Linear algebraic groups, armand borel

Graduate Texts in Mathematics 126 Editorial Board l.H Ewing F.W Gehring P.R Halmos Graduate Texts in Mathematics I 10 II 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 TAKEUTljZARING Introduction to Axiomatic Set Theory 2nd ed OXTOBY Measure and Category 2nd ed SCHAEFFER Topological Vector Spaces HILTONjSTAMMBACH A Course in Homological Algebra MAc LANE Categories for the Working Mathematician HUGHES/PIPER Projective Planes SERRE A Course in Arithmetic TAKEUTl/ZARING Axiomatic Set Theory HUMPHREYS Introduction to Lie Algebras and Representation Theory COHEN A Course in Simple Homotopy Theory CONWAY Functions of One Complex Variable 2nd ed BEALS Advanced Mathematical Analysis ANDERSON/FULLER Rings and Categories of Modules GOLUBITSKy/GUILLEMIN Stable Mappings and Their Singularities BERBERIAN Lectures in Functional Analysis and Operator Theory WINTER The Structure of Fields ROSENBLATT Random Processes 2nd ed HALMOS Measure Theory HALMOS A Hilbert Space Problem Book 2nd ed., revised HUSEMOLLER Fibre Bundles 2nd ed HUMPHREYS Linear Algebraic Groups BARNES/MACK An Algebraic Introduction to Mathematical Logic GREUB Linear Algebra 4th ed HOLMES Geometrio Functional Analysis and its Applications HEWITT/STROMBERG Real and Abstract Analysis MANES Algebraic Theories KELLEY General Topology ZARISKI/SAMUEL Commutative Algebra Vol I ZARISKI/SAMUEL Commutative Algebra Vol II JACOBSON Lectures in Abstract Algebra I: Basic Concepts JACOBSON Lectures in Abstract Algebra II: Linear Algebra JACOBSON Lectures in Abstract Algebra Ill: Theory of Fields and Galois Theory HIRSCH Differential Topology SPITZER Principles of Random Walk 2nd ed WERMER Banach Algebras and Several Complex Variables 2nd ed KELLEy/NAMIOKA et al Linear Topological Spaces MONK Mathematical Logic GRAUERT/FRITZSCHE Several Complex Variables ARVESON An Invitation to C*-Algebras KEMENy/SNELL/KNAPP Denumerable Markov Chains 2nd ed ApOSTOL Modular Functions and Dirichlet Series in Number Theory 2nd ed SERRE Linear Representations of Finite Groups GILLMAN/JERISON Rings of Continuous Functions KENDIG ElemenIary Algebraic Geometry LoEVE Probability Theory I 4th ed LOEVE Probability Theory II 4th ed MOISE Geometric Topology in Dimensions and continued after Index Armand Borel Linear Algebraic Groups Second Enlarged Edition Springer Science+Business Media, LLC Annand Borel School of Mathematics Institute for Advanced Study Princeton, New Jersey 08450 USA Editorial Board lH Ewing Department of Mathematics Indiana University Bloomington, IN 47401 USA F.W Gehring Department of Mathematics University of Michigan Ann Arbor, MI 48019 USA P.R Halmos Department of Mathematics Santa Clara University Santa Clara, CA 95093 USA First edition published by W.A Benjamin, Inc., 1969 Library of Congress Cataloging-in-Publication Data Borel, Armand Linear algebraic groups ! Armand Borel.-2nd enl ed p cm.-(Graduate texts in mathematics; 126) lncludes bibliographical references and indexes ISBN 978-1-4612-6954-0 ISBN 978-1-4612-0941-6 (eBook) DOI 10.1007/978-1-4612-0941-6 Linear algebraic groups Title II Series 1991 QA564.B58 512'.5-dc20 90·19774 Printed on acid-free paper © 1991 Springer Science+Business Media New York Originally published by Springer-Verlag New York Inc in 1991 Softcover reprint of the hardcover 2nd edition 1991 AlI rights reserved This work may not be translated or copied in whole or in par! without the written permission of the publisher (Springer Science+Business Media, LLC), except for brief excerpts in connection with reviews or scholarly analysis Use in connection with any form of information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed is forbidden The use of general descriptive names, trade names, trademarks etc., in this publication, even if the former are not especially identified, is not to be taken as a sign that such names, as understood by the Trade Marks and Merchandise Marks Act, may accordingly be used fteely by anyone Typeset by Thomson Press (India) Ltd., New Delhi 54 ISBN 978-1-4612-6954-0 Introduction to the First Edition These Notes aim at providing an introduction to the theory of linear algebraic groups