A.N Parshin I.R.Shafarevich (Eds.) Algebraic Geometry IV Linear Algebraic Groups Invariant Theory AA MATO003023430x # II Springer-Verlag
Berlin Heidelberg New York London Paris Tokyo Hong Kong Barcelona
Trang 2Encyclopaedia of Mathematical Sciences
Volume 55
Trang 3List of Editors, Authors and Translators Editor-in-Chief
R V Gamkrelidze, Russian Academy of Sciences, Steklov Mathematical Institute, ul Vavilova 42, 117966 Moscow, Institute for Scientific Information (VINITD, ul Usievicha 20a, 125219 Moscow, Russia Consulting Editors A.N Parshin, Steklov Mathematical Institute, ul Vavilova 42, 117966 Moscow, Russia I R Shafarevich, Steklov Mathematical Institute, ul Vavilova 42, 117966 Moscow, Russia Authors
V.L Popov, Chair of Algebra, Department of Mathematics, MIEM, Bolshoj Vuzovskij Per 3/12, 109028 Moscow, Russia
T A Springer, Mathematisch Instituut, Rijksuniversiteit Utrecht, Postbus 80.010, 3508 TA Utrecht, The Netherlands
E B Vinberg, Chair of Algebra, Moscow State University, 119899 Moscow, Russia
Translator of Part IT
Trang 5I Linear Algebraic Groups
T.A Springer
Contents
Introduction .- - eee ee ee een teens Historical Comments 0.0.0: c ec ee eee eee nee
Chapter 1 Linear Algebraic Groups over an Algebraically
Closed Field . . nee nen nee nee
§1I Recollections from Algebraic Geometry -. -
1.1 Affine Varletles Ặ eens 12 Morphisms - ch nh nh hen nh vờ 1.3 Some Topological Propertles - {nỉ 14 Tangent SpaCes chen nhe nh net 1.5 Properties of Morphisms_ -. -Ÿ 1.6 Non-Afline Varleles cà nh eee §2 Linear Algebraic Groups, Basic Definitions and Properties
2.1 The Definition of a Linear Algebralc Group
22 Some Basic Facts co Tnhh he 2.3 G-Spaces 2 cee te nee ees 2.4 The Lie Algebra of an Algebralc Oroup
2.5 QUOtEDfS nen nh nh nh hư §3 Structural Properties of Linear Algebraic Groups . -
3.1 Jordan Decomposition and Related Results
3.2 Diagonalizable Groups and Tori -
3.3 One-Dimensional Connected Groups -
3.4 Connected Solvable GỐroups_ -. . c se: 3.5 Parabolic Subgroups and Borel Subgroups -
3.6 Radicals, Semi-simple and Reductive Ốroups
§4 Reductive ỐTOUpS ch nen nh nh nh nh nhớ 4.1 Groups of Rank Ône eens 4.2 The Root Datum and the Root System .-
4.3 Basic Properties of Reductive OToups -
4.4 Existence and Uniqueness Theorems for Reductive Groups
4.5, Classification of Quasi-simple Linear Algebraic Groups
Trang 6Chapter 2 Linear Algebraic Groups over Arbitrary Ground Fields
§1 Recollections from Algebraic Geometry
1.1 F-Structures on Affine Varletles
1.2 F-Structures on Arbitrary Varletles
<< e en ees 1.4 Restriction of the Ground Field
§2 F-Groups, Basic Properties
2.1 Generaliies About F-Groups
2.2, Quotients M T ( MUỤ 2.3, FOFMS ẰẮẮ= eee eet eee e ee eees §3 Tort 0 eee eee 3.2 F-Toriin F-Groups 2.0 0.0.0 c cece cece eee ees 3.3 Split Toriin F-OTOUDS cu §4 Solvable OrOUDS 0Q teeters 4.1 Solvable Groups 0.00 c ccc cc eee eee ene eens 4.2 Sections =&a&a - -aớa 4.3 Elementary Ủnipotent Groups
4.4 Properties of Split Solvable Groups
4.5 Basic Results About Solvable F-Groups
§5 Reductive GrOUDS cece eee eects 5.1 Split Reductive GỐroups cà 5.2 Parabolic Subgroups cà 5.3 The Small Root System
5.4 The Groups G(F) 0 ajdđ31T— eens 5.5 The Spherical Tits Building of a Reductive F-Group
$6 Classification of Reductive F-OGroups
6.1 Isomorphism Theorem
6.2 Existence ŸŸÁŸ(Ả
Chapter 3 Spccial Fields .c §1 Lie Algebras of Algebraic Groups in Characteristic Zero
1.1, Algebraic Subalgebras
§2 Algebraic Groups and Lie GØroups
2.1 Locally Compact Fields
2.2 Real Lie OTOUDPS Qua §3 Linear Algebraic Groups over Finite Fields
3.1 Lang s Theorem and ¡ts Consequences
3.2 Finite Groups of Lie Type
3.3 Representations of Finite Groups of Lie Type
§4 Linear Algebraic Groups over Fields with a Valuation
Trang 7§5
I Linear Algebraic Groups 3 4.2 The Affine Building 0.0.0 2 c cece ee eee tees 104
4.3 Tits System, DecomposIlons cà 107
4.4 Local Flelds . - ch nh nh 108
Global Fields nh nh 109 5.1 AdeleGrOUpS Q0 he kh kg 109
5.2 Reduction Theory 0 cece cece eee ene eee 112
5.43 Finiteness Results SỐ nhe 115 5.4 Galois Cohomology co eee nh nhe 118
Trang 8Introduction
A linear algebraic group over an algebraically closed field k is a subgroup
of a group GL,(A) of invertible n x n-matrices with entries in k, whose elements
are precisely the solutions of a set of polynomial equations in the matrix coordi-
nates The present article contains a review of the theory of linear algebraic
groups
To develop the theory one needs tools from algebraic geometry The reader is assumed to have some familiarity with that subject
Chapter 1 of the article reviews the basic facts from the theory of linear algebraic groups over an algebraically closed field k This theory culminates in a classification of simple linear algebraic groups I have tried to explain carefully
the fundamental notions and results, to illustrate them with concrete examples,
and to give some idea of the methods of proof
There are several monographs about the material of this chapter ((B2], [Hu],
{Sp3]), where the interested reader can find more details about this material
Chapter 2 discusses the relative theory, where a field of definition comes into play This is, roughly, a subfield F of k such that the polynomial equations of the first line can be taken to have coefficients in F This relative theory is required, for example, if one wishes to deal with arithmetical questions involv-
ing algebraic groups
At the moment there do not exist monographs covering this theory, which
makes it less accesible I have tried to present a coherent picture, following the same lines as in Chapter 1
In Chapter 3 special features are discussed of the relative theory, for particu- lar fields of definition F, notably finite, local and global fields The aim of the
chapter is to show how the theory of algebraic groups is used in questions about
such special fields
There is a great abundance of material Because of limitations of space I have sometimes been quite sketchy.'
The references at the end of the article do not have the pretension of being complete But I hope that with the help of them a reader will be able to trace in
the literature further details, of he wishes to do so
A reference in the article to I, 2.3.4 (resp 2.3.4) refers to no 2.3.4 of Chapter 1 (resp of the same Chapter)
Historical Comments
By way of introduction to the subject of linear algebraic groups there follows
a brief review of anterior developments which have been incorporated, in some
1 This article was written in 1988 Today (in 1993) I would perhaps have written some parts differently
Trang 9I Linear Algebraic Groups 5
way or another, in the theory of linear algebraic groups, or which have influ- enced that theory
First there is the study of concrete linear groups Galois already introduced the group PGL,(IF,) of fractional invertible linear maps (z+ (az + b)/(cz + d)*) of the prime field IF, An extensive study of the general linear groups over such
a field (in any dimension) and related “classical” groups (like orthogonal ones)
was made by C Jordan in 1870 (in his book “Traité des substitutions”) This was continued by L.E Dickson around 1910 and by J Diedonné around 1950
These authors study group-theoretical questions, such as the determination of all normal subgroups, for classical groups
A landmark in this development is C Chevalley’s paper “Sur certains groupes simples” (Téhoku Math J., 1955, 14-66), in which Lie theory makes its
appearance He constructs, for any simple Lie algebra over the complex field,
a corresponding linear group over any field F and he discusses their group-
theoretical properties The standard classical groups are special cases
Incidentally, Jordan’s book — mentioned above — contains a version of
Jordan’s normal form of matrices The Jordan decomposition in linear algebraic
groups (see I, 3.1) is a descendant
Linear algebraic groups over the field of complex numbers appear in E
Picard’s work on Galois theory of linear differential equations (around 1885, see
his paper “Equations differentielles linéaires et les groupes algébriques de trans- formations”, Oeuvres IT, 117-131) An example of the questions studied by him is the following Consider an n‘" order homogeneous linear differential equation
in the complex plane
n n-1
OF yale) oes + + aol hf = 0, dz" dz
with polynomial coefficients a; One knows that the everywhere holomorphic solutions form an n-dimensional complex vector space, let (f;, , f,) be a basis Let L be the subfield of the field of meromorphic functions obtained by adjoin- ing to the field C(z) of rational functions the ƒ; and all their derivatives Denote by G the group of C(z)-linear automorphisms of L which commute with deriva- tion, This is the Galois group of the equation Picard’s aim is to develop a Galois theory If g € G there exist complex numbers (x;,(g)) such that
g.ƒ¡ = ` X„(8)ƒ
and Picard shows that the matrices (x;(ø)) e GL„(Œ) form a linear algebraic
group over C, isomorphic to G He seems to be the first to use a name like
“algebraic group”
This Galois theory was later algebraized and further developed by Ritt (around 1930) and Kolchin The work of the latter of 1948 (see his paper “On
certain concepts in the theory of algebraic matric groups”, Ann of Math 49, 771-789) contains results which are now basic ones in the theory of linear
Trang 10the Lie-Kolchin theorem (I, 3.4.1) which states that a connected solvable linear algebraic group can be triangulized This extends a result of Lie for complex
solvable Lie algebras In contrast to the latter result, the Lie-Kolchin theorem is true in any characteristic
A Weil’s work on Jacobians of algebraic curves (see his book “Variétés
