reductive groups

Reductive Groups

J.S. Milne

Version 1.00March 11, 2012

Eventually it is intended that these notes will provide a detailed exposition of the theory ofreductive algebraic groups (in about 300 pages). At present only the first chapter on splitreductive groups over arbitrary fields exists.

BibTeX information:

@misc{milneRG,

author={Milne, James S.},

title={Reductive groups},

year={2012},

note={Available at www.jmilne.org/math/}

}

v1.00 March 11, 2012 First version (77 pages).

Please send comments and corrections to me at the address on my websitehttp://www.jmilne.org/math/.

The photo is of a grotto on The Peak That Flew Here, Hangzhou, Zhejiang, China.

Copyright c 2012 J.S. Milne.

Single paper copies for noncommercial personal use may be made without explicit permis-sion from the copyright holder.

Table of Contents

Table of Contents 3Preface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

I Split reductive groups over fields 71 Review of algebraic groups . . . . . . . . . . . . . . . . . . . . . . . . . . 82 The root datum of a split reductive group . . . . . . . . . . . . . . . . . . . 153 Borel fixed point theorem and applications . . . . . . . . . . . . . . . . . . 384 Representations of split reductive groups . . . . . . . . . . . . . . . . . . . 515 Root data and their classification . . . . . . . . . . . . . . . . . . . . . . . 546 Construction of isogenies of split reductive groups: the isogeny theorem . . 627 Construction of split reductive groups: the existence theorem . . . . . . . . 64

II The Satake-Tits classification 691 Relative root systems and the anisotropic kernel. . . . . . . . . . . . . . . . 69

III Reductive group schemes 73

Bibliography 75

Index 77

3

PrefaceThe leading pioneer in the development of the theory ofalgebraic groups was C. Chevalley. Chevalleys principalreason for interest in algebraic groups was that they es-tablish a synthesis between the two main parts of grouptheory the theory of Lie groups and the theory of finitegroups. Chevalley classified the simple algebraic groupsover an algebraically closed field and proved the existenceof analogous groups over any field, in particular the finiteChevalley groups.

R.W. Carter.

Reductive algebraic groups (meaning group schemes) play a central role in many partsof mathematics, for example, in the Langlands program. Eventually it is intended that thesenotes will provide a detailed exposition of the theory of reductive algebraic groups. Atpresent only the first chapter on split reductive groups over arbitrary fields exists.

5

Notations; terminologyWe use the standard (Bourbaki) notations: N D f0;1;2; : : :g; Z D ring of integers; Q Dfield of rational numbers; RD field of real numbers; CD field of complex numbers; Fp DZ=pZD field with p elements, p a prime number. For integers m and n, mjn means thatm divides n, i.e., n 2mZ. Throughout the notes, p is a prime number, i.e., p D 2;3;5; : : :.

Throughout k is a commutative ring, and R always denotes a commutative k-algebra.Unadorned tensor products are over k. Notations from commutative algebra are as in myprimer CA (see below). When k is a field, ksep denotes a separable algebraic closure of kand kal an algebraic closure of k. The dual Homk-lin.V;k/ of a k-module V is denoted byV _. The transpose of a matrix M is denoted by M t .

When working with schemes of finite type over a field, we typically ignore the non-closed points. In other words, we work with max specs rather than prime specs, and pointmeans closed point.

We use the following conventions:X Y X is a subset of Y (not necessarily proper);X

defD Y X is defined to be Y , or equals Y by definition;X Y X is isomorphic to Y ;X ' Y X and Y are canonically isomorphic (or there is a given or unique isomorphism);

Passages designed to prevent the reader from falling into a possibly fatal error are sig-nalled by putting the symbolA in the margin.

ASIDES may be skipped; NOTES should be skipped (they are mainly reminders to theauthor).

ReferencesIn addition to the references listed at the end (and in footnotes), I shall refer to the followingof my notes (available on my website):

CA A Primer of Commutative Algebra (v2.22, 2011).AG Algebraic Geometry (v5.22, 2012).AGS Basic Theory of Affine Group Schemes (v1.00, 2012).LAG Lie Algebras, Algebraic Groups, and Lie Groups (v1.00, 2012).

The links to CA, AG, AGS, and LAG in the pdf file will work if the files are placed in thesame directory.

Also, I use the following abbreviations:

Bourbaki A Bourbaki, Alge`bre.Bourbaki AC Bourbaki, Alge`bre Commutative (IIV 1985; VVI 1975; VIIIIX 1983; X

1998).Bourbaki LIE Bourbaki, Groupes et Alge`bres de Lie (I 1972; IIIII 1972; IVVI 1981).Bourbaki TG Bourbaki, Topologie Generale.DG Demazure and Gabriel, Groupes Algebriques, Tome I, 1970.EGA Elements de Geometrie Algebrique, Grothendieck (avec Dieudonne).SGA 3 Schemas en Groupes (Seminaire de Geometrie Algebrique, 1962-64, Demazure,

Grothendieck, et al.); 2011 edition.monnnnn http://mathoverflow.net/questions/nnnnn/

6

CHAPTER ISplit reductive groups over fields

Split reductive groups1 are those containing a split maximal torus, i.e., a maximal torusisomorphic to Grm for some r . Every reductive group over a separably closed field is split.

To a reductive group G and a split maximal torus T we attach some combinatorial dataR.G;T /, called a root datum. Up to isomorphism, R.G;T / depends only on G, becauseany two split maximal tori are conjugate.

FromR.G;T / we are able to read off a great deal of information about the pair .G;T /,for example, the centre ofG, the structure of certain subgroups ofG, and the representationsof G.

The root datum ofR.G;T / determines .G;T / up to isomorphism, and every root datumarises from a pair .G;T /. Thus the split reductive groups over k are classified by the rootdata.

The root data have nothing to do with the ground field! In particular, we see that foreach reductive group G over kal, there is (up to isomorphism) exactly one split reductivegroup over k that becomes isomorphic to G over kal. However, there will in general bemany nonsplit groups, and so we are left with the problem of understanding them. This willoccupy Chapter II.

In 1 we review the basic theory of algebraic groups to fix terminology and to providea convenient reference. For proofs, we refer to AGS.

In 2 we define a root datum, and we explain how to attach a root datum to a splitreductive group.

In 3 we prove the Borel fixed point theorem and deduce some applications.In 4 we describe the finite-dimensional representations of a split reductive group in

terms of the root datum of the group.In 5, which is purely combinatorial, we classify the root data.In 6, following Steinberg 1999, we prove that isogenies of root data correspond to

isogenies of split reductive groups (isogeny theorem). In particular, the root datum of asplit reductive group determines it up to isomorphism.

In 7 we show that every root datum arises from a split reductive group. This allows usto define the Langlands dual of a split reductive group.

1In the theory of algebraic groups, and even in linear algebra, one often needs the ground field to bealgebraically closed in order to have enough eigenvalues and eigenvectors. By requiring that the group containsa split maximal torus, we ensure that there are enough eigenvalues without having to make an assumption onthe ground field.

7

8 I. Split reductive groups over fields

Throughout this chapter, k denotes the base field (an arbitrary field), and R denotesa commutative k-algebra (so for all R means for all k-algebras R). Unadorned tensorproducts are over k. All algebraic groups are affine, and semisimple group and reductivegroup mean semisimple algebraic group and reductive algebraic group. By a subgroupof an algebraic group, we mean an affine algebraic subgroup. By a representation of analgebraic group G we mean a homomorphism r WG! GLV with V finite-dimensional.

1 Review of algebraic groupsThis section summarizes what we need from AGS.

The notion of an algebraic group1.1 An affine algebraic group over a field k is a functorG from k-algebras to sets together

with a natural transformation mWGG!G such that(a) for all k-algebras R, m.R/ is a group structure on G.R/, and(b) G is representable by a finitely generated k-algebra.

A homomorphism .G;m/! .G0;m0/ of affine algebraic groups is a natural transformationuWG! G0 such that m0 .uu/D u m this just means that u.R/WG.R/! G0.R/ isa group homomorphism for all R. Since affine algebraic groups are the only ones we willbe concerned with, we refer to them as algebraic groups. The affine algebraic groups areexactly the algebraic groups that admit a faithful linear representation G ! GLV , and soare often called linear algebraic groups.

The condition (b) means that there exists a finitely generated k-algebra A together witha universal element a 2G.A/ such that the map

f 7!G.f /.a/WHom.A;R/!G.R/is a bijective for all R. The pair .A;a/ is uniquely determined up to a unique isomorphism,and A is called the coordinate ring O.G/ of G. Thus, for each g 2G.R/, there is a uniquehomomorphism O.G/!R sending a to g.

The natural transformation m corresponds to a comultiplication map WO.G/!O.G/O.G/ (a k-algebra homomorphism). To give a homomorphism uWG!H of al-gebraic groups over k is the same as giving a homomorphism u\WO.H/! O.G/ of k-algebras compatible with the comultiplication maps.

1.2 For example, let GLn denote the functor sending a k-algebra R to the set of invertiblenn matrices with entries in R. Matrix multiplication GLn.R/GLn.R/! GLn.R/ isnatural in R, and so defines a natural transformation mWGLnGLn! GLn satisfying (a).We can take O.G/ D kX;Y =.XY I / where X and Y are systems of n2 symbols andXY means matrix multiplication. In more detail,

O.G/D kX11; : : : ;Xnn;Y11; : : : ;YnnPj XijYjl il

1 i; l n

D kx11; : : : ;xnn;y11; : : : ;ynn:The universal element is the matrix x D .xij /1i;jn: for any invertible matrix C D .cij /with entries in a k-algebra R, there is a unique homomorphism of k-algebras O.G/! R

1. Review of algebraic groups 9

sending x to C . Define the determinant of a matrix by the usual formula,

det.aij /DX

2Sn sign./a1.1/ an.n/:

Then det is a homomorphism of algebraic groups

detWGLn!Gm; Gm defD GL1 :The trivial algebraic group is that with O.G/D k.

1.3 For an algebraic group .G;m/, we let jGj D SpecG. Then m endows jGj with thestructure of a group object in the category of schemes over k, i.e., jGj is an affine algebraicgroup scheme over k. It need not be reduced, i.e., O.G/ may have nilpotents. However, ifit is geometrically reduced, then it is smooth, and Cartiers theorem (AGS, VI, 9.3) showsthat it is always geometrically reduced when k has characteristic zero.

As we are working over fields, we usually ignore the nonclosed points in jGj. Thus,readers may regard jGj as an algebraic space in the sense of AG, Chapter 11. Moreover,those willing to take k to be algebraically closed of characteristic zero may regard jGj asan algebraic variety in the sense of AG, Chapters 1-10.

1.4 By a subgroup of an algebraic group, we mean an affine algebraic subgroup, i.e., arepresentable subfunctor H G such that H.R/ is a subgroup of G.R/ for all R. ThenO.H/ is a quotient ofO.G/. A subgroupH of G is normal ifH.R/ is normal in G.R/ forall R.

It is important to note that, with these definitions, the Noether isomorphism theoremshold for algebraic groups. For example, ifH and N are subgroups of an algebraic group Gand N is normal in G, then H and N are contained in a well-defined subgroup HN of G,H \N is a normal subgroup of H , and the natural map

H=H \N !HN=Nis an isomorphism of algebraic groups. Moreover, results such as the Schreier refinementtheorem2 hold with essentially the same proof as in the case of abstract groups.