over fields Their main objectives are to give some basic material over arbitrary fields (Chap I, II), and to discuss the structure of solvable and of reductive groups over algebraically closed fields (Chap III, IV) To complete the picture, they also include some rationality properties (§§15, 18) and some results on groups over finite fields (§ 16) and over fields of characteristic zero (§7) Apart from some knowledge of Lie algebras, the main prerequisite for these Notes is some familiarity with algebraic geometry In fact, comparatively little is actually needed Most of the notions and results frequently used in the Notes are summarized, a few with proofs, in a preliminary Chapter AG As a basic reference, we take Mumford's Notes [14], and have tried to be to some extent self-contained from there A few further results from algebraic geometry needed on some specific occasions will be recalled (with references) where used The point of view adopted here is essentially the set theoretic one: varieties are identified with their set of points over an algebraic closure of the groundfield (endowed with the Zariski-topology), however with some traces of the scheme point of view here and there These Notes are based on a course given at Columbia University in Spring, 1968, * at the suggestion of Hyman Bass Except for Chap V, added later, Notes were written up by H Bass, with some help from Michael Stein, and are reproduced here with few changes or additions He did this with marvelous efficiency, often expanding or improving the oral presentation In particular, the emphasis on dual numbers in §3 in his, and he wrote up Chapter AG, of which only a very brief survey had been given in the course It is a pleasure to thank him most warmly for his contributions, without which these Notes would hardly have come into being at this time I would also like to thank Miss P Murray for her careful and fast typing of the manuscript, and lE Humphreys, J.S Joel for their help in checking and proofreading it A Borel Princeton, February, 1969 *Lectures from May 7th on qualified as liberated class, under the sponsorship of the Students Strike Committee Space was generously made available on one occasion by the Union Theological Seminary Introduction to the Second Edition This is a revised and enlarged edition of the set of Notes: "Linear algebraic groups" published by Benjamin in 1969 The added material pertains mainly to rationality questions over arbitrary fields with, as a main goal, properties of the rational points of isotropic reductive groups Besides, a number of corrections, additions and changes to the original text have been made In particular: §3 on Lie algebras has been revised §6 on quotient spaces contains a brief discussion of categorical quotients The existence of a quotient by finite groups has been added to §6, that of a categorical quotient under the action of a torus to §8 In §11, the original proof of Chevalley's normalizer theorem has been replaced by an argument I found in 1973, (and is used in the books of Humphreys and Springer) In §14, some material on parabolic subgroups has been added §15, on split solvable groups now contains a proof of the existence of a rational point on any homogeneous space of a split solvable group, a theorem of Rosenlicht's proved in the first edition only for GL I and Ga §§ 19 to 24 are new The first one shows that in a connected solvable k-group, all Cartan k-subgroups are conjugate under G(k), a result also due to M Rosenlicht §§20, 21 are devoted to the so-called relative theory for isotropic reductive groups over a field k: Conjugacy theorems for minimal parabolic ksubgroups, maximal k-split tori, existence of a Tits system on G(k), rationality of the quotient of G by a parabolic k-subgroup and description of the closure of a Bruhat cell As a necessary complement, §22 discusses central isogenies §23 is devoted to examples and describes the Tits systems of many classical groups Finally, §24 surveys without proofs some main results on classifications and linear representations of semi-simple groups and, assuming Lie theory, relates the Tits system on the real points of a reductive group to the