abéliennes et courbes algébriques”, 1948) led him to a study of general algebraic groups, i.e algebraic varieties with a group structure given by morphisms in the sense of algebraic geometry His interest was primarily in abelian varieties, i.e connected algebraic groups which are projective varieties (in which the group structure is automatically commutative) Classically, abelian varieties over € were studied by transcendental methods which go back to Riemann
Weil (and others) established somewhat later basic general facts about quo-
tients of an arbitrary algebraic group by an algebraic subgroup They are indis- pensable ingredients for the theory of linear algebraic groups
The theory of linear algebraic groups was founded by A Borel in 1956 (“Groupes linéaires algébriques”, Ann of Math 64, 20-82) His work was com- pleted by Chevalley (“Classification des groupes de Lie algébriques”, Seminaire
Ecole Normale Supérieure, 1956-1958) In Borel’s work the influence of Kolchin’s work, alluded to above, is clearly visible Another essential element is the analogy with the theory of Lie groups, in its “global” form An infinitesimal approach to linear algebraic groups via Lie algebras is unsuitable in characteris- tic p > 0
Using the global methods of algebraic geometry, Borel established basic results, such as conjugacy theorems for maximal tori and Borel subgroups
(I, 3.5.3, 1 3.5.1) To obtain these he proves a fixed point theorem (I, 3.4.3), which
generalizes the Lie-Kolchin theorem, mentioned before Chevalley showed that analogues of results established in Lie theory with the help of the Lie algebra can be obtained with global methods (for example results about radicals, see I
4.2.6) The main result of his Séminaire is that the classification of simple linear
algebraic groups over an algebraically closed field of any characteristic, is com- pletely analogous to the classification of simple Lie algebras over the field of
complex numbers
In the work of Borel and Chevalley the influence of ideas and results from the
theory of Lie groups has been considerable Grosso modo, Chapter I of the
article is a review of the work of Borel and Chevalley
Finally, mention should be made of some generalizations of algebraic groups, which we have not — or hardly — touched upon First there are the group schemes,
studied extensively by Grothendieck and his collaborators (M Demazure and A Grothendieck, Schémas en groupes, Lect Notes in Math nos 151, 152, 153,
1970) In this article they appear in only a few places More recent generalizations are the quantum groups, which are algebraic groups in “non-commutative ge-
ometry” We have only given the definition (in 2.1.6) We have not said anything
Trang 11I Linear Algebraic Groups 7
Chapter 1
Linear Algebraic Groups over an Algebraically Closed Field
§1 Recollections from Algebraic Geometry
Some familiarity with algebraic geometry is assumed We shall recall a num- ber of basic notions and results For more details see [H], [Mu] or [Sp3]
1.1 Affine Varieties Let k be an algebraically closed field An affine alge-
braic variety X over k is determined by its algebra of regular functions k{X], a k-algebra of finite type, which is reduced i.e without non-zero nilpotent elements Such k-algebras are called affine X is the set of k-algebra homomorphisms k[X]—k For each ideal I of k[X], let (I) be the set of xe X such that
x(I) = 0 The sets ¥ {J} are the closed sets for a topology on X, the Zariski topology
The elements of kLX] define k-valued functions on X, the regular functions The affine variety defined by the polynomial algebra k[T,, , T,,] is affine n-space A", also denoted k”
1.2 Morphisms
1.2.1 If X and Y are affine varieties, a homomorphism of k-algebras @*: k[X] > kL Y] defines a map ø: Y > X, which is continuous Such maps are the
morphisms of affine k-varieties
1.2.2 A closed subset Y of the affine variety X has a canonical structure of
affine variety, with algebra k[Y] = k[X]/I, where I is the ideal of functions vanishing on Y Such a variety is a closed subvariety of X The corresponding morphism is a closed immersion
1.2.3 Next let ƒ e k[X ] — {0} and take k[Y ] = k[X ]; = k[X][71/4-ƒT), a localization of k[X], with @* the canonical homomorphism Then Y can be viewed as the open subset D(f) = {x € X |f(x) # 0} of X Such a set D(f), pro- vided with the k-algebra k[X],, is an affine variety Any open subset of X is a
union of finitely many open sets of the form D(f)
Example Let X = IM,(k), the space of n x n-matrices with entries in k, which is isomorphic to A" Let d(X) = det(X) be the determinant function The open set X, is the set of all invertible n x n-matrices
1.2.4 If X and Y are affine varieties, there exists a product variety X x Y,
with k[X x Y] = k[X]®,kLY]
Trang 121.3 Some Topological Properties Let X be an affine variety It has the noe- therian property: any family of closed subsets contains a minimal element (for
inclusion)
X is reducible if it is a union of two non-empty proper closed subsets, Other- wise X is irreducible Irreducibility is equivalent to: a non-empty open subset of X is dense
Also, X is irreducible if and only if the algebra k[X] is an integral domain In that case the quotient field of k[X] is denoted by k(X) Its transcendence degree over k is the dimension dim X of X
Any affine variety is the union of finitely many irreducible closed subsets, its
irreducible components, which are unique
1.4, Tangent Spaces If x is a point of the affine variety X, the homomor- phism x: k[X]—k defines a k[X]-module k,, with underlying vector space k The tangent space T,X of X at x is the k-vector space of k-derivations of kX]
in k,, ie linear maps D: kLX]— k such that D( fg) = f(x)(Dg) + (Df)g(x) If M,
is the maximal ideal Ker x of kLX] then T,X is isomorphic to the dual of
M,/(M,)? A morphism g: Y - X defines a map of tangent spaces (dq),: T, Y >
T,X, the differential of at y
If X is irreducible then xe X is smooth (or simple, or non-singular) if dim T,X = dim X The smooth points of X form a non-empty open subset We say that X is smooth if all its points are smooth
1.5 Properties of Morphisms
1.5.1 Let X and Y be irreducible affine varieties and g: Y ~ X a morphism It is said to be dominant if @Y is dense in X In that case »Y contains a non-
empty open subset of X
1.5.2 If g@ is dominant the defining homomorphism @*: kLX] — k[Y] is injec- tive Then @ is separable if the field k(Y) is a separable extension of g*k(X) This
is so if and only if there exists a simple point y € Y such that gy is simple and that (do),: T, Y > T,,X is surjective The set of such points of Y is open
1.5.3 If g is dominant there is a non-empty open subset U of Y such that the restriction of @ to U is an open map (ie the image of an open set is open) Moreover, U can be chosen such that for any closed irreducible subvariety X’ of X and any irreducible component Y’ of g7'X’ such that Y’n U ¥ @ we have dim Y’ — dim X’ = dim Y — dim X
1.5.4 If g is dominant and if for some ye Y the fiber p(y) is finite then dim Y = dim X
1.6 Non-Affine Varieties
Trang 13I Linear Algebraic Groups 9
such that the restriction of f to D(g) lies in KLD(g)] (see 1.2.3) Let @y(U) be the k-algebra of these functions We say that U is an affine open subset if O,(U) is an affine k-algebra, whose homomorphisms in k are precisely the evaluation maps fr f(x) for x € U
The intersection of two affine open subsets U, V is also affine open We have
K[U Š Ÿ] = k[U] @„>„¡ k[ VÌ
The Øy(U) for U open in X delne a sheaf Øy of k-algebras on X, which defines a ringed space (X, Øy)
1.6.2 A ringed space (X, @,) is called an algebraic variety (non necessarily affine) if X has a finite covering by open subsets (X;);.,; such that (a) for each i the restriction ringed space (X;, ©x|x,) is isomorphic to one of the kind described in 1.6.1, (b) for each pair (i, j) the intersection X, > X; is an affine open subset of X;,, for the structure of (a) and the k-algebra O,(X; 4 X;) is generated by @x(X,) and
Oy(X))
Morphisms of algebraic varieties are defined in an obvious way The notions
and results reviewed above for affine varieties carry over, as far as this makes sense We have the notions of open resp closed subvariety of an algebraic variety X.A locally closed subvariety of X is an open subvariety of a closed subvariety of X
1.6.3 Example Projective n-space IP” Here the underlying set X is the set of all lines in k”** Let X; be the set of those lines which have a basis vector (X15 0005 Xp-45 1, Xi445 -, X,) Then X; can be given a structure of affine algebraic
variety isomorphic to A" These structures can be glued together to give a
structure of algebraic variety on X If V is a finite dimensional vector space over k the set of lines in V has a structure of projective variety P(V), isomorphic to
P", where n + 1 = dim V
A projective variety is one which is isomorphic to a closed subvariety of some P" A quasi-projective variety is an open subvariety of a projective variety
Projective n-space can also be defined as the set of homomorphisms of kK[Tp, T,, -, 7, ] to kK[T] which are homogeneous for the standard gradings, closed sets being those sets of homomorphisms which annihilate homogeneous ideals in the first algebra
1.6.4 A variety X is complete if it has the following property: for any variety Y the projection map X x YY is closed Projective varieties are complete Affine varieties with infinitely many points are not complete
§2 Linear Algebraic Groups, Basic Definitions and Properties
2.1 The Definition of a Linear Algebraic Group
2.1.1 The most direct definition of the notion of a linear algebraic group — which however is non-intrinsic — is as follows For each integer n > 1 the group
