, focusing on their Borel subgroupsand their parabolic subgroups, and to prove the aforementioned results of Bate,Martin and Rohrle. In the study of parabolic subgroups, we will restrict ourselvesto the case where the group G is connected reductive.

We begin in Section 1 by introducing linear algebraic groups, giving severalexamples. We then define morphisms of algebraic groups, which are just morphismsof varieties that are also group homomorphisms, and we show that their kernelsand images are closed. We also investigate the notion of connectedness and giveexamples of connected linear algebraic groups.

In Section 2, we give the background in algebraic geometry that will be neededlater on. More specifically, we discuss tangent spaces, differentials, projective va-rieties, dimension theory and complete varieties, which play an important role inthe study of Borel subgroups in Section 4. However, the emphasis of this essay is

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on algebraic results, and not so much on the geometry of linear algebraic groups.So the treatment here is mostly expository, and contains almost no proofs.

Section 3 is split into three parts. In the first part, we discuss unipotent groups.These are linear algebraic groups which can be embedded into GLn as a groupof matrices whose only eigenvalue is 1. In order to study them, we introduce amultiplicative version of the well-known Jordan decomposition for endomorphismsof a finite dimensional vector space. This can be translated back to a decomposi-tion of elements of an arbitrary linear algebraic group into so-called unipotent andsemisimple parts. This leads to the definition of a unipotent group, and we thenprove that such a subgroup of GLn is conjugate to a group of upper triangularmatrices with 1’s along the diagonal (see Theorem 3.7 and Corollary 3.8).

In the second part, we introduce tori and outline some of their basic properties.A torus is a linear algebraic group isomorphic to the group of n × n diagonalmatrices for some n. An important concept needed to understand tori is the notionof a character. For an algebraic group G, a character of G is a morphism betweenG and the multiplicative group k∗. The set X(G) of characters of G is easily seento be an abelian group, and we show that it is finitely generated when G is a torus.

We finally investigate connected solvable groups in the third part. Our mainresult is the Lie-Kolchin theorem (Theorem 3.16), which asserts that a connectedsolvable subgroup of GLn has a common eigenvector. As an immediate consequence,we obtain that a connected solvable subgroup of GLn is conjugate to a subgroup ofthe group of all invertible upper triangular matrices (Corollary 3.17). Moreover, wegive without proof the result that a connected solvable group G is the semidirectproduct GuoT of its subgroup Gu of unipotent elements with a maximal torus T .

In Section 4, we study Borel subgroups of arbitrary linear algebraic groups. Todo so, we first study actions of algebraic groups. We also explain how to make aquotient G/H into a linear algebraic group when H is a closed normal subgroup ofG. Along the way, we will prove that linear algebraic groups can be embedded intosome GLn (see Corollary 4.8). Using these tools, we then show the Borel fixed pointtheorem (Theorem 4.13), which asserts that if a connected solvable group acts ona projective variety in a way so that the action is given by a morphism of varieties,then it must have a fixed point. This has important consequences. For instance,it implies that the Borel subgroups of a linear algebraic group are all conjugate(Proposition 4.14). We also introduce parabolic subgroups and show that they areconnected and self-normalising (see Corollary 4.28).

Section 5 is devoted to the study of connected reductive linear algebraic groups.The theory there is quite lengthy, and we do not have time for too much details, butwe try to explain all the steps required to obtain the structure of connected reductivegroups (Theorem 5.13). However, we do not prove the theorem, nor many of theresults that build up to it. The structure of these groups relies crucially on theirroot system, which itself is defined using Lie algebras. Therefore, after introducingreductive groups, we explain how to associate a Lie algebra to a linear algebraicgroup, and outline basic properties it must satisfy. This allows us to define theroot system of G with respect to a maximal torus T . We then work our way tothe structure theorem from there. Subsequently, we discuss a few basic facts aboutabstract root systems and state the result that a Borel subgroup B of G containingT defines a base for the root system. We finally obtain the very important Bruhatdecomposition (Theorem 5.26), which states that G can be decomposed as a disjoint

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union of double cosets of B. This decomposition is more generally true for any groupG with a BN-pair (see Definition 5.24), and it is in that context that we prove it.This result also allows us to deduce that the intersection of two Borel subgroups isconnected and contains a maximal torus (Corollary 5.27).

Because of lack of time, we do not discuss here the classification theorem ofChevalley, which gives a one-to-one correspondence between isomorphism classes ofsemisimple linear algebraic groups (these are connected reductive groups with noproper non-trivial closed connected solvable normal subgroups) and isomorphismclasses of root data, which are combinatorial objects that can be obtained from theroot system. One remarkable feature of this theorem is that it holds in arbitrarycharacteristic, and not only in characteristic 0 unlike the classification theoremfor semisimple Lie algebras. The reader is referred to [S, Chapters 9 and 10, inparticular Theorem 10.1.1] for a proof and exact statement of this theorem.

With the results of Section 5 at our disposal, we investigate parabolic subgroupsof connected reductive groups in Section 6. These have a nice structure which alsorelies crucially on the root system: they are uniquely determined by some subsetof the base of the root system which was obtained from the Borel subgroup theycontain. We prove again that these groups are closed, connected, self-normalisingsubgroups of G without appealing to the results in Section 4, and that two distinctparabolic subgroups containing the same Borel subgroup B are not conjugate. Wecarry on investigating the structure of parabolic subgroups by decomposing theminto a product of a unipotent group with a connected reductive group, called itsLevi complement. A Levi subgroup is then defined to be a conjugate of a Levi com-plement, and we show that Levi subgroups are precisely the centralisers of subtoriof G (Proposition 6.13). We then move on to the Borel-Tits theorem (Theorem6.15), which asserts that for a closed unipotent subgroup U of G contained in aBorel subgroup, there exists a parabolic subgroup P of G such that NG(U) ≤ Pand U is contained in a closed connected unipotent normal subgroup of P . Usingthis, we immediately obtain that given a maximal closed subgroup H of G, eitherH is reductive or H is parabolic (see Theorem 6.18).

Finally, we discuss G-complete reducibility in Section 7. A closed subgroup Hof G is G-completely reducible if whenever it is included in a parabolic subgroup ofG, it is actually included in a Levi subgroup of it. We also introduce the notionof a strongly reductive subgroup. A closed subgroup H of G is strongly reductiveif it is not contained in any parabolic subgroup of CG(S), where S is a maximaltorus of CG(H). We then show that a closed subgroup is strongly reductive if andonly if it is G-completely reducible (Theorem 7.7). This requires a few lemmas onthe intersection of two parabolic subgroups, which we state beforehand. Using atheorem of Martin [M], we deduce that a closed normal subgroup of a G-completelyreducible subgroup is itself G-completely reducible (Theorem 7.9), and using theBorel-Tits theorem, we show that the converse is not true in general.

In Sections 1-6, we follow the texts by Malle and Testerman [MT] and Humphreys[H], as well as Springer [S] or Borel [B] for some of the more geometrical results. InSection 7, our main reference is [BMR].

Acknowledgements

I am grateful to David Stewart for setting this essay and for kindly giving meadvice for a talk, based on this work, that I gave in the Part III Seminars.

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1. Basic definitions and examples

1.1. Introduction to Linear Algebraic Groups. Let k be an algebraicallyclosed field. Recall that a subset of kn is called an algebraic set if it is of theform

Z(I) = {(x1, . . . , xn) ∈ kn : f(x1, . . . , xn) = 0 for all f ∈ I}where I is an ideal in the polynomial ring k[T1, . . . , Tn]. Taking closed sets to bealgebraic sets defines a topolgy on kn, called the Zariski topology. An affine varietyis an algebraic set together with its induced Zariski topology.

Given an algebraic set X, we can define an ideal in k[T1, . . . , Tn] by

I(X) = {f ∈ k[T1, . . . , Tn] : f(x1, . . . , xn) = 0 for all (x1, . . . , xn) ∈ X}

The quotient k[X1, . . . , Xn]/I(X) is called the coordinate algebra or algebra of reg-ular functions on X, and is denoted by k[X]

If X ⊂ kn, Y ⊂ km are affine varieties, then the product X × Y naturally hasthe structure of an algebraic set in kn+m and thus is also an affine variety whenequipped with the Zariski topology (which in general is not the same as the producttopology). Note that k[X × Y ] ∼= k[X]⊗k k[Y ].

A map ϕ : X → Y is called a morphism of affine varieties if it can be definedby polynomial functions in the coordinates. Note that such maps are continuouswith respect to the Zariski topology. The morphism ϕ induces a k-algebra homo-morphism

ϕ∗ : k[Y ] −→ k[X]

f 7−→ f ◦ ϕ

We can now define our main object of study:

Definition 1.1. A linear algebraic group is an affine variety G equipped with agroup structure such that the group operations

µ : G×G −→ G, ι : G −→ G,

(g, h) 7−→ gh, g 7−→ g−1,

are morphisms of varieties.

Example 1.2. Let’s first look at several examples of linear algebraic groups:

(1) Ga = (k,+), the additive group of k. It’s clear that it satisfies the definition(it is the zero set of the zero polynomial), and we have k[Ga] = k[T ], theusual polynomial ring.

(2) Gm = (k∗,×), the multiplicative group of k∗. We can identify it with

{(x, y) ∈ k2 : xy = 1}

with componentwise multiplication by mapping x 7→ (x, x−1). This set isclearly a closed subset of k2, being the zero set of the polynomial T1T2− 1.Multiplication and inversion are clearly given by polynomials so it is a linearalgebraic group, and k[Gm] = k[T1, T2]/(T1T2 − 1) ∼= k[T, T−1].

(3) The general linear group GLn = {A ∈ Mn(k) : detA 6= 0} is also a linearalgebraic group. As for Gm, one way of seeing this is to identify it with

{(A, y) ∈ kn2

× k : detA · y = 1}

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via A 7→ (A,detA−1), with componentwise multiplication where we identify

kn2

with Mn(k) (and so multiplication in the first component is matrixmultiplication).

Since det is given by a polynomial in the matrix entries, this is a closed

subset of kn2+1. Multiplication is then clearly given by polynomials, and by

Cramer’s rule, so is inversion. We can also find the ring of regular functions:

k[GLn] = k[Tij , Y : 1 ≤ i, j ≤ n]/(det (Tij) · Y − 1)

∼= k[Tij : 1 ≤ i, j ≤ n]det (Tij)

the localisation of k[Tij : 1 ≤ i, j ≤ n] at the multiplicatively closed subsetgenerated by det (Tij).

(4) The special linear group SLn = {A ∈ GLn : detA = 1} is a closed subgroupof GLn, as det is given by a polynomial in the matrix entries, and so isalso a linear algebraic group. Its ring of regular functions is then clearlyk[Tij : 1 ≤ i, j ≤ n]/(det (Tij)− 1).

(5) Similarly, the following are closed subgroups of GL(n) and so linear alge-braic groups:• The group of invertible upper triangular matrices

Tn =

∗ . . . ∗

. . ....∗

∈ GLn

= {(aij) ∈ GLn : aij = 0 for i > j}.

• The group of upper triangular matrices with 1’s on the diagonal

Un =

1 . . . ∗

. . ....1

= {(aij) ∈ Tn : aii = 1 for 1 ≤ i ≤ n}.

• The group of invertible diagonal matrices

Dn =

{(∗ . . .

∗

)∈ GLn

}= {(aij) ∈ GL(n) : aij = 0 for i 6= j}.

Note that Dn∼= Gm × . . .×Gm︸ ︷︷ ︸

n times

(6) Let J2n =

(0 Kn

−Kn 0

)where Kn =

(0 1. ..

1 0

). The symplectic group in

dimension 2n is then defined to be

Sp2n = {A ∈ GL2n : AtJ2nA = J2n}.It is a closed subgroup of GL2n, thus a linear algebraic group. It is thegroup of linear transformations leaving invariant the non-degenerate skew-symmetric bilinear form given by J2n.

(7) Orthogonal groups are also linear algebraic groups. For simplicity, we as-sume char(k) 6= 2 (for the general case see [MT, section 1.2]). The orthog-onal group in dimension n is given by

GOn = {A ∈ GLn : AtKnA = Kn}.It is the group of linear transformations leaving invariant the non-degeneratesymmetric bilinear form given by Kn.

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Having defined linear algebraic groups, we then consider the maps between them.We require them to preserve both the geometrical structure and the group structureof algebraic groups.

Definition 1.3. Let G1, G2 be linear algebraic groups. A map ϕ : G1 → G2 is amorphism of algebraic groups if it is a group homomorphism and a morphism ofaffine varieties.

We would like for such maps to have nice images and kernels. In order to proveit, we first need a couple of basic results:

Lemma 1.4. Let U, V be two dense open sets of an algebraic group G. ThenG = U · V

Proof. Pick x ∈ G. From the definition of algebraic groups, inversion is a continousmap with continuous inverse (being its own inverse), and thus is a homeomorphim.Hence V −1 is also open dense. Similarly, multiplication by x is a homeomorphism,and thus xV −1 is open dense. Therefore U , being dense, must meet xV −1 and sox ∈ U · V . �

Recall that a subset X of a topological space Y is called locally closed if it isthe intersection of an open set with a closed set. Equivalently, X is locally closedif it is open in X. A subset of Y is called constructible if it is a finite union oflocally closed sets. It is a fact from algebraic geometry that morphisms of varietiesmap constructible sets to constructible sets (see [H, Theorem 4.4]). In particular,the image of a morphism is constructible. Also, it is a standard fact that if X is aconstructible subset of a variety, then it contains an open dense subset of X.

Proposition 1.5. Let H be a subgroup of G, H its closure. Then:

(i) H is a subgroup of G.(ii) If H is constructible, then H = H.

Proof. (i) Inversion being a homeomorphism, it’s easy to see that H−1

= H−1 = H.Similarly, for x ∈ H, multiplication by x is a homeomorphism and so xH = xH =H. Thus H ·H ⊂ H. Hence, for x ∈ H, Hx ⊂ H, and so Hx = Hx ⊂ H. ThereforeH is closed under inverses and multiplication, and so is a subgroup of G.

(ii) If H is constructible, then it contains an open dense subset U of H. Now,H is a linear algebraic group by (i), and so by Lemma 1.4, we have H = U · U ⊂H ·H = H. �

Corollary 1.6. Let ϕ : G1 → G2 be a morphism of algebraic groups. Then kerϕand ϕ(G1) are closed, and therefore are linear algebraic groups.

Proof. ϕ is continuous and kerϕ = ϕ−1({1}) is the inverse image of a closed set,so it is closed. Moreover, ϕ(G1) is a constructible subgroup of G2. By the previousproposition, it must be closed. �

It is clear that closed subgroups of GLn are linear algebraic groups. The followingimportant result shows that all linear algebraic groups arise in that way. We willprove it later, in Section 4, when we discuss quotients of algebraic groups.

Theorem 1.7. Any linear algebraic group can be embedded as a closed subgroupinto GLn for some n.

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1.2. Connectedness. The group structure of a linear algebraic groups allows usto know more about its geometrical structure. Recall that an affine variety X isirreducible if it cannot be written as a union U ∪ V of non-empty closed subsetsU, V . In general, an affine variety X can be written as

⋃ri=1Xi, for some r, where

the Xi are maximal irreducible subsets, called the irreducible components of X. Itis a fact that a morphism of varieties maps irreducible subsets to irreducible subsets(see [H, Prop 1.3A]). Also, if X and Y are irreducible, so is X × Y (see [H, Prop1.4]).

Also recall that a topological space X is connected if it cannot be written as adisjoint union UtV of non-empty closed subsets U, V . It is clear that an irreduciblevariety is connected, while the converse is not true in general. However, for algebraicgroups the converse does hold. More specifically:

Proposition 1.8. Let G be a linear algebraic group.

(i) The irreducible components of G are pairwise disjoint, and so are the con-nected components of G.

(ii) The irreducible component G◦ containing the identity is a closed normalsubgroup of finite index.

(iii) Any closed subgroup of G of finite index contains G◦.

Proof. (i) Let X, Y be irreducible components of G. Suppose X ∩ Y 6= ∅. Pickg ∈ X ∩ Y . Since multiplication by g−1 is an isomorphism of varieties, we knowthat g−1X and g−1Y are irreducible components, and we have 1 ∈ g−1X ∩ g−1Y .So without loss of generality (wlog), we may assume that 1 ∈ X ∩ Y . Now, sinceX ×Y is irreducible in G×G, it follows that µ(X ×Y ) = X ·Y is irreducible in G.Moreover, we have X ⊆ X · Y since 1 ∈ Y . By maximality of X, it must be thatX = X · Y , and similarly, we obtain Y = X · Y = X.

(ii) Since inversion is an isomorphism of varieties, (G◦)−1 is an irreducible com-ponent of G. As it contains 1, it must be G◦ by (i). Similarly, for h ∈ G◦, multipli-cation by h is an isomorphism of varieties and so hG◦ is an irreducible component,and it contains 1 since G◦ is closed under inverses. Therefore hG◦ = G◦. It fol-lows that for any g, h ∈ G◦, gh ∈ G◦. Hence G◦ is a subgroup. Also, for g ∈ G,conjugation by g is an isomorphism of varieties, so gG◦g−1 is again an irreduciblecomponent containing 1, and so it equals G◦. Thus G◦ is a normal subgroup.

For the last part, let X be an irreducible component of G. Pick g ∈ X. Wehave that g−1X is an irreducible component of G containing 1, and so it equalsG◦. Hence X = gG◦ and so all the irreducible components of G are cosets of G◦.It is clear that all cosets of G◦ are irreducible components, so since there are onlyfinitely many irreducible components of G, G◦ must have finite index.

(iii) Let H ≤ G be a closed subgroup of finite index. Then H◦ ≤ G◦ ≤ G andwe have [G : H◦] = [G : H] · [H : H◦], which is finite by (ii). Hence we can writeG◦ =

⊔gH◦, a finite disjoint union of cosets of H◦. Since G◦ is connected, it

follows that G◦ = H◦ ≤ H. �

We will therefore refer to the irreducible (or connected) components of G asthe components of G. An immediate consequence of the above proposition is thatϕ(G◦) = ϕ(G)◦ for any morphism of algebraic groups ϕ : G → H. Indeed, ϕ(G◦)is closed (by Corollary 1.6), connected (since G◦ is connected), contains 1 and hasfinite index in ϕ(G) by Proposition 1.8(ii) applied to G. The result follows byProposition 1.8(iii).

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Example 1.9. Let’s see which of our examples are connected. It is a well knownresult in algebraic geometry that a variety X is irreducible if and only if its ringof regular function k[X] is an integral domain. Therefore, we know that Ga, Gmand GLn are connected algebraic groups since their ring of regular functions wereintegral domains. Also, Dn is connected since it is a direct product of connectedalgebraic groups (namely n copies of Gm). On the other hand, it can be shownthat GOn is not connected (assuming char(k) 6= 2), with component at the identitySOn = GOn ∩ SLn, called the special orthogonal group.

It is also true that SLn, Tn and Un are connected. In order to show this, weneed a geometrical result (see [H, Proposition 7.5]):

Proposition 1.10. Let G be a linear algebraic group and fi : Xi → G, i ∈ I, afamily of morphisms from irreducible varieties Xi, such that 1 ∈ Yi = fi(Xi) forall i ∈ I. Then H = 〈Yi : i ∈ I〉 is a closed, connected subgroup of G. Moreover,for some finite sequence i1, . . . , in in I, H = Y ±1

i1· · ·Y ±1

in. In particular, if G is

generated by some family of closed connected subgroups, then it is connected.