similar notions introduced much earlier by E Cartan in a Lie theoretic framework Many corrections have been made to the text of the first edition and my thanks are due to J Humphreys, F.D Veldkamp, A.E Zalesski and V Platonov who pointed out most of them Vlll Introduction to the Second Edition I am also grateful to Mutsumi Saito, T Watanabe and especially G Prasad, who read a draft of the changes and additions and found an embarrassing number of misprints and minor inaccuracies I am also glad to acknowledge help received in the proofreading from H.P Kraft, who read parts of the proofs with great care and came up with a depressing list of corrections, and from D labon The first edition has been out of print for many years and the question of a reedition has been in the air for that much time After Addison-Wesley had acquired the rights to the Benjamin publications they decided not to proceed with one and released the publication rights to me I am grateful to SpringerVerlag to have offered over ten years ago to publish a reedition in whichever form I would want it and to several technical editors (starting with W Kaufmann-Biihler) and scientific editors for having periodically prodded me into getting on with this project I am solely to blame for the procrastination In preparing the typescript for the second edition, use was made to the extent possible of copies of the first one, whose typography was quite different from the one present techniques allow one to produce The insertions of corrections, changes and additions, which came in successive ways, presented serious problems in harmonization, pasting and cutting I am grateful to Irene Gaskill and Elly Gustafsson for having performed them with great skill I would also like to express my appreciation to Springer-Verlag for their handling ofthe publication and their patience in taking care of my desiderata A Borel Contents Introduction to the First Edition Introduction to the Second Edition Conventions and Notation V Vll Xl CHAPTER AG- Background Material From Algebraic Geometry §1 §2 §3 §4 §5 §6 §7 §8 §9 §1O §11 §12 §13 §14 §15 §16 § 17 §18 Some Topological Notions Some Facts from Field Theory Some Commutative Algebra Sheaves Affine K-Schemes, Prevarieties Products; Varieties Projective and Complete Varieties Rational Functions; Dominant Morphisms Dimension Images and Fibres of a Morphism k-structures on K -Schemes k-Structures on Varieties Separable points Galois Criteria for Rationality Derivations and Differentials Tangent Spaces Simple Points Normal Varieties References 10 11 14 17 19 20 20 21 23 26 29 32 36 40 42 45 CHAPTER I-General Notions Associated With Algebraic Groups § The Notion of an Algebraic Groups §2 Group Closure; Solvable and Nilpotent Groups 46 56 x Contents §3 The Lie Algebra of an Algebraic Group §4 Jordan Decomposition 62 79 CHAPTER II-Homogeneous Spaces §5 Semi-Invariants §6 Homogeneous Spaces §7 Algebraic Groups in Characteristic Zero 89 94 105 CHAPTER 111- Solvable Groups §8 Diagonalizable Groups and Tori §9 Conjugacy Classes and Centralizers of Semi-Simple Elements §10 Connected Solvable Groups 111 127 134 CHAPTER IV- Borel Subgroups; Reductive Groups § 11 §12 § 13 § 14 Borel Subgroups Cartan Subgroups; Regular Elements The Borel Subgroups Containing a Given Torus Root Systems and Bruhat Decomposition in Reductive Groups 147 159 163 179 CHAPTER V- Rationality Questions Split Solvable Groups and Subgroups Groups over Finite Fields Quotient of a Group by a Lie Subalgebra Cartan Subgroups over the Groundfield Unirationality Splitting of Reductive Groups § 19 Cartan Subgroups of Solvable Groups §20 Isotropic Reductive Groups §21 Relative Root System and Bruhat Decomposition for Isotropic Reductive Groups §22 Central Isogenies §23 Examples §24 Survey of Some Other Topics A Classification B Linear Representations C Real Reductive Groups References for Chapters I to V Index of Definition Index of Notation § IS §16 §17 §18 203 210 213 218 222 224 229 246 253 268 268 270 274 280 282 286 276 Rationality Questions v Write 9t!1) = (f n 9t!1) EB Po, where Po is the canonical complement We can find a complement PI to f n c in c which is invariant under Ad,K