Trang 14Definition A linear algebraic group G over k is a subgroup of some GL, which is a closed subset of GL,,
2.1.2 Examples of Linear Algebraic Groups
(a) GL, According to 1.2.3 we have k[GL,]=k[T,;, d*J, where d=det(T,,),
the T¡; being matrix variables GL, is a general linear group
We write GL, = G,, This is the multiplicative group, also written k* We have k[G,,] = k{T, T~*], the algebra of Laurent polynomials over k
We write SL, = {X e GL, |d(X) = 1}, this is the special linear group We have
k[SL,] = k[GL,]/(đ — 1) k[1;,]/(đ — 1)
(b) The subgroup of SL, consisting of the matrices
( })
with x e k is the additive group G,, also written k We have k[G,] = k[T] (c) Let Se GL, The X e GL, with XS(‘'X) = S$ form a linear algebraic group Instances are the various classical groups:
(i) Ifn = 2m is even and
0 1
S= ”
Ly 5)
we obtain the symplectic group Sp,
(ii) If char(k) 4 2 and S = 1, then G is the orthogonal group The special
orthogonal group is the intersection with SL,, it is a normal subgroup of index 2 We shall prefer to use another description of the orthogonal groups If n = 2m is even we denote by O, the group defined above, with 0 1, s-(? 0) and if n = 2m + 1 is odd we denote by O, the group obtained from 0 1, 0 s=|1, 0 0 0 0 1 Then O, is conjugate in GL, with the orthogonal group defined first We write SO, = 0,0 SL, (iii) If char(k) = 2 orthogonal groups are defined in another way If n = 2m is even put 0 1, ns ( 5)
and define O, to be the subgroup of GL, consisting of the matrices X such that
XT(X) + X is skew, ic symmetric with diagonal elements zero This is again
an algebraic group If n = 2m + 1 is odd the definition is similar, replacing T by
a larger matrix, as before
(d) The group of diagonal matrices in GL, (resp upper triangular matrices, resp
upper triangular matrices with diagonal elements one) is a linear algebraic group
Trang 15I Linear Algebraic Groups 11
2.1.3 The intrinsic definition of a linear algebraic group is as follows Definition A linear algebraic group G over k is an affine algebraic varie- ty with a group structure such that the group operations are morphisms of varieties
The group structure is defined by the product map 2: G x G > G ((x, y)> xy) and the inversion map 1: G > G (xt x7") and there is the neutral element e According to the definition z and : come from homomorphisms of k-algebras
*: KG] — k[G] ® k[G], #: k[G] > k[G],
moreover e is a homomorphism k[G] > k
The group axioms are expressed by the commutativity of the following dia- grams, Where A = k[G] and m: A @ A > A defines multiplication: R*@id 4a@A@A —— ABA «| A@A ‹- A n* A <4 4@A A@A 24 A@A my Xe “| id nm A e A, o\ “ A@A -——— A A@A <——— ABA id@i*
It is clear that an algebraic group in the sense of the definition of 2.1.1 is also one in the sense of 2.1.3 The converse is also true (see 2.3.4)
If one omits the adjectives “linear” and “affine” in the definition one obtains the definition of a general algebraic group We shall not need this notion 2.1.4, Examples (a) If G = GL, then k[G] = k[T;,, d~'], as before We have j}? 7* Tị; = hà Ti, © This moreover 1*T,, is the cofactor of T;, in (T;;) times d~' It is clear that e(T;;) = 6, (b) k[G,,] = k[T, T] and 2*T = T @ T, *T = T", e(T) = 1 (c) k{G,] = k[T] and n*T = T@14+1@T, e(T)=0
(d) Let V be a finite dimensional vector space over k with dual V* Put k[V] = Sym,(V*), the symmetric algebra of V* over k Then k[V] defines a
structure of algebraic variety on V, isomorphic to A”, where n = dim V
Trang 16independent of the choice of the basis, up to a constant) We write k[GL(V)] for this last algebra By transport of structure we obtain a linear algebraic group
GL(V), isomorphic to GL,,
2.1.5, Group Schemes Let R be a commutative ring (with unit) and A an R-algebra Assume that we are given algebra homomorphisms 2*: A > A @, A,
1: A A, e: A> R such that we have the properties of 2.1.3, in terms of com-
mutativity of diagrams We then say that A is an affine group scheme over R (this is a short-cut to the correct definition, which one can find in [DG] or [J]) Next let G be as in 2.1.3 and assume that k is an R-algebra We say that G has an R-structure if there is an affine group scheme A over R such that k[G] = A @xk and that the group operations for G are defined by x* @ id, 1* @ id, e @id We then denote the set of homomorphisms 4 > R by G(R) Then z* induces a product G(R) x G(R) G(R), which defines a group structure on G(R) This is the group of R-points of G
Examples
(a) The groups of examples (a), (b), (c) in 2.1.2 all have Z-structures
(b) Let n be an integer 21 Put A = Z[T]/(T" — 1) Then a*t =Â đt, i*t =
t"~1, e(t) = 1 (where ¢ is the image of t in A) define a structure of group scheme
This is the group scheme x, of n'" roots of unity
If k is a field of characteristic prime to n or zero then A @ 7k defines the finite linear algebraic group of n'" roots of unity in k* But if the characteristic of k divides n this k-algebra is not reduced, so cannot be the affine algebra of a linear algebraic group
(c) The following example is of a more general nature Assume that char(k) = p > 0 Then xt» x? defines a homomorphism of k into k (actually an automorphism), which makes k into a k-algebra denoted by Fk If A is any k-algebra define the k-algebra FA by FA = A @, Fk There is a k-homomor-
phism F: FA > A, sending a ® x to xa” If A is an affine algebra then F defines
an isomorphism of FA onto the subalgebra of A generated by all p™ powers a?
(a e 4)
If X is an affine variety over k define the affine variety FX by k[FX] = F(k[X]) We then have a morphism F: X > FX, the Frobenius morphism We also introduce the iterates F": X > F"X These define, in fact, functors on the category of affine k-varieties, with good properties
It follows that if G is a linear algebraic group over k then F"G has a structure of linear algebraic group such that the morphism F" is a group homomorphism Now let J, be the ideal in k[G] generated by the elements f”" — f?"(e) (fe k[G]) Then G, = k[G)/I, is a local ring, which has a structure of group scheme The G, are called the Frobenius kernels of G One can view G,, as the kernel of the homomorphism F” in the sense of group schemes
If G = k* then G, is isomorphic to the group scheme over k obtained from
Upn, as in example (b)
Trang 17I Linear Algebraic Groups 13
2.1.6 Quantum Groups We mention briefly another generalization of the
notion of algebraic group
Let k be as before Assume that A is a ~ not necessarily commutative — k-algebra and that n*: A + A@ A, 1*: A> A, e: A>k are linear maps with the properties of 2.1.3, z* and e being homomorphisms, and ;* an anti- homomorphisms Also assume that there exists a homomorphism (2°)* of the
opposite algebra A°, such that the same properties hold for A° and z*, (2°)*,
e In that situation A is a quantum group
Example We have k[SL,] = k[x, y, z, t\]/(xt — yz — 1) Let qe k* We de- fine the algebra A of the quantum group ,SL, as the algebra generated by elements x, y, z, t subject to the relations: xy = q !yx, xz = q Ìzx, yt = q ty,
zt = q l†Z, yz = 2ÿ, xt — q Ìyz = tx — qyz = Ì
Define z* by the same formulas as for k[SL; ]
We refer to [CP, Lu] for more details about quantum groups and for further references
2.2 Some Basic Facts
2.2.1 A homomorphism of algebraic groups @: G > H is a group homomor-
phism which is also a morphism of varieties The notion of isomorphism of algebraic groups is clear
H isa closed subgroup of G if it isa subgroup which is also a closed subvariety of the affine variety G in the sense of 1.2.2
If Gand H are algebraic groups the product variety G x H has a structure of algebraic group, the direct product of G and H
Examples of closed subgroups Let § be a subset of the algebraic group G Then the normalizer N(S) = N,(S) defined by
N(S) = {g € GlgSg™* = S}
and the centralizer Z(S) = Z,(S) defined by
Z(S) = {g € Gigsg™* = s for alls eS} are closed subgroups of G
If g € G we write Z(g) = Z,(g) = Z({g})
2.2.2 The algebraic group G is connected (for the Zariski topology) if and only if it is irreducible as an algebraic variety There is a unique irreducible component G° of G containing the neutral element It is a closed normal sub- group of finite index, the identity component of G
Examples
(a) GL,, SL,, G,,, G, are connected (the irreducibility is a direct consequence
of the definitions)
(b) Sp,,, is connected This can be shown by an application of the result of
2.2.4, using that Sp,,, is generated by elements of a simple form Similarly, SO,
Trang 18(c) The subgroup of upper triangular (resp diagonal) matrices in GL, is connected and solvable (resp commutative)
2.2.3 Let @: G > H be a homomorphism of algebraic groups Then Ker ¢ is a closed normal subgroup of G and Im @ is a closed subgroup of H
The last property follows by using 1.5.1 and the following lemma
Lemma Let U and V be two dense open subsets of the algebraic group G Then UV = G
2.2.4 Let (H;);-; be a family of closed irreducible subsets of the algebraic group G, which generate G as an abstract group Then G is connected and there is an integer n > 0 and (a(1), ., a(n)) € J” such that G = Hyg) Haw:
From this one infers that if H and K are closed subgroups of G, one of which is connected, the subgroup (H, K) generated by the commutators xyx7!y7! with x € H, ye K is closed and connected
It follows that if G is connected, the subgroups Y"G (resp @"G) defined by 2°G = G, 2"(G) = (2"~!Œ, B"'G) (resp @°G = G, $"(G) = (G, G" 'G)) are closed and connected
2.3 G-Spaces
2.3.1 Let G be an algebraic group An algebraic variety X (not necessarily
affine) is a G-space if there is a left action of the group G on the set X which is a
morphism of varieties a: G x X + X We write a(g, x) = g.x, so that (gh).x = g.(h.x),e.x = x(g,heG,x eX)
