Download Subgroups of Linear Algebraic Groups

Subgroups of Linear Algebraic Groups
Subgroups of Linear Algebraic Groups
Contents
Introduction
Acknowledgements
1. Basic definitions and examples
1.1. Introduction to Linear Algebraic Groups
1.2. Connectedness
2. Background in algebraic geometry
3. Tori, Unipotent and Connected Solvable Groups
3.1. Unipotent Groups
3.2. Tori
3.3. Connected Solvable Groups
4. Borel Subgroups
4.1. Actions of algebraic groups
4.2. The Borel fixed point theorem and consequences
5. Connected reductive groups
5.1. Reductive and semisimple groups
5.2. Lie algebras and root systems
5.3. Bruhat decomposition
6. Parabolic subgroups
6.1. Standard parabolic subgroups and the Levi decomposition
6.2. The Borel-Tits Theorem
7. G-complete reducibility
References
1
4
5
5
8
10
12
12
15
17
18
19
22
27
28
29
36
38
39
44
46
50
Introduction
Let G be a variety over an algebraically closed field k. We call G an algebraic
group if it is equipped with a group structure such that the multiplication map
G×G → G and the inversion map G → G are morphisms of varieties. Furthermore,
when G is an affine variety, it is called a linear algebraic group. For example, the
2
set Mn (k) of n × n matrices over k can be easily identified with k n so is an affine
variety. Now, since the determinant of a matrix is a polynomial in its entries, we see
that SLn is also an affine variety, and it’s easy to show that it is a linear algebraic
group. One can also show that GLn is a linear algebraic group.
Algebraic groups have applications to several areas of pure mathematics. For
instance, they are notably central to the Langlands program in Number Theory.
They can also be a good way to construct an important class of finite groups, called
finite groups of Lie type. These groups arise as fixed point sets of certain types
2
of endomorphisms of some linear algebraic group over k = Fq , where q is a prime
power. For example, most of the finite simple groups are finite groups of Lie type.
We will unfortunately not have time to discuss these applications, and the reader
is referred to [MT, Part III] for a detailed introduction to finite groups of Lie type.
The combination of the group structure with the variety structure on G forces
it to have some nice properties. For instance, a linear algebraic group is irreducible
as a variety if and only if it is connected (see Proposition 1.8). Moreover, any
linear algebraic group can be embedded as a closed subgroup of GLn for some n
(see Corollary 4.8). Thus linear algebraic groups can be viewed as certain groups
of matrices. However, the embedding into GLn is not canonical, and in general
we have no control over what it is. Therefore, our results will usually be stated
and proved in full generality, without assuming that our groups have already been
embedded into some GLn .
An important ingredient in studying linear algebraic groups is the notion of a
Borel subgroup, which is a maximal closed connected solvable subgroup. For example, the group of invertible upper triangular matrices is a Borel subgroup of GLn .
This naturally leads to the study of a larger class of subgroups, called parabolic
subgroups. These are closed subgroups which contain a Borel subgroup of G. When
the group G is connected and reductive, which means it has no non-trivial proper
closed connected unipotent normal subgroups, the structure of its parabolic subgroups is well understood. For instance, they can be expressed as a disjoint union
of double cosets of the Borel subgroup they contain (see Theorem 6.6). Parabolic
subgroups of connected reductive groups can then be used to generalise some familiar concepts from representation theory. In particular, Serre [Se] introduced the
notion of a G-completely reducible (G-cr for short) subgroup (see Definition 7.1),
which generalises the notion of a group acting completely reducibly on a vector
space V . Indeed, in the special case G = GL(V ) (V a finite dimensional k-vector
space), a closed subgroup H of G is G-cr if and only if V is a semisimple H-module.
A recent theorem of Bate, Martin and Röhrle [BMR] asserts that G-complete
reducibility is equivalent to the notion of strong reductivity, due to Richardson [R].
They then used this to show that a closed normal subgroup of a G-cr subgroup is
itself G-cr (see [BMR, Theorem 3.10]). In the case G = GL(V ), this reduces to a
well known result in Clifford theory: if V is a semisimple H-module and N H,
then V is a semisimple N -module. The purpose of this essay is to give an account
of the general theory of linear algebraic groups, focusing on their Borel subgroups
and their parabolic subgroups, and to prove the aforementioned results of Bate,
Martin and Röhrle. In the study of parabolic subgroups, we will restrict ourselves
to the case where the group G is connected reductive.