Let P = Po + PI' We claim that it is the (- I)-eigenspace of the differential of a Cartan involution First it clearly satisfies (1), hence p:k + p~k - p (kEf, pEp) is an involution of 1) Also, P is invariant under Ad~K, since Po is so automatically Then the above remarks show that p is the differential of an automorphism e' of Co x ~Ho having (KnCO) x (Kn~W) has its fixed point set, which commutes with K, acting by inner automorphisms Since EiJHO n CO is finite, the kernel of the canonical surjective morphism CO x ~ HO -> HO is finite, too, hence contained in K, and consequently pointwise fixed under e'; therefore e' goes down to HO and defines a Cartan involution eo of HO, again commuting with Int H " K It is then routine to check that eo extends to a Cartan involution of H with respect to K (b) In view of (I), a subspace of P is a subalgebra if and only if it is commutative It is known that the maximal commutative subalgebras of P are conjugate under KO, and that if a is one, then the eigenvalues of Ad X (X EO) are all real and I} is a direct sum of eigenspaces 1);, (AEO*), where 1);, = {Y Ell lad X( Y) '* = ie(X)' Y, (X EO)} The ie's for which I}; are the roots of I} with respect to o We have a = PI + ao, where 00 = Po n a and we can identify 0: to the space of linear forms on a which are zero on Pl' Let W = A·K"(O)/~ KO(O) Then Woperates in a natural way on a, leaving PI pointwise fixed, hence also on 0: The set

Ad G be the adjoint representation It is a central isogeny, whose differential is an isomorphism 278 v Rationality Questions Since the maximal R-split tori of Ad G are the images of the maximal R-split tori of G (22.7), we may further identify G with Ad G c GL(g) In suitable coordinates a is represented by diagonal matrices Therefore the smallest algebraic group d(a) whose Lie algebra contains a (cf 7.1) is a R-split torus If L(S)(R) =1= a, then S(R)/(exp a) is not compact, contradicting the fact that !!L H(a)/(exp a) is compact (see 24.7(1)) Hence a = L(S)(R) It now follows from the definitions that exHdex defines a bijection of R$(S, G) onto $(a, f)) As a consequence, the Weyl groups W(a, f)) and R W are the same By definition RW is the quotient A/'G(S)/!!LG(S), where S is a maximal R-split torus; by the above, we may assume that a = L(S)(R) By 21.2, we have ,A'"G(S) = AIG(S)(R)'!!L G(S), Now an element gEG(R) normalizes or centralizes S if and only if Ad g normalizes or centralizes s(R) = a We can therefore also write RW = -1" G(R)(a)/!!L G(R)(a) But this is equal to the Weyl group of $(a, f)) hence to JVKo(a)/!!L Ko(a) Therefore the inclusion A'" Ko(a) -> ,iV G(R)(a) induces a bijection of v-v Ko(a)/!!L Ko(a) onto ,IV' G(R)(a)/!!LG(R)(a), which completes the justification of the claim made at the beginning of this section It also follows that these quotients are equal to iVK(a)/!!LK(a) In this particular case, the groups W' and W defined in 24(b) are therefore equal, although G(R) is not necessarily connected 24.7 Two remarks on maximal compact subgroups We have used above a fact which has been known for a long time, but for which I not know a reference I shall take this opportunity to prove it as well as a sort of counterpart which has been familiar for an equally long time, has been used occasionally, but whose proof has not been published so far to my knowledge (i) We again adopt the framework of 24.6(a), fix K and write for K We want to prove e e Proposition Let U, V be subsets of p Assume there exists gEH such that Ad g(U) = V Then there exists kEK such that Ad k(u) = Adg(u)for all UE U We claim first it suffices to show that if X Ep and pEP are such that Ad p(X)Ep, then Ad p(X) = X Indeed, assume this to hold, let g, U, V be as in the statement and XEU We can write g=k'p (kEK,pEP) hence Adg(X)E V implies Adp(X)EAdk-J(V) C p, therefore Ad p(X) = X and Ad g(X) = Ad k(X) for all X E U Let now X Ep and pEP be such that Y = Ad p(X)Ep Since e is an automorphism we have Ad e(p)(de(X)) = de(Y) But de = - [d on p and e(p)=p-l, therefore Adp-J(-X)= -Adp(X) hence Ad p2(X)=X We have p=expZ, with ZEp and adZ diagonalizable (over R) in view of 23(e) and 8.15 If (Jc;) are its eigenvalues, then those of Ad p2 