Centralizers and normalizersLet H be a subgroup of an algebraic group G.

1.5 The centralizer C D CG.H/ of H in G is the subgroup R C.R/ of G such thatC.R/ consists of the elements of G.R/ centralizing3 H.R0/ in G.R0/ for all R-algebrasR0. When H is smooth, C.k/ consists of the elements of G.k/ centralizing H.ksep/ inG.ksep/. As CG.H/ need not be smooth, even when H and G are, this condition may notcharacterize CG.H/. Directly from its definition, one sees that the formation of CG.H/commutes with extensions of the base field. The centre ZG of G is CG.G/. See AGS, VII,6.

2Any two subnormal series of an algebraic group have equivalent refinements.3Let H be a subgroup of an abstract group G. An element g of G centralizes H (resp. normalizes H ) if

ghD hg for all h 2H (resp. gH DHg). The centralizer CG.H/ (resp. normalizer NG.H/) of H in G is theset of elements of G centralizing (resp. normalizing) H . They are both subgroups of G the normalizer isthe largest subgroup of G containing H as a normal subgroup.


1.6 The normalizer N D NG.H/ of H in G is the subgroup R N.R/ of G such thatN.R/ consists of the elements of G.R/ normalizing H.R0/ in G.R0/ for all R-algebrasR0. When H is smooth, N.k/ consists of the elements of G.k/ normalizing H.ksep/ inG.ksep/. Directly from its definition, one sees that the formation ofNG.H/ commutes withextensions of the base field. See AGS, VII, 6.

Finite algebraic groups; connected algebraic groups1.7 Recall that a k-algebra A is finite if it is finitely generated as a k-module, and it is

etale if it is a finite product of finite separable field extensions of k. An algebraic groupG is said to be finite (resp. etale) if O.G/ is a finite (resp. etale) k-algebra. An algebraicgroup G is finite if and only if G.kal/ is finite (AGS, XII, 1.6).

1.8 An algebraic group is said to be connected if it has no nontrivial etale quotient. Forany algebraic group G, there is an exact sequence

1!G!G! 0G! 1with G connected and 0G etale. The sequence is natural in G, and formation of thesequence commutes with extension of the base field. Moreover,

0.GG0/' 0G0G0:The coordinate ring O.0G/ of 0G is the largest etale subalgebra of O.G/.

1.9 An algebraic group is strongly connected if it has no nontrivial finite quotient. Asmooth algebraic group is strongly connected if it is connected (because a quotient of asmooth algebraic group is smooth, and smooth finite algebraic groups are etale). In partic-ular, connected and strongly connected coincide in characteristic zero.

Jordan decompositions in algebraic groups1.10 Recall that an endomorphism of a vector space V is said to be diagonalizable if

there exists a basis of V for which the matrix of is diagonal, and is said to be semisimpleif it becomes diagonalizable after an extension of the base field. An endomorphism isunipotent if its eigenvalues are all 1, in which case there exists a basis for V such that thematrix of lies in

Un.k/D

8=>; :

If is unipotent, then the eigenvalues of idV are all 0, and so it is nilpotent; conversely,if idV is nilpotent then is unipotent.

An automorphism of a vector space V over a perfect field k has a unique (Jordan)decomposition D s uD u s with s semisimple and u unipotent; moreover, eachof s and u can be expressed as a polynomial in (AGS, X, 2.2).


1.11 LetG be an algebraic group over a perfect field k. An element g ofG.k/ is said to besemisimple (resp. unipotent) if r.g/ is semisimple (resp. nilpotent) for all representationsr WG!GLV it suffices to check this for one faithful representation. Every g 2G.k/ hasa unique (Jordan) decomposition gD gs guD gugs with gs semisimple and gu unipotent(AGS, X, 2.8).

Diagonalizable groups1.12 The coordinate ring of Gm is kT;T 1 and its comultiplication map is

T 7! T T WO.Gm/!O.Gm/O.Gm/:Therefore, to give a homomorphism WG! Gm from an algebraic group G to Gm is thesame as giving an element u 2O.G/ (the image of T under \WO.Gm/!O.G/) such thatu is a unit in O.G/ and .u/D uu. Such elements are said to be group-like. Let

X.G/D Hom.G;Gm/' fu 2O.G/ j u is group-likeg,and let X.G/DX.Gksep/. The elements of X.G/ are called characters.

1.13 LetM be a finitely generated commutative (abstract) group, written multiplicatively.The functor

R Hom.M;R/ (homomorphisms of groups)is an algebraic group D.M/ with coordinate ring the group algebra of M :

kMD Pm2M amm j am 2 k ; PmammPn bnnDPm;nambnmn:Any set of generators for M generates kM as a k-algebra. The algebraic groups of theformD.M/ are said to be diagonalizable. Equivalently,G is diagonalizable if its coordinatering is spanned (as a k-vector space) by its group-like elements. If G is diagonalizable, sayG D D.M/, then it is possible to recover M from G as the set of group-like elementsin O.G/, i.e., M D X.G/. The functor M D.M/ is a contravariant equivalence ofcategories under which exact sequences correspond to exact sequences. (AGS, ChapterXIV.)

1.14 Let G be an algebraic group, and let .V;r/ be a representation of G. We say that Gacts on V through a character if r factors as

G!Gm 'Z.GLV / ,! GLV ,

or, equivalently, ifr.g/.v/D .g/v (1)

for all R, g 2G.R/, v 2 VR. When G is smooth, G acts on V through if (1) holds for allg 2G.ksep/, v 2 Vksep .4 For any representation r WT ! GLV of a diagonalizable group T ,

V DM

2X.T /V

where V is the subspace of V on which T acts through the character . See AGS, XIV,4.7.

4We have two homomorphism G! GLV that we know coincide on an algebraic subgroup H of G suchthat H.ksep/G.ksep/; this implies that Hksep DGksep when G is smooth.


Tori and groups of multiplicative type1.15 An algebraic group G is of multiplicative type if it becomes diagonalizable over an

extension field. It then becomes diagonalizable over ksep (AGS, XIV, 5.11). The functorG X.G/ is an equivalence from the category of groups of multiplicative type over k tothe category of finitely generated commutative groups equipped with a continuous actionof Gal.ksep=k/. Under the equivalence, exact sequences of algebraic groups correspond toexact sequence of modules.

1.16 (RIGIDITY) Every action5 of a connected algebraic group G on an algebraic groupH of multiplicative type is trivial (AGS, XIV, 6.1).

1.17 A smooth connected algebraic group of multiplicative type is called a torus. Thetori are exactly the algebraic groups G of multiplicative type such that X.G/ is torsion-free. Every algebraic group G of multiplicative type contains a largest subtorus, namely,the subgroup Gred such that X.Gred/D X.G/=ftorsiong.

Unipotent algebraic groups1.18 An algebraic group G is said to be unipotent if every nonzero representation of G

has a nonzero fixed vector, or, equivalently if, for every representation r WG! GLV , thereexists a basis of V for which r.G/ Un. A smooth algebraic group G is unipotent if andonly if G.ksep/ consists of unipotent elements (AGS, XV, 2.6).

Solvable algebraic groups1.19 Let G be an algebraic group. The derived group DG of G is the intersection of the

normal subgroups N such that G=N is commutative. It is a normal subgroup of G, andG=DG is the largest commutative quotient of G. When G is smooth, DG is the uniquesmooth subgroup of G such that .DG/.ksep/ is the derived group of G.ksep/. See AGS,XVI, 3.

1.20 Write D2G for the second derived group D.DG/ of G, D3G for the third derivedgroup D.D2G/ of G, and so on. An algebraic group G is said to be solvable if its derivedseries

G DG D2G terminates with 1.

1.21 A commutative algebraic group G over a perfect field has a unique decompositionG D Gs Gu with Gs of multiplicative type and Gu unipotent (in fact Gs and Gu are thelargest subgroups of their types) (AGS, XIV, 2.6).

1.22 A smooth connected solvable algebraic group over a perfect field k contains a uniqueconnected normal unipotent subgroup Gu such that G=Gu is of multiplicative type; theformation of Gu commutes with extension of the base field (AGS, XVI, 5.1).

5By an action of G on H we mean a natural transformation G H !H such that, for each R, G.R/H.R/!H.R/ is an action of the group G.R/ on H.R/.


The radical and unipotent radical1.23 Quotients and extensions of unipotent groups are unipotent. It follows that if H andN are unipotent subgroups of an algebraic group G with N normal, then HN is unipotent(because it is an extension of HN=N by N , and HN=N ' H=H \N ). Therefore, anymaximal normal unipotent subgroup U of G contains all other unipotent subgroups it isthe (unique) largest normal unipotent subgroup of G.

Similar statements hold with unipotent replaced by smooth, connected, or solv-able. Hence, among the smooth connected normal subgroups of an algebraic group G,there is a largest solvable one, called the radical RG of G, and a largest unipotent one,called the unipotent radical of G. The radical (resp. unipotent radical) of Gkal is called thegeometric radical (resp. geometric unipotent radical of G).

Clearly .RuG/kal RuGkal , but the two need not be equal unless k is perfect. For anyalgebraic group G, the radical (resp. unipotent radical) of G=RG (resp. G=RuG/ is trivial.

Semisimple and reductive algebraic groups1.24 A smooth connected algebraic group is semisimple (resp. reductive) if its geometric

radical (resp. its geometric unipotent radical) is trivial.Let G be a smooth connected algebraic group over a field k, and consider the smooth

connected normal subgroups ofG. IfG is semisimple (resp. reductive), thenG has no suchcommutative (resp. unipotent) subgroup; conversely, if G has no such commutative (resp.unipotent) subgroup, even over kal, then it is semisimple (resp. reductive).6

For example, GLn is reductive, but it is not semisimple because of the commutativesubgroup Dn (diagonal matrices). The algebraic group Tn of upper triangular matrices isnot reductive because of the subgroup Un (which is normal in Tn but not GLn).

1.25 A smooth connected algebraic group G is semisimple (resp. reductive), if Gk0 issemisimple (resp. reductive) for one extension field k0, in which case it is semisimple (resp.reductive) for all extension fields.

The structure of semisimple groups1.26 An algebraic group is simple (resp. almost-simple) if it is smooth, connected, non-

commutative, and every proper normal subgroup is trivial (resp. finite). For example, SLnis almost-simple for n > 1, and PSLn D SLn =n is simple. A simple algebraic group cannot be finite (because smooth connected finite algebraic groups are trivial).

Let N be a smooth subgroup of an algebraic group G. If N is minimal among thenonfinite normal subgroups of G, then either it is commutative or it is almost-simple; if Gis semisimple, then it is almost-simple.

1.27 An algebraic group G is said to be the almost-direct product of its algebraic sub-groups G1; : : : ;Gr if the map

.g1; : : : ;gr/ 7! g1 gr WG1 Gr !G6In characteristic p, there do exist algebraic groups having no smooth connected normal unipotent sub-

group over k, but acquiring such a subgroup over a purely inseparable extension such groups are said to bepseudo-reductive. The unipotent radical of a pseudo-reductive group is trivial, but not its geometric unipotentradical.


is a surjective homomorphism with finite kernel. In particular, this means that the Gi com-mute and each Gi is normal in G.