We know from linear algebra that SLn is generated by subgroups

Uij = {(akl) ∈ GLn : akk = 1, akl = 0 for (k, l) 6= (i, j)} (i 6= j)

of matrices with 1’s on the diagonal, arbitrary entry in the (i, j) position, and 0elsewhere. Similarly, Un is generated by the subgroups Uij for i < j. These sub-groups are all isomorphic to Ga, which is connected, so the above proposition givesus that SLn and Un are connected. Similarly, one can show that Tn is connected.

The above proposition has another useful consequence:

Proposition 1.11. Let H,K be subgroups of a linear algebraic group G, with Kclosed and connected. Then [H,K] is closed and connected.

Proof. For h ∈ H, define ϕh : K → G by g 7→ [h, g]. It is clearly a morphism, beinga composition of multiplication and inversion. Also, 1 = ϕh(1) for all h. Hence,we have that [H,K] = 〈ϕh(K) : h ∈ H〉 is closed and connected by Proposition1.10. �

Therefore closed connected subgroups behave well under taking commutators.In particular, if G is a connected linear algebraic group, then its derived subgroupG′ = [G,G] is a closed connected subgroup. Inductively, we then see that its nthderived subgroup is a closed, connected subgroup. With this in mind, we recall agroup-theoretic definition:

Definition 1.12. For a group G, define G(0) = G and G(i) = [G(i−1), G(i−1)] fori ≥ 1. We then obtain the derived series of G:

G = G(0) ≥ G(1) ≥ G(2) ≥ . . .

We say G is solvable (or soluble) if G(d) = 1 for some d. The smallest such d isthen called the derived length of G.

Similarly, one can define C0G = G and CiG = [Ci−1G,G] for i ≥ 1. We thendefine G to be nilpotent if CnG = 1 for some n.

Remark. If G is a nilpotent group and n is the largest integer such that CnG 6= 1,then CnG must commute with G so in particular Z(G) 6= 1.

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Example 1.13. Some of the algebraic groups we met are solvable. Indeed, Ga,Gm and Dn are obviously solvable since they are abelian. Also, Tn is solvable andUn is nilpotent. This can be shown by a direct calculation. It is easy to see that

T(1)n ⊂ Un and actually, they are equal (one can find generators of Un which are

commutators of elements of Tn). It’s then not too hard to see that

CmUn = {(aij) ∈ Un : aij = 0 for 0 < i− j ≤ m}

and so Un is nilpotent, and thus solvable. This then implies Tn is solvable.Ga, Gm, Dn, Tn and Un are all examples of connected solvable linear algebraic

groups. We will see later that such linear algebraic groups have a nice structure.

2. Background in algebraic geometry

Most of the results in this section will be stated without proof. Proofs can befound in [H, sections 1-6]. We need to recall a few facts from algebraic geometrywhich will be needed later on. In the theory of algebraic groups, we need to knowabout tangent spaces, projective varieties, complete varieties or the dimension ofvarieties.

Definition 2.1. For an affine variety X, we define the tangent space of X at x ∈ Xby

Tx(X) = {δ : k[X]→ k linear : δ(fg) = f(x)δ(g) + δ(f)g(x) for f, g ∈ k[X]}

the k-vector space of point derivations at x.

Having defined the tangent space, we can now define the differential of a mor-phism:

Definition 2.2. Let ϕ : X → Y be a morphism of affine varieties. The differentialdxϕ of ϕ at x ∈ X is the map dxϕ : Tx(X)→ Tϕ(x)(Y ) defined by dxϕ(δ) = δ ◦ ϕ∗for δ ∈ Tx(X).

Taking differentials behave functorially (see [H, 5.4]):

Proposition 2.3. Let ϕ : X → Y and ψ : Y → Z be morphisms of affine varieties,and x ∈ C. Then dx(ψ ◦ ϕ)x = dϕ(x)ψ ◦ dxϕ.

In Section 4, we will consider actions of algebraic groups on varieties other thanjust affine varieties. To this end, we recall facts about projective varieties:

Definition 2.4. Projective n-space Pn is defined to be the set of equivalence classesof kn+1 \ {0, 0, . . . , 0} relative to the equivalence relation

{x0, x1, . . . , xn} ∼ {y0, y1, . . . , yn} ⇐⇒ ∃λ ∈ k∗ such that yi = λxi for all i

For a k-vector space V of dimension n + 1, we can identify Pn with the set of all1-dimensional subspaces of V , usually denoted by P(V ).

We can define a Zariski topology on Pn: define the closed sets to be the sets givenby the vanishing of some collection of homogeneous polynomials in k[T0, . . . , Tn]. Aprojective variety is a closed subset of some Pn, equipped with the induced topology.A quasi-projective variety is an open subset of a projective variety. Projectivevarieties are clearly quasi-projective.

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In general, a variety is defined to be a pair (X,OX), where X is a topologicalspace and OX is a sheaf of functions on X, such that X has a finite open cover

⋃Ui

with each (Ui,OX |Ui) isomorphic to an affine variety. Moreover we also require thatthe diagonal ∆ = {(x, x) : x ∈ X} is closed in X ×X. Morphisms are then definedto be continuous maps which preserve the sheaf of functions. For more details see[H, section 2]. In practice, our varieties will always be affine or quasi-projective.

A useful example of a projective variety is the flag variety of a finite dimensionalvector space V . A flag of V is a chain 0 = V0 ⊂ V1 ⊂ . . . ⊂ Vk = V where allinclusions are strict. A full flag is one where dimVi+1 = dimVi + 1 for all i, i.e onewhere k = dimV . The flag variety is defined to be the set of all full flags of V . Itcan indeed be given the structure of a projective variety (see [H, 1.8]).

We now recall the notion of dimension:

Definition 2.5. For an irreducible variety X, its ring of regular function k[X] isan integral domain, so we can take k(X) to be its field of fractions. We define thedimension of X to be the the transcendence degree of k(X) over k. Equivalently, itis the maximal length of a chain of prime ideals in k[X]. In general, for a reduciblevariety X, we define dimX = max{dimXi : 1 ≤ i ≤ r} where the Xi are theirreducible components of X.

Note that for a linear algebraic group G, dimG = dimG◦ since the componentsof G are the cosets of G◦, which are all isomorphic as varieties to G◦ and so all havethe same dimension. In particular dimG = 0 if and only if G is finite: a connectedspace of dimension 0 is just a point, so G◦ = 1 and since G is a union of finitelymany cosets of G◦, it is finite.

The following proposition shows how dimension behaves well with respect tomorphisms (see [H, Theorem 4.3]):

Proposition 2.6. Let ϕ : X → Y be a morphism of irreducible varieties with ϕ(X)dense in Y . Then there exists a non-empty open subset U ⊆ Y with U ⊆ ϕ(X)such that

dimϕ−1(y) = dimX − dimY for all y ∈ U

We deduce a ‘rank-nullity’ result for morphisms of algebraic groups:

Corollary 2.7. Let ϕ : G1 → G2 be a morphism of linear algebraic groups. Then

dimϕ(G1) + dim kerϕ = dimG1

Proof. Every fiber ϕ−1(y) is a coset of kerϕ, and thus has the same dimension.Apply Proposition 2.6 with X = G◦1 and Y = ϕ(G1)◦. �

Example 2.8. We find the dimension of some algebraic groups:

(1) dimGa = dimGm = 1 since clearly k(Ga) = k(Gm) = k(T ). An easyinductive argument using Corollary 2.7 shows that dimDn = n for all n.

(2) dim GLn = n2 since the field of fractions of k[Tij : 1 ≤ i, j ≤ n]detTij isk(Tij : 1 ≤ i, j ≤ n).

(3) dim SLn = n2−1 using the previous two examples and Corollary 2.7 appliedto the surjective morphism det : GLn → Gm.

We conclude our discussion of dimension with the following result, which will beuseful for inductive arguments:

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Proposition 2.9. If Y is a proper, closed subset of an irreducible variety X, thendimY < dimX.

Proof. Let Y1 ⊆ Y be an irreducible component. The inclusion ϕ : Y1 ↪→ X inducesa surjective k-algebra homomorphism ϕ∗ : k[X] → k[Y1]. Since Y1 is irreducible,k[Y1] is an integral domain and so kerϕ∗ is a prime ideal in k[X], non-zero sinceY1 ⊂ X is proper. Then any chain of prime ideals in k[Y1] lifts to a chain of primeideals in k[X] through kerϕ∗, hence of greater length since X is irreducible and so0 is a prime ideal in the integral domain k[X]. �

We now move on to complete varieties. These will be useful when we will studyactions of algebraic groups on projective varieties.

Definition 2.10. A variety X is complete if, for any variety Y , the projectionmorphism X × Y → Y is a closed map, i.e maps closed sets to closed sets. Clearly,a closed subvariety of a complete variety is complete.

We summarize all the results we’ll need in the following proposition (see [H,section 6]):

Proposition 2.11. (i) A projective variety is complete.(ii) A complete quasi-projective variety is projective.(iii) A complete affine variety has dimension 0.(iv) If ϕ : X → Y is a morphism of varieties and X is complete, then ϕ(X) is

closed in Y , and complete.

3. Tori, Unipotent and Connected Solvable Groups

Having established some basic facts about linear algebraic groups, a first questionwe could ask is the following: what do one-dimensional linear algebraic groups looklike? It turns out that any one-dimensional connected linear algebraic group isisomorphic to either Ga or Gm. This seemingly innocent result is actually quitedifficult to show and we will not prove it here (see [H, section 20] for a proof). Thistells us that one dimensional connected groups are all abelian, and so solvable.Connected solvable groups are quite important to the general theory, as we shallsee later, and so we study them in this section. We first start by two particularexamples of such groups: unipotent groups and tori.

3.1. Unipotent Groups. Recall the additive Jordan decomposition for endomor-phisms: if V is a finite dimensional k-vector space and α ∈ End(V ), then thereexists unique s, n ∈ End(V ) such that s is semisimple, i.e diagonalisable, n is nilpo-tent, α = s + n and sn = ns. Moreover s and n are both polynomials in α withconstant coefficient equal to zero.

Definition 3.1. An endomorphism u ∈ End(V ) is unipotent if u− 1 is nilpotent.Equivalently, u is unipotent if the only eigenvalue of u is 1.

There is also a multiplicative version of the Jordan decomposition:

Proposition 3.2. For g ∈ GL(V ), there exists s, u ∈ GL(V ) such that g = su =us, where u is unipotent and s is semisimple.

Proof. From the additive decomposition, we can write g = s + n with s, n asdescribed above. Since g is invertible, so is s and so we may define u = 1 + s−1n.

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As n is nilpotent and sn = ns, we have that u− 1 = s−1n is nilpotent. Therefore uis unipotent and su = s+ n = g. If g = su = us is any such decomposition, whereu = 1 + n with n nilpotent and commuting with s, then g = s + sn is the uniqueadditive Jordan decomposition, and u, s are therefore uniquely determined. �

We can then transfer this definition to an arbitrary linear algebraic group:

Theorem 3.3. (Jordan decomposition) Let G be a linear algebraic group.

(i) For any embedding ρ of G into some GL(V ) and for any g ∈ G, there existsunique gs, gu ∈ G such that g = gugs = gsgu, where ρ(gs) is semisimpleand ρ(gu) is unipotent.

(ii) The decomposition g = gugs = gsgu is independent of the chosen embedding.(iii) Let ϕ : G1 → G2 be a morphism of algebraic groups. Then ϕ(gs) = ϕ(g)s

and ϕ(gu) = ϕ(g)u.

We won’t give a proof of this result (see [H, Theorem 15.3]), but we give here animportant step which we will need later on. Given a linear algebraic group G, eachx ∈ G defines a morphism G → G given by g 7→ gx, which induces a k[G]-algebrahomomorphism ρx : k[G]→ k[G] defined by ρx(f)(g) = f(gx) for f ∈ k[G], g ∈ G.This defines an action of G on k[G].

Proposition 3.4. Let G be a linear algebraic group and V a finite dimensionalsubspace of k[G]. Then there exists a finite dimensional G-invariant subspace Xcontaining V . In particular, k[G] is a union of finite dimensional G-invariantsubspaces. Moreover, the restriction of any such finite dimensional subspace Xaffords a morphism of algebraic groups ρ : G→ GL(X).

Proof. It’s enough to prove this for V = 〈f〉, a one-dimensional subspace. Recallthat multiplication gives a morphism µ : G×G→ G and that k[G×G] is isomorphicto k[G]⊗kk[G]. Therefore write µ∗(f) =

∑i∈I fi⊗gi, so that ρx(f) =

∑i∈I gi(x)fi.

Hence the finite dimensional subspace generated by {fi : i ∈ I} contains ρx(f) forall x ∈ G. It follows that the subspace X generated by {ρx(f) : x ∈ G} is containedin it and so is finite dimensional. It is clearly G-invariant and it contains V so thefirst part follows. For the last part, we see from the above construction that thecoordinates of ρx inX are polynomial functions in x. Therefore the map x 7→ (ρx)|Xaffords a morphism of algebraic groups G→ GL(X). �

An endomorphism x of a vector space V is called locally finite if V is a unionof finite dimensional x-stable subspaces. Proposition 3.4 shows ρx is locally finite.One can show that locally finite endomorphisms have a Jordan decomposition inthe sense of Proposition 3.2. The idea of the proof of Theorem 3.3 is to use theunipotent and semisimple parts of ρg, for g ∈ G, to construct gu and gs.

Definition 3.5. Let G be a linear algebraic group. The decomposition g = gugs =gsgu in Theorem 3.3 is called the Jordan decomposition of g ∈ G and g is calledsemisimple (respectively unipotent) if g = gs (respectively g = gu). We write

Gu = {g ∈ G : g is unipotent}Gs = {g ∈ G : g is semisimple}

for the subsets of unipotent and semisimple elements of G. If G = Gu, then we sayG is a unipotent group. Note that Gu is a closed subset since the set of unipotentelements of GLn is closed, given by the polynomial (T − 1)n.

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Remarks. (i) Observe that the subgroup generated by Gu is a characteristicsubgroup of G where by characteristic, we mean here that it is preservedunder all algebraic group automorphisms. Indeed, if α ∈ Aut(G), take anyembedding ρ of G into some GLn. Then ρ ◦ α is another embedding of Ginto GLn, and by Theorem 3.3(ii), we have that the image of Gu is stillunipotent, thus Gαu ⊆ Gu.

(ii) A group G for which G = Gs is not called semisimple. Semisimple groupshave a different definition (see Section 5).

Example 3.6. It’s clear that Un is unipotent for any n and that for G = Tn,

Gu = Un. Note also that Ga ∼= U2 =

{(1 ∗0 1

)}is a unipotent group. We also

met groups where G = Gs, for example Gm or more generally Dn for n ≥ 1.

As said above, Un and more generally subgroups of Un are unipotent subgroups ofGLn. It turns out that all unipotent subgroups of GLn are conjugate to a subgroupof Un. To get there, we first need the following result:

Theorem 3.7. Let G be a unipotent subgroup of GL(V ) for some non-zero finitedimensional vector space V . Then G has a common eigenvector in V .

Proof. Identify V with kn where n = dimV . We use induction on n. The resultis obvious if dimV = 1 (every v ∈ V is a common eigenvector of G), so assumedimV > 1. Suppose V has a proper non-zero subspace W stable under G. Thenby choosing appropriate bases, we may assume that

G ≤{(

∗ ∗0 ∗

)}More specifically, every element g ∈ G can be written in the form(

ϕ(g) ∗0 ψ(g)

)where ϕ : G → GL(W ) is the canonical restriction morphism, and ψ : G →GL(V/W ). Now ϕ(G) is also unipotent, so by induction hypothesis there exists acommon eigenvector v ∈W ⊂ V for G.

Therefore we may assume that V is an irreducible G-module. We need thefollowing theorem of Burnside (see [L, XVII, section 3]): if R is a subalgebra ofEnd(V ) which acts irreducibly on V , then R = End(V ).

Now, the assumption that G is unipotent implies Tr(x) = Tr(1) = dimV for allx ∈ G. Writing x as 1 + n with n nilpotent, we have for all y ∈ G:

Tr(y) = Tr(xy) = Tr(y + ny) = Tr(y) + Tr(ny).

Therefore Tr(ny) = 0. Now, the k-linear combinations of the elements of G mustalso satisfy this. These form a subalgebra R of End(V ), which acts irreducibly onV since G does. Burnside’s theorem then implies that for all y ∈ End(V ) and forall x = 1 + n ∈ G, Tr(ny) = 0. Taking y to be the standard unit matrices Eij , wesee that we must have n = 0 (by Eij , we mean the matrix whose (i, j)th entry is 1and all other entries are 0). Hence G = 1 and since V is irreducible, dimV = 1, acontradiction. �

Corollary 3.8. If G ≤ GLn is a unipotent group, then G is conjugate to a subgroupof Un. Since Un is nilpotent (see Example 1.13), it follows that G is nilpotent, andso solvable.

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Proof. By Theorem 3.7, G has a common eigenvector v ∈ V = kn. Let V1 = 〈v〉.Then G acts on V/V1, the image of G in GL(V/V1) being again unipotent. Inductionon dimV then allows us to construct a basis of V with respect to which elementsof G are represented by upper triangular matrices. Since they are also unipotent,it follows that these matrices are in Un. �

This result can be seen as a generalisation of the fact that p-groups are nilpotent.Indeed, if char(k) = p > 0, then an endomorphism u is unipotent if and only if

upf

= 1 for some f ≥ 1 since for some large enough f , we require that upf − 1 =

(u− 1)pf

= 0.

3.2. Tori.

Definition 3.9. A torus is a linear algebraic group isomorphic to Dn for somen ≥ 0.

It turns out that an important concept in studying tori is their characters.

Definition 3.10. For G a linear algebraic group, a character of G is a morphismof algebraic groups χ : G → Gm. The set of characters of G is denoted by X(G).Note that it can be considered as a subset of k[G].

A cocharacter of G is a morphism of algebraic groups γ : Gm → G. The set ofcocharacters is denoted by Y (G).

X(G) is clearly an abelian group with respect to

(χ1 + χ2)(g) = χ1(g)χ2(g) for χ1, χ2 ∈ X(G), g ∈ G.

Similarly, if G is commutative then Y (G) is an abelian group with respect to

(γ1 + γ2)(x) = γ1(x)γ2(x) for γ1, γ2 ∈ Y (G), x ∈ Gm.

In particular, since we use the additive notation, we will denote by 0 the charactermapping everything to 1, and similarly for cocharacters.

Given χ ∈ X(G) and γ ∈ Y (G), χ ◦ γ ∈ End(Gm). Now, an element f ∈End(Gm) belongs to k[T, T−1]. If f is of the form aTn for some a ∈ k∗, n ∈ Z,then since f is a group homomorphism, we must have f = Tn as f(1) = 1. If f isnot of this form then we can find n < m in Z such that f(T ) =

∑mi=n aiT

i withai ∈ k, an, am 6= 0. But since f is a group homomorphism, we must have that1 = f(T )f(T−1). Expanding f(T )f(T−1), we see that the coefficient of Tm−n isnon-zero, a contradiction.