In this situation the orbit of x € X is the set G x and the isotropy group of x is the closed subgroup
G, = {ge Glg.x = x}
If G, = G then x is a fixed point of G in X The set of fixed points is closed
X is a homogeneous space for G if G acts transitively on_X, ie if there is only
one orbit A homogeneous space is smooth (as a consequence of the fact that X
has smooth points, see 1.4) In particular, G is smooth
The notion of a morphism of G-spaces or equivariant morphism of G-spaces is Clear
The following useful lemma should be noted
Lemma An orbit is locally closed, i.e is open in its closure
The proof follows from the property of 1.5.1
2.3.2 Now assume in the situation of 2.3.1 that G is a linear algebraic group and that X is affine Then the action « is defined by a homomorphism œ*: k[X ]— k[G] © kLX] That « is a group action is tantamount to
(a) the homomorphisms z* @ id and id @ ø* of k[G] @k[X] to k[G]@
k[G] @ kLX] coincide;
Trang 19I Linear Algebraic Groups 15
Here x* and e are as in 2.1.3
This algebraic definition of an action leads to a definition of an action of a group scheme over a ring R on an affine scheme over R, compare 2.1,5
2.3.3 Examples
(a) Let V be a finite dimensional vector space over k A homomorphism of algebraic groups g: G > GL(V) is called a rational representation of G in V In
that case G acts on V by g.v = o(g)v (g € G, ve V) Such an action is called
linear
(b) Let G =G,, Giving a G-action on the affine variety X is tantamount to giving a Z-grading on the algebra k[X], i.e a direct sum decomposition k[X]= @ KLX],,, meZ into vector spaces such that for m,ne Z K[X] yn K[X]; c R[X ]„+„- The connection between grading and action is given by atf= T"™@f if fe k[X],-
(c) Let G = G, and assume char(k) = 0 Giving a G-action on the affine vari-
ety X is now equivalent to giving a k-derivation D of the ring K[X] such that any element of k[X] is annihilated by some power of D (D is the vector field on X defining the action)
The connection with the action is given by
3*/= 3) T"@(ml)ˆ'D"ƒ m20
If char(k) > 0 the algebraic description is somewhat more involved
2.3.4 Let Gand X be as in 2.3.2 For g € G define a linear map a(g) of the (in
general infinite dimensional) vector space kX] by
o(g) f(x) = f(g x)
Then o is a representation of the abstract group G in k[X]
Proposition There is an increasing sequence (V,),>1 of finite dimensional sub-
spaces of k[X] whose union is kX] such that (a) o(g)V, = V, forgéG, n> 1,
(b) the homomorphism G > GL(V,) defined by o is a rational representation of G foralin> 1,
A statement equivalent to (a) is: every element f of k[X] lies in a finite
dimensional subspace of kL.X] which is o(G)-invariant To prove this statement
write a*f = } h, @ f, and observe that all o(g)f lie in the subspace of V spanned by the ƒ,
Trang 20one sees that an algebraic group in the sense of 2.1.3 is also one in the sense of 2.1.1
If o is a representation of G in a vector space V such that V is the union of an
increasing sequence (V,) of finite dimensional subspaces with the properties (a)
and (b) of the proposition we say that o is a rational representation of G in V
Example Let V = k[G] For g € G define linear maps A(g), p(g) of V by
gfx) = f(g *x), — p(g) F(x) = (x9),
these are the left- and right translations by g Then 4 and p are rational repre-
sentations of G in k[G] For arbitrary g, h e G we have A(g)p(h) = p(h)A(g)
2.4 The Lie Algebra of an Algebraic Group G is a linear algebraic group
2.4.1 Denote by T the k[G]-module of k-derivations of k[G], i.e the k- linear maps D of k[G] into itself such that for f| g € kLX]
D( fg) = f(Dg) + (Df)g-
This is a Lie algebra over k, with product given by [D, D’] = DD’ — D'D If
p =Char(k) > 0 it is a restricted Lie algebra (or p-Lie algebra) This means
that there is a p'" power operation (in this case: Di+ D”) with certain formal
properties (see for example [DG, p 275 ])
The Lie algebra L(G) of G is the subalgebra of T of the derivations com-
muting with all left translations A(g) It is a restricted Lie algebra if p > 0 Moreover, G operates linearly or L(G) via right translations (D> p(g)Dp(g)"')
On the other hand denote by g the tangent space T,G at the neutral element
(see 1.4) For géG the corresponding inner automorphism fixes e and thus
defines a linear map Ad(g) of g: its differential at e Proposition There is an isomorphism of k[G]-modules F:T— k[G] ®ys such that Fi(g)F* = Ä(g) @id, Fp(g)F* = p(g)@ Ad(g) As a consequence one obtains that F induces a bijection f: L(G) > g, such that #{p(g)Dp(ø} `) = Ad(g)( f(D)
We shall identify L(G) and g via f and view thus g as a (restricted) Lie algebra, the Lie algebra of G It also follows that Ad is a rational representation of G in
g, the adjoint representation
Trang 21I Linear Algebraic Groups 17
2.4.2 Họ: G > H is a homomorphism of linear algebraic groups the differ- ential dg of @ at e is a homorphism of (restricted) Lie algebras In particular, if G is a closed subgroup of H and @ the inclusion map, then do identifies g with a (restricted) subalgebra of b
The following characterization of isomorphisms is useful: @ is an isomor-
phism if and only if it is an isomorphism of abstract groups and dg is an isomorphism of Lie algebras (if char(k) = 0 the second condition can even be
omitted)
More generally, let : X — Y be an equivariant morphism of homogeneous G-spaces Then y is an isomorphism of varieties if and only if it is bijective and at some point x € X the tangent map (dy),: T,X — T,,.Y is bijective The second condition can be rephrased as: wy is separable (see 1.5.2)
2.4.3 Examples
(a) Let G = GL, Denote by gl, the Lie algebra of ail n x n-matrices with entries in k, the Lie product being given by [X, Y] = X Y — YX (with the struc- ture of restricted Lie algebra defined by the usual p' power if p > 0) If X = (x;;) € gl, then
Dy Ti; = =*È Tip Xnj
defines a derivation of k[G] = k[T,;, d~'] (see 2.1.2 (a)) which commutes with
all left translations Hence X +> Dy is a linear map gl, > L(G) It is an isomor- phism of Lie algebras It follows that the Lie algebra of GL, (resp GL(V)) is isomorphic to gl, (resp the Lie algebra of endomorphisms End(V)) The Lie algebra of SL, then corresponds to the subalgebra sl, of gl, formed by the
matrices with trace zero
(b) G = G,, This is the case n = 1 of the previous example We see that L(G) d of k[G] = k[T, T~'] We have X? = X is spanned by the derivation X = Tar (if p > 0) d (c) G = G, Now L(G) is spanned by the derivation aT of k[G] =k[T] We have X? = 0 (if p > 0)
(d) Let s be an automorphism of G (in the sense of algebraic groups), of finite
order prime to char(k) Denote by
G, = {g € G\s(g) = g}
the fixed point group of s Then G, is a closed subgroup of G and its Lie algebra
is the subalgebra g, of g defined by
Trang 22This applies to the examples of 2.1.2 (c) if G, is Sp, or O,, when char(k) # 2
Then G = GL, and
s(X) = SÚX) "§ `,
S being as in these examples
We see that then the Lie algebra g, is
g, = {X egl,|'X = —SXS“}
(e) Let ¢: GGL, be a rational representation of G Put 9(g) = (f;,(9)), with ƒ¡; € k[G) Then for X € g (=T.G) we have de(X) = (Xf;;)
(f) The differential of the adjoint representation is the linear map ad: g >
End(g) given by
(ad X)(Y) = [X, Y] (X, Y eg)
2.4.4, The tangent space of a variety at a point is a “first order infinitesimal object” One can also introduce higher order objects
An example of such an object, in the case of a linear algebraic group G, is the
algebra of distributions D = Dg, a k-algebra (in general non-commutative), a
finer object than the Lie algebra Its definition is as follows For details see [DG, Ch 2, 6.1] Let M = {fe k[G]|f(e) = 0}, a maximal ideal in k[G] Then for n > 0 we have that k[G]/M""? is a finite dimensional vector space over k We denote by D, its dual There is an obvious injection D, @ D„.; Put
D = lim D, —
Using the homomorphism z* defining the group structure one defines a struc- ture of k-algebra on D, such that for m,n > 0 Đạ D, C Đạ„.„ It is clear that q={XeD,|X(U = 0) One shows that if X, Y eq we have [X, Y] = XY — YX, these products being taken in D The product structure on k[G] induces, moreover, a coalgebra structure on D
2.5 Quotients G is a linear algebraic group (the theorem of 2.5.1 is also true for general algebraic groups, however)
2.5.1 Let H be a closed subgroup of G
Definition A quotient of G by H is a homogeneous space G/H for G together with a base point a e G/H, whose isotropy group 1s H, such that for any pair (X, b) of a homogeneous space together with a point be X whose isotropy
group contains H, there exists a unique G-equivariant morphism g: G/H > X
Trang 23I Linear Algebraic Groups 19
We have the following theorem
Theorem A quotient (G/H, a) exists It is unique up to G-isomorphism
The proof requires some work We refer to the literature for the algebro- geometric details It should perhaps be remarked that a result like the theorem is more delicate than the analogous result for Lie groups (as algebraic varieties are “more rigid” than C-varieties)
To describe the quotient G/H, the following result of Chevailey is useful
Proposition There is a rational representation »:G—GL(V) and a one-
dimensional subspace L € V such that
H = {g€Glo(g)L = L},
bh = {X egldo(X)L c L} Here b 1s the Lie algebra of H
In the situation of the proposition, let x denote the point of the projective space IP(V) defined by L There is a morphism of G-spaces G/H — P(V), sending g.a to e(g).x, which induces an isomorphism of G/H onto the G-orbit of x in P(V) This implies that G/H is a quasi-projective variety (see the lemma of 2.3.1) We may (and shall) view the points of G/H as cosets gH, the point a corre- sponding to H We denote by z the morphism G > G/H with zg = gH