are exp 2A i In particular, the eigenspace for the eigenvalue of Ad p2 is the zero eigenspace for ad Z, and also the I-eigenspace for Ad p Since X is fixed under Ad p2, it is then also fixed under Ad p and it commutes with Z From this we see that V.24 Survey of Some Other Topics 279 (1) as was asserted at the end of 24.6(b) (ii) Let now H be any Lie group with finite component group We have already recalled part of the theorem on maximal compact subgroups It is also known that G/K is homeomorphic to euclidean space In fact, this can be made much more precise (cf [Ho:lII, §3, N° 2]): Given K, there exists a K-invariant subspace m of f), a diffeomorphism qJ of m onto a closedK-invariant submanifold M of H, which is K-equivariant, K acting by the adjoint representation on Ill, by inner automorphisms on M, such that qJ x Id:(X,k)l +qJ(X)'k (XEIlI,kEK) is a diffeomorphism of m x K onto H, which is K-equivariant, K acting by the adjoint representation on Ill, by left translations on K and H As a counterpart to the previous proposition, we have the Proposition Let U and V be subsets of K Assume there exists gEG such that 9U = V Then there exists kEK such that k X =9x for all XEU Proof First consider a maximal compact subgroup Land mEM such that mK = L (which can always be found, by the conjugacy of maximal compact subgroups and the decomposition H = M· K) We claim that m centralizes LnK Let xELnK Then m-1'x'm=YEK hence m'y= x'm = x·m·x-1·x and therefore x = y and m = x'm'x- by the uniqueness of the decomposition H = M· K Applied to L = K, this shows in particular (2) 1"H(K) =(1 HK)n M· K Let now g, U, V be as in the proposition Then V c K n K We have just proved the existence of mEM centralizing 9K n K, in particular V, such that "'K = K The element m- also centralizes V, whence (3) The element m- normalizes K, hence can be written by (2) as (4) We have then, by (3) and (4): (5) which proves the proposition 280 Rationality Questions v References for Chapters I to V A Borel, Groupes linea ires algebriques Annals of Math (2) 64 (1956), 20-82 A Borel and T.A Springer, Rationality properties of linear algebraic groups Proc Symp Pure Math., vol IX (1966), 26-32 A Borel and T.A Springer, Rationality properties of linear algebraic groups /1, Tohoku Math Jour (2) 20 (1968) 443-487 A Borel et J Tits, Groupes reductifs Publ Math I.H.E.S 27 (1965), 55-150 A Borel et J Tits, "Complements a l'article: Groupes reductifs," Publ Math I.H.E.S 41 (1972),253-276 A Borel et J Tits, "Homomorphismes "abstraits" de groupes a1gebriques simples," Annals of Math 97 (1973), 499-571 A Borel et J Tits, 'Theoremes de structure et de conjugaison pour les groupes algebriques lineaires." CR Acad Sci Paris 287 (1978), 55-57 ~ N Bourbaki, Groupes et algebres de Lie Chapitre I: Algebres de Lie, Hermann Paris (1960) N Bourbaki, Groupes et algebres de Lie, Chap IV, V, VI, 2eme edition, Masson, Paris 1981 10 N Bourbaki, Groupes et algcbres de Lie Chap VII VIII, Hermann Paris 1975 11 N Bourbaki, Algebre Commutative, Chap 5-7, Hermann, Paris 1964 12 C Chevalley, Theorie des groupes de Lie (a) Tome II: Groupes algebriques, Paris 1951, (b) Tome III: Groupes algebriques, Hermann Paris 1955 13 C Chevalley, Seminaire sur la classification des groupes de Lie algebriques (Mimeographed Notes), Paris 1956-58 14 M Demazure et P Gabriel, Groupes algebriques, Tome I, Masson, Paris 1970 15 M Demazure et A Grothendieck, Schemas en groupes I, II, III, Springer L.N.M 151, 152, 153, Springer Verlag 1970 16 G Fano, Sulle varieta algebriche can un gruppo cOlllinuo non integrabile di transformazioni proiellive in se Mem Reale Accad d Sci di Torino (2) 46 (1896), 187-218 17 J Humphreys, Linear algebraic groups, Grad Text Math 21, Springer-Verlag 1975 18 N Jacobson, Lie algebras Interscience Tracts in pure and applied math 10, New York 1962, Interscience Pub! 19 E Kolchin, Algebraic matric groups and the Picard- Vessiot theory of homogeneous linear differential equations Annals of Math (2) 49 (1948), 1-42 20 K Kolchin, On certain concepts in the theory of algebraic matric groups Ibid 774-789 21 S Lang, Algebraic groups overfinitefields Amer Jour Math 87 (1965),555-563 22 D Mumford and