Therefore, we just proved that End(Gm) = {t 7→ tj : j ∈ Z} ∼= Z. In particular,for χ ∈ X(G) and γ ∈ Y (G), ∃〈χ, γ〉 ∈ Z such that χ ◦ γ : t 7→ t〈χ,γ〉. This gives usa map 〈 , 〉 : X(G)× Y (G)→ Z.

Note that having established what End(Gm) is, it’s easy to see what the char-acters of Dn are. Indeed, writing elements of Dn as g = diag(t1, . . . , tn), we candefine a character χi : g 7→ ti. It’s quite easy to see that X(Dn) is generated bythe χi, a typical element being of the form χa11 . . . χann for some a1, . . . , an ∈ Z. SoX(Dn) ∼= Zn. Similarly, we see that for γ ∈ Y (Dn), composing with the projectionon the ith diagonal element gives an endomorphism of Gm which is therefore of theform t 7→ tdi . Thus γ is of the form t 7→ diag(td1 , . . . , tdn).

Using this, it can easily be shown that the map 〈 , 〉 : X × Y → Z is a perfectpairing, that is, any homomorphism X → Z is of the form χ 7→ 〈χ, γ〉 for some

16

γ ∈ Y , and any homomorphism Y → Z is of the form γ 7→ 〈χ, γ〉 for some χ ∈ X,where X = X(T ) and Y = Y (T ) for a torus T (see [MT, Prop 3.6]).

Now, given any torus T , we can identify T with Dn for some n and so X(T ) ∼=Zn is a finitely generated abelian group. Therefore it makes sense to talk aboutcharacters being linearly independent.

Definition 3.11. Let T be a torus, H ≤ T a subgroup and X1 ≤ X(T ) a subgroupof the character group. Then we define

H⊥ = {χ ∈ X(T ) : χ(h) = 1 for all h ∈ H},a subgroup of X(T ). We also define

X⊥1 = {t ∈ T : χ(t) = 1 for all χ ∈ X1} =⋂χ∈X1

kerχ,

a closed subgroup of T

It is a fact that given linearly independent characters χ1, . . . , χn ∈ X(T ) andelements c1, . . . , cn ∈ Gm, there exists t ∈ T such that χi(t) = ci for 1 ≤ i ≤ n (see[H, Lemma 16.2C]). Using this, we can show:

Proposition 3.12. Let T be a torus, H ≤ T a subgroup and X1 ≤ X(T ). Then:

(i) If H1 ≤ H is a subgroup of finite index, then H⊥1 /H⊥ is finite.

(ii) (X⊥1 )⊥/X1 is finite.

Proof. (i) By the structure theorem for finitely generated abelian groups, it’s enoughto show that all the elements of H⊥1 /H

⊥ have finite order. Let h1, . . . , hr be a com-plete list of cosets representatives for H1 in H. Then every h ∈ H is equal to higfor some i and g ∈ H1. Take χ ∈ H⊥1 . Then χ takes only finitely many values onH, namely χ(h1), . . . , χ(hr). So χ(H) is a finite subgroup of k∗, say of order n ≥ 1.It follows that nχ ∈ H⊥ as required.

(ii) If (X⊥1 )⊥ = X1 we’re done. Otherwise pick χ ∈ (X⊥1 )⊥ \ X1. Again it’senough to show nχ ∈ X1 for some n ≥ 1. Suppose not and aim for a contradiction.Since X1 ≤ X(T ) ∼= Zn for some n, X1

∼= Zr for some r ≤ n and it has a basisχ1, . . . , χr. Then we have that χ, χ1, . . . , χr are linearly independent and thereforethere exists t ∈ T such that χi(t) = 1 (1 ≤ i ≤ r) but χ(t) 6= 1. Hence t ∈ X⊥1 butt /∈ kerχ, contradicting the assumption that χ ∈ (X⊥1 )⊥. �

Using the properties of characters, one can show (see [H, 16]):

Proposition 3.13. Any closed subgroup of Dn is a torus.

We finally state a result about the “rigidity” of tori (see [H, Corollary 16.3] fora proof):

Theorem 3.14. Let G be a linear algebraic group and T ≤ G a torus. ThenNG(T )◦ = CG(T )◦.

It follows that the quotient NG(T )/CG(T ) is finite since NG(T )◦ ≤ NG(T ) hasfinite index and NG(T )◦ ≤ CG(T ) by the theorem.

As an example, take G = GLn and T = Dn. It’s clear that T = CG(T ) andmoreover NG(T ) = M , the set of monomial matrices (i.e matrices with exactlyone non-zero entry on each row). Then since Dn ≤ M has finite index and isconnected, we must have that Dn = M◦ by Proposition 1.8(c). So indeed we haveNG(T )◦ = CG(T )◦ and here NG(T )/CG(T ) ∼= Sn is a symmetric group.

17

3.3. Connected Solvable Groups. We wish to get a result similar to Theorem3.7 for connected solvable groups. This is the analogue of Lie’s theorem for Liealgebras, except Lie’s theorem only holds in characteristic 0 while here we don’tmake any assumptions on char(k). In the proof we will need the following lemma(see [H, Prop 15.4] for a proof):

Lemma 3.15. Let M ⊂ GLn be a commuting set of matrices, then M is trigonalis-able (i.e we can find a basis with respect to which all elements of M are representedby upper triangular matrices).

Theorem 3.16. (Lie-Kolchin) Let G be a connected solvable subgroup of GL(V ),with V 6= 0 finite dimensional. Then G has a common eigenvector in V , i.e V hasa one-dimensional subspace which G stabilises.

Proof. It is a fact that if G is solvable, then so is G (it’s not difficult to see that

G(i) = G(i)

for all i, see [B, 2.4]), so we may assume that G is closed in GL(V ).Write V = kn for some n ≥ 1. We argue by induction on n and on the derivedlength d of G.

If n = 1, the result is trivial. So suppose n > 1. If d = 1, then G is commutative,and by Lemma 3.15 we have that G has a common eigenvector. So assume d > 1.

Suppose first that there is a proper 0 6= W < V which is stabilised by G. Thenas in the proof of Theorem 3.7, by choosing appropriate bases, we may write anyg ∈ G in the form (

ϕ(g) ∗0 ψ(g)

)where ϕ : G → GL(W ) is the canonical restriction morphism, and ψ : G →GL(V/W ). Now, ϕ(G) is connected solvable and acts on W with dimW < dimV .So by induction hypothesis there is v ∈W < V such that v is a common eigenvectorof G.

The only case left to consider is if G acts irreducibly on V . Assume this holds.We again aim for a contradiction.

Let G′ = [G,G]. It is closed and connected by Proposition 1.11, and obviouslysolvable with derived length d−1. Hence by induction hypothesis there is a commoneigenvector v ∈ V for G′. Note that gv is also a common eigenvector of G′ for anyg ∈ G, since G′ is normal in G: for h ∈ G′, g−1hg ∈ G′, so g−1hgv = λv for someλ ∈ k, and therefore h(gv) = λgv.

Let W denote the non-zero subspace of V spanned by the common eigenvectorsof G′. By the above W is G-invariant. Since G acts irreducibly on V , it followsthat W = V . Hence V has a basis consisting of common eigenvectors of G′. So theelements of G′ are diagonal matrices with respect to that basis, and therefore G′ iscommutative.

Now, for fixed h ∈ G′, all conjugates ghg−1, for g ∈ G, are in G′ and henceare diagonal with the same eigenvalues as h. Therefore, there are only finitelymany possibilities for ghg−1. So let ϕh : G → G′ be the morphism g 7→ ghg−1

(it is a morphism since multiplication and taking inverses are morphisms). Theimage ϕh(G) is finite by the above discussion, and connected since G is connected.Therefore we must have ϕh(G) = {h}, i.e h ∈ Z(G). Hence G′ ≤ Z(G).

Now, every element of Z(G) commutes with G in its action on V , so by Schur’slemma they are represented by scalar multiples of the identity. So elements of G′ arescalar multiples of the identity and they must have determinant 1 since commutators

18

have determinant 1. Therefore there are only finitely many possibilities for elementsof G′ (namely λ.In where λn = 1). As G′ is connected it follows that G′ = 1, andso G is commutative, contradicting d > 1. �

Corollary 3.17. Let G be a connected, solvable subgroup of GLn. Then G is con-jugate to a subgroup of Tn, the linear algebraic group of upper triangular matrices.

Proof. Completely similar to Corollary 3.8. �

We discuss here one application of this result. There is a natural split exactsequence

1 −→ Un −→ Tnπ−→ Dn −→ 1

where π is the morphism t1 ∗. . .

0 tn

7→t1 0

. . .0 tn

Let G ≤ Tn be a closed connected subgroup. The restriction of π to G has kernelGu = G∩Un, a closed normal subgroup. The image T = π(G) is a closed connectedsubgroup of Dn, hence a torus by Proposition 3.13. So we get an exact sequence

1 −→ Gu −→ Gπ−→ T −→ 1

Since T is abelian, it follows that [G,G] ≤ Gu. This proves most of part (i) in thefollowing structure theorem for connected solvable groups (see [H, Theorem 19.3and Prop 19.4] for a full proof):

Theorem 3.18. Let G be a connected solvable linear algebraic group. Then:

(i) Gu is a closed, connected, normal subgroup of G and [G,G] ≤ Gu.(ii) If T is a maximal torus of G, then all maximal tori are conjugate to T and

G = Gu o T . Moreover, NG(T ) = CG(T )

By maximal torus, we mean a subtorus of G which isn’t contained in any othersubtorus. Also, the semidirect product of two algebraic groups G and H is con-structed in the same way as for abstract groups, except we require that G acts as agroup of algebraic group automorphisms of H. Analogously, as for abstract groups,a linear algebraic group G is the semidirect product of the closed subgroups H,Kif H �G, H ∩K = 1 and the product map H oK → G is an isomorphism of linearalgebraic groups. The omitted proof of Theorem 3.18 has the following corollary:

Corollary 3.19. Let G be a connected solvable group. Then any semisimple ele-ment of G lies in a maximal torus.

4. Borel Subgroups

Having studied connected solvable linear algebraic groups, we now consider con-nected solvable subgroups of a linear algebraic group G. This leads to the definitionof a Borel subgroup.

Definition 4.1. A subgroup B ≤ G is called a Borel subgroup if it is a maximalclosed, connected, solvable subgroup.

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Example 4.2. From Corollary 3.17, we see that a connected solvable subgroup ofGLn is conjugate to a subgroup of Tn, and we saw that Tn is a closed, connected,solvable subgroup of GLn. Thus, we see that Tn is a Borel subgroup of GLn.Moreover, if B is a Borel subgroup of GLn, then by the above it is conjugate to asubgroup of Tn. By maximality, this conjugate must equal the whole of Tn. Thusall Borel subgroups of GLn are conjugate.

Clearly, Borel subgroups always exist: just take a closed connected solvablesubgroup of maximal dimension. Our first main tool to study Borel subgroups isthe Borel fixed point theorem, a result about the fixed points in the action of aconnected solvable algebraic groups on a projective variety. We therefore start byconsidering actions of algebraic groups.

4.1. Actions of algebraic groups.

Definition 4.3. Let G be a linear algebraic group, and let X be a variety. We saythat X is a G-space if there exists a group action

G×X −→ X

(g, x) 7−→ g.x

of G on X which is also a morphism of varieties. If the action of G on X is transitive,X is said to be homogeneous.

A morphism ϕ : X → Y is called a morphism of G-spaces if ϕ(g.x) = g.ϕ(x) forall g ∈ G, x ∈ X.

Example 4.4. An easy example of a G-space is G itself. Indeed, the action ofG on itself by conjugation satisfies the above definition. Also, suppose V is afinite dimensional vector space. A rational representation of G is a morphismϕ : G → GL(V ), and V is then called a kG-module. In this situation V is aG-space (we can identify it with the affine variety kn, where n = dimV ) via theaction (g, v) 7→ ϕ(g)v. The associated projective space P(V ) is also a G-space viathe action (g, 〈v〉) 7→ 〈ϕ(g)v〉

Now, we look at some easy consequences of the definition:

Proposition 4.5. Let X be a G-space.

(i) For every x ∈ X the stabiliser Gx = {g ∈ G : g.x = x} is a closed subgroupof G.

(ii) The set XG = {x ∈ X : g.x = x for all g ∈ G} is closed in X.(iii) For Y,Z ⊆ X, with Z closed, the transporter

TranG(Y,Z) = {g ∈ G : g · Y ⊆ Z}

is a closed subset of G.(iv) For a closed subgroup H ≤ G, NG(H) = {g ∈ G : gHg−1 = H} and

CG(H) = {g ∈ G : ghg−1 = h for all h ∈ H} are closed.

Proof. (i) By definition of G-spaces, for x ∈ X the map

ϕx : G −→ X

g 7−→ g.x

is a morphism of varieties, and so Gx = ϕ−1x ({x}) is closed as {x} is closed.

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(ii) For g ∈ G, the map

ψ : X −→ X ×Xx 7−→ (x, g.x)

is a morphism of varieties by definition of G-spaces, and hence the set

Xg = {x ∈ X : g.x = x} = ψ−1(∆)

is closed in X since X is a variety (recall ∆ = {(x, x) : x ∈ X}). Thus XG =⋂g∈GX

g is closed.

(iii) The orbit map ϕx in the proof of (i) being a morphism, and Z being closedin X, we have that

TranG(Y, Z) =⋂y∈Y

ϕ−1y (Z)

is an intersection of closed sets, and therefore is closed.(iv) Let G act on itself by conjugation. Then CG(H) =

⋂h∈H Gh is closed by

(i). Moreover, we have NG(H) ⊆ TranG(H,H). Now, if gHg−1 ⊆ H, then wemust have equality since gHg−1 is a closed subgroup of dimension equal to dimH.Therefore NG(H) = TranG(H,H), which is closed by (iii). �

The next obvious step is to consider orbits. However, in general, these are notnecessarily closed. We can still say some things about them. Firstly, we can observethat if G is irreducible then so is any orbit G.x, x ∈ X, since it is the image ofG under the morphism g 7→ g.x. Secondly, even though orbits are not closed ingeneral, the orbits of minimal dimension are closed:

Proposition 4.6. Let X be a non-empty G-space. Then

(i) Every orbit G.x is open in its closure.(ii) Orbits of minimal dimension are closed.

Proof. (i) The orbit map

G −→ X

g 7−→ g.x

is a morphism of varieties with image G.x, which is constructible by the discussionpreceding Proposition 1.5, so contains an open dense subset Y of its closure. AsG.x is the union of G-translates of Y , the result follows.

(ii) Note that for x ∈ X, g ∈ G, g.G.x is closed and contains G.x. ThereforeG.x ⊆ g.G.x. Similarly G.x ⊆ g−1.G.x. Applying multiplication by g, we see thatG.x = g.G.x. The element g was arbitrary, so we have that G.x is a union ofG-orbits.

Pick x ∈ X such that dimG.x is minimal. Suppose G.x isn’t closed. ThenG.x\G.x is a (non-empty) union of G-orbits, which is closed in G.x by (i). We aimto show it has dimension smaller than dimG.x, which would give a contradiction.If Y ⊆ G.x is an irreducible component intersecting G.x, then (G.x \ G.x) ∩ Y isa proper closed subset of Y , thus of strictly smaller dimension by Proposition 2.9.The result then follows from the definition of dimension. �

When a group acts transitively on some set, we expect this set to correspondto some set of cosets G/H for some H ≤ G. In order to translate this into thecontext of G-spaces, we first need to give G/H the structure of a variety when Gis an algebraic group. We begin by the following theorem of Chevalley:

21

Theorem 4.7. (Chevalley) Let H ≤ G be a closed subgroup. Then there exists arational representation ϕ : G → GL(V ) and a one-dimensional subspace W ≤ Vsuch that H = {g ∈ G : ϕ(g)W = W}.Proof. The ideal I � k[G] of functions vanishing on H is finitely generated (as k[G]is Noetherian), say I = (Fj : j ∈ J) for some finite set J . By Proposition 3.4,there exists a finite dimensional G-invariant subspace X of k[G] containing the Fj ,and a corresponding morphism ρ : G → GL(X). Then M = X ∩ I is H-invariant.Conversely, if x ∈ G is such that ρx(M) = M , then since M generates I, we have

ρx(I) = ρx(M)ρx(k[G]) = Mk[G] = I

and it follows that all functions in I vanish at x, and thus x ∈ H. Therefore wehave H = {x ∈ G : ρx(M) = M}.

If d = dimM , we set V = ∧dX, the dth exterior power of X, with ϕ : G →GL(V ) the rational representation induced by the natural G-action. Then the one-dimensional subspace W = ∧dM is ϕ(H)-invariant. Now assume that ϕ(g)W = Wfor some g ∈ G. Let w1, . . . , wd be a basis for M , and v1, . . . , vd be a basis forρg(M).

By assumption, we have ϕ(g)(w1∧ . . .∧wd) ∈W , but by the choice of the secondbasis it must be a multiple of v1 ∧ . . . ∧ vd. It follows that each vi ∈ M and soρg(M) = M . Hence g ∈ H and W has the required properties. �

An immediate consequence of Theorem 4.7 is the embedding theorem in 1.7:

Corollary 4.8. Any linear algebraic group can be embedded as a closed subgroupof GLn for some n

Proof. Choose H = 1 in Theorem 4.7. We then obtain a faithful (i.e injective)rational representation ρ : G → GL(V ) ∼= GLn where n = dimV , since ker ρ ≤ H.Therefore we have an embedding of G into GLn as a closed subgroup. �

We now give the structure of a variety to a set of cosets G/H. Let H ≤ Gbe a closed subgroup, and V be as in Theorem 4.7. Let v ∈ P(V ) be the pointcorresponding to the line 〈v〉 in V stabilised by H. Set X = G.v ⊆ P(V ). Then Xis a homogeneous G-space and the action gives a surjective map ϕ : G→ X definedby g 7→ g.v, with fibers the cosets of H. This induces a bijection ϕ : G/H → X.

Using this bijection, we endow G/H with the structure of a variety, X being avariety. Indeed, X ⊆ P(V ) is closed so is a projective variety. Also, X being anorbit in a G-space, it is open in its closure by Proposition 4.6(i). Therefore, X isa quasi-projective variety. We call G/H, endowed with its structure of a variety,the quotient space of G by H. By construction, the natural map π : G → G/His a morphism of varieties, and it can be shown that the topology on G/H is thequotient topology obtained from π (see [H,12.1]). Moreover, the variety structureon G/H satisfies a universal property, and so is independent of the chosen rationalrepresentation (again, see [H, 12.1]). By Proposition 2.6 applied to π, we obtain:

Proposition 4.9. Let H ≤ G be a closed subgroup of a linear algebraic group G.Then dimG/H = dimG− dimH.

In the case when H �G, more can be said about the structure of G/H:

Proposition 4.10. Let H be a closed normal subgroup of G, then we can endowG/H with the structure of an affine variety and it is then a linear algebraic groupwith its usual group structure.