1.28 An almost-direct product of almost-simple algebraic groups is obviously semisimple.The converse is also true. Thus, the semisimple algebraic groups are exactly those thatare the almost-direct product of their almost-simple subgroups (called its almost-simplefactors).

From this it follows quotients of semisimple algebraic groups are semisimple. A smoothconnected normal subgroup of a semisimple algebraic group is a product of the almost-simple factors that it contains, and is therefore also semisimple.

Clearly DG D G when G is almost-simple, and so this is also true for all semisim-ple algebraic groups. In other words, a semisimple algebraic group has no commutativequotients. Moreover, its centre is a finite group of multiplicative type.

The structure of reductive groupsTHEOREM 1.29 If G is reductive, then its radical RG is a torus, and .RG/kal D RGkal .The centre ZG of G is a group of multiplicative type whose largest subtorus is RG. Thederived group DG of G is semisimple, ZG\DG is the centre of DG, and G DRG DG.Therefore G is the almost-direct product of a torus RG and a semisimple group DG:

1!RG\DG!RGDG!G! 1: (2)

For example, the centre of SLn is ZG D n and its radical is 1. The centre of GLn andits radical both equal Dn, its derived group is SLn, and the sequence (2) is

1! n! DnSLn! GLn! 1:PROOF. Because G is reductive, (1.21) shows that ZGkal is a group of multiplicative type.As ZGkal D .ZG/kal , this implies that ZG itself is of multiplicative type.

BecauseG is reductive, (1.22) shows thatRGkal is of multiplicative type. As .RG/kal RGkal , RG itself is of multiplicative type, and as it is smooth and connected, it is a torus.Rigidity (1.16) implies that the action of G on RG by inner automorphisms is trivial, andso RG ZG. Hence RG .ZG/red, but clearly .ZG/red RG, and so

RG D .ZG/red. (3)Now

.RG/kal(3)D .ZG/redkal D .ZGkal/red (3)D RGkal .

This completes the proof of the first two statements of the theorem.We next show that the algebraic group ZG\DG is finite. For this, we may replace k

with its algebraic closure. Then it suffices to show that the group

.ZG\DG/.k/D .ZG/.k/\ .DG/.k/is finite (see 1.7). Since .RG/.k/ is of finite index in .ZG/.k/, it suffices to show that.RG/.k/\ .DG/.k/ is finite. Choose a faithful representation G! GLV , and regard Gas an algebraic subgroup of GLV . Because RG is diagonalizable, V is a direct sum

V D V1 Vr

2. The root datum of a split reductive group 15

of eigenspaces for the action of RG (see 1.14). When we choose bases for the Vi , .RG/.k/consists of the matrices 0B@A1 0 00 : : : 0

0 0 Ar

1CAwith Ai of the form diag.i .t/; : : : ;i .t//, t 2 k. As i j for i j , we see that thecentralizer of .RG/.k/ in GL.V / consists of the matrices of this shape but with the Aiarbitrary. Because .DG/.k/ consists of commutators, its elements have determinant 1.But SL.Vi / contains only finitely many scalar matrices diag.ai ; : : : ;ai /, and so .RG/.k/\.DG/.k/ is finite.

Note that RG DG is a normal subgroup of G. The quotient G=.RG DG/ is semisim-ple because .G=.RG DG//kal is a quotient of Gkal=RGkal , which is semisimple. On theother hand, G=.RG DG/ is commutative because it is a quotient of G=DG. Therefore itis trivial, and so

G DRG DG:Now the homomorphism

DG!G=RGis surjective with finite kernel RG\DG ZG\DG. As G=RG is semisimple, so also isDG.

CertainlyZ.G/\DGZ.DG/, but, becauseGDRG DG andRGZG,Z.DG/Z.G/. This completes the proof of the theorem. 2

2 The root datum of a split reductive groupIn this section, we define split reductive groups and root data, and we explain how to attacha root datum to a split reductive group.

Split toriA split torus is a smooth connected diagonalizable algebraic group. Under the equivalenceof categories M D.M/ (see 1.13), the split tori correspond to torsion-free commuta-tive groups M of finite rank. The choice of an isomorphism M ' Zr determines an iso-morphism D.M/ ' D.Z/r D Grm, and so the split tori are exactly the algebraic groupsisomorphic to finite products of copies of Gm.

A torus is a smooth connected group of multiplicative type. A torus becomes diagonal-izable, hence a split torus, over a finite separable extension of the base field (1.15).

A subtorus of GLV is split if and only if it is contained in Dn for some choice of basisof V (apply 1.14). In particular, a subtorus T of GLn is split if and only if there exists aP 2 GLn.k/ such that P T P1 Dn.

Consider for example

T D

a b

b aa2Cb2 0

GL2 :

The characteristic polynomial of such a matrix is

X22aXCa2Cb2 D .X a/2Cb2


and so its eigenvalues areD abp1:

It is easy to see that T is split (i.e., diagonalizable over k) if and only if 1 is a square in k:The only group-like elements in O.Gm/D kT;T 1 are the powers T n, n 2 Z, of T .

Therefore, the only homomorphisms Gm!Gm are the maps t 7! tn for n 2 Z, and soEnd.Gm/' Z:

For a split torus T , we set

X.T /D Hom.T;Gm/D group of characters of T;X.T /D Hom.Gm;T /D group of cocharacters of T:

There is a pairing

h ; iWX.T /X.T /! End.Gm/' Z; h;i D : (4)Thus

..t//D t h;i for t 2Gm.R/DR:Both X.T / and X.T / are free abelian groups of rank equal to the dimension of T , andthe pairing h ; i realizes each as the dual of the other.

For example, let

T D Dn D

8=>; :

Then X.T / has basis 1; : : : ;n, where

i .diag.a1; : : : ;an//D ai ;and X.T / has basis 1; : : : ;n, where

i .t/D diag.1; : : : ;it ; : : : ;1/:

Note that

hj ;i i D1 if i D j0 if i j ;

i.e.,

j .i .t//Dt if i D j1 if i j :

Some confusion is caused by the fact that we write X.T / and X.T / as additivegroups. For example, if aD diag.a1;a2;a3/, then

.52C73/.a/D 2.a/53.a/7 D a52a73:For this reason, some authors use an exponential notation .a/ D a. With this notation,the preceding equation becomes

a52C73 D a52a73 D a52a73.


Split reductive groupsLetG be a reductive group over a field k, and let T be a torus inG. Any torus containing Tcentralizes it, and so if T is equal to its own centralizer, then it is maximal. Later we shallsee that the converse is also true (2.16c), and so

a torus in a reductive group is maximal if and only if it is equal to its owncentralizer.

Since the formation of centralizers commutes with extension of the base field, we see thata maximal tori in reductive groups remain maximal after extension of the base field.7

For example, Dn is a maximal torus in GLn because it is equal to own centralizer.Specifically, let A 2Mn.R/; if Ei iAD AEi i then aij D 0D aj i for all j i , and so Amust be diagonal if it commutes with all the Ei i .

A reductive group is split if it contains a split maximal torus. A reductive group overa separably closed field is automatically split, as all tori over such a field are split. As wediscuss below, for any reductive group G over a separably closed field k and subfield k0 ofk, there exists a split reductive group G0 over k0; unique up to isomorphism, that becomesisomorphic to G over k.

Example: The general linear group

2.1 The group GLn is a split reductive group (over any field) with split maximal torus Dn.On the other hand, let H be the quaternion algebra over R. As an R-vector space, H hasbasis 1; i;j; ij , and the multiplication is determined by

i2 D1; j 2 D1; ij Dj i:It is a division algebra with centre R. There is an algebraic group G over R such that

G.R/D .RRH/

for all R-algebras R (AGS, I, 4.3). In particular, G.R/ D H. As CRH M2.C/, Gbecomes isomorphic to GL2 over C, but as an algebraic group over R it is not split, becauseits derived group G0 is the subgroup of elements of norm 1, and as G0.R/ is compact, itcant contain a split torus.

Example: The special linear group

2.2 The group SLn is a split semisimple group, with split maximal torus the diagonalmatrices of determinant 1.

Example: The orthogonal groups

2.3 Let k be a field of characteristic 2. A quadratic form on a k-vector space V is amapping qWV ! k such that q.x/ D .x;x/ for some symmetric bilinear form on V .Then

q.xCy/D q.x/Cq.y/C2.x;y/7An important theorem of Grothendieck shows that this last statement is true for all smooth connected

affine algebraic groups (SGA 3, XIV, 1.1).


and so q determines uniquely. A quadratic space is a pair .V;q/ consisting of a vectorspace and a quadratic form. For example,

qWk2! k;.x;y/ 7! 2xy;is a quadratic form on the vector space k2, and a quadratic space isomorphic to .k2;q/ iscalled a hyperbolic plane. A subspace W of V is

isotropic if q.x/D 0 for some nonzero x 2W; totally isotropic if q.x/D 0 for all x 2W , anisotropic if it is not isotropic (so q.x/ is zero only for x D 0).

The Witt index of .V;q/ is the maximum dimension of a totally isotropic subspace. Forexample, if the Witt index of a hyperbolic plane is 1. If the Witt index is r , then V is anorthogonal sum

V DH1 ? : : :?Hr ? V1 (Witt decomposition)where each Hi is a hyperbolic plane and V1 is anisotropic (Witt decomposition, ALA, I,18.9). The associated algebraic group SO.q/ is split if and only if its Witt index is as largeas possible.

(a) Case dimV D n is even, say, n D 2r . When the Witt index is as large as possiblethere is a basis for which the matrix8 of the form is

0 I

I 0

, and so

q.x1; : : : ;xn/D x1xrC1C Cxrx2r :Note that the subspace of vectors

.; : : : ; r;0; : : : ;0/is totally isotropic. The algebraic subgroup consisting of the diagonal matrices of the form

diag.a1; : : : ;ar ;a11 ; : : : ;a1r /

is a split maximal torus in SO.q/.(b) Case dimV D n is odd, say, nD 2rC1. When the Witt index is as large as possible

there is a basis for which the matrix of the form is

0@1 0 00 0 I0 I 0

1A, and soq.x0;x1; : : : ;xn/D x20Cx1xrC1C Cxrx2r :

The algebraic subgroup consisting of the diagonal matrices of the form

diag.1;a1; : : : ;ar ;a11 ; : : : ;a1r /

is a split maximal torus in SO.q/.Notice that any two nondegenerate quadratic spaces with largest Witt index and the

same dimension are isomorphic. In the rest of the notes, Ill refer to these groups as thesplit SOns.

8Recall that SO.q/ consists of the automorphs of this matrix with determinant 1, i.e., SO.q/.R/ consists

of the nn matrices A with entries in R and determinant 1 such that At0 I

I 0

AD

0 I

I 0

:


Example: the symplectic group

2.4 Let V D k2n, and let be the skew-symmetric form with matrix

0 I

I 0

, so

.Ex; Ey/D x1ynC1C Cxny2nxnC1y1 x2nyn:The corresponding symplectic group Spn is split, and the algebraic subgroup consisting ofthe diagonal matrices of the form

diag.a1; : : : ;ar ;a11 ; : : : ;a1r /

is a split maximal torus in Spn.