Then x is a separable morphism (1.5.2), as a consequence of the proposition Moreover, x is universally open, i.e for any variety X the morphism
nxid:G xX >G/Hx X
is open (this is true, more generally, for any equivariant morphism of homo- geneous spaces) This last property has the following consequence
Lemma For any closed subvariety X of G the set {(yH, z)|y‘zye X} is
closed in G/H x G
2.5.2 Normal Subgroups Next let H be a closed normal subgroup of the
linear algebraic group G
Proposition The quotient variety G/H has a structure of linear algebraic
group
This follows by elementary arguments from the proposition in 2.5.1 One has the usual properties of quotient groups We shall not spell them out
The Lie algebra of G/H can be identified with the quotient g/h of the Lie algebra g by the normal subalgebra b
§3 Structural Properties of Linear Algebraic Groups
Trang 243.1 Jordan Decomposition and Related Results
3.1.1 Jordan Decomposition of Linear Maps Let V be a vector space over k and aa linear map of V We say that ais locally finite if there is an increasing sequence of finite dimensional subspaces of V stabilized by a, whose union
is V Moreover, a is semi-simple if V has a basis consisting of eigenvectors for a
and a is locally unipotent if any vector of V is annihilated by a power of a — 1 If V is finite dimensional this means that all eigenvalues of a equal 1
Proposition Let a be an invertible locally finite linear map of V There are
unique invertible linear maps a, and a, such that a, is semi-simple, a, is locally unipotent and a = a,d, = a„d,
This follows from the theory of Jordan normal forms a, (resp a,) is the
semi-simple (resp unipotent) part of a and the decomposition a = a,a, is the
Jordan decomposition If W is an a-stable subspace of V then a,, a, stabilize W and induce the semi-simple resp unipotent part of the restriction a|y Similarly for V/W This is an analogous additive Jordan decomposition of a locally finite linear map a: a = a, + a, with a,a, = a,a,, where now a is locally nilpotent, ie a+ 1 is locally unipotent
3.1.2 Jordan Decomposition in G Recall that if g € G the right translation map p(qg) of k[G] is locally finite (see 2.3.4)
Theorem (Jordan decomposition in G) (i) If g € G there exist unique elements
Ys» Ju in G such that p(g,) = P(9)s, P(Gu) = p(g)„ and 0;g„ = Gus}
(ii) If @: GH is a homomorphism of linear algebraic groups then @(g,) =
@(g),, (Gu) = P(D)us
(iti) If V is a finite dimensional vector space and G = GL(V) then g, and g,, are as in 3.1.1
g, and g, are the semi-simple and unipotent part of g
It follows from the theorem that if G is a closed subgroup of some GL, the semi-simple and unipotent parts of g € G, viewed as an element of GL,, lie in G and coincide with the elements of the theorem
It should be noticed that the maps gt» g, and gt» g, of G into itself are in general not morphisms of varieties (as one sees for example in SL,)
The following technical property of semi-simple elements is needed in the
theory
Lemma Let s € G be semi-simple The Lie algebra of the centralizer Z,(s) is the subalgebra Z,(s) of the Lie algebra g defined by
Z,(s) = {X € g|Ad(s)X = X}
The proof involves the fact that the morphism G — G sending g to gsg™'s™' is separable, if s is semi-simple (see [Sp3, 4.4])
There is an analogue of the theorem for the Lie algebra g It states that an
Trang 25I Linear Algebraic Groups 21
3.1.3 Unipotent and Abelian Groups G is unipotent if all its elements are unipotent
Proposition 1 Let G be a subgroup of GL, consisting of unipotent matrices
There is x € GL, such that xGx™ consists of upper triangular matrices
This is really a result from linear algebra It applies to unipotent algebraic
groups and it has the following consequence
Corollary A unipotent linear algebraic group is nilpotent as an abstract group
For the group of unipotent upper triangular matrices in GL, is nilpotent Notice that if char(k) = p > 0, G is unipotent if and only if all its elements have p-power order
An application of the well-known result that a commuting set of linear maps of a finite dimensional vector space can be simultaneously triangulized is the following result
Proposition 2 Assume G to be abelian
(i) The sets G, and G, of semi-simple resp unipotent elements of G are closed
subgroups;
(ii) The product map G, x G, > G is an isomorphism of algebraic groups; (iii) G is connected if and only if G, and G, are connected
3.2 Diagonalizable Groups and Tori
3.2.1 G is called diagonalizable if it is abelian and all its elements are semi-
simple
If G is an arbitrary linear algebraic group we denote by X*(G) the group of homomorphisms of algebraic groups G > G,, and by X,,(G) the set of homomor- phisms of algebraic groups G,,— G If G is abelian we also view X,(G) as a group The elements of X*(G) are characters of G, those of X,,(G) cocharacters Notice that X*(G) is a subset of k[G]
Proposition, The following properties of a linear algebraic group are equivalent: (a) G is diagonalizable;
(b) X*(G) is an abelian group of finite type and X*(G) is a k-basis of kLG];
(c) Any rational representation of G is a direct sum of one-dimensional ones A main ingredient in the proof is Dedekind’s theorem about linear indepen- dence of characters of a group It follows that a diagonalizable group can be viewed as a group consisting of diagonal matrices in some GL, It also follows
from part (b) of the theorem that if G is diagonalizable, the algebra k[G] is
isomorphic to the group algebra k[X*(G)] Examples
(a) G=G,, Nowk[G] = k[T, T~'], where T is the function on G = k* with
T(x) = x The characters of G are the T” (me Z), hence X*(G) = Z It also
Trang 26(b) G = G, Now X*(G) = X,(G) = 0, since the elements of G, are unipotent (see part (ii) of the theorem in 3.1.2)
(c) Gis the group of n™ roots of unity in G,, (n prime to the characteristic p) Now k[G] = k[T]/(T" — 1), which is also the group algebra of the cyclic group of order n
3.2.2 Tori A connected diagonalizable group G is called a torus (or alge- braic torus), These play an important rdle in the theory of linear algebraic groups, which is similar to the réle of the “classical” tori in the theory of com- pact Lie groups
Proposition G is a torus if and only if G is isomorphic to some (G,,)" This is a consequence of property (b) of the previous proposition
Corollary In a torus the set of points of finite order is dense
Let T be a torus Write X = X*(T), XY = X,(T) These are free abelian
groups of rank equal to dim T, as a consequence of the proposition We always write their group structure additively X and X” are in duality via a pairing
< , > defined by
x(x" (t)) = tr"?
forxe X,x” EX’, tek*
3.2.3 Rigidity of Tori Tori do not have “non-trivial algebraic families of endomorphisms” This is the content of the following result
Proposition Let T be a torus and X a connected algebraic variety Assume that p is a morphism X x T > T such that for each x e X the map tr w(x, t) is a homomorphism Then (x, t) is independent of x
This is a consequence of the density of points of finite order It should be noticed that the automorphism group of an n-dimensional torus is isomorphic to the group GL,(Z) of integral n x n-matrices with an integral inverse (this is the automorphism group of the character group) In particular, the automor-
phism group of G,, has order 2
3.2.4 Weights Let T be a torus and assumes it acts linearly in the finite dimensional vector space V (in the sense of 2.3.3 (a)) The weights of T in V are the characters x € X*(T) such that the space
V, = {ve V|[t.v = x(t)v for all te T}
is non-zero The V, are called the weight spaces for the action and their non-zero vectors are weight vectors By the proposition of 3.2.1, V is the direct sum of the weight spaces V,
3.2.5 Linearizable Torus Actions Let T be a torus and assume that X is a projective T-variety We also assume that X is a closed subvariety of IP" and that there is a rational representation g of T in k"*! such that the following holds Let x = kv (a line in k"*") lie in X Then for t e T we have
Trang 27I Linear Algebraic Groups 23
In this situation we say that the T-action on X is linearizable First assume that T= k* If xe X we say that lim,.ot.x (resp lim,_,, t.x) exists and equals ye X if the morphism k* + X sending t to t.x (resp t™'.x) extends to a mor- phism k > X sending 0 to y (in fact, this definition makes sense for a k*-action
on any variety)
Lemma If T = k* then lim,_,o t.x and lim, ,, t.x exist for all x € X
It suffices to prove this when X = P", in which case the result becomes obvi-
ous if one diagonalizes the representation ~ The lemma leads to the following
fixed point theorem for a linearizable action of an arbitrary torus
Theorem If dim X > 0 (resp dim X > 1) then T has at least two (resp three) fixed points in X
Let (e;) be a basis of k”*! such that there exist characters x; of T with ele, =x(e, (te T)
Choose a cocharacter xY of T such that <x,, x’) # <x;, x”) if x; # x, Then the fixed points of T in IP" are the same as those of the one-dimensional subtorus
x’ (k*) This reduces the proof to the case T = k*, in which case the lemma
implies the existence of at least two fixed points if dim X > 0 The rest requires a bit more work
3.3 One-Dimensional Connected Groups
Theorem A one-dimensional connected linear algebraic group G is isomorphic to either G, or G,,