22

The way to prove this is to show that there exists a rational representationψ : G → GL(W ) such that H = kerψ. We can then identify G/H with a closedsubgroup of GL(W ), thus a linear algebraic group. The idea is to use Theorem 4.7to obtain a rational representation ϕ : G → GL(V ) and a line L in V stabilisedby H. Since H acts on L by scalar multiplication, there is an associated characterχ0 ∈ X(H). Consider then the sum of all non-zero subspaces Vχ (χ ∈ X(H)),where

Vχ = {v ∈ V : ϕ(h)v = χ(h)v for all h ∈ H}.It can easily be shown (see [H, 11.4]) that this sum is direct (and so only finitelymany of the Vχ are non-zero), and that G permutes the Vχ, H being normal in G.Therefore, we may replace V by this sum. Let W ≤ End(V ) be the subspace ofendomorphisms which stabilise each of the Vχ. The fact that G stabilises each Vχwill produce a rational representation ψ : G→ GL(W ) with the required property(see [H, Theorem 11.5] for the details).

We can now define a new example of a linear algebraic group: taking G = GLnand H = Z(G) = {tIn : t ∈ k∗} ∼= Gm, the projective general linear group PGLn isthe quotient G/H, a linear algebraic group of dimension n2 − 1.

Being able to take quotients by a normal subgroup is a useful tool. For example,it is used in the omitted proof of Theorem 3.18(i). As an illustration of its uses,we show the following elementary result about centers of nilpotent linear algebraicgroups:

Proposition 4.11. Let G be a connected nilpotent linear algebraic group of positivedimension and H ≤ G a proper closed subgroup. Then Z(G) has positive dimensionand dimH < dimNG(H).

Proof. All CiG are connected by Proposition 1.11, so by the remark following Defini-tion 1.12, Z(G) contains a non-trivial connected CnG and so has positive dimension.For the last part, use induction on dimG. If dimG = 1 then by Proposition 2.9,dimH = 0 and so H is finite. But we must also have Z(G) = G since Z(G) is a non-trivial closed subgroup of positive dimension (it is the intersection of the centralisersof all the elements of G, hence closed). Thus G is abelian and NG(H) = G.

So assume dimG > 1 and let Z = Z(G)◦. If Z ⊆ H, then G/Z is connectednilpotent, has smaller dimension than G and is non-trivial, so by induction hypoth-esis the image H/Z has smaller dimension than its normaliser NG(H)/Z. ThusdimH < dimNG(H). If Z is not included in H, then HZ is a subgroup of NG(H)of larger dimension than H. �

4.2. The Borel fixed point theorem and consequences. We finally turn tothe fixed point theorem. In order to show it, we need the following geometricalresult on complete G-spaces (see [H, Lemma 21.1]):

Lemma 4.12. Let X, Y be two irreducible, homogeneous G-spaces and let ϕ : X →Y be a bijective morphism of G-spaces. If Y is complete, then X is complete.

Theorem 4.13. (Borel fixed point theorem) Let G be a connected, solvable linearalgebraic group, acting on a non-empty projective G-space X. Then G has a fixedpoint, i.e ∃x ∈ X such that g.x = x for all g ∈ G.

Proof. We argue by induction on dimG. If dimG = 0 then G = 1 as G is connectedand the result is trivial. So suppose dimG ≥ 1.

23

Let G′ = [G,G]; it is a closed, connected, solvable subgroup of G by Proposition1.11. Since G is solvable, by definition G′ must be a proper subgroup. ThereforedimG′ < dimG and by induction hypothesis there is x ∈ X such that x is fixed byG′. Let Y denote the (non-empty) set of fixed points of G′.

By Proposition 4.5(ii), Y = XG′ is closed and so also projective. Also, G sta-bilises Y since G′ � G. Indeed, for y ∈ Y , g ∈ G, h ∈ G′, we have [h, g]y = y,i.e (h−1g−1hg)y = y, and so h(gy) = g(hy) = gy. Thus gy is fixed by G′ and sobelongs in Y . Therefore, we may replace X by Y .

Now, all stabilisers Gx contain G′ and so are normal subgroups of G. Indeed,Gx/G

′ ≤ G/G′ is normal since G/G′ is abelian.So G/Gx is an affine variety by Proposition 4.10. Also, there exists x ∈ X with

G.x closed in X by Proposition 4.6(ii), and so G.x is projective. Then the canonicalmorphism G/Gx → G.x is bijective, with G/Gx affine and G.x projective. Alsonote that since G is irreducible, so are G/Gx and G.x.

Therefore we can apply Lemma 4.12 (recall that G.x is complete by Proposition2.11(i)) to obtain that G/Gx is complete, and therefore has dimension 0 by Propo-sition 2.11(iii). It must then be that G/Gx = 1 since it’s connected, i.e Gx = G asrequired. �

Remarks. (i) Since a closed subset of a complete variety is also a completevariety, the same result still holds when X is a complete G-space, withexactly the same proof. In practice though, we will only use it in theprojective case.

(ii) This gives a short proof of the Lie-Kolchin theorem: if G ≤ GL(V ) thenG naturally acts on P(V ), which then has the structure of a G-space. If Gis connected solvable, we therefore know it must have a fixed point, i.e Gfixes a line in V . In other words, it has an eigenvector in V .

We now discuss the applications of this theorem. Using a similar argument tothe proof of the Borel fixed point theorem, we obtain that all Borel subgroups of alinear algebraic group are conjugate (something we had already noticed in the caseof GLn):

Proposition 4.14. Let B be a Borel subgroup of a linear algebraic group G.

(i) If G is connected, then G/B is a projective variety.(ii) All other Borel subgroups are conjugate to B.

Proof. (i) Let S be a Borel subgroup of maximal dimension. Embed G into someGL(V ). By Theorem 4.7, there is a one-dimensional subspace V1 whose stabiliserin G is precisely S. Thanks to the Lie-Kolchin theorem, we know that the inducedaction of S on V/V1 is trigonalisable. So there is a full flag 0 = V0 ⊂ V1 ⊂ . . . ⊂Vn = V stabilised by S, call it f . In fact, by choice of V1, S is the stabiliser of theflag f . Therefore the induced morphism from G/S into the orbit of f in the flagvariety is bijective.

But, the stabiliser H of any full flag is solvable. Indeed, if H is the stabiliserof 0 = V0 ⊂ V1 ⊂ . . . ⊂ Vn = V , then H is conjugate to a subgroup of Tn, andso is solvable. Therefore, by considering H◦ we must have that dimH ≤ dimS bychoice of S. Hence f has an orbit of smallest possible dimension. It is thereforeclosed by Proposition 4.6, and so projective. The same argument as in the proof ofTheorem 4.13 then gives that G/S is complete. Since it is quasi-projective, it mustthen be projective by Proposition 2.11(ii).

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Now our Borel subgroup B acts on G/S via h : gS 7→ hgS. By Theorem 4.13,there exists g ∈ G such that BgS = gS, i.e g−1Bg ⊆ S. By maximality of B,we must have equality. So B is conjugate to S (similarly any Borel subgroup isconjugate to S) and the assertion follows.

(ii) As Borel subgroups are connected and contain 1, they all lie in G◦, and theyare clearly Borel subgroups of G◦. The result then follows from the proof of (i). �

Actually, the fact that a quotient G/H is projective, where G is connected, isa property somehow characterised by Borel subgroups. Indeed, if H ≤ G is aclosed subgroup such that G/H is projective, then taking a Borel subgroup B, wesee that it acts on G/H as in the proof of Proposition 4.14. We then get to thesame conclusion that Bg ≤ H for some g ∈ G, and it follows that H contains aBorel subgroup. Conversely, if H is a closed subgroup containing a Borel subgroupB, then we have a surjective morphism G/B → G/H with G/B projective andthus complete, forcing the image G/H to be complete and thus projective (byProposition 2.11(ii) and (iv)). We have therefore proved the following:

Proposition 4.15. Let G be a connected linear algebraic group and H ≤ G a closedsubgroup. Then G/H is projective if and only if H contains a Borel subgroup.

Definition 4.16. A closed subgroup P of a connected linear algebraic group Gis called parabolic if G/P is projective. By Proposition 4.15, a closed subgroup isparabolic if and only if it contains a Borel subgroup.

Example 4.17. Take G = GLn and B = Tn, a Borel subgroup of G. Let v1, . . . , vnbe the standard basis for kn. The stabiliser of any flag of the form 〈v1, . . . , vi(1)〉 ⊂〈v1, . . . , vi(2)〉 ⊂ . . . clearly is a closed subgroup containing B, and so is a parabolicsubgroup. For n = 3, we obtain two parabolic subgroups in this way, consisting ofmatrices of one of the forms * *

0 0 ∗

or

∗ ∗0

*0

We can use Borel subgroups to generalise known facts on connected solvable

groups to arbitrary linear algebraic groups. For instance:

Corollary 4.18. The maximal tori of G are conjugate.

Proof. Let T1, T2 be maximal tori of G. Since a torus is a connected solvablesubgroup, we can find Borel subgroups B1, B2 such that Bi contains Ti (i = 1, 2).By Proposition 4.14(ii), there exists g ∈ G such that Bg1 = B2. So T g1 and T2 aremaximal tori of B2, which is a connected solvable group. By Theorem 3.18(ii), theyare conjugate. �

The proof essentially shows that the maximal tori of G are the maximal tori ofits Borel subgroups: if T ≤ B is a maximal torus of B, and T1 ≤ G is a maximaltorus of G, then some conjugate of T1 is in B, and therefore conjugate to T , forcingT to be a maximal torus of G. We can now define the following:

Definition 4.19. The rank of a linear algebraic group G is the dimension of amaximal torus of G, denoted by rk(G). This is well-defined by Corollary 4.18.

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Example 4.20. (i) For G = GLn, we saw that B = Tn is a Borel subgroup,and T = Dn is obviously a torus. If T1 ≥ T is a maximal torus, then T1

is abelian so centralises Dn. But CG(Dn) = Dn (take any diagonal matrixwith distinct eigenvalues, its centraliser will be Dn) so T = T1 is a maximaltorus. Therefore GLn has rank n.

(ii) Similarly, for G = SLn, we have a Borel subgroup B = SLn ∩ Tn and amaximal torus T = SLn ∩Dn, so SLn has rank n− 1.

(iii) For G = SO2n, we have a torus

T =

t1. . .tnt−1n. . .t−11

∈ Dn : ti ∈ k∗

≤ SO2n

isomorphic to Dn, hence of dimension n. So rk(SO2n) ≥ n. If T1 ≥ Tis a maximal torus, then pick s ∈ T with distinct eigenvalues. Since T1 isabelian and contains T , we have T1 ≤ CGL2n

(s) = D2n. On the other hand,it can be easily checked from the definition of SO2n that SO2n ∩D2n = T .Therefore T = T1 is a maximal torus and rk(SO2n) = n.

We can deduce some further results about Borel subgroup. Since a Borel sub-group of G is contained in G◦, and is clearly a Borel subgroup of G◦, we will assumefor the rest of this section that the group G is connected.

Proposition 4.21. Let B ≤ G be a Borel subgroup. Then:

(i) An automorphism of G which fixes B pointwise must be the identity.(ii) Z(G)◦ ⊆ Z(B) ⊆ CG(B) = Z(G)

Proof. (i) Let σ be such an automorphism. Then the morphism ϕ : G→ G definedby x 7→ σ(x)x−1 sends B to 1, so factors through G/B. Since G/B is projective,and so complete, we have that the image of ϕ in G is closed by Proposition 2.11(iv),and so affine. But by Proposition 2.11(iv) it must also be complete, forcing it tohave dimension 0. But ϕ(G) is connected (as G is), so ϕ(G) = 1 as required.

(ii) Z(G)◦ is a closed, connected, solvable subgroup of G so is contained in someBorel subgroup S. By Proposition 4.14(ii), there exists g ∈ G such that Sg = B.But conjugation leaves Z(G)◦ invariant, so it must be contained in B. The firstinclusion follows. The second inclusion is obvious, and so is Z(G) ⊆ CG(B). Finally,for x ∈ CG(B), conjugation by x satisfies the assumptions of (i), and so we musthave x ∈ Z(G). �

Corollary 4.22. If G contains a nilpotent Borel subgroup B (so in particular ifB = Bu or Bs), then G is nilpotent (i.e G = B).

Proof. Argue by induction on dimG. If dimG = 0 the result is trivial. So assumedimG ≥ 1. If B = 1, then by the previous proposition, Z(G) = CG(1) = G. SoG is abelian and therefore nilpotent. Therefore, we may assume dimB > 0. ByProposition 4.11, Z(B) has positive dimension. But we know from the previousproposition that Z(G)◦ ⊆ Z(B) ⊆ Z(G). So we can consider the lower dimen-sional group G/Z(G)◦, with a nilpotent Borel subgroup B/Z(G)◦. By inductionhypothesis, these are equal, which forces G = B. �

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Using this, we obtain a result on centralisers of tori:

Proposition 4.23. Let T ≤ G be a maximal torus, C = CG(T )◦. Then C isnilpotent and T is its unique maximal torus. Moreover C = NG(C)◦.

Proof. First, T is central in C, so by Corollary 4.18, it is the unique maximal torusof C. Now, let B be a Borel subgroup of C, containing T . Then B/T ∼= Bu isunipotent so nilpotent by Theorem 3.18(ii) and Corollary 3.8 (which we apply byfirst embedding B in some GLn). Since T is abelian, it must be that B is nilpotent.So C is nilpotent by Corollary 4.22. For the last part, the inclusion C ⊆ NG(C)◦ isobvious since C is connected. Conversely, T is clearly normal in NG(C)◦ (as it’s theunique maximal torus of C), and so must be central by Theorem 3.14. It followsthat NG(C)◦ ⊆ C. �

Corollary 4.22 can be applied more generally to show that maximal tori are non-trivial unless the group G is unipotent. Indeed, if G is solvable, either G = Guis unipotent or G has a non-trivial maximal torus by Theorem 3.18. If it’s non-solvable, then a maximal torus would be contained in a Borel subgroup B, andwould clearly be a maximal torus of B. If it was trivial, then B would be unipotentby Theorem 3.18, and so nilpotent. Thus G would be nilpotent, contradicting ourassumption that G is non-solvable.

Now, we saw that all Borel subgroups are conjugate. It turns out that given aBorel subgroup B, its conjugates cover G (see [H, 22.2] for a proof):

Theorem 4.24. Let G be a connected linear algebraic group, B ≤ G a Borelsubgroup. Then G =

⋃g∈G g

−1Bg

Note that for G = GLn, B = Tn, Theorem 4.24 follows from the existence ofJordan normal form (which follows from k = k). Note also that since k = k, wehave GLn = SLn · Z(GLn) (given A ∈ GLn, we can pick a scalar matrix whosedeterminant is the inverse of detA), and so this also applies to SLn. An immediateconsequence of Theorem 4.24 is the following:

Corollary 4.25. Let G be a connected linear algebraic group.

(i) Every semisimple element of G lies in a maximal torus.(ii) Every unipotent element of G lies in a closed connected unipotent subgroup.(iii) The maximal closed, connected, unipotent subgroups of G are all conjugate

and are of the form Bu, where B is a Borel subgroup of G.

Proof. (i) Every g ∈ G lies in some Borel subgroup by Theorem 4.24. In particular,if g is semisimple, then it lies in some Borel subgroup B, and so in a maximal torusof it by Corollary 3.19.

(ii) If g is unipotent in G, then g lies in some Borel subgroup B, and morespecifically in Bu, which is closed and connected by Theorem 3.18.

(iii) If U ≤ G is a closed, connected, unipotent subgroup of G, then it must besolvable by Corollary 3.8. So it is contained in a Borel subgroup B, and hence inBu. �

We now return to centralisers of tori. Using the Borel fixed point theorem andTheorem 4.24, we obtain:

Theorem 4.26. Let S be a torus in G. Then CG(S) is connected.

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Sketch of proof. This result can be shown in the case where G is solvable (see [H,Prop 19.4]). So we aim to reduce the problem to the solvable case. To do so, givenx ∈ CG(S), we want to find a Borel subgroup containing both S and x. Then thiswill force x to lie in CB(S)◦ ⊆ CG(S)◦ as required.

By Theorem 4.24, x lies in a conjugate of a given Borel subgroup B. Moreover,x acts on G/B via gB 7→ xgB, so the fixed point set

X = {gB ∈ G/B : g−1xg ∈ B}

is non-empty (as x lies in some conjugate of B) and closed by the proof of Propo-sition 4.5(ii). Hence X is a projective variety (as G/B is projective) and S acts onit by left multiplication. By the Borel fixed point theorem, there is a fixed point,i.e g−1Sg ≤ B for some gB ∈ X. This gives the required Borel subgroup. �

Theorem 4.24 tells us that the conjugates of a Borel subgroup cover G, and so inparticular unless G is solvable, Borel subgroups are not normal subgroups. Indeed,they are even self-normalising. If N = NG(B), then B is clearly a Borel subgroupof N◦. Since B is normal in N◦, it follows that B = N◦ by Theorem 4.24. Thefollowing stronger assertion requires much more work and won’t be proved here (see[H, 23.2] for a proof):

Theorem 4.27. Let B be a Borel subgroup of G. Then B = NG(B).

Note again that this is easy to see for G = GLn, with Borel subgroup Tn. Thistheorem allows us to uncover some properties of parabolic subgroups of G:

Corollary 4.28. Let P be a parabolic subgroup of G. Then P = NG(P ). Inparticular, P is connected.

Proof. By definition, P contains some Borel subgroup B. Actually, B is evidentlya Borel subgroup of P ◦. Let x ∈ NG(P ). Then both B and xBx−1 are Borelsubgroups of P ◦, so they are conjugate by some y ∈ P ◦ (by Proposition 4.14).Hence, xy ∈ NG(B) = B (by Theorem 4.27). But then we have xy ∈ P ◦. Sincey ∈ P ◦, we must have x ∈ P ◦. It follows that P ◦ = P = NG(P ). �

Corollary 4.29. Let P , Q be parabolic subgroups of G, both of which include aBorel subgroup B. If P , Q are conjugate in G, then P = Q.

Proof. Write Q = x−1Px. Then B and x−1Bx are both Borel subgroups of Q, andso are conjugate in Q. Hence there exists y ∈ Q such that Bxy = B. Thereforexy ∈ NG(B) = B ⊆ Q (by the theorem), and again this forces x ∈ Q. It followsthat P = Q. �

This implies that the number of conjugacy classes of parabolic subgroups of Gis the number of parabolic subgroups which contain a given Borel subgroup B. Wewill prove these results again in section 6 for connected reductive groups withoutappealing to Theorem 4.27.

5. Connected reductive groups

In the next section we will investigate the parabolic subgroups of particular kindsof linear algebraic groups, called connected reductive groups. We therefore startby discussing the properties of those groups which will be needed to develop thestructure of their parabolic subgroups.

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5.1. Reductive and semisimple groups. We first observe the following conse-quence of Proposition 1.10: if N,N ′ are closed connected subgroups of a linearalgebraic group G such that NN ′ is a group, then NN ′ = 〈N,N ′〉 is a closed con-nected subgroup of G. In particular, if N,N ′ � G are closed, connected, solvablenormal subgroups, then NN ′ is also a closed, connected, solvable normal subgroup.Thus we may define the following:

Definition 5.1. The maximal closed, connected, solvable normal subgroup of alinear algebraic group G is called the radical R(G) of G.