J Fogarty, Geometric Invariant Theory, 2nd enlarged edition, Ergeb Math u Grenzgeb 34, Springer-Verlag 1982 23 D Mumford, The red book of varieties and schemes, Springer L.N.M 1358, Springer-Verlag 1988 24 T Qno, Arithmetic of algebraic tori Annals of Math (2) 74 (1961), 101-139 25 M Rosenlicht, Some basic theorems on algebraic groups Amer Jour Math 78 (1956),401-443 26 M Rosenlicht, Some rationality questions on algebraic groups Annali di Mat (IV) 43 (1957),25-50 Additional References for §§23 and 24 281 27 M Rosenlicht, On quotient varieties and the affine embedding of certain homogeneous spaces Trans Amer Math Soc 101 (1961),211-223 28 M Rosenlicht, "Questions of rationality for solvable algebraic groups over nonperfectfields" Annali di Mat (IV) 61 (1962), 97-120 29 J.-P Serre, Groupes algebriques et corps de classes, Hermann, Paris 1959 30 J.-P Serre, Cohomologie galoisienne Lect Notes Math 5, Springer-Verlag 1965 31 J.-P Serre, Algebres de Lie semi-simples complexes Benjamin New York 1966 32 T.A Springer, Linear algebraic groups, Progress in Math 9, Birkhiiuser 1981 33 A Weil, On algebraic groups and homogeneous spaces ArneI' Jour Math 77 (1955),493-512 Additional references for §§23 and 24: [Bl] A Borel, Representations lineaires et espaces homogenes kiihleriens des groupes simples compacts, (1954), Coil Papers 1, 392-396, Springer-Verlag 1983 [B2] A Borel, Properties and linear representations of Chevalley groups, in Seminar on algebraic groups and related finite groups, Lect Notes Math 131, 1-55, Springer-Verlag 1955 [B3] A Borel, Linear representations of semi-simple algebraic groups, Proc Symp Pure Math 29, Amer Math Soc (1975), 421-439 [BS] A Borel et J- P Serre, Theoremes de finitude en cohomologie galoisienne, Comm Math Hely 39 (1964),111-164 [C] e CheYalley, Certains schbnas de groupes simples, Sem Bourbaki 1960/61, Exp 219 [D] J Dieudonne, La geometrie des groupes c1assiques, Erg d Math u.i Grenzgeb (N.F.) 5, Springer-Verlag 1955 [He] S Helgason, Differential geometry and symmetric spaces, Acad Press 1962 [Ho] G Hochschild, The structure of Lie groups, Holden-Day, San Francisco 1965 [Hu] Humphreys, Ordinary and modular representations of Cheyalley groups, Lect Notes Math 528, Springer-Verlag 1976 [lM] N Iwahori and H Matsumoto, On some Bruhm decompositions and the structure of the Heeke ring oj p-adic Chevalley groups, Publ Math I.H.E.S 25 (1965), 5-48 [1] 1.e Jantzen, Representations of algebraic groups, Acad Press 1987 [Ta] M Takeuchi, A hyperalgebraic proof of the isomorphism and isogen)' theorems for reductive groups, J Algebra 85 (1983),179-196 [Ti I] Tits, Classification of algebraic semisimple groups, Proc Symp Pur Math IX, ArneI' Math Soc 1966, 32-62 [Ti2] J Tits, Formes quadratiques, groupes orthogonaux et alqebres de Clifford, Iny Math (1968), 19-41 [Ti3] Tits, Representations lineaires irreductibles d'un groupe reduct if sur un corps quelconque, J f reine u angew Math 247 (1971), 196-220 [W] A Weil, Algebras with involution and the classical groups, J Indian Math Soc 24 (1960), 589-623 Index of Definition Action, AG.2.4 closed, 1.8 free, 1.8 principal, 1.8 Additive group G., 1.6 Adjoint representation Ad, 3.5 Admissible scalar product, 14.7,21.1 Affine k-algebra, AG.5.2 Affine K-schemes, AG.5.2, AG.5.3, AG.5.4, AG.5.5 Affine k-varieties, AG.12.1 Affine morphism, AG.6.5 Affine space, AG.7.1 Affine variety, AG.5.3, AG.5.4 Algebraic curves, AG.18.5 Algebraic group, 1.1 Algebraic Lie algebra, §7 Algebraic transformation space, 1.7 Almost direct product of groups, xi Anisotropic (torus), H.14 Annihilator, AG.3.1, AGJ.5 Antiautomorphism, 23.7 Antihermitian form, 23.8 Base change, AG.15.8 for fields, AG.2.1 Basis of a root system, 14.7 Birationality, AG.8.2 Boolean algebra homomorphism, AG.1.3 Borel subalgebra, 14.16 Borel subgroup, 11.1, 23.4, 24.3 Borel-Weil theorem, 24.4 Bruhat decomposition, 14.11 Canonical Cartan involution, 24.6 Cartan involution, 24.6 Cartan-Malcev theorem, 24.6 Cartan subgroup, 11.13 Categorical quotient, 6.16 Cellular decomposition (of G/B), 14.11 Centralizer, 1.7, 23.4, 24.6 Character (of an algebraic group), 5.2 Characteristic exponent, AG.2.2 Classification over K, 24.1, 