The Lie algebra of an algebraic groupWe assume that the reader is familiar with the notion of a Lie algebra (LAG, I, 1).

Let k" defD kX=.X2/ be the ring of dual numbers. There are homomorphisms of k-algebras

k! k"! k; a 7! aC0"; aCb" 7! a;and so, for any algebraic group G over k, there are homomorphisms

G.k/!G.k"/!G.k/:The Lie algebra Lie.G/ of G is defined to be the kernel of the second homomorphism:

Lie.G/D Ker.G.k"/!G.k//:Following the usual convention, we often denote the Lie algebra of an algebraic group bythe corresponding fraktur letter.

For example,

GLn.k"/D fACB" j A 2 GLn.k/, B 2Mn.k/g;and so

gln D fI CB" j B 2Mn.k/g'Mn.k/:

Let V be a vector space, and let V "D k"V D V V". ThenEndk"-lin.V "/D fC" j ; 2 Endk-lin.V /g (5)

where C" acts on V " according to the rule.C"/.xCy"/D .x/C ..y/C.x//":

Therefore

GLV .k"/D GL.V "/D fC" j 2 Aut.V /, 2 End.V /g,


and so

glV D fidV C" j 2 End.V /g' End.V /:

Lie is a functor. For example, a representation r WG ! GLV of G on V defines ahomomorphism dr Wg! End.V /, which can be described as follows: an element g ofG.k"/ is mapped by r to an automorphism r.g/ of V "; if g lies Lie.G/, then r.g/ DidV C" for some 2 End.V /, and dr.g/D .

The groupG.k"/ acts on itself by conjugation, and this action preserves g. It thereforeinduces an action of G.k/G.k"/ on g,

G.k/g! g:The same construction gives an action

G.R/gR! gR (6)of G.R/ on gR DRg for every k-algebra R, and hence a representation

AdWG! GLgof G on the vector space g. On applying Lie, we get a homomorphism

adWg! Endk-lin.g/: (7)We define the bracket on g by the formula

ad.x/.y/D x;y:For example, when G D GLV , the action (6) is

; 7! 1WGL.VR/End.VR/! End.VR/.In particular, an element idV C" of glV GL.V "/ acts on End.V "/ as the endomor-phism

xCy" 7! .idV C"/ .xCy"/ .idV "/D xCy"C . xx /".In other words, idV C" acts on End.V "/ as idV C" where is the endomorphism ofEnd.V / such that .x/D xx (see (5) with End.V / for V ). Hence, with the usualidentifications,

ad./.x/D xx ; ;x 2 glV D End.V /,and the bracket on glV is

;D :The above constructions are natural inG, and so, for any algebraic subgroupG of GLn,

the homomorphisms Ad and ad and the bracket for G are induced by those on GLn.

PROPOSITION 2.5 Let H be a smooth algebraic subgroup of a connected algebraic groupG. If LieH D LieG then H DG; in particular G is smooth.

PROOF. For an algebraic groupG, dimLieG dimG, with equality ifG is smooth. There-fore

dimH D dimLieH D dimLieG dimG:Now dimH dimG because H is a subgroup of G, and so dimH D dimG. This impliesthat H DG. 2


The roots of a split reductive groupLet .G;T / be a split reductive group, and let

AdWG! GLg; gD Lie.G/;be the adjoint representation. In particular, T acts on g, and because T is diagonalizable,

gD g0M

g

where g0 is the subspace on which T acts trivially, and g is the subspace on which T actsthrough the nontrivial character (see 1.14). The nonzero occurring in this decomposi-tion are called the roots of .G;T /. They form a finite subset RDR.G;T / of X.T /.9

Example: GL2

2.6 We take T be the split maximal torus

T Dx1 0

0 x2

x1x2 0

:

ThenX.T /D Z1Z2

where a1Cb2 is the characterdiag.x1;x2/ 7! diag.x1;x2/a1Cb2 D xa1xb2 :

The Lie algebra g of GL2 is gl2 DM2.k/ with A;B D AB BA;and T acts on g byconjugation,

x1 0

0 x2

a b

c d

x11 00 x12

D

a x1x2b

x2x1c d

!:

Write Eij for the matrix with a 1 in the ij th-position, and zeros elsewhere. Then T actstrivially on g0 D kE11C kE22, through the character D 1 2 on g D kE12, andthrough the character D 21 on g D kE21.

Thus, RD f;g with D 12. When we use 1 and 2 to identify X.T / withZZ, R becomes identified with f.e1 e2/g:

Example: SL2

2.7 We take T to be the split torus

T Dx 0

0 x1

:

ThenX.T /D Z

9There are several different notations used for the roots, R.G;T /, .G;T /, and .G;T / all seem tobe used, often by the same author. Conrad et al. 2010 write R D .G;T / in 3.2.2, p. 94, and R.G;T / D.X.T /;.G;T /;X.T /;.G;T /_/ in 3.2.5, p. 96.


where is the character diag.x;x1/ 7! x. The Lie algebra g of SL2 is

sl2 Da b

c d

2M2.k/

aCd D 0

;

and T acts on g by conjugation,x 0

0 x1

a b

c a

x1 00 x

D

a x2b

x2c a

Therefore, the roots are D 2 and D2. When we use to identify X.T / with Z,R becomes identified with f2;2g:

Example: PGL2

2.8 Recall that this is the quotient of GL2 by its centre, PGL2 D GL2 =Gm. For all localk-algebras R, PGL2.R/D GL2.R/=R. We take T to be the split maximal torus

T Dx1 0

0 x2

x1x2 0

x 0

0 x

x 0

:

ThenX.T /D Z

where is the character diag.x1;x2/ 7! x1=x2. The Lie algebra g of PGL2 isgD pgl2 D gl2=fscalar matricesg;

and T acts on g by conjugation:x1 0

0 x2

a b

c d

x11 00 x12

D

a x1x2b

x2x1c d

!:

Therefore, the roots are D and D. When we use to identify X.T / with Z, Rbecomes identified with f1;1g.

Example: GLn

2.9 We take T to be the split maximal torus

T D Dn D(

x1 0

:::0 xn

! x1 xn 0

):

ThenX.T /D

M1inZi

where i is the character diag.x1; : : : ;xn/ 7! xi . The Lie algebra g of gln isgln DMn.k/ with A;BD AB BA;


and T acts on g by conjugation:

x1 0

:::0 xn

!0B@a11 a1n::: aij

::::::

:::an1 ann

1CA0@x11 0:::

0 x1n

1AD0BBB@

a11 x1xn a1n::: xi

xjaij

:::

::::::

xnx1an1 ann

1CCCA :Write Eij for the matrix with a 1 in the ij th-position, and zeros elsewhere. Then T acts

trivially on g0 D kE11C C kEnn and through the character ij defD i j on gij DkEij . Therefore

RD fij j 1 i;j n; i j g:When we use the i to identify X.T / with Zn, then R becomes identified with

fei ej j 1 i;j n; i j gwhere e1; : : : ; en is the standard basis for Zn.

The root datum of a split reductive groupDEFINITION 2.10 A root datum is a quadrupleRD .X;R;X_;R_/ where10 X;X_ are free Z-modules of finite rank in duality by a pairing h ; iWX X_! Z, R;R_ are finite subsets of X and X_ in bijection by a map $ _,

satisfying the following conditions

(rd1) h;_i D 2 for all 2R;(rd2) s.R/R for all 2R; where s is the homomorphism X !X defined by

s.x/D xhx;_i; x 2X , 2R;(rd3) the group of automorphisms W.R/ of X generated by the s for 2R is finite.

Note that (rd1) implies thats./D;

and that the converse holds if 0. Moreover, because s./D,s.s.x//D s.xhx;_i/D .xhx;_i/hx;_is./D x;

i.e.,s2 D 1:

Clearly, also s.x/D x if hx;_iD 0. Thus, s should be considered an abstract reflectionin the hyperplane orthogonal to _.

10More accurately, it is an ordered sextuple,

.X;X_;h ; i;R;R_;R!R_/;but everyone says quadruple. Or, as Brian Conrad points out, it is quintuple because the bijection R! R_ isuniquely determined by everything else (Conrad et al. 2010, 3.2.4).


The elements of R and R_ are called the roots and coroots of the root datum (and _is the coroot of ). The group W DW.R/ of automorphisms of X generated by the s for 2R is called the Weyl group of the root datum.

We want to attach to each split reductive group .G;T / a root datumR.G;T / withX D X.T /;R D roots;X_ D X.T / with the pairing X.T /X.T /! Z in (4), p. 16R_ D coroots (to be defined).

Example: .SL2;T / with T the subgroup of diagonal matrices:

2.11 In this case,

X DX.T /D Z; Wdiag.x;x1/ 7! xX_ DX.T /D Z; W t 7! diag.t; t1/;

and

RD f;g D f2;2gR_ D f_;_g D f;g:

Note thatt

7! diag.t; t1/ 27! t2and so h;_i D 2 in fact, we had only one choice for _. As always,

s./D; s./D etc., and so s.R/R. Finally, s has order 2, and so the Weyl group W.R/D f1;sg isfinite. HenceR.SL2;T / is a root system, isomorphic to

.Z;f2;2g;Z;f1;1g/with the canonical pairing hx;yi D xy and the bijection 2$ 1, 2$1.

Example: (PGL2;T / with T the subgroup of diagonal matrices.

2.12 In this case,

X DX.T /D Z; Wdiag.x1;x2/ 7! x1=x2X_ DX.T /D Z; W t 7! diag.t;1/;

and

RD f;g D f;gR_ D f_;_g D f2;2g:

Note thatt27! diag.t2;1/ 7! t2

and so h;_i D 2. One checks thatR.PGL2;T / is a root system, isomorphic to.Z;f1;1g;Z;f2;2g/

with the canonical pairing hx;yi D xy and the bijection 1$ 2, 1$2.


Root data of rank 1.

2.13 If is a root, then so also is , and there exists an _ such that h;_i D 2.It follows immediately, that R.SL2;T / and R.PGL2;T / are the only two root data withX D Z and R nonempty. There is also the root datum

.Z;;;Z;;/;which is attached to the split reductive group .Gm;Gm/.

Example: .GLn;Dn/.

2.14 In this case

X DX.Dn/DM

iZi ; i Wdiag.x1; : : : ;xn/ 7! xi

X_ DX.Dn/DM

iZi ; i W t 7! diag.1; : : : ;1;

it ;1; : : : ;1/

and

RD fij j i j g; ij D i j ;R_ D f_ij j i j g; _ij D i j :

Note that

tij7! diag.1; : : : ; it ; : : : ;

j

t1; : : :/ij7! t2

and so hij ;_ij i D 2. Moreover, s.R/R for all 2R. We have, for example,

sij .ij /Dijsij .ik/D ik hik;_ij iij

D ik hi ;i iij .if k i;j )D i k .i j /D jk

sij .kl/D kl .if k i;j , l i;j ).Finally, let E.ij / be the permutation matrix in which the i th and j th rows have beenswapped. The action

A 7!E.ij / A E.ij /1ofEij on GLn by inner automorphisms stabilizes T and swaps xi and xj . Therefore, it actson X D X.T / as sij . This shows that the group generated by the sij is isomorphic tothe subgroup of GLn generated by the E.ij /, which is isomorphic to Sn. In particular, Wis finite. HenceR.GLn;Dn/ is a root datum, isomorphic to

.Zn;fei ej j i j g;Zn;fei ej j i j gequipped with the standard pairing hei ; ej i D ij and the bijection .ei ej /_ D ei ej .Here, as usual, e1; : : : ; en is the standard basis for Zn.