By Theorem 3.18, we see that R(G)u is a closed, connected, normal unipotentsubgroup of R(G). It actually is a characteristic subgroup of it, and so sinceR(G) � G, we have that R(G)u � G. Since unipotent groups are nilpotent and sosolvable by Corollary 3.8, we have that any closed, connected, unipotent normalsubgroup of G is contained in R(G) and so in R(G)u. Therefore R(G)u is themaximal closed, connected, unipotent normal subgroup of G, called the unipotentradical of G, usually denoted by Ru(G).

Definition 5.2. A linear algebraic group G is called reductive if Ru(G) = 1. It iscalled semisimple if it is connected and R(G) = 1. Clearly, a semisimple group isreductive. Also, if G is connected, G/R(G) is semisimple and G/Ru(G) is reductive.

The radical of G is given by its Borel subgroups:

Proposition 5.3. Let G be connected. Then:

(i) R(G) = (⋂B B)

◦.

(ii) if G is reductive, then R(G) = Z(G)◦.

Proof. (i) R(G) is a connected solvable subgroup, so contained in some Borel sub-group B. If H is another Borel subgroup, then we know B and H are conjugate, sayH = gBg−1. Then R(G) = gR(G)g−1 ⊆ gBg−1 = H since R(G) � G. ThereforeR(G) ⊆

⋂B B and as it is connected, we have R(G) ⊆ (

⋂B B)

◦.

Conversely, let H = (⋂B B)

◦. H is clearly a closed, connected subgroup, and

it’s solvable since it’s a subgroup of some Borel subgroup B. Moreover, H � Gsince any conjugate of H is still contained in all Borel subgroups (as they are allconjugate), and is connected as conjugation is an automorphism of algebraic group,so is contained in H. Thus H is a closed, connected, solvable normal subgroup ofG and so by maximality of R(G), we get the other inclusion.

(ii) Clearly, Z(G)◦ ⊆ R(G) since it’s a closed, connected, solvable normal sub-group. Now, R(G) = R(G)uoT for some maximal torus T by Theorem 3.18. SinceG is reductive, R(G)u = Ru(G) = 1 and so R(G) = T is a torus. Therefore, byTheorem 3.14, the group G/CG(R(G)) = NG(R(G))/CG(R(G)) is finite. Hence,CG(R(G)) contains G◦ = G and so equals G. Thus R(G) is contained in Z(G) andsince it is connected, we have R(G) ⊆ Z(G)◦. �

This proposition tells us that the only difference between connected reductivegroups and semisimple groups is that connected reductive groups may have a non-trivial centre.

Example 5.4. (1) If G is connected solvable, then by Theorem 3.18, Ru(G) =Gu and R(G) = G. So a connected solvable reductive group is a torus anda semisimple solvable group is trivial.

29

(2) G = GLn is reductive. Indeed, let T−n be the Borel subgroup consisting ofinvertible lower triangular matrices. Then, by Proposition 5.3(i), we have

R(G) ≤ Tn ∩ T−n = Dn

and thus Ru(G) = R(G)u = 1. However, G is not semisimple. Indeed,Z(GLn) = {tIn : t ∈ k∗} ∼= Gm is connected so by Proposition 5.3(ii),R(GLn) = Z(GLn) is non trivial. On the other hand, we see that PGLn issemisimple.

(3) The same argument as in (2) shows that G = SLn is reductive. Moreover,

R(G) = Z(G)◦ = ({tIn : t ∈ k∗} ∩ SLn)◦ = ({tIn : tn = 1})◦ = 1

again by Proposition 5.3(ii). Therefore G is semisimple.

5.2. Lie algebras and root systems. We can associate to connected reductivegroups a root system, which will turn out to be essential to their structure. In orderto define this root system, we first need to associate a Lie algebra to a given linearalgebraic group.

Let G be a connected linear algebraic group, and A = k[G] its ring of regularfunctions. A k-linear map D : A → A satisfying D(fg) = fD(g) + D(f)g forall f, g ∈ A is called a derivation of A. We denote by Derk(A) the set of allderivations of A. It is an easy calculation to show that if D1, D2 ∈ Derk(A), thenD1 ◦D2 −D2 ◦D1 = [D1, D2] ∈ Derk(A). So we can give Derk(A) the structure ofa Lie algebra.

Now, for x ∈ G, we have an action λx : A→ A on A = k[G] given by (λx.f)(g) =f(x−1g) for f ∈ A, g ∈ G. We can now give the following definition:

Definition 5.5. The Lie algebra of G is the subspace

Lie(G) = {D ∈ Derk(A) : Dλx = λxD for all x ∈ G}of left invariant derivations of A = k[G], a Lie subalgebra of Derk(A). In thegeneral case where G is not necessarily connected, we define Lie(G) to be Lie(G◦).

There is a second, more geometric definition. Recall the definition of tangentspaces from Section 2. Note that since G acts on itself homogeneously by lefttranslation, the tangent space at any g ∈ G is naturally isomorphic to T1(G). Sowe will consider only T1(G). We can define the following map:

θ : Lie(G) −→ T1(G)

D 7−→ (f 7→ D(f)(1))

This is a k-linear map. It can be shown to be an isomorphism (see [H, Theorem9.1]). So we can use this map to give T1(G) the structure of a Lie algebra. Withthis alternative definition, one can show (see [S, Corollary 4.4.6] and [H, Prop 5.1]):

Theorem 5.6. Let G, G1, G2 be linear algebraic groups. Then:

(i) dimG = dimG◦ = dim (Lie(G))(ii) Lie(G1 ×G2) ∼= Lie(G1)⊕ Lie(G2) as Lie algebras

Example 5.7. We show that Lie(GLn) ∼= gln, the algebra of n × n matrices.Indeed, define

f : gln −→ Derk(k[G])

X 7−→ DX

30

where DX is the derivation defined by DX(Tij) =∑nl=1 TilXlj . A simple calculation

gives that

λg(DXTij) =∑l,m

g−1imTmlXlj = DX(λgTij).

where g−1ij denotes the (i, j)th entry of g−1. Hence DX is left invariant, and the map

is well defined. It’s easy to see that it is a Lie algebra homomorphism, and that itskernel is zero, so it’s injective. By Theorem 5.6(i), dim Lie(GLn) = n2 = dim gln,so the map is surjective and we have a Lie algebra isomorphism.

Now, morphisms of algebraic groups give rise to Lie algebra homomorphismsbetween their Lie algebras in a natural way (see [H, Theorem 9.1]):

Proposition 5.8. Let ϕ : G1 → G2 be a morphism of algebraic groups. Thendϕ : Lie(G1)→ Lie(G2) is a Lie algebra homomorphism.

Having obtained the Lie algebra of a linear algebraic group G, we can thennaturally consider the Lie algebras of its closed subgroups as Lie subalgebras ofLie(G). If H ≤ G is a closed subgroup, defined by an ideal I � k[G], then wehave k[H] = k[G]/I. So, any D ∈ Lie(G) with DI ⊆ I defines in a natural way aderivation of k[H], and similarly any δ ∈ T1(G) with δI = 0 will define an elementof T1(H). We then have (see [H, Lemma 9.4, Corollary 10.4A and Theorem 11.5]):

Theorem 5.9. Let H ≤ G be a closed subgroup with ideal I � k[G]. Then:

(i) Lie(H) = {D ∈ Lie(G) : DI ⊆ I} and T1(H) = {δ ∈ T1(G) : δI = 0}.(ii) If H � G is a closed normal subgroup, then Lie(H) is an ideal of Lie(G).

Moreover, the differential of the projection G→ G/H is the canonical pro-jection onto Lie(G)/Lie(H) and this induces an isomorphism Lie(G/H) ∼=Lie(G)/Lie(H).

It is a standard fact that Lie(SLn) = sln, the Lie subalgebra of gln consisting oftraceless n×n matrices over k (see for example [H, 9.4]). The calculation eventuallycomes down to showing that differentiating the determinant gives you the trace.Similarly, the Lie algebra of Dn is the Lie subalgebra of gln consisting of n × ndiagonal matrices.

One of the uses of the Lie algebra of an algebraic group G is that it allows us todefine a rational representation G → GL(Lie(G)) in the following way: for x ∈ G,define Intx : G → G by Intx(y) = xyx−1. Let Adx = dIntx : Lie(G) → Lie(G).This defines a map Ad : G → GL(Lie(G)) given by x 7→ Adx, called the adjointrepresentation of G.

It can be shown (see [H, Prop 10.3 and Theorem 10.4]) that Ad defines a rationalrepresentation of G and that ad = dAd satisfies ad(X)(Y ) = [X,Y ] for X,Y ∈Lie(G). In the special case of G = GLn, Lie(G) = gln, a direct calculation (see[MT, Example 7.13]) shows that Ad g for GLn is just matrix conjugation by g ongln. Moreover, it’s not hard to see in general that if H ≤ G is closed, then Lie(H),as defined in Theorem 5.9, is AdG(H)-invariant and that AdG |H = AdH on Lie(H).So for any closed subgroup of GLn, Ad is also given by matrix conjugation.

We now apply this to our study of reductive groups. Let G be a non-trivialreductive group, and T ≤ G be a maximal torus. Since G is not unipotent, dimT ≥1 by our discussion following Proposition 4.23. Write g for Lie(G). The image ofT under the adjoint representation AdT ≤ GL(g) is a set of commuting semi-simple elements, so can be simultaneously diagonalised (the proof is contained in

31

[H, Lemma 15.4] along with the one of Lemma 3.15). For χ ∈ X(T ), we write

gχ = {v ∈ g : (Ad t)(v) = χ(t)v for all t ∈ T}

for the common T -eigenspaces. Then g =⊕

χ∈X(T ) gχ. Using this decomposition,

we define

Φ(G) = {χ ∈ X(T ) : χ 6= 0, gχ 6= 0},the set of roots of G with respect to T , which we will usually simply denote by Φ.We also define W = NG(T )/CG(T ), the Weyl group of G with respect to T (whichwe will sometimes denote by WG(T )). We already saw in Theorem 3.14 that thisgroup is finite.

If w = nCG(T ) ∈ W , then we can define tw = n−1tn for t ∈ T (this clearlydoesn’t depend on the choice of n). We can then use this to define actions of theWeyl group on the character group X(T ) and on the cocharacter group Y (T ) inthe following way: for all w ∈W,χ ∈ X(T ), γ ∈ Y (T ), t ∈ T and c ∈ Gm, we define

(w.χ)(t) = χ(tw)

(w.γ)(c) = γ(c)w−1

It’s easy to see that these actions are faithful and are compatible with the pairing〈 , 〉 : X(T )× Y (T )→ Z in the sense that

〈w.χ, γ〉 = 〈χ,w−1.γ〉

Moreover, let w ∈W , say w = nCG(T ), and let α ∈ Φ, v ∈ gα and t ∈ T . We have

(Ad tAdn)(v) = (AdnAd tw)(v)

= Adn(α(tw)v)

= α(tw) Adn(v).

This shows that (Adn)(gα) ⊆ gw.α. In particular, if gα 6= 0 then gw.α 6= 0. Thuswe see that Φ is W -stable.

Example 5.10. We illustrate this with the examples of GLn and SLn. Let G =GLn and T = Dn. We know that the adjoint action of G on gln is by conjugationand that Lie(T ) is the set of diagonal matrices in gln. We let Eij be the standardbasis elements of gln (its (i, j)th entry is 1 and all other entries are 0). We have fori 6= j

Ad (diag(t1, . . . , tn))(Eij) = diag(t1, . . . , tn)Eijdiag(t−11 , . . . , t−1

n ) = tit−1j Eij ,

and so we see that the character χij : T → Gm defined by diag(t1, . . . , tn) 7→ tit−1j

is a root of G. Moreover, gln∼= Lie(T )⊕

⊕i 6=j〈Eij〉 so we have

Φ = {χij : i 6= j}.

All gα, α ∈ Φ, are one dimensional and we already saw in the remark followingTheorem 3.14 that the Weyl group W is isomorphic to the symmetric group Sn.

Similarly, for G = SLn, with maximal torus T = Dn ∩ SLn, we obtain roots χijdefined as above for i 6= j. Indeed, sln is generated by the Eij , i 6= j, and Lie(T ),the set of traceless diagonal matrices. So the above calculations still go throughand we have Φ = {χij : i 6= j}. Note that the gα, α ∈ Φ, are again one-dimensional.

To carry on further, we need the following result (see [H, Theorem 26.1]):

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Theorem 5.11. Let G be a connected linear algebraic group, and T ≤ G a maximaltorus. Then Ru(G) is the identity component of the intersection of the unipotentparts of the Borel subgroups containing T .

We also need the fact that for G connected, S a subtorus of G, then all Borelsubgroups of CG(S) are of the form B ∩ CG(S) for a Borel subgroup B of G (see[H, Corollary 22.4]). We then obtain:

Corollary 5.12. Let G be connected reductive. If S ≤ G is a subtorus, thenCG(S) is also connected reductive. Moreover, if T ≤ G is a maximal torus, thenCG(T ) = T .

Proof. We already know that CG(S) is connected by Theorem 4.26. Let T ≤ Gbe maximal torus containing S. Then T ≤ CG(S) since T is abelian, and by thetheorem applied to H = CG(S) and to G, we have

Ru(H) =⋂T≤B

(B ∩H)u ≤⋂T≤B

Bu = Ru(G) = 1

since G is reductive. Thus CG(S) is reductive.Finally, putting S = T , we see that CG(T ) is connected reductive. But it’s

also nilpotent by Proposition 4.23. Then by Example 5.4(1), we have that T =CG(T ). �

Therefore we see that the Weyl group is equal to NG(T )/T . Now let α ∈ Φ.Define Tα = (kerα)◦ ≤ T , a subtorus of T of codimension 1 (by rank-nullity appliedto α). Also define Cα = CG(Tα), a connected reductive group by Corollary 5.12,and Gα = [Cα, Cα], which can be shown to be a dimension 3 semisimple group ofrank 1 isomorphic to either SL2 or PGL2 (see [H, Corollary 25.3] and [S, Theorem7.2.4])

It is a fact that G = 〈Cα | α ∈ Φ〉, and that the Cα are non-solvable. Moreoverwe have that Lie(Cα) =Lie(T )⊕gα⊕g−α with dim gα = 1 (see [MT, Prop 8.15 and8.16]). Using this, one can obtain the following structure theorem for connectedreductive groups (see [MT, Theorem 8.17 and Corollary 8.22] for a proof):

Theorem 5.13. Let G be connected reductive, T ≤ G a maximal torus, g =Lie(G)and Φ = Φ(G). Then:

(1) g =Lie(T )⊕⊕

α∈Φ gα with dim gα = 1 for all α ∈ Φ.(2) dimG = dim Lie(G) = |Φ|+ rk(G).(3) For all α ∈ Φ, there exists a morphism of algebraic groups uα : Ga → G

inducing an isomorphism onto Uα = im(uα). Moreover, Lie(Uα) = gα andwe have tuα(c)t−1 = uα(α(t)c) for all t ∈ T and c ∈ Ga. Then Uα is theunique one-dimensional connected unipotent subgroup of G normalised byT such that Lie(Uα) = gα.

(4) For w = nT ∈W , we have nUαn−1 = Uw.α.

(5) Gα = 〈Uα, U−α〉.(6) G = 〈T,Uα | α ∈ Φ〉.(7) Z(G) =

⋂α∈Φ kerα.

(8) G = [G,G]R(G) = [G,G]Z(G)◦

The one-dimensional subgroups Uα are called the root subgroups of G. For eachα ∈ Φ, since Tα lies in the centre of Cα, it is a normal subgroup of it and so we canform the quotient Cα = Cα/Tα. The canonical projection Cα → Cα then induces

33

a bijection WCα(T ) ∼= WCα(T/Tα). Now Cα is a rank 1 non-solvable connected

linear algebraic group, and it can be shown that such groups have Weyl group oforder 2 (see [H, Theorem 25.3]). Denote by sα the unique non-trivial element ofthat Weyl group (which has order 2). Since CG(T ) = CCα(T ) = T by Corollary5.12, we can naturally view WCα(T ) as a subgroup of W , and so sα as an elementof W .

Example 5.14. If we take G = SL2, then we have

sl2 = (sl2)0 ⊕ (sl2)α ⊕ (sl2)−α

where α :

(c

c−1

)7→ c2 following Example 5.10. Its Weyl group has order 2, with

sα being represented by the matrix

(0 1−1 0

). Here, Uα = U2, U−α =

{(1 0∗ 1

)}and indeed, SL2 is generated by these.

Similarly, if we take G = PGL2, then G has a maximal torus T = D2 containedin a Borel subgroup B = T2, where we write X for the image of some X ⊆ GL2 viathe projection GL2 → PGL2. It can easily be shown that for char(k) 6= 2, PGL2

has its Lie algebra isomorphic to sl2. We then get the same decomposition of sl2 as

above, with α :

(c

1

)7→ c (which corresponds to our χ12 earlier), Uα = U2 and

sα having preimage

(0 11 0

).

Now, furthermore, for each α ∈ Φ, there is a unique α∨ ∈ Y (T ) such thatsα.χ = χ − 〈χ, α∨〉α for all χ ∈ X(T ). We then have 〈α, α∨〉 = 2 and α∨ iscalled the coroot corresponding to α (see [MT, Lemma 8.19]). The elements sα arereflections on the real vector space X(T ) ⊗Z R, which we usually denote by ER,that is to say they have an eigenvector with eigenvalue -1 (namely α) and theyfix a hyperplane of ER pointwise (namely the kernel of χ 7→ 〈χ, α∨〉). Moreover,W = 〈sα | α ∈ Φ〉 (this is proved for example in [S, Theorem 8.2.8]). All these factslead to the definition of a root system:

Definition 5.15. A subset Φ of a finite-dimensional real vector space E is calledan (abstract) root system if:

(R1) Φ is finite, 0 /∈ Φ, 〈Φ〉 = E.(R2) if c ∈ R is such that α, cα ∈ Φ, then c = ±1.(R3) for each α ∈ Φ there exists a reflection sα ∈ GL(E) along α stabilising Φ.(R4) for α, β ∈ Φ, sα.β − β is an integral multiple of α.

The group W = 〈sα|α ∈ Φ〉 is called the Weyl group of Φ. The dimension of E iscalled the rank of Φ.

Since Φ is finite, generates E and is stable under W , it follows that the Weylgroup of a root system is always finite. For a connected reductive group G, its setof roots Φ forms a root system of 〈Φ〉R when viewed as a subset of ER, with Weylgroup W . Indeed, (R1) and (R3) are clear from what we’ve already said. (R2)follows from the fact that the only roots of Cα with respect to T are ±α. For aproof of (R4), see [MT, Lemma 15.4 and Example 15.5].

We now recall a few standard facts about abstract root systems (see [MT, Ap-pendix A] for proofs). A subset ∆ ⊆ Φ is called a base of Φ if it is a vector space

34

basis of E such that any β ∈ Φ can be written as∑α∈∆ cαα, where cα ∈ Z for all

α ∈ ∆ and either all cα ≥ 0 or all cα ≤ 0. Then the set

Φ+ = {β ∈ Φ : β =∑α∈∆

cαα with cα ≥ 0}

is called the system of positive roots of Φ with respect to ∆. Also define Φ− = −Φ+.