24.2 Classification theorem, 24.1 Closed immersions, AG.5.6 Closed set of roots, 14.7 Closed subvariety, AG.14.4 Combinatorial dimension, AG.1.4, AG.3.2, AG.3.4 Comorphism, xi Complementary root, 8.17 Completely reducible (representation), 8.19 Complete variety, AG.7.4 Complex semi-simple Lie algebra representations, 24.3 Conjugacy class, 9.1 Conjugate variety, AG.14.3 Complete variety, AG.7.4 Connected components, AG.1.2 Constructible set, AG.!.3 Convolution (right or left), 3.4 Cross section, 6.13 Defect of Q, 23.5 !i-action, 24.5 Density, AG.1.2 Index of Definition Derived series {~iG}, 2.4 Descending central series {~iG}, 2.4 Diagonalizable group, 8.2 Diagonalizable group split over k, 8.2 Diagonal map (of a Hopf algebra), 1.2 Diagonal torus, 23.4 Diagram, 24.1 Diffeomorphism, 24.7 Differential, AG.16.1 Differential criteria, AG.2.3 Dimension, 23.7 Dimension of a variety, AG.9.1, AG.9.2, AG.9.3, AG.IO.! Direct spanning, 14.3 Division algebra, 23.7 with involution, 23.7 Dual numbers, AG.16.2 Dynkin diagram, 14.7 Endomorphism nilpotent, 4.1 semi-simple, 4.1 unipotent, 4.1 Epimorphism, AG.3.5, AG.!2.1 £-O"-hermitian forms, 23.8, 23.9 -£-T-hermetian forms, 23.8, 23.9 Fibre, AG.10.1, AG.l3.2 Field extension, AG.2.1 Flag (rational over k), 15.3 Flag variety, 10.3 Frobenius morphism, §16 Frobenius isogeny, 16.1,24.1 Full ring of fractions, AG.3.!, AG.3.3 Function field, AG.8.l Fundamental highest weights, 24.3 Functor of points, AG.!3.l Galois actions, AG.§ 14 k-structure defined by, AG.14.2 on k-varieties, AG.14.3 on vector spaces, AG.14.! General linear group GL n , 1.6 Generic points, AG.13.5 Geometric reductivity, 24.4 Grassmannian, 10.3 Group algebraic (defined over k), 1.1 283 anisotropic over k, 20.1 diagonalizable (and split over k), 8.2 isotropic over k, 20.1 isotropy, 1.7 linear algebraic, 1.6 nilpotent, 2.4 reductive, 11.21 reductive, k-split, 18.6 semi-simple, 11.21 solvable, 2.4 solvable, k-split, 15.1 trigonalizable (over k), 4.6 Group closure, 2.1 Hopf algebra, 1.2 Hypersurfaces, AG.9.2 Ideal, AG.3.2, AG.3.3, AG.3.4 Idempotents, AG.2.5 Image, AG.IO.l Index of a quadratic form, 23.5 Integral closure, AG.3.6 Integral extensions, AG.3.6 Involutions, 23.7, 23.9 Irreducible components, AG.l.l, AG.1.2, AG.1.3, AG.3.4, AG.3.8, AG.6.4 defined over kS' AG.l2.3 Irreducible root system, 14.7 Isogeny, 16.1, ~22, 23.6 central, 22.3, 22.11 quasi-central, 22.3 Isogeny (of diagrams), 24.1 Isotropy group, AG.2.5, 1.7 Jordan decomposition additive, 4.2 multiplicative, 4.2 in an affine group, 4.4 in the Lie algebra of an affine group, 4.4 k-algebra, AG.5.3, AG.5.4 k-derivations, AG.~ 15, AG.16.1 k-forms of G, 24.2 k-group, 1.1 Killing-Cartan classification, 24.1 k-index of a k-form, 24.2 k-morphic action, 1.7 284 Index of Definition k-morphism, AG.I!.3, AG.14.5 1.1 k-rank, 21.1, 23.1 Krull dimension, AG.604 of A, AG.304 K-scheme, AG.5.3, AG.504, AG.16.3 K -space, AG 5.1 k-splitdiagonalizable group, 8.2, 23.2 reductive group, 18.6 solvable group, 15.1 k-structures, AG.14.2 on k-algebras, AG.II.2 on K-schemes, AG.11.3 on Vector spaces, AG.II.I k-varieties, AG.14.3, AG.1404, AG.14.5 Levi subgroup, 13.22 Lie algebra (restricted), 3.1 Lie algebra of an algebraic group, 3.3 complex semi-simple representations, 24.3 Lie-Kolchin theorem, 10.5 Linear algebraic group, 1.6 locally trivial fibration, 6.13 Linear representations of semi-simple groups 2404 Localization AG.).I AG.15.5 Locally closed sets AG.I.) Local ring, AG.3.l, AG.3.2 on a variety, AG.604 Maximal compact subgroups, 24.7 Morphism of algebraic groups, AG.5.1, AG.10.1, AG.10.2, AG.10.3, 1.\ dominant, AG.8.2, AG.I)o4 Nilpotent elements, AG.2.1, AG.3.3, AG.5.3 Nilpotent endomorphism 4.1 Nilpotent group, 204 Nil radical, AG.3.3, AG.12.1 Noetherian spaces, AG.I.2, AG.304 AG.3.5, AG.3.7, AG.3.9, AG.5.3 Noether normalization, AG.3.7 Non-degenerate quadratic form, 23.5 Normalization, AG.18.2 Normalizer, 1.7 Normal varieties AG.