This is a first classification result, basic for the structure theory of linear algebraic groups The proof is non-trivial A sketch of a geometric argument is as follows (filling in the details requires some care) View a group as in the theorem as an open subset of a smooth projective curve C Then C must have an infinite group of automorphisms fixing one point, which can only be if C is isomorphic to P! The complement of G must then consist of one or two points, and G is isomorphic to G, resp G,, (see [B2, 10.9])
There are more elementary proofs There is a fairly easy reduction to the case that G is unipotent, char(k) = p > 0 and G? = e, in which case one has to show
that G ~ G, (see [Hu, no 20]) 3.4 Connected Solvable Groups
3.4.1 Assume now G to be connected and solvable, as an abstract group It follows from 2.2.4 that the higher commutator subgroups 2”G are closed con- nected subgroups
Trang 28In contrast to the similar result of proposition 1 in 3.1.3 this is not a result in linear algebra, the connectedness assumption is needed
We sketch a proof It suffices to show: if moreover G operates irreducibly in
V =k" then n = 1 By induction on dim G one may assume that the elements of
the commutator group G’ (=%'G) have a common eigenvector in V Using the
connectedness of G’ and the fact that G’ < SL, one concludes that G’ is trivial,
so that G is abelian In this case it is clear that n = 1 (see 3.1.3) Corollary 1 Let G be connected and solvable
(i) The set G, of unipotent elements of G is a connected, normal, unipotent,
closed subgroup containing the commutator subgroup; (ii) G/G, is a torus;
(iti) If, moreover, G is nilpotent the set G, of semi-simple elements is a torus and the product map G, x G, > G is an isomorphism of algebraic groups
Corollary 2 G has a normal series G = Gy > G, 2>+*' > G, = e, such that all
quotients G;/G,,, are isomorphic to either G, or G,,
The proof uses the classification of one dimensional groups
3.4.2 Basic Results on Connected Solvable Groups Theorem Let G be connected and solvable (i) Two maximal tori of G are conjugate;
(ii) Every semi-simple element of G lies in a maximal torus; (H1) The centralizer of a semi-simple element of G is connected;
(iv) Let T be a maximal torus of G The product map T x G, > G is an isomor-
phism of varieties
Here a maximal torus of G means a subtorus of G not strictly contained in a larger one It is clear that these exist The proof of the theorem proceeds by induction on dim G The theorem is due to Borel The following fixed point theorem is also due to him
3.4.3 Borel’s Fixed Point Theorem
Theorem Let G be connected and solvable and let X be a complete G-variety Then G has fixed points in X
Sketch of a proof: Using corollary 2 of 3.4.1 one reduces to the case that G is either G, or G,, If x € X is not a fixed point the orbit G.x is an affine variety, being the image of G under a finite morphism Then G x can not be closed in X and the finite complement of G x in its closure must consist of fixed points
Example Let G be a (connected, solvable) closed subgroup of GL, Take for
X the variety of all complete flags in V = k", i.e of all sequences W¡ c W; c - c V, = V of subspaces of V, with dim V, = i This is a projective, hence complete, variety, on which GL, acts So G acts on X and by Borel’s fixed point theorem there is a flag fixed by G This statement is another formulation of the Lie-
Trang 29I Linear Algebraic Groups 25
3.5 Parabolic Subgroups and Borel Subgroups
3.5.1 Now G is again an arbitrary linear algebraic group
Definition A closed subgroup P of G is parabolic if the quotient variety G/P is complete
If P is parabolic, G/P is even a projective variety, by the remarks made in 2.5.1
Lemma Let P and Q be closed subgroups of G with P > Q If P is parabolic in G and Q is parabolic in P then Q is parabolic in G
This follows from the definitions, using the universal openness property of 2.5.1
Proposition If G is non-solvable it contains proper parabolic subgroups
Assume that G is a closed subgroup of some GL(V) Using the lemma of 2.3.1 one sees that there is x € P(V) such that the orbit G x is projective The isotropy group P of x is then parabolic If P = G we pass to V, = V/x and continue in the same manner The proposition then emerges
Definition A Borel subgroup of G is a closed, connected, solvable subgroup which is maximal for these properties
Theorem (i) A closed subgroup of G is parabolic if and only if it contains a
Borel subgroup In particular, if B is a Borel subgroup then G/B is projective; (ii) Two Borel subgroups of G are conjugate
The if-part of (i) follows from the lemma and the proposition The other
statements come from Borel’s fixed point theorem We call G/B a flag manifold (the name is explained by the next example) By the conjugacy of Borel sub- groups, it is independent of the choice of B, up to isomorphism
3.5.2 Examples
(a) G = GL, Then G operates linearly in V =k" Let X be the variety of all complete flags in V (see the example in 3.4,3) This is a homogeneous space for G Let (e;) be the canonical basis of V and denote by W, the subspace spanned by e;,. , é; The isotropy group in G of the flag (W,, , W,) is the group B of
upper triangular matrices in G, which is connected and solvable (2.2.2) Since X
is a projective variety it follows from part (i) of the previous theorem that Bis a Borel subgroup By part (ii) the Borel subgroups of G are the isotropy groups of the complete flags, i.e the subgroups of G which are upper triangular with respect to some basis of V
Next let (V,, , V,) be an arbitrary flag in V, ie a sequence of distinct sub-
Trang 30(b) G = Sp,,, Let V = k?” and define an alternating bilinear form on V by
(x, y) = * (XiVm+i — Xm+idis
for x = (x,), y = (),) € V Then
SPom = {g € GL,|(g x, ø y) = (x, y) for x, ye V}, as follows from the definition of Sp,,, (2.1.2)
A subspace W of V is totally isotropic for this bilinear form if (x, y) = 0 for all x, y € W One knows that then dim W < m Similarly to the case of GL,, the Borel subgroups of G are the isotropy groups in G of complete isotropic flags, ie flags (V,, , V,,) with all V, totally isotropic, and dim V; = i The parabolic subgroups are the isotropy groups of arbitrary isotropic flags (as follows again
from the results in 4.3.4),
(c) G = SO, (char(k) 4 2) Now define a symmetric bilinear form on V = k" by m «Xx, y> = » (Xi Vm+¡ + Xm+i}i) ifn = 2m is even, i=1 respectively
(x, ¥> = DP (Omit XmaiV) + Xme1 ifn = 2m + 1 is odd i=1
The G is the subgroup of SL, fixing this form One has again a description of
Borel subgroups and parabolic subgroups involving isotropic flags
3.5.3 Maximal Tori and Cartan Subgroups A maximal torus in an arbitrary linear algebraic group G is defined as in 3.4.2
Theorem Two maximal tori of G are conjugate
This follows from the conjugacy of Borel subgroups and the conjugacy of maximal tori in solvable groups The dimension of a maximal torus of G is the rank of G
Let T be a maximal torus, C = Z,(T) its centralizer and N = N,(T) its nor-
malizer C is called a Cartan subgroup
Lemma (i) C is a normal subgroup of N and the quotient W = N/C is finite; (ii) The identity component C° is nilpotent;
(iii) Let X = {(xC°, y)e G/C° x G|x"lyx e C9} The image of the projection
map X — G contains a non-empty open subset of G
(i) follows from the rigidity of tori (3.2.3) and is in fact true for an arbitrary
subtorus of G
(ii) follows by observing that C°/T is unipotent The set X of (iii) is closed in
G/C° x G (by the lemma in 2.5.1) The assertion of (iii) then will follow if one shows that there is g € G lying in only finitely many conjugates of C (by 1.5.4,
Trang 31Pad 15 sự 3 27
Z,(g) = Z,(T), Such elements exist, for an arbitrary subtorus T of G (to prove this it suffices to consider the case G = GL,) The finite group W is the Weyl group of (G, T) It operates faithfully on the torus T
3.5.4 Examples
(a) G=GL, Let T be the subgroup of diagonal matrices Then clearly Z,(T) = T, which shows that T is a maximal torus, coinciding with its Cartan subgroup The normalizer N is the subgroup whose matrices have in each row and each column only one non-zero entry The Weyl group is isomorphic to the symmetric group S,, the operation on T corresponding to permutation of diago- nal elements
The intersection T SL, is a maximal torus in SL, and its normalizer is NaSL,, The corresponding Weyl group is again isomorphic to S,
(b) G = Sp,,, We use the description of 3.5.2 (b) Let (e;) be the canonical basis of k?" The subgroup T of G consisting of elements diagonal with respect to this basis is a maximal torus of G The elements of T are given by
t(€;) = Xjej, tm+i) = Xi Cm+i (1 <i <m), where (X,, , X,) € (K*)" We have Z,(T) = T
The Weyl group W is now isomorphic to the semi-direct product of the symmetric group S,, and the elementary abelian 2-group {1, —1}”, the first group operating on the second one by permutation of coordinates This semi- direct product is a hyperoctahedral group W operates on T in the following manner: if we identify te T with a set (x,, , x,,} in (k*)” then the elements of S,, permute the x; and the elements of {1, —1}" send (x,, ,x,,) into (xit, , x2), where e, = +1
(c) G = SO,,,4,(char(k) # 2) The description is as in 3.5.2 (c) A maximal torus T in G is given by the following elements
{ie = Xie), U(Cism) = Xi 6m¿¡ (1 <i < Mm),
t(Com+1) = Cam+i-
We have Z,(T) = T The Weyl group is as in (b) and the action on T is similar
(d) G = SO,,,(char(k) 4 2) View SO,,, as a subgroup of SO,,,,, in an obvi- ous way Then the maximal torus T of SO,,,,, is also one for SO,,, The Weyl
group is now the semi-direct product of S,, and the subgroup of {1, —1}” of the
elements ¢,, , &, With €, .¢,, = 1 This is normal subgroup of index two of the Weyl group of the previous example
3.5.5 Further Properties of Borel Groups, Applications G is a connected linear algebraic group
Theorem 1 Every element of G lies in a Borel subgroup