Proposition 5.16. Any abstract root system Φ has a base, and the Weyl group Wacts simply-transitively on the set of bases of Φ. Moreover, given a base ∆, we haveW = 〈sα | α ∈ ∆〉 and for every α ∈ Φ, there exists w ∈W such that wα ∈ ∆.

So we see that a root system can be recovered from a base. Therefore a rootsystem is completely determined by a base. Now, returning to connected reductivegroups, we see that a Borel subgroup B determines a base in the following way:

Theorem 5.17. Let G be connected reductive, T ≤ G a maximal torus, containedin some Borel subgroup B. Then:

(i) There exists a base ∆ of Φ such that

B = T ·∏α∈Φ+

Uα

for any fixed order of the factors Uα. Moreover, we have uniqueness of theexpression with respect to the product in the fixed order.

(ii) G = 〈T,Uα | α ∈ ±∆〉

Sketch of proof. For (i), see [S, Prop 8.2.1]. For the proof of (ii), note that Gα =[Cα, Cα] = 〈Uα, U−α〉 contains a preimage of sα. Indeed, Gα is normalised by Tby Theorem 5.13(3) and (5), so GαT is a group. It is therefore equal to 〈T,Gα〉,a closed connected subgroup by Proposition 1.10. Since GαT contains U±α and T ,its Lie algebra must be equal to Lie(Cα) by the discussion preceding Theorem 5.13.Therefore, by Theorem 5.6(i), we have that dimGαT = dimCα and so Cα = GαT .It follows that if nα ∈ NCα(T ) is a preimage for sα, then we can write it as gt withg ∈ Gα and t ∈ T , and we see that g is also a preimage of sα.

Thus we have that H = 〈Uα | α ∈ ±∆〉 contains preimages of all w ∈ W sinceW = 〈sα | α ∈ ∆〉. Hence Uβ ≤ H for all β ∈ Φ by Proposition 5.16 and Theorem5.13(4), and the result follows by Theorem 5.13(6). �

Definition 5.18. The base ∆ obtained in Theorem 5.17 is called the set of simpleroots with respect to T ≤ B, and the elements sα, α ∈ ∆, are called simplereflections.

Example 5.19. Carrying on with our example of SLn, with maximal torus T =Dn ∩ SLn, we had Φ = {χij : i 6= j} where χij : diag(t1, . . . , tn) 7→ tit

−1j . A base of

Φ is given by

∆ = {χi,i+1 : 1 ≤ i < n}since we have, in additive notation, χij = χi,i+1 + χi+1,i+2 + . . . χj−1,j for i < jand χij = −χji for i > j. So Φ+ = {χij : i < j}. Moreover, for i 6= j, we haveUχij = Uij where

Uij = In + 〈Eij〉since Uij is clearly an isomorphic copy of Ga so is a one-dimensional closed con-nected unipotent subgroup of SLn, with T acting on it in the way described in

35

Theorem 15.3(3). Let B = Tn ∩ SLn, a Borel subgroup containing T . Then clearlyB = T ·

∏i<j Uij . So we indeed have

B = T ·∏α∈Φ+

Uα

as in the theorem. Finally, the algorithm of Gaussian elimination show that SLn isgenerated by the Uij , i 6= j, and diagonal matrices.

We now use Theorem 5.17 to understand the structure of B some more. First,fix isomorphisms uα : Ga → Uα for all α ∈ Φ.

Proposition 5.20. Let G be connected reductive, B ≥ T a Borel subgroup, and letU ≤ Ru(B) be a T -stable subgroup. Then U is the product of the root subgroups Uαthat it contains, and so is closed and connected. In particular, the Uα, α ∈ Φ+, arethe minimal non-trivial T -stable subgroups of Ru(B).

Proof. First note that if uα(c) ∈ U for some α ∈ Φ+ and c 6= 0, then tuα(c)t−1 =uα(α(t)c) ∈ U for all t ∈ T , and so Uα ≤ U since α 6= 0. Also observe that since U isunipotent, it is nilpotent and so U ′ = [U,U ] is a T -stable closed, connected subgroupof smaller dimension. Therefore we can argue by induction on the dimension of U .If U has dimension 0 then the result is trivial.

Suppose U has dimension at least 1. By induction hypothesis, U ′ is the productof the Uα it contains. Assume the result fails. By Theorem 5.17(i), there exists1 6= u ∈ U of the form u =

∏α∈M uα(cα) where M ⊆ Φ+ such that Uα * U for all

α ∈ M . Take u such that |M | is minimal. By the above, we must have |M | > 1.Take distinct β, γ ∈M . Then β and γ are linearly independent (since they’re bothin Φ+) and so we must have kerβ 6= ker γ. Indeed, if we had kerβ = ker γ then γand β would give two different isomorphisms T/ kerβ ∼= Gm, and composing thosewould give an endomorphism of Gm, namely the one mapping β(t) 7→ γ(t) for allt ∈ T . Since we know endomorphisms of Gm are given by raising to some integerpower, this would give a linear dependence between β and γ.

Thus ∃t ∈ T with, say, β(t) = 1 but γ(t) 6= 1. We then have, modulo U ′,

tut−1u−1 ≡∏α∈M

uα(α(t)cα) ·∏α∈M

uα(−cα) ≡∏α∈M

uα((α(t)− 1)cα)

which does not involve uβ , but which does still involve a non-trivial element ofUγ . So we have a non-trivial product of smaller length, contradicting our choice ofM . �

Corollary 5.21. Let G be connected reductive with maximal torus T and H ≤ Ga connected reductive subgroup normalised by T . Then H = 〈T ∩H,Uα | Uα ≤ H〉.

Proof. Note that HT is a connected reductive subgroup of G. Let BH be a Borelsubgroup of HT containing T . It is contained in some Borel subgroup B of G. LetU = Ru(BH). By Proposition 5.20, U =

∏α∈M Uα for some subset M ⊆ Φ. Then,

by Theorem 5.17, HT = 〈T,Uα | α ∈ ±M〉. Now, by Theorem 5.13(8), we haveU ≤ [HT,HT ] ≤ H and the result follows. �

We also have the following result which describes how elements inRu(B) multiply(see [H, Lemma 32.5]):

36

Proposition 5.22. (Commutator formula) Given a connected reductive group Gwith root system Φ, take a fixed total ordering of Φ compatible with addition. Thenthere exists integers cmnαβ such that the morphisms uα can be chosen so that for allroots α 6= β we have

[uα(t), uβ(u)] =∏

m,n>0

umα+nβ(cmnαβ tmun) for all t, u ∈ k

where the product is taken over all integers m,n > 0 such that mα+nβ ∈ Φ, takenaccording to the chosen ordering.

This indeed even allows us to know how elements of B multiply, since we knowhow elements of T multiply with elements of Ru(B) thanks to Theorem 5.13(3).

5.3. Bruhat decomposition. We now discuss how to decompose a connectedreductive group G into double cosets of B. We have from Theorem 5.17 thatRu(B) =

∏α∈Φ+ Uα. For w ∈ W = NG(T )/T , write w for an arbitrarily fixed

preimage of w in NG(T ). Note that the double coset BwB is independent of thechoice of preimage. We first consider how double cosets multiply:

Lemma 5.23. Let α ∈ ∆ be a simple root with corresponding simple reflection s.Then for all w ∈W we have

BwB ·BsB ⊆ BwsB ∪BwB

Proof. It’s easy to check directly that in Gα ∼= SL2 or PGL2, we have

s(Uα \ {1})s ⊆ TUαsUα ⊆ BsBby using Examples 5.14 and 5.19, and hence sUαs ⊆ BsB ∪B - (∗)

Now, by Theorem 15.7, we have B = T ·Uα ·∏β∈Φ+\{α} Uβ . Recall from Theorem

5.13(4) that vUβ v = Uvβ for all v ∈W . Also, observe that sα.β ∈ Φ+ for all α 6= β ∈Φ+. Indeed, write β =

∑γ∈∆ cγγ with cγ ≥ 0 for all γ. Then sα.β = β − 〈β, α∨〉α

so all the coefficients of sα.β in its expansion with respect to ∆ are positive, exceptmaybe for the coefficient of α. If sα.β did belong to Φ−, then this would forcecγ = 0 for all γ 6= α and so β = α, contradicting our assumption on β.

Thus, thanks to these facts, we know that Uβ sB = sUsβB = sB for all β 6= αand so we have that BsB = UαsB. Now assume first that wα ∈ Φ+. Then

BwB ·BsB = BwBsB = BwUαsB = BUwαwsB = BwsB.

On the other hand, if wα ∈ Φ− then wsα = w(−α) = −wα ∈ Φ+. So, writingv = ws, we have by the previous that BvB · BsB = BvsB = BwB. Therefore,putting all these together, we have

BwB ·BsB = BvsBsB ⊆ Bv(BsB ∪B) = BvBsB ∪BvB = BwsB ∪BwBwhere the inclusion follows from (∗). �

We now define an important combinatorial group theoretical definition which isuseful in the study of connected reductive groups:

Definition 5.24. A pair B,N of subgroups of a group G is called a BN-pair if thefollowing holds:

(BN1) G is generated by B and N .(BN2) B ∩N is a normal subgroup of N .(BN3) The group W = N/(B ∩N) is generated by a set S of involutions.

37

(BN4) If s ∈ N is a preimage of s ∈ S under the natural projection N → W andn ∈ N , then BnB ·BsB ⊆ BnsB ∪BnB.

(BN5) For s defined as above, sBs 6= B.

The group W is called the Weyl group of the pair.

It doesn’t come to much of a surprise that using a Borel subgroup B of a con-nected reductive group G, we can obtain a BN-pair for G:

Theorem 5.25. Let G be a connected reductive algebraic group with Borel subgroupB, and let N = NG(T ) for some maximal torus T ≤ B. Then B,N form a BN-pairin G whose Weyl group is the usual Weyl group of G.

Proof. By Theorem 3.18(ii) and Corollary 5.12, we have B ∩ N = NB(T ) =CB(T ) = T , which is normal in N . Thus we have (BN2). Then W = N/(B ∩N)is the usual Weyl group of G, generated by the set S of simple reflections, giv-ing (BN3). Recall that G = 〈T,Uα | α ∈ Φ〉 and B = 〈T,Uα | α ∈ Φ+〉. Forα ∈ Φ+, we have U−α = sαUαsα ≤ 〈B,N〉, and so (BN1) follows. (BN4) is justLemma 5.23. Finally, writing α for the root corresponding to a simple reflections ∈ S, sBs contains sUαs = U−α, which is not contained in B, since else we wouldhave Gα = 〈Uα, U−α〉 ≤ B, contradicting the fact that B is solvable (as Gα isnon-solvable). So we have (BN5). �

Theorem 5.26. (Bruhat decomposition) Let G be a group with a BN-pair. Then

G =⊔w∈W

BwB

for any choice of preimages w ∈ N mapping to w ∈W = N/(B ∩N).

Proof. We first show that BvB ∩ BwB = ∅ if w 6= v. Write v as a product ofminimal length `(v) in the generators from S. Argue by induction on `(v), withwlog `(v) ≤ `(w). If `(v) = 0, then v = 1. If B ∩ BwB 6= ∅ then w ∈ B ∩ N (asw ∈ N) and so w = 1 = v. So assume `(v) > 0 and suppose BvB∩BwB 6= ∅. Thenclearly BvB = BwB. Now, we may write v = v′s where s ∈ S and `(v′) < `(v).Then

Bv′s = Bv ⊆ BvB = BwB

Thus, by (BN4), this gives

Bv′ ⊆ BwBs ⊆ BwsB ∪BwB

So either Bv′B = BwsB or Bv′B = BwB. By induction hypothesis this givesv′ = ws or v′ = w. But `(v) ≤ `(w) so v′ 6= w and we must have v′ = ws.Therefore v = v′s = w.

Now pick x, y ∈⊔w∈W BwB, say x = b1vb2 and y = b′1wb

′2 for bi, b

′i ∈ B and

v, w ∈ W . Then an easy induction on `(w) shows that xy ∈⊔w∈W BwB. Indeed,

if `(w) = 1 this is given by (BN4). If `(w) > 1, writing w = w′s with s ∈ S and

`(w′) < `(w), we obtain by induction hypothesis that b1vb2b′1w′ ∈⊔w∈W BwB, say

it equals bub′ for some b, b′ ∈ B and u ∈ W . So xy = bub′sb′2, which then is in ourdisjoint union by (BN4).

Therefore⊔w∈W BwB is closed under multiplication, and it is clearly closed

under inverses and contains 1, so it is a subgroup. Since it contains B and N , itmust be all of G by (BN1). �

38

Thus we see that connected reductive groups are disjoint unions of double cosetsof one of their Borel subgroups thanks to Theorems 5.25 and 5.26. This result hasan immediate corollary which was not obvious at all until now:

Corollary 5.27. Let G be reductive. Then the intersection of any two Borel sub-groups of G is connected and contains a maximal torus.

Proof. Borel subgroups and tori are contained in G◦, so we may assume G is con-nected. Let B,B′ be Borel subgroups. We know they are conjugate by Proposition4.14, say B′ = gBg−1. Choose a maximal torus T ≤ B with Weyl group W . ByTheorem 5.26, g = b1wb2 for some b1, b2 ∈ B and w ∈ NG(T ). Then

B′ = gBg−1 = b1wb2Bb−12 w−1b−1

1 = b1wBw−1b−1

1 .

Since w ∈ NG(T ), we have wT w−1 = T and so b1wT w−1b−1

1 = b1Tb−11 ≤ B. But

by the above b1Tb−11 = b1wT w

−1b−11 ≤ B′, so we have b1Tb

−11 ≤ B∩B′ as required.

Now, write T ′ = b1Tb−11 . Then B = T ′U where U = Ru(B). We then have

B ∩B′ = T ′H where H = B ∩B′ ∩U , a closed T ′-stable subgroup. By Proposition5.20, H is generated by the Uα it contains and so is connected by Proposition 1.10.Hence, B ∩B′ is also connected thanks to Proposition 1.10. �

Actually, for a connected reductive group G, Theorem 5.26 can be strengthenedin the following way: for any w ∈W , we can define a subgroup

U−w =∏

α∈Φ+,w.α∈Φ−

Uα

(it is a subgroup by the commutator formula), and it can be shown that every g ∈ Gcan be written uniquely as g = uwb where b ∈ B, w ∈W and u ∈ U−w with respectto the positive system Φ+ ⊆ Φ determined by T ≤ B (see [H, Thm 28.3 and 28.4]).

Now, since W acts simply-transitively on the set of bases, there exists a uniquew0 ∈ W such that w0(∆) = −∆ (moreover w2

0 = 1 since it stabilises ∆). We canthen use the strengthened version of 5.26 to show the following:

Lemma 5.28. Bw0 ∩B = T

Proof. Write U = Ru(B). Let g ∈ Bw0 ∩ B, so g = w0utw0 = t′u′ for someu, u′ ∈ U , t, t′ ∈ T following Theorem 5.17. Then

w0−1t′u′ = uw0t

w0 .

By definition of w0, we have that U−w0= U , so by the uniqueness in the strengthened

version of Theorem 5.26, we see that u′ = 1 = u. Therefore g = t′ ∈ T . Conversely,T ⊆ Bw0 ∩B as w0 normalises T . �

Bw0 is called the Borel subgroup opposite to B, and is denoted by B−.

6. Parabolic subgroups

We now use the results from the previous section to investigate parabolic sub-groups of connected reductive groups. It turns out that they can be entirely de-scribed using the root system. We will also prove the very beautiful Borel-Titstheorem which has applications to the study of maximal subgroups.

39

6.1. Standard parabolic subgroups and the Levi decomposition. We keepusing the same notation as in the previous section: let G be a connected reductivelinear algebraic group, T ≤ G a maximal torus contained in a Borel subgroup B,Φ the root system of G with Weyl group W = N/T where N = NG(T ), ∆ the setof simple roots with respect to T ≤ B and S = {sα | α ∈ ∆} the set of simplereflections. Recall S generates W .

For I ⊆ S, let WI = 〈s ∈ I〉. WI is called a standard parabolic subgroup of W . Aparabolic subgroup of W is a subgroup conjugate to some WI . Let ∆I = {α | sα ∈ I}and ΦI = Φ∩

∑α∈∆I

Zα, the corresponding parabolic subsystem of roots. It is fairly

easy then to show that ΦI is a root system in 〈ΦI〉R with base ∆I and Weyl groupWI , by using Proposition 5.16 and the fact that Φ is a root system with base ∆.

We can then translate this to subgroups of G. Indeed, for I ⊆ S, define

PI = BWIB =⊔

w∈WI

BwB.

Clearly, for I = ∅ and I = S, we get B and G respectively by Theorem 5.26. Moregenerally, PI is always a subgroup of G containing B. Indeed, it contains 1, andis clearly closed under inverses. Moreover, by Lemma 5.23, BwB · BsB ⊆ PI forw ∈ WI and s ∈ I. An easy induction similar to the one in the proof of Theorem5.26 then gives that PI is closed under multiplication. So PI is a subgroup, and itclearly contains B.

Proposition 6.1. For I ⊆ S, PI = 〈T,Uα | α ∈ Φ+ ∪ ΦI〉. In particular, PI is aclosed, connected, self-normalising subgroup of G containing B.

Proof. Let H = 〈T,Uα | α ∈ Φ+ ∪ ΦI〉 and write Φ±I = ΦI ∩ Φ±. Since B ⊆ PI ,it follows that Uα ⊆ PI for all α ∈ Φ+. Also, ΦI is a root system with Weylgroup WI , so WI acts simply-transitively on the set of bases by Proposition 5.16,and thus there exists a unique w ∈ WI such that w(∆I) = −∆I . In particular,w(Φ+

I ) = Φ−I . Hence, for all β ∈ Φ−I , there exists α ∈ Φ+I such that β = wα. But

then Uβ = Uwα = wUαw−1 ⊆ PI , thus proving H ⊆ PI .

Conversely, note that for α ∈ ∆ with corresponding simple reflection s ∈ S, wemay choose a preimage s ∈ 〈Uα, U−α〉 (this is the same argument as in the proofof Theorem 5.17). So H contains preimages of all s ∈ I, and thus of all w ∈ WI .Therefore, since B ≤ H, it follows that PI ⊆ H and so we have PI = H as required.

The fact that PI is a closed connected subgroup then follows from Proposition1.10. Since it contains B, it is a parabolic subgroup, and so is self-normalising byCorollary 4.28. �

Definition 6.2. The PI , I ⊆ S, are called standard parabolic subgroups.

We wish to show that all parabolic subgroups are conjugate to some PI . Inwhat follows, we will make use of the fact from the proof of Theorem 5.26 thatBvB = BwB ⇐⇒ v = w for v, w ∈ W . We begin by making the statement ofLemma 5.23 more precise.

Lemma 6.3. (i) If s ∈ S, w ∈ W with `(ws) ≥ `(w) then BwB · BsB ⊆BwsB

(ii) If s ∈ S, w ∈W with `(ws) ≤ `(w) then BwB ·BsB ∩BwB 6= ∅

Proof. (i) Use induction on `(w). If `(w) = 0, the result is trivial. So suppose`(w) ≥ 1 and write w = s′w′ with s′ ∈ S and w′ ∈ W with `(w′) = `(w) − 1.