Definition of the corootsIn the above examples we wrote down the coroots without giving any idea of how to find(or even define) them. In this subsection, we remedy this.

For G equal to SL2 or PGL2 there is no problem: each group has only one root , andthere is a unique choice for the coroot _. In the general case, we show that each root arises as the root of a copy of SL2 or PGL2 inside G, and we can take _ to be thecorresponding coroot of the copy.

Semisimple groups of rank 0 or 1

Let G be a reductive group over k. Since any two maximal tori in G become conjugateover kal (see 3.22 below), they have the same dimension. The rank of G is defined to thecommon dimension of its maximal tori.

PROPOSITION 2.15 (a) Every semisimple group of rank 0 is trivial.(b) Every split semisimple group of rank 1 is isomorphic to .SL2;T / or .PGL2;T / with

T as in (2.11) or (2.12).

PROOF. (a) Let G be a semisimple group of rank 0. We may assume that k is algebraicallyclosed. Any subgroup H of G such that all the elements of H.k/ are unipotent is itselfunipotent (1.18), hence solvable, and hence trivial (G being semisimple). In particular, ifGis nontrivial, not all the elements ofG.k/ are unipotent, and so it contains a semisimple ele-ment (1.11). The smallest algebraic subgroup H of G such that H.k/ contains the elementis commutative, and therefore decomposes into Hs Hu (see 1.21) where Hs.k/ consistsof semisimple elements andHu.k/ of unipotent elements. AgainHuD 0. If all semisimpleelements of G.k/ are of finite order, then G is finite, and hence trivial (being connected).If G.k/ contains a semisimple element of infinite order, then H contains nontrivial torus,contradicting the fact that G has rank 0.

(b) One shows that G contains a solvable subgroup B such that G=B P1. From thisone gets a nontrivial homomorphism G! Aut.P1/' PGL2. See Theorem 3.40 below (orSpringer 1998, 7.2.4 in the case that k is algebraically closed). 2

Centralizers and normalizers of tori

PROPOSITION 2.16 Let T be a torus in a reductive group G.

(a) The centralizer CG.T / of T in G is a reductive group; in particular, it is smooth andconnected.

(b) The identity component of the normalizer NG.T / of T in G is CG.T /; therefore,NG.T /=CG.T / is a finite etale group.

(c) The torus T is maximal if and only if T D CG.T /:

PROOF. (a) We defer the proof to the next section cf. 3.44 (note that CG.T / is shown tobe smooth in AGS, XIV, 7.3, and so the classical proof, which shows that CG.T /red isreductive, in fact shows that CG.T / is reductive).

(b) Certainly NG.T / CG.T /DCG.T /. But NG.T /=CG.T / acts faithfully on T ,and so it is trivial by rigidity (1.16). Now

NG.T /=CG.T /DNG.T /=NG.T / D 0.NG.T //


with 0.NG.T // etale (see 1.8).(c) Certainly, if CG.T / D T , then T is maximal because any torus containing T is

contained in CG.T /. Conversely, if T is a maximal torus in G, then it is a maximal torus inC

defD CG.T /. As C is reductive, its radical RC is a torus. Clearly RC T , and so equalsit. Hence C=T is a semisimple group. It has rank 0, because a nontrivial torus in C=Twould correspond to a torus in C properly containing T , and so it is trivial (2.15a). ThusCG.T /D T . 2

The Weyl group of .G;T / is defined to be

W.G;T /DNG.T /.k/=CG.T /.k/:We shall see later that W.G;T / equals to the Weyl group of the root datum of .G;T /. Inparticular, it doesnt change when we extend the field, and so

W.G;T /DW.Gkal ;Tkal/DNG.T /.kal/=CG.T /.kal/D .NG.T /=CG.T //.kal/:As

W.G;T / .NG.T /=CG.T //.k/;we see that NG.T /=CG.T / is a constant finite algebraic group, equal to the Weyl group ofthe root datum of .G;T /.

NOTES Perhaps prove (a) by tannakian means. We may suppose that k is algebraically closed.To give a representation of T on a vector space V amounts to giving a gradation of V , and theelements of G centralizing T are those that preserve these gradations. From this one can see thatthe representations of the centralizer are semisimple and satisfy the condition for it to be stronglyconnected.

Example: .SL2;T / with T the subgroup of diagonal elements.

2.17 In this case, CG.T /D T but

NG.T /Da 0

0 a1[

0 a1a 0

:

Therefore W.G;T /D f1;sg where s is represented by the matrix nD0 1

1 0

. Note that

n

a 0

0 a1n1 D

0 1

1 0

a 0

0 a1

0 11 0

Da1 00 a

,

and sosdiag.a;a1/D diag.a1;a/:

Example: .GLn;Dn/.

2.18 In this case, CG.T / D T but NG.T / contains the permutation matrices (those ob-tained from I by permuting the rows). For example, let E.ij / be the matrix obtained fromI by interchanging the i th and j th rows. Then

E.ij / diag. ai aj / E.ij /1 D diag. aj ai /:


More generally, let be a permutation of f1; : : : ;ng, and let E./ be the matrix obtainedby using to permute the rows. Then 7! E./ is an isomorphism from Sn onto the setof permutation matrices, and conjugating a diagonal matrix by E./ simply permutes thediagonal entries. The E./ form a set of representatives for CG.T /.k/ in NG.T /.k/, andso W.G;T /' Sn.

Definition of the coroot of a root

LEMMA 2.19 Let .G;T / be a split reductive group. The action of W.G;T / on X.T /stabilizes R.

PROOF. Let s 2W.G;T /, and let n 2G.k/ represent s. Then s acts on X.T / (on the left)by

.s/.t/D .n1tn/; t 2 T .kal/:Let be a root. Then, for x 2 .g/kal and t 2 T .kal/,

t .nx/D n.n1tn/x D s..s1ts/x/D .s1ts/sx;and so T acts on sg through the character s, which must therefore be a root. 2

THEOREM 2.20 Let .G;T / be a split reductive group, and let be a root of .G;T /.

(a) There exists a unique subgroup U of G isomorphic to Ga such that, for any isomor-phism uWGa! U,

t u.a/ t1 D u..t/a/, all t 2 T .R/, a 2G.R/:(b) Let T D Ker./, and let G be centralizer of T in G. Then W.G;T / contains

exactly one nontrivial element s, and there is a unique _ 2X.T / such thats.x/D xhx;_i; for all x 2X.T /: (8)

Moreover, h;_i D 2.(c) The algebraic group G is generated by T , U, and U.

The cocharacter _ is called the coroot of , and the group U in (a) is called the rootgroup of . Thus the root group of is the unique copy of Ga in G that is normalized byT and such that T acts on it through .

We prove 2.20 after first giving a consequence and some examples.

COROLLARY 2.21 For any split reductive group .G;T /, the system

R.G;T /D .X.T /;R;X.T /;R_/with R_ D f_ j 2Rg and the map 7! _WR!R_ determined by (8) is a root datum.

PROOF. Condition (rd1) holds by (b). The s attached to lies in W.G;T / W.G;T /,and so stabilizes R by the lemma. Finally, all s lie in the Weyl group W.G;T /, and sothey generate a finite group (in fact, the generate exactly W.G;T /; see 3.43). 2


NOTES Better statement: Let .G;T / be a split reductive group, and let be a root of .G;T /. Thenthere exists a unique homomorphism of (affine) algebraic groups11

uWga !Gsuch that

t u.a/ t1 D u..t/a/for all R, t 2 T .R/, a 2G.R/, and Lie.u/ is the given inclusion g! g.

Example: .SL2;T /.

2.22 Let be the root 2. Then T D 1 and G D G. The unique s 1 in W.G;T / isrepresented by

0 1

1 0;

and the unique _ for which (8) holds is .

Example: .GLn;Dn/.

2.23 Let G D GLn, and let D 12 D 12. ThenT D fdiag.x;x;x3; : : : ;xn/ j xxx3 : : :xn 1g

and G consists of the invertible matrices of the form0BBBBB@ 0 0 0 00 0 0

: : ::::

0 0 0

1CCCCCA :Clearly

n D

0BBBBB@0 1 0 0

1 0 0 0

0 0 1 0: : :

:::

0 0 0 1

1CCCCCArepresents the unique nontrivial element s of W.G;T /. It acts on T by

diag.x1;x2;x3; : : : ;xn/ 7! diag.x2;x1;x3; : : : ;xn/:For x Dm11C Cmnn,

sx Dm21Cm12Cm33C CmnnD xhx;12i.12/:

Thus (8), p. 28, holds if and only if _ is taken to be 12.In general, the coroot _ij of ij is

t 7! diag.1; : : : ;1; it ;1; : : : ;1;j

t1;1; : : : ;1/:Clearly hij ;_ij i D ij _ij D 2.

11Recall that, for a finite-dimensional vector space V , Va is the algebraic group R RV .


Proof of Theorem 2.20.

The semisimple rank of a reductive group is the rank of its derived group (i.e., the dimensionof a maximal torus in DG).LEMMA 2.24 Let be a root of the split reductive group .G;T /, let T DKer./, and letG be centralizer of T in G. Then .G;T / is a split reductive group of semisimple rank 1.

PROOF. The group G is reductive by (2.16a), and the torus T is maximal in G becauseit is maximal in G. As T is split, this shows that .G;T / is a split reductive group. Thereis an isogeny

DG!G=TIas G=T has rank 1, so also does DG. 2

LEMMA 2.25 Let .G;T / be a split reductive group of semisimple rank 1, and let 2R.G;T /. Then there exists a homomorphism uWGa!G such that

t u.x/ t1 D u..t/x/ for all t 2 T .R/, x 2G.R/: (9)The image of u is independent of the choice of u.

PROOF. Let u be a homomorphism satisfying (9). Then

t u.x/ t1 u.x/1 D u...t/1/x/;and so the image of u lies in DG. Thus, we may suppose that G is semisimple, and that.G;T / is as in (2.11) or (2.12). A homomorphism u satisfying (9) has image a subgroup Uof G such that Lie.U/D g. There is only one such smooth connected subgroup, namely,U2. 2

This proves (a) of Proposition 2.20, and the remaining statements can be proved bylooking at each of the two cases (2.11) or (2.12).

Alternatively, to avoid a case-by-case argument, apply the following lemma.