Let B be a Borel subgroup and put Y = {(xB, y) e G/B x G|x"!yx e B}
Since G/B is complete the projection morphism z: Y > G is closed From part (iii) of the lemma in 3.5.3 one then infers that x is surjective Here one needs the
TM [HH 3430xX
Trang 32Corollary 1 Every semi-simple element of G lies in a maximal torus This follows from the corresponding result for solvable groups (3.4.2 (ii)) Corollary 2 The centralizer of a subtorus of G is connected In particular:
Cartan subgroups are connected
If S is a subtorus and x € Z,(S) then an application of the fixed point theorem of 3.4.3 to S and the closed subvariety (B being a Borel group) {yB|y"'xy e B} of G/B shows that there is a Borel group containing both S and x Then use
3.4.2 (iii),
Proposition Let S be a subtorus of G and B a Borel group containing S Then
Z,(S) 0 B is a Borel subgroup of Z,(S) The proof uses the rigidity of tori (3.2.3) Theorem 2 A parabolic subgroup is connected
Recall that G is assumed to be connected An equivalent statement is: if B is a Borel group then B = N,(B) This is a crucial result, first proved by Chevalley
See [B2, 11.16]
Corollary (i) Let T be a maximal torus of G The number of Borel groups
containing T equals the order |W| of the Weyl group of (G, T),
(ii) If |W| = 1 then G is solvable If |W| = 2 the flag manifolds G/B are one-
dimensional
It also follows from theorem 2 that one may view a flag manifold G/B as the variety of all Borel groups of G
3.6 Radicals, Semi-simple and Reductive Groups
Definition The radical Rad(G) (resp the unipotent radical R,(G)) of G is the maximal closed, connected, solvable (resp unipotent), normal subgroup of G G
is semi-simple (resp reductive) when Rad(G) (resp R,(G)) is trivial
It is easy to see that radicals exist Clearly R,(G) < Rad(G) A semi-simple group is reductive
Using Borel’s fixed point theorem one sees that Rad(G) is the identity compo- nent of the intersection of all Borel groups of G From the rigidity of tori (3.2.3) one concludes that in a reductive group G the radical Rad(G) is a torus in the center of the identity component G°
The structure theory of reductive groups is a central part of the theory of linear algebraic groups It will be reviewed in the next section,
§4 Reductive Groups
Trang 33I Linear Algebraic Groups 29
4.1.1 Assume that G is non-solvable of rank one (ie dim T = 1) Since the automorphism group of G,, has order 2 (3.2.3) the Weyl group W of (G, T) has order 2 (by the corollary of theorem 2 in 3.5.5) Let n be an element in N,(T) — Z,(T) One shows that if B is a Borel subgroup of G containing T we have that G is the union of B and BnB (this is an example of a Bruhat decomposition, see 4.3.2), Moreover, as a consequence of the same corollary and the theorem in 3.2.5 we find that dim G/B = 1, which implies that dim(B/B na nBn”') = 1 Exploiting these facts further one arrives at the follow- ing basic result (details can be found in [Sp3, 8.2]
Theorem Let G be a connected, semi-simple linear algebraic group of rank one Then G is isomorphic to either SL, or PGL
Here PGL, is the quotient of GL, by the one-dimensional torus of scalar
multiplications
A somewhat weaker result can be proved in an algebro-geometric manner (see [B2, 13.13]) One first shows that the smooth projective curve G/B must be isomorphic to P', as it has an infinite group of automorphisms fixing a point Then one uses Liiroth’s theorem, according to which the automorphism group of P! is isomorphic to PGL,
4.1.2 Roots Now let G and T be arbitrary We denote by X and X” the character (resp cocharacter) group of T (see 3.2.1) Let R < X be the (finite) set of non-zero weights of T in the Lie algebra g of G, T acting via the adjoint representation (see 3.2.4) For œe R iet T, be the identity component of the kernel of the character x Then T, is a subtorus of T of codimension 1 The centralizer G, = Z,(T,) is a connected closed subgroup of G (by corollary 2 of
theorem 1 in 3.5.5)
If xe R and G, is non-solvable then @ is called a root of G relative to T Let R=R(G, T) < X be the set of roots If W = Ng(T)/Z,(T) is the Weyl group of
(G, T) then W acts on T and stabilizes R
For « € R choose n, € Ng (T) — Zg(T) and let s, be the element of W defined by nạ
Lemma 1 There is a unique wŸY XY such that for xe X, xY e XY we haue {ie =x— (x, a" a,
Sy-X° =x’ — Ca,x’ >a”, Moreover, <a, wY > = 2 and s? = 1
If ais a root then G/Rad(G,) is semi-simple of rank one Using the theorem of 4.1.1 the lemma is obtained via explicit checks in SL, and PGL,
Trang 34to f2 resp f~? The corresponding coroots send t € k* to the above matrix resp,
its inverse For both roots we may take 0 1
n, =
—1 0
The next lemma shows that G can be built up from the groups G,,
Lemma 2 G is generated by the G,, ae R
This follows from the observation that the Lie algebra of G is spanned by
those of the G, (« € R), which follows from the lemma in 3.1.2 4.2 The Root Datum and the Root System
4.2.1 A root datum is a quadruple ¥ = (X, R, X“, R’) where X and XY are
free abelian groups, in duality by a pairing < , >, R and R” are finite subsets of X resp X”, such that there is a bijection a-+a” of R onto R” For aeR
define endomorphisms s, of X resp X ’ by the formulas of lemma 1 in 4.1.2 The
following axioms are imposed: (RD 1) [fae R then (a, a” > = 2;
(RD 2) [fae Rthens,R = R,s,RY = RY
It follows that s? = 1 and s,a = —a We call s, the reflection defined by « The group of automorphisms of X generated by them is called the Weyl group of 7 It is a finite group It is clear that we have introduced a root datum in 4.1.2 We
denote it by ¥(G, T)
Proposition W(¥(G, T)) is isomorphic to Ng(T)/Z,(T)
The second group was introduced in 3.5.3 If Y is as above the dual root
đatum 1s (XY, RY, X, R)
4.2.2, The Root System Let ¥ be an arbitrary root datum Assume that
R # @ Denote by Q the subgroup of X generated by R and put V = 0 @,R Then R (=R ® 1) is a root system in V in the usual sense (see e.g [Bo2, p 142])
This means that:
(1) R is finite and generates V, moreover 0 ¢ R;
(2) If aE R there is «’ in the dual VY of V such that <a, a” > = 2 and that
the reflection s, (defined as before) stabilizes R;
(3) If a, Be R then a’ () is an integer
In the case that ¥ = ¥(G, T) we have that R = @ if and only if G is solvable
So if G is non-solvable we have a root system R = R(G, T) of G relative to T It
has the further property of being reduced, ic if ae R, ce Q and cae R, then c = +1 (a consequence of the theorem of 4.1.1)
4.2.3, Let R be a root system in the real vector space V As before, its Weyl group is the finite group generated by the reflections s, (« € R)
Trang 35I Linear Algebraic Groups 31 n> 0, n; > 0, equals zero Given a system of positive roots R* there exists a
unique subset D of R* consisting of linearly independent elements such that every root in R® is a positive integral linear combination of the roots in D The set D is called the basis of R defined by R* The set S = {s,|a € D} then gener-
ates the Wey! group W of R In fact, (W, S) is a Coxeter system For these facts see [Bo2, Ch VI]
4.2.4, Examples
(a) G = GL,, Let T be the torus of diagonal matrices Define characters «;, of
T by
a, (diag(t,, ,t,)) = tư q <s i, j S h, i # j)
Then R is the set of «,; The corresponding groups G, are non-solvable, so R = R The groups G,/Ker « are all isomorphic to PGL, Now a,j is the homo-
morphism k* > G with ;;(¢) a diagonal matrix with i" entry t, j" entry t~' and
all other diagonal entries 1
The root datum Y(G, T) = (X, R, XY, R”) is given by X = XY = Z" with standard pairing, R = RY = {e, — e,|i #j}, where (e,) is the canonical basis
The Weyl group is isomorphic to the symmetric group S, (compare 3.5.4 (a)) The set R* = {e, —- e|1 <i <j <n} is a system of positive roots The corre- sponding basis is D = {e; — e,,,|1 <i<n-— 1}
(b) G=SL,, T again being the torus of diagonal matrices Now the root
datum of (G, T) is (X,, R,, Xj’, Ry), where (X, X’ being as in (a)) X,=X/@(e, + + 8#), XxX, = Heenan) eX” y Xx, = of i=1 If x: X > X, is the canonical map then R, = aR and RY = RY < X, (R and RY as in (a))
For G = PGL, the root datum is the dual of the one for SL,
(c) G = Sp,,, (m = 2) With the maximal torus T described in 3.5.4 (b) one obtains using 2.4.3(d) the root datum (X, R, X ’, RY), where again X = XY = Z"
and R={+2e, te tell <ij<niF ji, RY = {te, te + ell <i, j<n,i#j} A system of positive roots for R is {2e;, e; + e|1 <i <j <n}, the corresponding basis being {e; — e,,,|1 <i<n— 1}vV {2e,} Similar data for RY are obtained by replacing the roots 2e; by e; (1 <i <n)
(d) G = SỐ;„„¡ (char(k) # 2, n > 2) The root datum is the dual of the one of the previous example
(ec) G = SO,,, (char(k) # 2, n>2) We now obtain the root datum (X, R,
XY, RY) with X = XY =Z",R=RY =( +e, + ei Fj} A system of positive roots is {e,te,|1 <i<j<n}, with corresponding basis {e,—e,,,|1<i<n—1l}uU
{ent + en}:
The root systems associated to the root data of examples (a), (c), (d), (e) are
Trang 364.2.5 Borel Groups and Systems of Positive Roots We keep the notations of 4.1.2 If x is a root then the group G,/Rad(G,) is semi-simple of rank one It follows that
L(G,/R,(G,)) = L(Zg(T)/Zg(T) 0 R,(Gz)) ® KX, @ kX _,,
where X, and X_, are weight vectors for « and —«, respectively Now let B be a Borel subgroup containing T By 3.5.5, proposition, B 4 G, is a Borel subgroup in G, Using the theorem of 4.1.1 we see that
L(BOG,/R,(G,)) = L(Zg(T)/Zg(T) > R,AG)) OEX,,
where f is either « or —« (the left hand side being viewed as a subspace of L(G,/R,(G,)) We denote by R*(B) the set of these £, « running through the roots of (G, T)
Proposition R*(B) is a system of positive roots in R The proof uses information about SL, ({Sp3, p 197])