40

Suppose the result is false, and aim for a contradiction. By Lemma 5.23, BwB ·BsB ∩BwB 6= ∅. So in particular wBs ∩BwB 6= ∅. Multiplying on the left by s′,we get w′Bs ∩ s′BwB 6= ∅.

But `(w′s) ≥ `(s′w′s) − 1 = `(ws) − 1 ≥ `(w) − 1 = `(w′) so by induction

hypothesis, Bw′B ·BsB ⊆ Bw′sB and thus Bw′sB ∩ s′BwB 6= ∅. Now, note thatif we take inverses in Lemma 5.23, using the fact that s2 = 1, we obtain that forall s ∈ S, w ∈W ,

BsB ·BwB ⊆ B ˙swB ∪BwB

and applying this, we get that

s′BwB ⊆ B ˙s′wB ∪BwB = Bw′B ∪BwB.

It follows that Bw′sB intersects one of these double cosets, and hence either w′s =w′ (but this would imply s = 1, a contradiction) or w′s = w. So we must havew′s = w and therefore w′ = ws. But this is a contradiction since `(w′) < `(w) ≤`(ws) by assumption.

(ii) By Lemma 5.23, sBs ⊆ BsB ∪ B. Therefore, by (BN5), sBs ∩ BsB 6= ∅.Multiplying on the left by ws, we get wBs ∩ wsBsB 6= ∅. But `(ws2) = `(w) ≥`(ws) so by (i), BwsB · BsB ⊆ BwB. Hence, wBs ∩ BwB 6= ∅ and the resultfollows. �

Lemma 6.4. Let w = s1 · · · sr, with si ∈ S, be a shortest expression for w, so thatr = `(w) and let I = {s1, . . . , sr} ⊆ S. Then 〈B, w〉 = 〈B, w−1Bw〉 = PI .

Proof. We show by induction on r that s1, . . . , sr ∈ 〈B, w−1Bw〉. Let w′ = wsr,so `(w′) < `(w). By Lemma 6.3(ii), BwB · BsrB ∩ BwB 6= ∅. In particular,sr ∈ Bw−1BwB, which lies in 〈B, w−1Bw〉. So if r = 1, we’re done. If r > 1, byinduction hypothesis s1, . . . , sr−1 ∈ 〈B, w′−1Bw′〉 = 〈B, srw−1Bwsr〉 which lies in〈B, w−1Bw〉 since sr does. So the group generated by B and w−1Bw contains apreimage of all elements of I, and hence of all elements of WI . Therefore it containsPI . But we have 〈B, w−1Bw〉 ⊆ 〈B, w〉 ⊆ PI , giving equality throughout. �

This lemma has the following consequences:

Lemma 6.5. (i) S is precisely the set of those w ∈W for which B ∪BwB isa group.

(ii) Let w ∈W and I, J ⊆ S. If w−1PIw ⊆ PJ then w ∈ PJ . In particular, forany I ⊆ S, we have NG(PI) = PI .

Proof. (i) If I = {s}, s ∈ S, then WI = {1, s} and so B ∪ BsB = PI is a group.Conversely, if B ∪ BwB is a group and w = s1 · · · sr is a shortest expression, then{s1, . . . , sr} ⊆ 〈B, w〉 = B ∪ BwB by Lemma 6.4. Thus Bs1B ⊆ B ∪ BwB. Sinces1 /∈ B, we must have Bs1B∩BwB 6= ∅ and therefore s1 = w. It follows that r = 1and w ∈ S.

(ii) We have that both B and w−1Bw lie in PJ . Write a shortest expressionw = s1 · · · sr and let I ′ = {s1, . . . , sr} ⊆ S. By Lemma 6.4, PI′ ⊆ PJ , and so PJcontains a preimage of all elements in WI′ . From the double coset decompositionof PJ and the fact that BvB = BwB ⇐⇒ v = w for all v, w ∈ W , it followsthat WI′ ⊆ WJ and so w ∈ WJ and w ∈ PJ . The fact that NG(PI) = PI is thenimmediate. �

41

Remark. We had already proved that NG(PI) = PI , but this time we didn’t appealto Corollary 4.28, which itself depended on Theorem 4.27. Actually, since any Borelsubgroup B is evidently of the form PI , we just proved Theorem 4.27 for connectedreductive groups.

We can now prove that every parabolic subgroup is conjugate to some PI :

Theorem 6.6. (i) The subgroups of G containing B are all of the form PIfor some I ⊆ S. In particular all parabolic subgroups are conjugate to somePI .

(ii) The PI are mutually non-conjugate. In particular, PI = PJ implies I = J .

Proof. (i) Let P be a subgroup of G containing B. Any element of P can be writtenin the form g = bwb′ for some b, b′ ∈ B, w ∈W by Theorem 5.26. Thus w ∈ P andso P can be written as a union of double cosets P =

⊔w∈M BwB for some M ⊆W .

Let I be the set of all s ∈ S appearing in the shortest expressions of elements ofM . Clearly P ⊆ PI and by Lemma 6.4, we have PI ⊆ P . For the last part, anyparabolic subgroup P contains a Borel subgroup B′, which is conjugate to B bysome g ∈ G. Thus P g contains B and so equals PI for some I ⊆ S.

(ii) Suppose PI is conjugate to PJ , say PI = P gJ for some g ∈ G. Write g = bwb′

for some b, b′ ∈ B, w ∈W . Then PI = P wJ and so PI = PJ by Lemma 6.5(ii). Usingthe same argument as in the proof of Lemma 6.5(ii), we therefore have WI = WJ .But then, by Lemma 6.5(i), I and J are both the set of elements w ∈ WI = WJ

such that B ∪BwB is a group, ie I = J . �

Remark. More generally, PI ⊆ PJ implies I ⊆ J . Indeed, we then have WI ⊆ WJ ,and so by Lemma 6.5(i), we have I ⊆ J .

Example 6.7. LetG = SLn. Following Examples 5.14 and 5.19, we have a maximaltorus T = Dn ∩ SLn contained in the Borel subgroup B = Tn ∩ SLn, root systemΦ = {χij : 1 ≤ i, j ≤ n, i 6= j}, base ∆ = {χi,i+1 : 1 ≤ i < n} and set of positiveroots Φ+ = {χij : i < j}. We also have Uij = In + 〈Eij〉 and

si =

Ii−1

0 1−1 0

In−i−1

T ∈ NG(T )

is the simple reflection corresponding to χi,i+1. We have W ∼= Sn under the iso-morphism given by si 7→ (i i + 1). So we may identify S = {si : i < n} with{(1 2), . . . , (n− 1 n)}. Let I = S \ {sa} for some a < n. Then

WI = 〈si | a 6= i < n〉 ∼= Sa × Sn−aand ΦI = {χij : i, j ≤ a or i, j ≥ a}. Then by Proposition 6.1 it’s not hard to see

that PI =

{(A ∗0 B

): A ∈ GLa, B ∈ GLn−a

}∩ SLn. This parabolic subgroup is

the stabiliser of the subspace of kn given by its first a basis vectors. In general, aspointed out in Example 4.17, stabilisers of flags are parabolic subgroups.

Now that we know what parabolic subgroups of G look like, we can decomposethem furthermore in the following way: for I ⊆ S, define

UI =∏

α∈Φ+\ΦI

Uα = 〈Uα | α ∈ Φ+ \ ΦI〉

42

(the product is a subgroup thanks to the commutator formula) and let

LI = 〈T,Uα | α ∈ ΦI〉.

We can then show that LI is a complement to UI in PI :

Proposition 6.8. Let I ⊆ S. Then Ru(PI) = UI and LI is complement to UI , soPI = UI o LI . In particular, LI is reductive with root system ΦI . Moreover, allclosed complements to UI are conjugate to LI in PI and LI = CG(Z(LI)

◦).

Proof. Thanks to the commutator formula (Proposition 5.22), we have that UI�PIand PI = 〈LI , UI〉 by Proposition 6.1. Now let Z = (

⋂α∈ΦI

kerα)◦ ≤ T , a subtorus

of T , and let L = CG(Z). It’s clear from Theorem 5.13(3) that LI ≤ L.Conversely, L is connected reductive by Corollary 5.12 so by Corollary 5.21 it

is generated by T and the Uβ it contains. Suppose Uβ ≤ L. Then β is trivialon Z again thanks to Theorem 5.13(3) so β ∈ Z⊥. Now, Z has finite index in⋂α∈ΦI

kerα = 〈ΦI〉⊥, so thanks to Proposition 3.12(i) we know that some multiple

of β lies in (〈ΦI〉⊥)⊥. Furthermore, some multiple of β lies in 〈ΦI〉 by Proposition3.12(ii). So, by (R2), β ∈ ΦI and thus L = LI .

Therefore LI∩UI is a normal unipotent subgroup of the reductive group LI = L,hence trivial. So PI = UI o LI . As LI is reductive, we also have Z(LI)

◦ = Z (byTheorem 5.13(7)), so LI = CG(Z(LI)

◦) and Z is a maximal torus in R(PI) = ZUI .Now, suppose L′ is another closed complement to UI in PI . Then L′ ∼= LI as

abstract groups (since they’re both isomorphic to PI/UI as abstract groups), andso if we let Z ′ = Z(L′)◦, we have Z ′ ∼= Z(LI)

◦ = Z. As Ru(PI) = UI , we haveRu(L′) ∼= Ru(PI/UI) = 1 and so Ru(Z ′) = 1. It follows that Z ′ is a connectedsolvable reductive group and so a torus (by Example 5.4). Now, Dn 6∼= Dm asabstract groups for n 6= m (by counting the number of elements of fixed finiteorder), so Z ′ and Z must have the same dimension. Hence Z ′ is a maximal torusof R(PI). Maximal tori of an algebraic group are conjugate (by Corollary 4.18), soZ and Z ′ are conjugate, and thus L′ = CG(Z ′) (which holds by the first part) isconjugate to L = CG(Z). �

Definition 6.9. The decomposition PI = UI oLI is called the Levi decompositionof PI , and LI is called the standard Levi complement or Levi factor of PI . A Levisubgroup of G is a subgroup conjugate to some LI .

Remark. The proof of Proposition 6.8 shows that for any Levi subgroup L, we haveL = CG(Z(L)◦).

Corollary 6.10. For any I ⊆ S, we have PI = NG(UI).

Proof. Let N = NG(UI). Clearly PI ≤ N . Thus N is a parabolic subgroup of G,so equal to PJ for some J ⊇ I in S. Suppose J 6= I. Then let s = sα ∈ J \ I. Wehave s ∈ N and Uα ≤ UI (as α ∈ Φ+ \ ΦI). Therefore U−α = sUαs

−1 ≤ UI , whichis a contradiction. So J = I and PI = N . �

Example 6.11. If we takeG = SLn as in Example 6.7, we had a parabolic subgroupPI with I = S \ {sa}. Then since ΦI = {χij : i, j < a or i, j > a}, it follows thatits standard Levi complement is

LI =

{(A 00 B

): A ∈ GLa, B ∈ GLn−a

}∩ SLn

43

and so we see LI is conjugate to LJ where J = S \ {sn−a}. Thus, unlike forparabolic subgroups, Levi subgroups for different subsets I of S can be conjugate.

In a more general context, for a flag 0 = V0 ⊆ V1 ⊆ . . . ⊆ Vr = kn withdimVi/Vi−1 = ni, its stabiliser in GLn is a parabolic subgroup whose Levi comple-ment is the subgroup of block diagonal matrices isomorphic to GLn1

× . . .×GLnr(where we identify GLni with GL(Vi/Vi−1)). For example if r = 3 we get a parabolicsubgroup

P =

A1 ∗ ∗

0 A2 ∗0 0 Ar

: Ai ∈ GLni

with Levi complement

L =

A1 0 0

0 A2 00 0 A3

: Ai ∈ GLni

We see that stabilisers of flags give typical examples of parabolic subgroups

of GLn or SLn. Using the Levi decomposition, one can show that all parabolicsubgroups of SLn are of that form (see [MT, Prop 12.13]).

From Proposition 6.8, we see that any Levi subgroup is equal to CG(Z) for sometorus Z. We want to show that the converse holds. Recall that a parabolic subgroupof W is a subgroup conjugate to some WI . We have the following basic result aboutabstract root systems (see [MT, Corollary A.29]):

Lemma 6.12. Let Φ be an abstract root system of E with Weyl group W , and letM ⊆ E. Then the pointwise stabiliser CW (M) of M in W is a parabolic subgroup.Conversely, any parabolic subgroup H ≤W is the centraliser of its fixed point space,ie H = CW (EH).

Proposition 6.13. Let G be connected reductive, Z ≤ G a torus. Then CG(Z) isa Levi subgroup of G.

Proof. Let C = CG(Z). By Corollary 5.12, C is connected reductive. A Borelsubgroup BC of C is contained in a Borel subgroup B of G. Since Z ≤ Z(C)◦ ≤BC ≤ B (where the second inclusion follows from Proposition 5.3(i)), Z is containedin a maximal torus T of B (and so of G). Clearly T ≤ C so by Theorem 5.13,C = 〈T, Vα | α ∈ ΦC〉 where ΦC is the root system of C with respect to T and theVα are the root subgroups of C.

For α ∈ ΦC , 0 6= Lie(Vα) ⊆ gα and so α ∈ Φ where Φ is the root system of Gwith respect to T . Since all the gα have dimension 1, we have Lie(Vα) = gα and soby Theorem 5.13(3), Vα = Uα, the root subgroup of G with respect to α.

By Theorem 5.13(3), Uα ≤ C if and only if Z ≤ kerα and so C = 〈T,Uα | Z ≤kerα〉. Therefore ΦC = Φ ∩ 〈α : Z ≤ kerα〉, the intersection of Φ with a subspaceof 〈Φ〉R. By Lemma 6.12, ΦC is then a parabolic subsystem of Φ, and therefore Cis the Levi complement of the corresponding parabolic subgroup of G. �

We introduce one further concept related to Levi subgroups. Recall from Lemma5.28 that given a Borel subgroup B of G, containing a maximal torus T , there existsa unique Borel subgroup B−, called the Borel subgroup opposite to B, such thatB ∩ B− = T . This is an example of a more general construction. Indeed, given aparabolic subgroup P , and a Levi subgroup L of P , there exists a unique parabolicsubgroup P− of G such that L is a Levi subgroup of P− and P ∩ P− = L. The

44

subgroup P− is called the opposite parabolic subgroup to P with respect to L. Sinceany parabolic subgroup is conjugate to a standard parabolic subgroup PI for someI ⊆ S and any Levi subgroup of PI is conjugate to LI , it’s enough to show that P−

exists for P = PI and L = LI . Then just define P−I = 〈T,Uα | −α ∈ ΦI ∪Φ+〉 (or

in the double coset notation, P−I = B−WIB−). It is clear that P−I is a parabolic

subgroup, and its intersection with P is LI . Finally, if P− is an opposite parabolicsubgroup to PI with respect to LI , then its intersection withRu(P ) =

∏α∈Φ+\ΦI Uα

is trivial and so it must equal P−I .Note that then we have Ru(P )∩Ru(P−) = 1 for any parabolic subgroup P with

Levi L, where P− is the opposite to P with respect to L, since it holds for P = PI .

6.2. The Borel-Tits Theorem. We now aim to show a fundamental result onnormaliser of unipotent subgroups. We still assume G is a connected reductivelinear algebraic group. A key step in the proof is the following result:

Proposition 6.14. Let V be a closed unipotent subgroup of G, and N = NG(V ).Assume that V lies in some Borel subgroup of G, and that Ru(N) ⊆ V (equivalentlyV ◦ = Ru(N)). Then N is a parabolic subgroup of G, with V = Ru(N).

Assume for now that Proposition 6.14 is true. Then we get:

Theorem 6.15. (Borel-Tits) Let U ≤ G be a closed unipotent subgroup whichlies in a Borel subgroup of G. Then there exists a parabolic subgroup P of G withU ≤ Ru(P ) and NG(U) ≤ P .

Proof. Set U0 = U and define inductively Ni+1 = NG(Ui) and Ui+1 = Ui ·Ru(Ni+1)for i ≥ 0. Then U = U0 ≤ U1 ≤ . . . and N1 ≤ N2 ≤ . . . are chains of closedsubgroups. We claim each Ui is contained in a Borel subgroup of G. This holds fori = 0, so assume Ui ≤ B for i ≥ 0 and some Borel subgroup B of G.

We let Ui act on the projective variety G/B by left multiplication. Then, sinceUi ≤ B, the set X of fixed points is a non-empty closed subset, thus also projective.X is stabilised by the connected solvable group Ru(Ni+1) ≤ NG(Ui), so by the Borelfixed point theorem (Theorem 4.13), there exists gB, for some g ∈ G, which is fixedby Ru(Ni+1) and by Ui. So Ui+1 = Ui · Ru(Ni+1) fixes gB and hence Ui+1 iscontained in gBg−1 as claimed.

Now,

Ui+1/Ui = Ui ·Ru(Ni+1)/Ui ∼= Ru(Ni+1)/(Ui ∩Ru(Ni+1)),

so since Ru(Ni+1) is connected, we have that Ui/Ui+1 is connected. Thus eitherdim (Ui+1/Ui) ≥ 1 or Ui+1 = Ui. Since G has finite dimension, the sequenceU = U0 ≤ U1 ≤ . . . must terminate, ie there is some l ≥ 0 such that for all k ≥ 0,Ul = Uk+l. Therefore we also have Nl+1 = Nl+k+1 for all k ≥ 0.

Let V = Ul and P = Nl+1 = NG(V ). Then we know V lies in some Borelsubgroup of G, and Ru(P ) ⊆ V . By Proposition 6.14, P is a parabolic subgroupand V = Ru(P ), so U ≤ V = Ru(P ) and NG(U) = N1 ≤ Nl+1 = P . �

Since a closed connected unipotent subgroup is a solvable, it lies in some Borelsubgroup. Thus we have the following immediate corollary:

Corollary 6.16. Let U ≤ G be a closed connected unipotent subgroup. Then thereexists a parabolic subgroup P of G with U ≤ Ru(P ) and NG(U) ≤ P .

Now, if we consider inclusions of connected reductive groups, we get the following:

45

Corollary 6.17. Let H be a closed connected reductive subgroup. Let PH be aparabolic subgroup of H. Then there exists a parabolic subgroup PG of G withRu(PH) ≤ Ru(PG) and PH ≤ PG.

Proof. Ru(PH) is a closed connected unipotent subgroup, so by Corollary 6.16there exists a parabolic subgroup PG of G such that PH ≤ NG(Ru(PH)) ≤ PG andRu(PH) ≤ Ru(PG). �

We now complete the proof of Theorem 6.15:

Proof of Proposition 6.14. Let B be a Borel subgroup containing V , and S =(B ∩ N)◦. Then S lies in some Borel subgroup B1 of N , and choose a subgroupnB1n

−1, n ∈ N , such that Ru(B1 ∩ nB1n−1) = Ru(N) (we can do this by taking

an opposite subgroup, as defined after Lemma 5.28, of the image of B1 in theconnected reductive group N◦/Ru(N)). Let B′ = nBn−1.