LEMMA 2.26 Let G be the subgroup of G of semisimple rank 1 generated by T , U, andU, and let n 2G represent the nontrivial element of the Weyl group of .G;T /. Thenthere is a unique _ 2X_ such that

nD h;_i all 2X: (10)

PROOF. Let 2 X D X.T /. We first show that there exists a G-module V such thatV 0. To see this, regard as an element of O.T /, and let f be an element of O.G/that restricts to it. Let V be any finite-dimensional subspace of O.G/ containing f andstable under G. For v 2 V ,

u.a/v DX

i0aivi ; some vi 2 V; (11)

because u.a/ is a polynomial in a. If v 2 V for some 2 X , then vi 2 V Ci. Thisis a simple calculation using (11) and the definition of U. We have that V 0 and thatPi2ZVCi is invariant under T , U, and U, and hence also under G and n. Thus

w D C i for some i 2 Z, and the map 7! i defines an _ 2 X_ for which (10)holds; and clearly _ is unique. 2


It follows that h;_i D 2, either because w D or because w2 D 1, both ofwhich hold because the semisimple rank of G is 1. If we extend scalars Z!R and X_ isidentified with X via any n-invariant positive definite bilinear form, then _ takes on thefamiliar form .2=.;// so that the lemma implies that important fact that 2.;/=.;/2Z for any two roots ;.

NOTES Following SGA 3, XX, 1.3, define an elementary k-system to be a triple .G;T;/ where Gis a reductive group of semisimple rank 1, T is a maximal torus inG, and is a roots of .G;T /. Firstprove everything for such a triple. For example, there exists a unique homomorphism uWga ! Gsuch that

t u.a/ t1 D u..t/a/for all R, t 2 T .R/, a 2 G.R/, and Lie.u/ is the given inclusion g ! g. Then deduce Theorem2.20. At present, the proof of Theorem 2.20 is too sketchy.

First applications of root dataComputing the centre of a reductive group

We explain how to compute the centre of a reductive group from its root datum.

PROPOSITION 2.27 (a) Every maximal torus T in a reductive algebraic group G containsthe centre Z.G/ of G.

(b) The kernel of AdWT ! GLg is Z.G/.

PROOF. (a) Clearly Z.G/ CG.T /, but (see 2.16) CG.T /D T .(b) When k has characteristic zero, the kernel of AdWG ! GLg is Z.G/ for any con-

nected algebraic group (see LAG), and so the kernel of Ad jT isZ.G/\T DZ.G/. In gen-eral, Ker.Ad/=Z.G/ is a unipotent group (AGS, XV, 3.6), and so the image of Ker.Ad jT /in Ker.Ad/=Z.G/ is trivial (AGS, XV, 2.17), which implies that Ker.Ad jT /Z.G/. Thereverse inclusion follows from (a). 2

From the proposition,

Z.G/D Ker.Ad jT /D\

2RKer./:

For example,

Z.GLn/D\

ij Ker.i j /D(

x1 0

:::0 xn

! xi D xj if i j

)\GLn

'GmIZ.SL2/D Ker.2/D

x 00 x1

j x2 D 1' 2;

Z.PGL2/D Ker./D 1:

On applying X to the exact sequence

0!Z.G/! T t 7!..t//!Y

2RGm (12)


we get (1.15) an exact sequenceM2RZ

.m/ 7!Pm!X.T /!X.Z.G//! 0;and so

X.Z.G//D X.T /

fsubgroup generated by Rg (13)

For example,

X.Z.GLn//' Zn=X

ij Z.ei ej /' Z (by .ai / 7!Pai /I

X.Z.SL2//' Z=.2/IX.Z.PGL2//' Z=ZD 0:

Semisimple and toral root data

It is possible to determine whether a reductive group is semisimple or a torus from its rootdatum.

DEFINITION 2.28 A root datum is semisimple if the subgroup of X generated by R is offinite index.

PROPOSITION 2.29 A split reductive group is semisimple if and only if its root datum issemisimple.

PROOF. A reductive group is semisimple if and only if its centre is finite, and so this followsfrom (13). 2

DEFINITION 2.30 A root datum is toral if R is empty.

PROPOSITION 2.31 A split reductive group is a torus if and only if its root datum is toral.

PROOF. If the root datum is toral, then (13) shows that ZG D T . Hence DG has rank 0,and so it is trivial (2.15a). Therefore

G(1.29)D ZG DG D T .

Conversely, if G is a torus, then the adjoint representation is trivial and so gD g0. 2

The root data of the classical split semisimple groupsWe compute the root datum attached to each of the classical split semisimple groups. Ineach case the strategy is the same. We work with a convenient form of the group G in GLn.We first compute the weights of the split maximal torus of G on gln, and then check thateach nonzero weight occurs in g (in fact, with multiplicity 1). Then for each we find thegroup G centralizing T, and use it to find the coroot _.


Example (An): SLnC1.

Take T to be the maximal torus of diagonal matrices

diag.t1; : : : ; tnC1/; t1 tnC1 0:Then

X.T /DLi ZiZ; i Wdiag.t1; : : : ; tnC1/ 7! tiDPiX.T /D

Paii 2Li Zi jPai D 0 ; Xaii W t 7! diag.ta1 ; : : : ; tan/;

with the pairing such thathj ;Pi aii i D aj :

Write xi for the class of i in X.T /. Then T acts trivially on the set g0 of diagonalmatrices in g, and it acts through the character ij

defD xi xj on kEij , i j (cf. p.25).Therefore

RD fij j 1 i;j nC1; i j g:It remains to compute the coroots. Consider, for example, the root D 12. Then G

in (2.23) consists of the matrices of the form0BBBBB@ 0 0 0 00 0 0

: : ::::

0 0 0

1CCCCCAwith determinant 1. As in (2.20), W.G;T /D f1;sg where s acts on T by interchangingthe first two coordinates. Let DPnC1iD1 ai xi 2X.T /. Then

s./D a2 x1Ca1 x2CPnC1iD3 ai xiD h;12i.x1 x2/:

In other words,s12./D h;_12i12

with _12 D 12, which proves that 12 is the coroot of 12.When the ordered index set f1;2; : : : ;nC1g is replaced with an unordered set, we find

that everything is symmetric between the roots, and so the coroot of ij is

_ij D i jfor all i j .

Example (Bn): SO2nC1 :

Consider the symmetric bilinear form on k2nC1,

.Ex; Ey/D 2x0y0Cx1ynC1CxnC1y1C Cxny2nCx2nyn


Then SO2nC1defD SO./ consists of the 2nC 1 2nC 1 matrices A of determinant 1 such

that.AEx;A Ey/D .Ex; Ey/;

i.e., such that

At

0@1 0 00 0 I0 I 0

1AAD0@1 0 00 0 I0 I 0

1A :The Lie algebra of SO2nC1 consists of the 2nC12nC1 matrices A of trace 0 such that

.AEx; Ey/C.Ex;A Ey/D 0;(AGS, XI, 4.8), i.e., such that

At

0@1 0 00 0 I0 I 0

1AC0@1 0 00 0 I0 I 0

1AAD 0:Take T to be the maximal torus of diagonal matrices

diag.1; t1; : : : ; tn; t11 ; : : : ; t1n /

Then

X.T /DM

1inZi ; i Wdiag.1; t1; : : : ; tn; t11 ; : : : ; t

1n / 7! ti

X.T /DM

1inZi ; i W t 7! diag.1; : : : ;iC1t ; : : : ;1/

with the pairing h ; i such thathi ;j i D ij :

All the charactersi ; i j ; i j

occur as roots, and their coroots are, respectively,

2i ; i j ; i j:

Example (Cn): Sp2n.

Consider the skew symmetric bilinear form k2nk2n! k;.Ex; Ey/D x1ynC1xnC1y1C Cxny2nx2nyn:

Then Sp2n consists of the 2n2n matrices A such that.AEx;A Ey/D .Ex; Ey/;

i.e., such that

At

0 I

I 0AD

0 I

I 0:


The Lie algebra of Spn consists of the 2n2n matrices A such that.AEx; Ey/C.Ex;A Ey/D 0;

i.e., such that

At

0 I

I 0C

0 I

I 0AD 0:

Take T to be the maximal torus of diagonal matrices

diag.t1; : : : ; tn; t11 ; : : : ; t1n /:

Then

X.T /DM

1inZi ; i Wdiag.t1; : : : ; tn; t11 ; : : : ; t

1n / 7! ti

X.T /DM

1inZi ; i W t 7! diag.1; : : : ;it ; : : : ;1/

with the obvious pairing h ; i. All the characters2i ; i j ; i j

occur as roots, and their coroots are, respectively,

i ; i j ; i j:

Example (Dn): SO2n.

Consider the symmetric bilinear form k2nk2n! k;.Ex; Ey/D x1ynC1CxnC1y1C Cxny2nCx2ny2n:

Then SOn D SO./ consists of the nn matrices A of determinant 1 such that.AEx;A Ey/D .Ex; Ey/;

i.e., such that

At0 I

I 0

AD

0 I

I 0

:

The Lie algebra of SOn consists of the nn matrices A of trace 0 such that.AEx; Ey/C.Ex;A Ey/D 0;

i.e., such that

At0 I

I 0

C0 I

I 0

AD 0:

When we write the matrix asA B

C D

, then this last condition becomes

ACDt D 0; C CC t D 0; BCB t D 0:Take T to be the maximal torus of diagonal matrices


diag.t1; : : : ; tn; t11 ; : : : ; t1n /

and let i , 1 i r , be the characterdiag.t1; : : : ; tn; t11 ; : : : ; t1n / 7! ti :

All the charactersi j ; i j

occur, and their coroots are, respectively,

i j ; i j:REMARK 2.32 The subscript on An, Bn, Cn, Dn denotes the rank of the group, i.e., thedimension of a maximal torus.

The main theoremsLet .G;T / be a split reductive group, with root datumR.G;T /:THEOREM 2.33 Let T 0 be a split maximal torus in G. Then T 0 is conjugate to T by anelement of G.k/.

PROOF. See 3.22, 3.23 below. 2

EXAMPLE 2.34 Let G D GLV , and let T be a split torus. A split torus is (by definition)diagonalizable, i.e., there exists a basis for V such that T Dn. Since T is maximal, itequals Dn. This proves the theorem for GLV since any two bases are conjugate by anelement of GLV .k/.

It follows that the root datum attached to .G;T / depends only on G (up to isomor-phism).

THEOREM 2.35 (EXISTENCE) Every reduced root datum arises from a split reductive group.G;T /.

PROOF. See Section 7 below (or Springer 1998, 16.5). 2

A root datum is reduced if the only multiples of a root that can also be a root are and .THEOREM 2.36 (ISOMORPHISM) Every isomorphism R.G;T /!R.G0;T 0/ of root dataarises from an isomorphism .G;T /! .G0;T 0/.

PROOF. See Section 6 below or Springer 1998, 16.3.2. 2


In fact, with the appropriate definitions, every isogeny of root data (or even epimor-phism of root data) arises from an isogeny (or epimorphism) of split reductive groups.G;T /! .G0;T 0/.

Later we shall define the notion of a base for a root datum. If bases are fixed for .G;T /and .G0;T 0/, then ' can be chosen to send one base onto the other, and it is then unique upto composition with a homomorphism inn.t/ such that t 2 T .kal/ and .t/ 2 k for all .NOTES Add pinnings (epinglages) cf. mo17594.

NOTES Things we didnt prove in this section:

Every split semisimple group of rank 1 is isomorphic to SL2 or PGL2 (2.15). The centralizer of a torus in a reductive group is a reductive group (2.16).