4.2.6 Characterization of the Unipotent Radical, Applications The next re-
sult, due to Chevalley, is a basic one
Theorem The unipotent radical of G is the identity component of the intersec-
tion of the Borel subgroups containing T
Besides the previous proposition, the proof uses the following facts: (a) The
Borel subgroups of G containing T generate G (a consequence of lemma 2 in
4.1.2), (b) Let R be a root system and R* a system of positive roots If « and f are linearly independent roots there is an element w of the Weyl group such that wa and w8 both lie in R*
Corollary 1 If S is a subtorus of G then R,(Z,(S)) < R,(G) In particular, if G is reductive then so is Z,(S)
See 3.5.5, proposition
Corollary 2 If G is reductive then Z,(T) = T Hence Cartan subgroups and maximal tori coincide
Using the theorem of 4.1.1 we now deduce the following properties Proposition Let G be reductive, let R be the root system of (G, T) (i) The roots of R are the non-zero weights of T in the Lie algebra of G; (ii) For any « € R there exists an isomorphism x, of G, onto a closed subgroup
X, of G such that forte T,a ek,
tx,(a)t! = x,(a(t)a);
(iii) If Bis a Borel subgroup containing T then « € R*(B) if and only if X, < B The X, are the root subgroups of G associated to T
Example G = GL,, The roots were described in 4.2.4(a) Let x,; = x,,, We
Trang 37I Linear Algebraic Groups 33
4,3 Basic Properties of Reductive Groups G is a non-trivial connected re- ductive linear algebraic group, T is a maximal torus of G and B a Borel sub- group containing T We put R* = R*(B) and we write « > 0 if «e R* For the results to be reviewed below see [Sp3, Ch 10]
4.3.1 Properties of Borel Groups
Proposition 1 Let («,, ,%,) be the roots in R*, in some order The mor-
phism @: (G,)" > R,B with p(ay, ., &,) = Xq,(41)°** Xq,,(An) is an isomorphism of varieties
Here the x, are as in 4.2.6 It follows that R,(B) is generated by the root groups X, with « > 0 Also, dim R,(B) = |R*| =n
Corollary Let a, Be R,a # +f There exist elements c, 4; ; € k, defined if i, j
are integers >0 with ia + jB € R, such that
x„(4)x;(b)x„(a)”" = xạ;(b) Il Xiatjp(Ca,p:i.j4 D’),
the product being taken in some preassigned order
Theorem Let R* be an arbitrary system of positive roots in R
(i) T and the X, with ae R* generate a Borel subgroup; _
(ii) There is a unique element w of the Weyl group such that R* = w(R*) The basis D defined by R* can be characterized as follows in terms of Borel
groups
Proposition 2 Let B be a Borel subgroup containing T There exists « € D with R*(B) = s,(R*) if and only if dim B 7 B = dim B — 1
43.2 Bruhat’s Lemma Recall that S = {s,|a € D} generates the Weyl group
W of R For w e W define the length I(w) relative to S to be the minimal integer
h such that w can be written as a product of h elements of S Further define for
weW
R(w) = {« e Riwae —R*}
Then the number of elements of R(w) equals /(w) There is a unique element wo
of W with maximal length We have
Wo(R*) = —R*
and I(w)) = |R*| Put U = R,B and let U,, be the subgroup of U generated by the root subgroups X, with a e R(w) If we W = N,(T)/T we denote by wa representative of w in N,(T) We write C(w) = BwB, a double coset in G
Theorem (Bruhat’s lemma) (i) G is the disjoint union of the C(w), we W;
(ii) Each C(w) is a locally closed subvariety of G The map U,, x B —> G send- ing (u, b) to uwb is an isomorphism of varieties;
(iii) The closure C(w) is a union of certain C(w’)
Trang 38Corollary For s eS, we W we have
C(s)C(w) = C(sw) if l(sw) > I(w), C(s)C(w) = C(sw) U C(w) if sw) < !(w)
This implies that (G, B, N, S) is a Tits system in the sense of [Bo2, Ch IV]
It follows from the theorem that dim C(w) = [(w) + dim B It also follows that there is one open C(w), namely C(w,.) Hence dim G = dim C(wo) = đim 7 + | RỊ Define an order on W by: w’ < wif C(w’) < C(w) (the Bruhat order) It can be shown that if w=s, 8, (h= I(w)), then we have w’ < w if and only if w’ equals a subproduct, obtained by deleting some of the s;,
4.3.3, Schubert Varieties Let X = G/B, the flag manifold of G and denote for
we W by X,, the image of C(w) in X This is a Bruhat cell The closure S,, = X,, is a Schubert variety These are interesting projective varieties, in general singu- lar The results of 4.3.2 then imply:
Proposition (i) X is the disjoint union of the X,,, w € W; (ii) X,, is a locally closed subvariety of X, isomorphic to k'™:
(ili) S, = Uw <w Xw
In particular, X = wew X,, is a “paving of X by affine spaces” One can view the Bruhat cells in X as the B-orbits (or the U-orbits) in X, the groups
acting via left translations A useful variant of the decomposition (i) (and of the corresponding decomposition of G) is as follows Consider X x X, with the
diagonal G-action defined by g.(x, x’) = (g.x, g.x’)(x, x’ € X) Let O(w) be the G-orbit containing (B, w B) Then the O(w) are the G-orbits in X x X and one
has properties similar to those of the previous proposition (and the theorem of 4.3.2) Recall that we may view the points of X to be the Borel groups of G (see
3.5.5) We say that an ordered pair (B,, B,) of Borel subgroups is in position
we W if (B,, B) € O(w) If B, = x;Bx;' (i = 1, 2) then this is so if and only if Xị'x; e BwB
Example G = GL, We view X as the variety of complete flags in V = k"
Trang 39I Linear Algebraic Groups 35
4.3.4, Parabolic Subgroups We first notice that any parabolic subgroup of G
is conjugate to a unique one containing B (by the theorem in 3.5.1 and theorem
2 in 3.5.5)
Let D be the basis of R defined by R* For any subset J < D we denote by R, < R the set of roots which are linear combinations of roots in J Put Ry = R, 0 R* Then R, is a root system (in the appropriate vector space) in which R; is a system of positive roots, with corresponding basis I The Weyl group of R,
is the subgroup W, of W generated by the reflections s, with « € I
Theorem (i) P; = wew, C(w) is a parabolic subgroup of G containing B;
(ii) If P is a parabolic subgroup containing B there is a unique subset I of D such that P = P,;
(iii) R,(P,) is generated by the root groups X, with a € R* — R;;
(iv) Let L, be the subgroup of G generated by T and the X, with a € R, Then
L, is a closed subgroup of P, which is connected, reductive The product map L, x R,(P,;) > P, is an isomorphism of varieties
The theorem provides an explicit description of parabolic groups Notice that Pg = B, Pp = G For I = D — {a} with a € D we obtain the maximal para-
bolic subgroups containing B
If P is a parabolic subgroup of G a Levi subgroup of P is a closed subgroup L
of P such that the product map L x R,(P)— P is an isomorphism Two Levi
groups are conjugate in P Examples
(a) G =GL, Take B (resp T) to be the subgroup of upper triangular (resp
diagonal) matrices According to 4.2.4 (a) the basis D consists of the characters
x, of T given by
%(điag(f, ,f„)) = tan (l<i<n-— 1)
We identify D with {1, 2, , 2 — 1} Let J be a subset of D and write its comple- ment as
D_—]-= {ai, đa + đ;, ,di +''" + ass},
with a; > 0 Now the description of P, coming from (iii) and (iv) of the theorem
(using the description of the root groups X,, see 4.2.6) shows that P, consists of upper triangular block matrices, with blocks along the diagonal of consecutive SIZ©S đị, đ;, , đy, Where
đị +; +'''+dc_ ¡ +úy¿=ñh
Then L„ 1s isomorphic to GL„, x GL„, x -:: x GL„ and R„(P¡) consists of the matrices in P, where the diagonal blocks contain identity matrices Incidentally, it should be noticed that for different J the subgroups L,; may be conjugate in G, which implies that a group may be a Levi subgroup for several parabolic
subgroups
The quotient variety G/P, (I as above) is isomorphic to the variety of all flags
Trang 40In particular, if P, is a maximal parabolic subgroup then G/P, is isomorphic to the Grassmannian G,,,, of all a-dimensional subspaces of k”, for some a
(b) Another way to show that the parabolic subgroups of G containing B
{in the situation of the previous example) are the groups of upper triangular block matrices containing B, is to observe that the number of these parabolic groups equals 2”~', by the theorem One then checks that the number of differ-
ent groups of block matrices also equals that number This argument can also be used to get the description of the parabolic subgroups in the other classical
groups, mentioned in 3.5.2
There is another useful description of parabolic subgroups Let A be a cocharacter of G, ic a homomorphism of algebraic groups k* > G Define a
k*-action on G by
t.x =At)xdA(t)* (te k*,x eG)
Proposition (i) The set of x éG such that lim, 9 t.x exists is a parabolic
subgroup P(A) of G The centralizer of Im / is a Levi subgroup of P(A) Moreover,
fim tax = eh
1-0
R,(PQ)) = * eG (ii) Any parabolic subgroup of G is a P(A)
The limit occurring in the proposition is as in 3.2.5 The proof comes fairly easily from the above description of parabolic subgroups
4.3.5 Generalized Schubert Varieties With the previous notations, it is clear from the theorem in 4.3.2 that we have a double coset decomposition
G =\) BwP,
This can be made more precise, as a consequence of general results about Tits systems [Bo2, p 28] We only give a result on generalized Schubert varieties G/P, With the notations of 4.3.4 put
W, = {we W(I(ws,) > I(w) for a € I}
Then the product map W/ x W, > W is bijective [loc cit., p 37] Let Y = G/P,
and denote by Y,, the image of BwP, in Y
Proposition (i) Y is the disjoint union of the Y,,, w é Wy;
(ii) Y,(w € W) is a locally closed subvariety of Y, isomorphic to k'™,
Example Y = P"~! This is a Grassmannian of example (a) in 4.3.4 With
the notations of that example we have I = {2, 3, ,n — 1} and W, is the sub- group of W = S, fixing 1,so We~ S,_, Now W is the set of permutations wS,
with