By choice of B′, we have Ru(N ∩ B ∩ B′) ⊆ Ru(N) = V ◦. On the other hand,n normalises V so V lies in the unipotent part of the solvable group N ∩ B ∩ B′and therefore V ◦ = Ru(N ∩ B ∩ B′). Now, we know from Corollary 5.27 thatB ∩ B′ is connected solvable, so by Theorem 3.18 V ⊆ Ru(B ∩ B′). If V hadsmaller dimension than Ru(B ∩ B′), then by Proposition 4.11 the normaliser ofV in that group would have larger dimension than V . But that normaliser is justN∩Ru(B∩B′) (as N = NG(V )), whose connected component is a closed connectedsolvable normal subgroup of N ∩B ∩B′, hence contained in Ru(N ∩B ∩B′) = V ◦,giving a contradiction. Therefore V = Ru(B ∩ B′) and so V is connected andB ∩ B′ ⊆ N . Since B ∩ B′ contains a maximal torus T of G by Corollary 5.27, itfollows that N has maximal rank in G.

We now use the root system of G with respect to T to show that N is parabolic.We let ∆ be the base of Φ determined by B as in Theorem 5.17 and S be the setof simple reflection. Also define Ψ to be the set of roots of N with respect to T .These are the roots α ∈ Φ such that Uα ⊆ V and the roots ±β corresponding tothe roots of the connected reductive group N◦/V with respect to the image of T .Take α ∈ ∆. If Uα ⊆ V , then α ∈ Ψ. Otherwise, Uα ⊆ B but Uα * V , so since

V = Ru(B ∩B′), it follows that Uα * B′ and thus U−α ⊆ B′. Let B′′ = sαBsα−1.

Recall from the proof of Lemma 5.23 that sα.β ∈ Φ+ for all α 6= β ∈ Φ+, andhence the root subgroups in B′′ are the root subgroups in B except for Uα, whichis replaced by U−α. Therefore U−α normalises B′ and Ru(B ∩B′′), and hence alsoRu(B ∩ B′ ∩ B′′) = Ru(B ∩ B′) = V . Thus −α ∈ Ψ and since U−α * V , we musthave that ±α ∈ Ψ.

So we proved that ∆ ⊆ Ψ. So B ⊆ N and N is a parabolic subgroup, morespecifically the standard parabolic subgroup PI where I = {sα ∈ S : Uα * V }.

�

Using the Borel-Tits theorem, we get the following striking result:

Theorem 6.18. Let G be connected reductive.

(i) Suppose H ≤ G is a maximal proper closed subgroup. Then either H◦ isreductive or H is a parabolic subgroup.

(ii) Let U ≤ G be a unipotent subgroup. Then U lies in some Borel subgroupof G.

Proof. (i) Suppose H◦ is not reductive. Then Ru(H◦) is a non-trivial connectedunipotent subgroup of G. By Corollary 6.16, there exists a parabolic subgroup P of

46

G such that Ru(H◦) ≤ Ru(P ) and NG(Ru(H◦)) ≤ P . Thus H ≤ NG(Ru(H◦)) ≤P . By maximality of H, we must have H = P .

(ii) We prove this by induction on dimG. If dimG = 1, then G ∼= Ga orGm and so G is solvable. Thus G is a Borel subgroup containing U . So assumedimG > 1. Since U is unipotent, it is nilpotent by Corollary 3.8 and so Z(U) 6= 1

(see Definition 1.12). Then pick 1 6= u ∈ Z(U) and let U1 = 〈u〉, the closure of thesubgroup generated by u. We have U ≤ NG(〈u〉) ≤ NG(U1).

By Theorem 4.24, u belongs to some Borel subgroup B of G. So 〈u〉 ≤ B andit follows that U1 ≤ B. More specifically, U1 ≤ Ru(B) = Bu by Theorem 3.18.By the Borel-Tits theorem, there exists a parabolic subgroup P of G such thatU ≤ NG(U1) ≤ P and U1 ≤ Ru(P ).

But 1 6= U1, so Ru(P ) 6= 1 and hence P 6= G. Also, the group P/Ru(P ) isconnected reductive and has smaller dimension than dimG. The image of U inthis quotient, namely U · Ru(P )/Ru(P ), is unipotent by Theorem 3.3(iii), so byinduction hypothesis, it is contained in a Borel subgroup of P/Ru(P ). This Borelsubgroup is of the form H/Ru(P ) where H is a closed connected solvable of G withRu(P ) ≤ H. Let B1 be a Borel subgroup of G with B1 ≤ P and Ru(P ) ≤ B1.Then B1/Ru(P ) is a closed connected solvable subgroup of P/Ru(P ) and so lies ina conjugate of H/Ru(P ). Then B1 ≤ Hg for some g ∈ G and by maximality of B1,we have B1 = Hg. So H is a Borel subgroup containing U as required. �

Therefore we see that Theorem 6.15 holds for arbitrary closed, unipotent sub-groups of G. Theorem 6.18(ii) also has the consequence that maximal unipotentsubgroups of G are precisely unipotent radicals of Borel subgroups.

The Borel-Tits theorem can also be used to investigate the representations ofthe Levi complement of a parabolic subgroup of G, but we don’t discuss this here,see [MT, sections 15, 16 and 17].

7. G-complete reducibility

In this section, we follow the work of Bate, Martin and Rohrle [BMR]. In rep-resentation theory, a finite dimensional representation V of a group H is said tobe completely reducible if whenever 0 6= W < V is a proper H-invariant subspace,then it has an H-invariant complement W ′ so that V = W ⊕W ′. If we identify Vwith kn for some n, W with km for some m < n, and H with some subgroup ofGLn, the above definition gives that whenever, after changing basis, we can writeelements of H as elements of

P =

{(A ∗0 B

): A ∈ GLm, B ∈ GLn−m

},

then in fact they are, again up to a change of basis, elements of

L =

{(A 00 B

): A ∈ GLm, B ∈ GLn−m

}.

Note that P is a parabolic subgroup of GLn and L is a Levi subgroup of it. Thismotivates the following definition due to Serre [Se]:

Definition 7.1. Let G be a connected reductive linear algebraic group, and H ≤ Ga closed subgroup. H is said to be G-completely reducible (G-cr) if whenever it iscontained in a parabolic subgroup of G, it is actually contained in a Levi subgroupof it.

47

The idea here was to try to extend results about representations of algebraicgroups ρ : H → GL(V ) to the more general case of a morphism H → G where G isan arbitrary connected reductive group (see [Se]). Now, G itself is obviously G-crsince it is a parabolic subgroup which is its own Levi subgroup. More generally, anyclosed subgroup H which isn’t contained in any proper parabolic subgroup of G isG-cr. Following Serre [Se], H is said to be G-irreducible (G-irr) in that case. As forG-complete reducibility, this reduces to a familiar concept in representation theoryfor the special case G = GL(V ), namely here that V is an irreducible representationof H. Serre also defined a closed subgroup to be G-indecomposable (G-ind) if itis not contained in any Levi subgroup of a proper parabolic subgroup of G. Thisunsurprisingly also reduces to a well-known notion from representation theory inthe case G = GL(V ). Indeed, in that case, H being G-ind just says that V is anindecomposable H-module (i.e it is a representation of H which cannot be writtenas a direct sum of two proper non-trivial subrepresentations).

As in the previous two sections, we assume throughout that our linear alge-braic group G is connected reductive. Recall from Corollary 5.12 that centralisersof subtori of G are also connected reductive. We make the following importantdefinition due to Richardson [R]:

Definition 7.2. A closed subgroup H of G is said to be strongly reductive if it isnot contained in any proper parabolic subgroup of CG(S), where S is a maximaltorus of CG(H). Equivalently, H is strongly reductive if it is CG(S)-irr.

Remark. This is independent of the choice of S, since all maximal tori of CG(H)are conjugate.

Maximal tori of G, for instance, are strongly reductive since they equal theirown centraliser by Corollary 5.12. Richardson introduced the notion of strongreductivity in order to study the action of G on Gn by simultaneous conjugation,and in particular to study the closed orbits of that action (we shall not discuss thishere, see [R], in particular Theorem 16.4). One basic property of strong reductivityis the following:

Lemma 7.3. If H is a strongly reductive subgroup of G, then H◦ is reductive.

Proof. Let U = Ru(H). Suppose for a contradiction that U 6= 1. Let S be amaximal torus of CG(H). Recall that CG(S) is connected reductive. Then by theBorel-Tits theorem (Theorem 6.15) applied to U in CG(S), there exists a parabolicsubgroup of CG(S) containing H, contradicting strong reductivity. �

It turns out that the concepts of strong reductivity and G-complete reducibilitycoincide. In order to show this, we need a few results on parabolic subgroups. Thefirst one is due to Borel and Tits, and determines what the parabolic subgroups ofG contained in a fixed parabolic subgroup P of G are (see [BT, Prop 4.4]):

Proposition 7.4. Let P, P ′ be parabolic subgroups of G, and L ≤ P a Levi sub-group.

(i) (P ∩P ′) ·Ru(P ) is a parabolic subgroup of G. It equals P if and only if P ′

contains a Levi subgroup of P . It is a Borel subgroup of G if P ′ is a Borelsubgroup.

(ii) The parabolic subgroups of G contained in P are the semidirect productsof the parabolic subgroups of L with Ru(P ). Two parabolic subgroups of G

48

contained in P are conjugate if and only if their intersections with L areconjugate in L.

The following lemma tells us some more about the intersection of two parabolicsubgroups (see [BMR, Lemma 6.2(iii)]):

Lemma 7.5. Let P and Q be two parabolic subgroups of G, with Levi subgroupsLP and LQ respectively, such that LP ∩LQ contains a maximal torus T of G. Then

P ∩Q = (LP ∩ LQ) · (LP ∩Ru(Q)) · (Ru(P ) ∩ LQ) · (Ru(P ) ∩Ru(Q))

where Ru(P ∩Q) is the product of the last three factors.

We finally require the following result about parabolic subgroups of a connectedreductive subgroup of G, which strengthens Corollary 6.17 (see [BMR, Corollary2.5]):

Lemma 7.6. Let H be a connected reductive subgroup of G. If Q is a parabolicsubgroup of H with Levi subgroup LH , then there exists a parabolic subgroup Pof G, and a Levi subgroup L of P such that P ∩ H = Q, L ∩ H = LH andRu(P ) ∩H = Ru(Q).

Note that we already knew this in the special case of Borel subgroups of cen-tralisers of tori (see the discussion preceding Corollary 5.12). We can now provethe main theorem of this section:

Theorem 7.7. (Bate, Martin, Rohrle) Let H ≤ G be a closed subgroup. Then His G-completely reducible if and only if H is strongly reductive.

Proof. Suppose H is G-cr, and let S be a maximal torus of CG(H). Assume thatH is contained in some proper parabolic subgroup Q of CG(S), and aim for acontradiction. By Lemma 7.6 and Corollary 5.12, there exists a parabolic subgroupP of G such that Q = CG(S) ∩ P . Since S lies in Z(CG(S))◦, it must lie in allBorel subgroups of CG(S) by Proposition 5.3 and so it lies in Q ⊆ P . Now, as His G-cr, we have H ⊆ L for some Levi subgroup L of P . Moreover, T = Z(L)◦ isa torus of CP (H) (it is a subtorus of L by Theorem 5.13(7), which applies sinceL is connected reductive). Recall from Proposition 6.8 that L = CG(T ). Now,S is a maximal torus of CP (H) (as S ⊆ P ), so some conjugate of T lies in S byCorollary 4.18, say gTg−1 ⊆ S for some g ∈ CP (H). Thus, CG(S) ⊆ CG(gTg−1) =gCG(T )g−1 = gLg−1 ⊆ P , and so Q = CG(S), contradicting the assumption thatQ was a proper subgroup. Therefore, H is strongly reductive.

Conversely, suppose H is strongly reductive. Let S be a maximal torus of CG(H),and L = CG(S). By Proposition 6.13, L is a Levi subgroup of some parabolicsubgroup Q of G. Since H is strongly reductive, it is not contained in any properparabolic subgroup of L, and thus by Proposition 7.4(ii) we have that Q is a minimalparabolic subgroup of G containing H. Now, let P be a parabolic subgroup of Gcontaining H. We want to show that H is contained in a Levi subgroup of P . Wehave H ⊆ P ∩ Q. If P ′ ⊆ P is a parabolic subgroup of G, and if M ′ is a Levisubgroup of P ′, then there exists a Levi subgroup M of P such that M ′ ⊆M (thisis clear for standard parabolic subgroups, and the general case then follows easily).Hence, we may assume wlog that P is also minimal among parabolic subgroupscontaining H. Now, by Proposition 7.4(i), (P ∩Q) ·Ru(P ) is a parabolic subgroupof G containing H which is contained in P , so by minimality of P we must haveP = (P ∩Q) ·Ru(P ). Similarly, we have Q = (P ∩Q) ·Ru(Q). It follows from the

49

same proposition that P contains a Levi subgroup MQ of Q, and Q contains a Levisubgroup MP of P .

Claim. P ∩Q contains a common Levi subgroup of both P and Q.

Indeed, fix Levi subgroups LP and LQ of P and Q respectively, such that LP∩LQcontains a maximal torus of G. Since P ∩ Q contains a maximal torus T , say, ofG by Corollary 5.27, we can take LP and LQ to be Levi subgroups containing T .Then by Lemma 7.5, we have

P ∩Q = (LP ∩ LQ) · (LP ∩Ru(Q)) · (Ru(P ) ∩ LQ) · (Ru(P ) ∩Ru(Q))

where Ru(P ∩ Q) is the product of the last three factors. Now, MP is connectedreductive so MP ∩ Ru(P ∩ Q) is trivial. Therefore there is a bijective morphismfrom MP onto a subgroup of LP ∩LQ. Now, dimLP = dimMP since they are bothLevi subgroups of P , so this forces LP ≤ LQ. By the same argument applied toMQ, we get LQ ≤ LP . Hence P ∩Q contains a common Levi subgroup M of bothP and Q as claimed, namely M = LP = LQ.

Let P− be the unique opposite parabolic subgroup to P in G with respect toM , so that P ∩ P− = M . The set of roots of G with respect to T decomposes asthe disjoint union Ψ(M) ∪Ψ(Ru(P ) ∪Ψ(Ru(P−)) (this is easy to see when P is astandard parabolic subgroup). Since Ru(Q) ∩M is trivial, it follows that we have

Ru(Q) = (Ru(Q) ∩Ru(P−)) · (Ru(Q) ∩Ru(P )).

Since L and M are Levi subgroups of Q, there exists x ∈ Q such that xMx−1 = L.Actually, x can be chosen to be in Ru(Q) since Q = Ru(Q) · M . So we canwrite x = yz where y ∈ Ru(Q) ∩ Ru(P−) and z ∈ Ru(Q) ∩ Ru(P ) by the abovedecomposition. Since z ∈ P ∩Q, we have zMz−1 ⊆ P ∩Q. Replacing M by zMz−1

if necessary, we may therefore assume that z = 1. But then, as y ∈ P−, we havethat L = yMy−1 ⊆ P− and so H lies in P−. Therefore H ⊆ P ∩ P− = M asrequired, and so H is G-cr. �

Corollary 7.8. Let H be a closed subgroup of G. Then the following are equivalent:

(i) H is strongly reductive.(ii) H is G-cr.(iii) H is CG(S)-irr for a maximal torus S of CG(H).(iv) for every parabolic subgroup P of G which is minimal among parabolic sub-

groups containing H, the subgroup H is L-irr for some Levi subgroup L ofP .

(v) there exists a parabolic subgroup P of G which is minimal among parabolicsubgroups containing H such that H is L-irr for some Levi subgroup L ofP .

Proof. By definition of strong reductivity, (i)⇐⇒ (iii). The equivalence of (i) and(ii) is Theorem 7.7, and (iv) ⇒ (v) is obvious. Now, given a maximal torus S ofCG(H), CG(S) is a Levi subgroup of some parabolic subgroup P ofG by Proposition6.13. If H is CG(S)-irr, then P is a minimal parabolic subgroup with respect tocontaining H by Proposition 7.4(ii), giving (iii) ⇒ (v). The proof of Theorem 7.7shows that (v)⇒ (ii). Finally, if H is G-cr, then H is contained in a Levi subgroupL of any parabolic subgroup P of G, minimal with respect to containing H. So His L-irr by Proposition 7.4(ii), giving (ii) ⇒ (iv). �

50

Therefore we see that the study of G-cr subgroups reduces to the study of L-irrsubgroups for a Levi subgroup L of G. We now use these results to extend Cliffordtheory to the context of G-complete reducibility. A famous result in Clifford theoryasserts that if V is a completely reducible representation of a group H, and if N�H,then V is a completely reducible representation of N . Equivalently, a semisimpleH-module is also semisimple as an N -module. Now, Martin showed that if H ≤ Gis a strongly reductive subgroup, and N �H is a closed normal subgroup, then Nis also strongly reductive (see [M, Theorem 2]). The proof uses geometrical ideasalong the line of the theory developed by Richardson [R]. Using this result, andTheorem 7.7, we immediately get:

Theorem 7.9. Let H be a closed subgroup of G, and N a closed normal subgroupof H. If H is G-cr, then so is N . In particular, H◦ is G-cr.

Therefore we can indeed extend Clifford’s theorem to the context of G-completereducibility, since Theorem 7.9 reduces to it in the case G = GL(V ). Note that theconverse of Theorem 7.9 is not true in general. Indeed, take H to be a non-trivialfinite unipotent subgroup of G. Then by the Borel-Tits theorem, there exists aparabolic subgroup P of G such that H ⊆ Ru(P ). Therefore H is not G-cr, whileon the other hand H◦ = 1 trivially is. However, it can be shown that the conversedoes hold with the further assumption that CG(N) is contained in H (see [BMR,Theorem 3.14]).

References

[B] A. Borel, Linear Algebraic Groups. Graduate Texts in Mathematics, 126. Springer, 1991.

[BMR] M.Bate, B. Martin, G. Rohrle, A geometric approach to complete reducibility. Invent.math. 161 (2005), 177-218.

[BT] A. Borel, J. Tits, Groupes Reductifs. Publ. Math.. Inst. Hautes Etud. Sci. 27 (1965), 55-150.

[H] J. Humphreys, Linear Algebraic Groups. Graduate Texts in Mathematics, 21. Springer-

Verlag, New York, 1975.[L] S.Lang, Algebra. Addison-Wesley, Reading, Mass., 1965.

[M] B. Martin, A normal subgroup of a strongly reductive group is strongly reductive. J. Algebra

265 (2003), 669-674.[MT] G. Malle, D. Testerman, Linear Algebraic Groups and Finite Groups of Lie Type. Cambridge

Studies in Advanced Mathematics, 133. Cambridge University Press, Cambridge, 2011.

[R] R. W. Richardson, Conjugacy classes of n-tuples in Lie algebras and algebraic groups. DukeMath. J. 57 (1988), 1-35.

[S] T. A. Springer, Linear Algebraic Groups. Second Edition. Progress in Mathematics, 9.

Birkhauser, Boston, 1998.[Se] J.-P. Serre, Complete Reducibilite. Seminaire Bourbaki, 56eme annee, 2003-2004, no 932.