3 Borel fixed point theorem and applicationsBorel showed that, by using some algebraic geometry, for example, by using the complete-ness of algebraic varieties to replace compactness, it is possible to carry over some of thearguments from Lie groups to algebraic groups. In this section, we explain this. In thisversion of the notes, we often assume that k is algebraically closed.

Brief review of algebraic geometryWe need the notion of an algebraic variety (not necessarily affine). To keep things simple, Iuse the conventions of my notes AG. Thus, an algebraic variety over k is topological spaceX together with a sheaf OX such that OX .U / is a k-algebra of functions U ! k; it isrequired that X admits a finite open covering X DSUi such that, for each i , .Ui ;OX jUi /is isomorphic to SpmAi for some affine k-algebraAi ; finally,X is required to be separated.

3.1 A projective variety is a variety that can be realized as a closed subvariety of someprojective space Pn. Every closed subvariety of a projective variety is projective.

3.2 Let V be a vector space of dimension n over k.(a) The set P.V / of one-dimensional subspaces of V is in a natural way a projective

variety: in fact the choice of a basis for V defines a bijection P.V /$ Pn1.(b) Let Gd .V / be the set of d -dimensional subspaces of V . When we fix a basis for V ,

the choice of a basis for S determines a dnmatrixA.S/whose rows are the coordinates ofthe basis elements. Changing the basis for S multipliesA.S/ on the left by an invertible d d matrix. Thus, the family of d d minors of A.S/ is determined by S up to multiplicationby a nonzero constant, and so defines a point P.S/ of P.

nd/1: One shows that the map

S 7! P.S/ is a bijection of Gd .V / onto a closed subset of P.nd/1 (called a Grassmannvariety; AG 6.26). For a d -dimensional subspace S of V , the tangent space

TS .Gd .V //' Hom.S;V=S/(ibid. 6.29).

(c) For any sequence of integers n > dr > dr1 > > d1 > 0 the set of flagsVr V1

with Vi a subspace of V of dimension di has a natural structure of a projective algebraicvariety (called a flag variety; AG p. 133).

3.3 An algebraic variety X is said to be complete if, for all algebraic varieties T , theprojection map X T ! T is closed (AG 7.1). A closed subvariety of a complete varietyis complete. Every projective variety is complete (AG 7.7). If X is complete, then itsimage under any regular map X ! Y is closed and complete (AG 7.3). An affine variety iscomplete if and only if it has dimension zero, and so is a finite set of points (AG 7.5).

3.4 A regular map f WX! S is proper if, for all regular maps T ! S , the mapXS T !T is closed. If f WX ! S is proper, then, for any complete subvariety Z of X , the imagef Z of Z in S is complete (AG 8.26); moreover, X is complete if S is complete (AG 8.24).Finite maps are proper because they are closed (AG 8.7) and the base change of a finite mapis finite.

38

3. Borel fixed point theorem and applications 39

3.5 A regular map 'WY ! X is said to be dominant if its image is dense in X . If 'is dominant, then the map f 7! ' f WOX .X/! OY .Y / is injective, and so, when Xand Y are irreducible, ' defines a homomorphism k.X/! k.Y / of the fields of rationalfunctions. A dominant map Y ! X of irreducible varieties is said to be separable whenk.Y / is separably generated over k.X/, i.e., it is a finite separable extension of a purelytranscendental extension of k.X/. A regular map 'WY ! X of irreducible varieties isdominant and separable if and only if there exists a nonsingular point y 2 Y such x D '.y/is nonsingular and the map d'WTy.Y /! Tx.X/ is surjective, in which case, the set of suchpoints y is open (apply the rank theorem, AG 5.32).

3.6 A bijective regular map of algebraic varieties need not be an isomorphism. For ex-ample, x 7! xpWA1! A1 in characteristic p corresponds to the map of k-algebras T 7!T pWkT ! kT , which is not an isomorphism, and

t 7! .t2; t3/WA1! fy2 D x3g A2

corresponds to the map kt2; t3 ,! kt , which is not an isomorphism. In the first example,the map is not separable, and in the second the curve y2D x3 is not normal. Every separablebijective map 'WY ! X with X normal is an isomorphism (AG 10.12 shows that ' isbirational, and AG 8.19 then shows that it is an isomorphism).

3.7 The set of nonsingular points of a variety is dense and open (AG 5.18). Therefore, analgebraic variety on which an algebraic group acts transitively by regular maps is nonsin-gular (cf. AG 5.20). As a nonsingular point is normal (i.e., OX;x is integrally closed; seeCA 17.5), the same statements hold with nonsingular replaced by normal.12

QuotientsIn AGS, VIII, 17.5 we defined the quotient of an algebraic group G by a normal algebraicsubgroupN . Now we need to consider the quotient ofG by an arbitrary algebraic subgroupH .

In this section, we assume that k is algebraically closed, and we consider only reduced(hence smooth) algebraic groups.

3.8 Let G be a smooth algebraic group. An action of G on a variety X is a regular mapG X ! X such that the underlying map of sets is an action of the abstract group G onthe set X . Every orbit for the action is open in its closure, and every orbit of minimumdimension is closed. In particular, each orbit is a subvariety of X and there exist closedorbits. The isotropy group Go at a point o of X is the inverse image of o under the regularmap g 7! goWG!X , and so is an affine algebraic subgroup of G). (AG 10.6.)

3.9 Let H be a smooth subgroup of the smooth algebraic group G. A quotient of G byH is an algebraic variety X together with a transitive action GX ! X of G and a point

12The proof that nonsingular points are normal is quite difficult. It is possible to avoid it for some applica-tions by showing directly that the set of normal points in an algebraic variety is open and dense (Springer 1998,5.2.11).


o fixed by H having the following universal property: for any variety X 0 with an action ofG and point o0 of X 0 fixed by H , the map go 7! go0WX.k/!X 0.k/ is regular:

G X

X 0:

g 7! go

g 7! go0

Clearly, a quotient is unique (up to a unique isomorphism) if it exists.

3.10 LetH be a smooth subgroup of a connected smooth algebraic groupG, and considera transitive action G X ! X of G on a variety X . Suppose that there exists a point o inX such that the isotropy group Go at o is H . The pair .X;o/ is a quotient of G by H if andonly if the map

g 7! g oWG '!Xis separable. To see this, note that the fibres of the map ' are the conjugates of H . Inparticular, they all have dimension dimH , and so

dimX D dimGdimH(AG 10.9). The map

.d'/eWTeG! TeXcontains TeH in its kernel. As dimTeG D dimG and dimTeH D dimH (because G andH are smooth), we see that .d'/e is surjective if and only if its kernel is exactly TeH .Therefore .X;o/ is a quotient of G by H if Ker.d'/e D Lie.H/ (by 3.5).

PROPOSITION 3.11 A quotient exists for every smooth subgroup H of a smooth algebraicgroup G. It is a quasi-projective algebraic variety, and the isotropy group of the distin-guished point is H .

PROOF. As in the case of a normal subgroup, a key tool in the proof is Chevalleys theorem(AGS, VIII, 13.1): there exists a representation G!GLV and a one-dimensional subspaceL in V such that H is the stabilizer of L. The action of G on V defines a action

GP.V /! P.V /of the algebraic group G on the algebraic variety P.V /. Let X be the orbit of L in P.V /,i.e., X D G L. Then X is a subvariety of P.V /, and so it is quasi-projective (becauseP.V /) is projective). Let ' be the map g 7! gLWG! X . One can show that the kernel of.d'/e is h, and so the map is separable. This implies that X is a quotient of G by H . 2

We let G=H denote the quotient of G by H . Because H is the isotropy group at thedistinguished point,

.G=H/.k/DG.k/=H.k/:


3.12 The proof of the proposition shows that, for any representation .V;r/ of G and lineL, the orbit of L in P.V / is a quotient of G by the stabilizer of L in G. For example, letG D GL2 and H D T2 D f. 0 /g. Then G acts on k2, and H is the subgroup fixing theline LD f.0 /g. Then G=H is isomorphic to the orbit of L, but G acts transitively on theset of lines, and so G=H ' P1.

3.13 When H is normal in G, this construction of G=H agrees with that in (AGS, VIII,8.1). In particular, when H is normal, G=H is affine.

3.14 LetGX!X be a transitive action of a smooth algebraicG on a varietyX , and leto be a point of X . Let H be the isotropy group at o. The universal property of the quotientshows that the map g 7! go factors through G=H . The resulting map G=H ! X is finiteand purely inseparable.

ASIDE 3.15 Quotients exist under much more general hypotheses. Let G be an affine algebraicgroup scheme over an arbitrary field k, and let H be an affine subgroup scheme. Then then thesheaf associated with the flat presheaf R G.R/=H.R/ is represented by an algebraic schemeG=H over k. See DG, III, 3, 5.4, p. 341.

For any homomorphism of k-algebras R! R0, the map G.R/=H.R/! G.R0/=H.R0/ is in-jective. Therefore, the statement means that there exists a (unique) scheme G=H of finite type overk and a morphism WG!G=H such that,

for all k-algebras R, the nonempty fibres of the map .R/WG.R/! .G=H/.R/ are cosets ofH.R/;

for all k-algebras R and x 2 .G=H/.R/, there exists a finitely generated faithfully flat R-algebra R0 and a y 2G.R0/ lifting the image of x in .G=H/.R0/.

The variety G=H constructed in (3.11) has these properties (cf. the proof of AGS, VII, 7.4), and soagrees with the more general object.

The Borel fixed point theoremTHEOREM 3.16 When a smooth connected solvable algebraic group acts on an algebraicvariety, no orbit contains a complete subvariety of dimension > 0.

PROOF. LetG be as in the statement of the theorem. If the theorem fails forG acting onX ,then it fails for Gkal acting on Xkal , and so we may suppose that k is algebraically closed.We use induction on the dimension of G.

It suffices to prove the theorem for G acting on G=H where H is a smooth connectedsubgroup of G (because any orbit of G acting on a variety has a finite covering by such avariety G=H (3.14), and the inverse image of a complete subvariety under a finite map iscomplete (3.4)).

Fix a smooth connected subgroup H of G with H G. Either H maps onto G=DGor it doesnt. In the first case, DG acts transitively on G=H , and the statement followsby induction.13 In the second case, the image of H in G=DG is a proper normal sub-group, and we let N denote the proper normal subgroup of G containing DG and such that

13DG is again smooth and connected by AGS, XVI, 3.5.


G=N ' Im.H/=DG (N exists by the Correspondence Theorem AGS, IX, 5.1, it is con-nected by AGS, XIII, 3.11, and it is smooth by AGS, XVII, 1.1). Consider the quotient map WG=H ! G=N . Let Z be a complete subvariety of G=H , which we may assume to beconnected. Because N is normal, G=N is affine (see 3.13), and so the image of Z in G=Nis a point (see 3.3). Therefore Z is contained in one of the fibres of the map , but theseare all isomorphic to N=H , and so we can conclude by induction again. 2

THEOREM 3.17 (BOREL FIXED POINT THEOREM) A smooth connected solvable algebraicgroup acting on a complete variety over an algebraically closed field always has a fixedpoint.