§18 Nullstellensatz, AG.3.8 One-parameter group (multiplicative), 8.6 Open immersion, AG.5.6 Open map, AG.1804 Opposite Borel subgroup, 14.1 Opposite parabolic subgroups, 14.20 Opposition involution, 24.3 Orthogonal groups, 1.6(7),2304,23.6,23.9 in characteristic two, 23.6 Parabolic subgroup, 11.2 p-isogeny, 24.1 Polynomial rings, AG.15.2, AG.15.) Positive roots, 14.7 Presheaves, AGo4.1 Prevariety, AG.5.3 Principal open set, AG.304 Products of open subschemes, AG.6.1 Products of varieties, AG.9.3 Projective spaces, AG.7.2 Projective varieties AG.7J, AG.704 Quadratic forms, 2304 in characteristic two, 2).5 Quasi-coherent modules, AG.5.5 Quasi-compactness, AG.I.2 Quasi-projective variety, AG.7.3 Quotient morphism (over k), 6.1 Quotient (of V by G), 6.3 Radical, 11.21 Rank (of an algebraic group), \2.2 Rational functions, AG.8.1 Rationality questions for rcpresentations, 24.5 Rational representation, 1.6 Rational varieties, AG.13.7 Real reductive groups, 24.6 Reduced rings, AG.2.l, AGJ.3 Reduced root system, 14.7 24.3 Reductive group, 11.21 Reflection, 13.13, 14.7 Regular clement, 12.2 Regular element in a Lie algebra, 18.1 Regular functions, AG.6.3 Regular torus, 13.1 Residue class rings, AG.15.3 Restrictions, AGo4.!, AGo4.2, AGo4.3 Ring of fractions, AG.2.5 Index of Definition Root (of G relative to T), 8.17 Root group, 23.6 Root outside a subgroup, 8.17 Root system, 14.7 R-split torus, 24.6 Semi-direct product, 1.11 Semi-simple anisotropic kernel, 24.2 Semi-simple element in an affine group, 4.5 Semi-simple endomorphism, 4.1 Semi-simple group, 11.21 Semi-simple rank, 13.13 Separable extensions, AG.2.2, AG.2.5 Separable field extensions, AG.15.6 Separable points, AG.13.1 AG.13.2 Separating transcendence basis, AG.2J, AGJ.7 Sheafification, AG.4.3 Sheaves, AGo4.2, AGo4.3 Simple points, AG.§17 Simple roots, 14.8 Singular element, 12.2 Singular subspaces, 23.5 Smooth varieties, AG.17.1 Solvable group, 2.4 Special set of roots, 14.5 Stability group, 1.7 Stalk, AGo4.I, AGo4.3, AG.5.1 Subgroup Borel, 11.1 Cartan, 11.13 parabolic, 11.2 Subschemes defined over k, AG.lIo4 Subvarieties, AG.6.3 defined over k, AG.12.2 Support of a module, AG.3.5 Symmetric algebra, AG.16.3 Symplectic basis, 23.2 Symplectic form, 23.8 285 Symplectic group SPz", 1.6, 23.3, 23.9 Tangent bundle, AG.16.2 lemma, AG.15.9 Tangent spaces, AG.§16 Tensor products, AG.16.7 Tits system, §23, 24.6 Torus, 8.5 anisotropic, 8.14 regular, semi-regular, singular, 13.1 split over k, 8.2 Translation (right or left), 1.9 Transporter, 1.7 Trigonalizable (over k), 4.6 Unipotent element in an affine group, 4.5 Unipotent endomorphism, 4.1 Unipotent group, 4.8 Unipotent radical, 11.21 Unique factorization domain, AG.3.9 Unirational varieties, AG.13.7 Unitary groups, 23.9 Universal k-derivation, AG.15.1 Varieties, AG.6.2 Weight (of a torus), 5.2 Weight (of a root system), 24.1 Weyl chamber (algebraic group), 13.9 Weyl chamber (root system), 14.7 Weyl group (of an algebraic group), 11.19 Weyl-group (of a root system), 14.7 Weyl-module, 2404 Zariski dense subset, AG.13.5, AG.13.7 Zariski tangent space, AG.16.1 Zariski topology, AG.304, AG.6.6, AG.8.2 Index of Notation AO (opposite algebra of A), 23.7 D+, D-, 23.7 *-action (r operation in [Ti \]), 24.2 D- C (eigenspace), 23.9 Ad,3.5 ~o, sf(M) (M subset of an algebraic group), 2.1 sd(M) (M subset of an algebraic Lie algebra of char 0), 7.2 I a(M) (M subset of an algebraic Lie algebra of char 0), 7.2 a (homomorphism), AG.5.2, AG.8.2 AG.5.2 70° (comorphism of 70- 1, 24.2 k~(set of simple roots), 24.2 t1 action, 24.2 t1 set, 23.4 (dalx, AG.16.1 An' 23.9 70', Mset of simple roots), 24.2 70), AG.6.3 AG.8.2 ann, AG.3.1 Derk(A, Mj, AG.15.1 dim V, AG.9.1 dim X, AG.1.4, AG.3.4, AG.3.8, AG.3.9, AG.10.\ dimxX, AG.\.4 e,,(xj, AG.14.3 ex, AG.14.3 A p , AG.3.\, AG.3.4 e:\ AG.16.2 Aut P(E), 24.4 Ei., 24.4 {I.}, AG.\3.4 F' (sheafification of F}, AG.4.3 F G , AG.2.4 C[G],24.1 F k ·, AG.2.\ C6'( G), C6'i( G), 9)i( G), 2.4 F x ' AGA.\ c.(rr), 24.4 G (for SU(F)), 23.9 coker (fj, AG.3.5, AG.4.3 GO, 1.2 Index of Notation 287 G a ,1.6 MorK.a1g(A, B)k, AG.l1.2 Gad' 24.1 /1, 24.1, 24.6 G r ,24.1 nil X, 18.1 GsC' 24.1 V G (M),1.7 Gal (k'lk), AG.14.2 n(g), 18.1 GIR,6.8 02m GLn, 1.6(2) O(Q),23.6 GLn(D), 23.2 QA/K, GL v , GL(V), 1.6(8) p (characteristic exponent of k), AG.2.2 gin' 23.4 Po, P, PI' 24.6 G/m, 17.2 Pn, AG.7.2 g,3.3 PGL ,1O.8 gl(E) (E vector space), 3.1 &., 24.4, 24.5 W,24.6 &~, Hom k , AG.7.1 P(