PROOF. According to (3.8), the action has a closed orbit, which is complete (see 3.3), andhence is a finite set of points (Theorem 3.16). As the group is connected, so is the orbit. 2

REMARK 3.18 It is possible to recover the Lie-Kolchin theorem (AGS, XVI, 4.7) from theBorel fixed point theorem. Let G be a smooth connected solvable subgroup of GLV , andlet X be the collection of maximal flags in V (i.e., the flags corresponding to the sequencedimV D n > n 1 > > 1 > 0). As noted in (3.2), this has a natural structure of aprojective variety, and G acts on it by a regular map

g;F 7! gF WGX !Xwhere

g.Vn Vn1 /D gVn gVn1 :According to the theorem, there is a fixed point, i.e., a maximal flag such that gF D F forall g 2G.k/. Relative to a basis e1; : : : ; en adapted to the flag,14 G Tn.

ASIDE 3.19 The improvement, Theorem 3.16 of Borels fixed point theorem, is from Allcock 2009.For the standard proof, see AG, 10.8.

Borel subgroupsThroughout this subsection, G is a smooth connected algebraic group.

DEFINITION 3.20 A Borel subgroup of G is a smooth subgroup B such that Bkal is amaximal smooth connected solvable subgroup of Gkal .

For example, T2defD f. 0 /g is a Borel subgroup of GL2 (it is certainly connected and

solvable, and the only connected subgroup properly containing it is GL2, which isnt solv-able).

THEOREM 3.21 Let G be a smooth connected algebraic group.

If B is a Borel subgroup of G, then G=B is projective. Any two Borel subgroups of G are conjugate by an element of kal.14That is, such that e1; : : : ; ei is a basis of Vi .


PROOF. We may assume k D kal.We first prove that G=B is projective when B is a Borel subgroup of largest possible

dimension. Apply the theorem of Chevalley (AGS, VIII, 13.1) to obtain a representationG! GLV and a one-dimensional subspace L such that B is the subgroup fixing L. ThenB acts on V=L, and the Lie-Kolchin theorem gives us a maximal flag in V=L stabilized byB . On pulling this back to V , we get a maximal flag,

F WV D Vn Vn1 V1 D L 0in V . Not only does B stabilize F , but (because of our choice of V1), B is the isotropygroup at F , and so the map G=B ! B F is finite (see 3.14). This shows that, when welet G act on the variety of maximal flags, G F is the orbit of smallest dimension, becausefor any other maximal flag F 0, the stabilizer H of F 0 is a solvable algebraic subgroup ofdimension at most that of B , and so

dimG F 0 D dimGdimH dimGdimB D dimG F:ThereforeG F is a closed (3.8), and hence projective, subvariety of the variety of maximalflags in V . The map G=B ! G F is finite, and so G=B is complete (see 3.4). As it isquasi-projective, this implies that it is projective.

To complete the proof of the theorems, it remains to show that for any Borel subgroupsB and B 0 with B of largest possible dimension, B 0 gBg1 for some g 2 G.k/ (becausethe maximality of B 0 will then imply that B 0 D gBg1). Let B 0 act on G=B by b0;gB 7!b0gB . The Borel fixed point theorem shows that there is a fixed point, i.e., for some g 2G.k/, B 0gB gB . Then B 0g gB , and so B 0 gBg1 as required. 2

THEOREM 3.22 Let G be a smooth connected algebraic group. All maximal tori in G areconjugate by an element of G.kal/.

PROOF. Let T and T 0 be maximal tori. Being smooth, connected, and solvable, they arecontained in Borel subgroups, say T B , T 0 B 0. For some g 2G, gB 0g1 D B , and sogT 0g1 B . Now T and gT 0g1 are maximal tori in the B , and we can apply the theoremfor connected solvable groups (Springer 1998, 6.3.5; eventually, AGS, XVI, 7.1). 2

REMARK 3.23 We mention two stronger results.

(a) (Grothendieck): Let G be a smooth affine algebraic group over a separably closedfield k. Then any two maximal tori in G are conjugate by an element of G.k/. InConrad et al. 2010, Appendix A, 2.10, p. 401, it is explained how to deduce this fromthe similar statement with k algebraically closed.

(b) (Borel-Tits): Let G be a smooth affine algebraic group over a field k. Then any twomaximal split tori in G are conjugate by an element of G.k/. In Conrad et al. 2010,Appendix C, 2.3, p. 506, it is explained how to deduce this from the statement thatmaximal tori in solvable groups are G.k/-conjugate.

THEOREM 3.24 For any Borel subgroup B of G, G DSg2G.kal/gBg1.SKETCH OF PROOF. Show that every element x of G is contained in a connected solvablesubgroup of G (sometimes the identity component of the closure of the group generated byx is such a group), and hence in a Borel subgroup, which is conjugate to B (3.21). 2


THEOREM 3.25 For any torus T in G, CG.T / is connected.

PROOF. We may assume that k is algebraically closed. Let x 2 CG.T /.k/, and let B bea Borel subgroup of G. Then x is contained in a connected solvable subgroup of G (see3.24), and so the Borel fixed point theorem shows that the subset X of G=B of cosets gBsuch that xgB D gB is nonempty. It is also closed, being the subset where the regular mapsgB 7! xgB and gB 7! gB agree. As T commutes with x, it stabilizes X , and anotherapplication of the Borel fixed point theorem shows that it has a fixed point in X . In otherwords, there exists a g 2G such that

xgB D gBTgB D gB:

Thus, both x and T lie in gBg1 and we know that the theorem holds for connected solvablegroups (Springer 1998, 6.3.5; eventually, AGS, XVI, 7.2). Therefore x 2 CG.T /. 2

Parabolic subgroupsIn this subsection, assume that k is algebraically closed. Throughout, G is a smooth con-nected algebraic group.

DEFINITION 3.26 An algebraic subgroup P of G is parabolic if G=P is projective.

THEOREM 3.27 Let G be a smooth connected algebraic group. An algebraic subgroup Pof G is parabolic if and only if it contains a Borel subgroup.

PROOF. H) : Let B be a Borel subgroup of G. According to the Borel fixed pointtheorem, the action of B on G=P has a fixed point, i.e., there exists a g 2 G such thatBgP D gP . Then Bg gP and g1Bg P .(H : Suppose P contains the Borel subgroup B . Then there is quotient map G=B!

G=P . Recall thatG=P is quasi-projective, i.e., can be realized as a locally closed subvarietyof PN for some N . Because G=B is projective, the composite G=B ! G=P ! PN hasclosed image (see 3.4), but this image is G=P , which is therefore projective. 2

COROLLARY 3.28 Every connected solvable parabolic algebraic subgroup of a connectedalgebraic group is a Borel subgroup.

PROOF. Because it is parabolic it contains a Borel subgroup, which, being maximal amongconnected solvable groups, must equal it. 2

Examples of Borel and parabolic subgroupsExample: GLV

Let G D GLV with V of dimension n. Let F be a maximal flagF WVn1 V1


and let G.F / be the stabilizer of F , so

G.F /.R/D fg 2 GL.V R/ j g .Vi R/ Vi R for all ig:ThenG.F / is connected and solvable (because the choice of a basis adapted to F defines anisomorphismG.F /!Tn), and GLV =G.F / is projective (because GL.V / acts transitivelyon the space of all maximal flags in V ). Therefore, G.F / is a Borel subgroup (3.28). Forg 2 GL.V /,

G.gF /D g G.F / g1:Since all Borel subgroups are conjugate, we see that the Borel subgroups of GLV are pre-cisely the groups of the form G.F / with F a maximal flag.

Now consider G.F / with F a (not necessarily maximal) flag. Clearly F can be re-fined to a maximal flag F 0, and G.F / contains the Borel subgroup G.F 0/. Therefore it isparabolic. Later well see that all parabolic subgroups of GLV are of this form.

Example: SO2n

Let V be a vector space of dimension 2n, and let be a nondegenerate symmetric bilinearform on V with Witt index n. By a totally isotropic flag we mean a flag Vi Vi1 such that each Vi is totally isotropic. We say that such a flag is maximal if it has themaximum length n.

LetF WVn Vn1 V1

be such a flag, and choose a basis e1; : : : ; en for Vn such that Vi D he1; : : : ; ei i. Thenhe2; : : : ; eni? contains Vn and has dimension15 nC 1, and so it contains an x such thathe1;xi 0. Scale x so that he1;xiD 1, and define enC1D x 12.x;x/e1. Then .enC1; enC1/D0 and .e1; enC1/D 1. Continuing in this fashion, we obtain a basis e1; : : : ; en; enC1; : : : ; e2nfor which the matrix of is

0 I

I 0

.

Now let F 0 be a second such flag, and choose a similar basis e01; : : : ; e0n for it. Then thelinear map ei 7! e0i is orthogonal, and maps F onto F 0. Thus O./ acts transitively on theset X of maximal totally isotropic flags of V . One shows that X is closed (for the Zariskitopology) in the flag variety consisting of all maximal flags Vn V1, and is thereforea projective variety. It may fall into two connected components which are the orbits ofSO./.16

Let G D SO./. The stabilizer G.F / of any totally isotropic flag is a parabolic sub-group, and one shows as in the preceding case that the Borel subgroups are exactly thestabilizers of maximal totally isotropic flags.

15Recall that in a nondegenerate quadratic space .V;/,

dimW CdimW ? D dimV:

16Let .V;/ be a hyperbolic plane with its standard basis e1; e2. Then the flags

F1W he1iF2W he2i

fall into different SO./ orbits.


Example: Sp2nAgain the stabilizers of totally isotropic flags are parabolic subgroups, and the Borel sub-groups are exactly the stabilizers of maximal totally isotropic flags.

Example: SO2nC1Same as the last two cases.

EXERCISE I-1 Write out a proof that the Borel subgroups of SO2n, Sp2n, and SO2nC1 arethose indicated above.

Smooth connected algebraic groups of rank oneLet k be an algebraically closed field. In this section, we prove (following Allcock 2009)that every semisimple group of rank 1 over k is isogenous to PGL2 (Proposition 2.15).

Preliminaries

Let G be a smooth connected algebraic group, and let B be a Borel subgroup of G. Thefollowing are consequences of the completeness of the flag variety G=B .

3.29 All Borel subgroups, and all maximal tori, are conjugate (see 3.21; in fact, for anymaximal torus T , NG.T / acts transitively on the Borel subgroups containing T ).

3.30 The group G is solvable if one of its Borel subgroups is commutative.

3.31 The connected centralizer CG.T / of any maximal torus T lies in every Borel sub-group containing T .

3.32 The normalizer NG.B/ of a Borel subgroup B contains B as a subgroup of finiteindex (and therefore is equal to its own normalizer).

3.33 The centralizer of a torus T in G has dimension equal dimgT .

The proof uses that CG.T / is smooth (see 2.16).

PROPOSITION 3.34 Let .V;r/ be a representation of a torus T . In any closed subvariety ofdimension d in P.V / stable under T , there are at least d C1 points fixed by T .

PROOF. (Borel 1991, IV, 13.5). Let Y be a closed subvariety of P.V / of dimension d . AsT is connected, it leaves stable each irreducible component of Y , and so we may supposethat Y is irreducible. We use induction on the dimension of Y . If dimY D 0, t

reductive groups

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