We begin in Section 1 by introducing linear algebraic groups, giving several
examples. We then define morphisms of algebraic groups, which are just morphisms
of varieties that are also group homomorphisms, and we show that their kernels
and images are closed. We also investigate the notion of connectedness and give
examples of connected linear algebraic groups.
In Section 2, we give the background in algebraic geometry that will be needed
later on. More specifically, we discuss tangent spaces, differentials, projective varieties, dimension theory and complete varieties, which play an important role in
the study of Borel subgroups in Section 4. However, the emphasis of this essay is
3
on algebraic results, and not so much on the geometry of linear algebraic groups.
So the treatment here is mostly expository, and contains almost no proofs.
Section 3 is split into three parts. In the first part, we discuss unipotent groups.
These are linear algebraic groups which can be embedded into GLn as a group
of matrices whose only eigenvalue is 1. In order to study them, we introduce a
multiplicative version of the well-known Jordan decomposition for endomorphisms
of a finite dimensional vector space. This can be translated back to a decomposition of elements of an arbitrary linear algebraic group into so-called unipotent and
semisimple parts. This leads to the definition of a unipotent group, and we then
prove that such a subgroup of GLn is conjugate to a group of upper triangular
matrices with 1’s along the diagonal (see Theorem 3.7 and Corollary 3.8).
In the second part, we introduce tori and outline some of their basic properties.
A torus is a linear algebraic group isomorphic to the group of n × n diagonal
matrices for some n. An important concept needed to understand tori is the notion
of a character. For an algebraic group G, a character of G is a morphism between
G and the multiplicative group k ∗ . The set X(G) of characters of G is easily seen
to be an abelian group, and we show that it is finitely generated when G is a torus.
We finally investigate connected solvable groups in the third part. Our main
result is the Lie-Kolchin theorem (Theorem 3.16), which asserts that a connected
solvable subgroup of GLn has a common eigenvector. As an immediate consequence,
we obtain that a connected solvable subgroup of GLn is conjugate to a subgroup of
the group of all invertible upper triangular matrices (Corollary 3.17). Moreover, we
give without proof the result that a connected solvable group G is the semidirect
product Gu o T of its subgroup Gu of unipotent elements with a maximal torus T .
In Section 4, we study Borel subgroups of arbitrary linear algebraic groups. To
do so, we first study actions of algebraic groups. We also explain how to make a
quotient G/H into a linear algebraic group when H is a closed normal subgroup of
G. Along the way, we will prove that linear algebraic groups can be embedded into
some GLn (see Corollary 4.8). Using these tools, we then show the Borel fixed point
theorem (Theorem 4.13), which asserts that if a connected solvable group acts on
a projective variety in a way so that the action is given by a morphism of varieties,
then it must have a fixed point. This has important consequences. For instance,
it implies that the Borel subgroups of a linear algebraic group are all conjugate
(Proposition 4.14). We also introduce parabolic subgroups and show that they are
connected and self-normalising (see Corollary 4.28).
Section 5 is devoted to the study of connected reductive linear algebraic groups.
The theory there is quite lengthy, and we do not have time for too much details, but
we try to explain all the steps required to obtain the structure of connected reductive
groups (Theorem 5.13). However, we do not prove the theorem, nor many of the
results that build up to it. The structure of these groups relies crucially on their
root system, which itself is defined using Lie algebras. Therefore, after introducing
reductive groups, we explain how to associate a Lie algebra to a linear algebraic
group, and outline basic properties it must satisfy. This allows us to define the
root system of G with respect to a maximal torus T . We then work our way to
the structure theorem from there. Subsequently, we discuss a few basic facts about
abstract root systems and state the result that a Borel subgroup B of G containing
T defines a base for the root system. We finally obtain the very important Bruhat
decomposition (Theorem 5.26), which states that G can be decomposed as a disjoint
4
union of double cosets of B. This decomposition is more generally true for any group
G with a BN-pair (see Definition 5.24), and it is in that context that we prove it.
This result also allows us to deduce that the intersection of two Borel subgroups is
connected and contains a maximal torus (Corollary 5.27).
Because of lack of time, we do not discuss here the classification theorem of
Chevalley, which gives a one-to-one correspondence between isomorphism classes of
semisimple linear algebraic groups (these are connected reductive groups with no
proper non-trivial closed connected solvable normal subgroups) and isomorphism
classes of root data, which are combinatorial objects that can be obtained from the
root system. One remarkable feature of this theorem is that it holds in arbitrary
characteristic, and not only in characteristic 0 unlike the classification theorem
for semisimple Lie algebras. The reader is referred to [S, Chapters 9 and 10, in
particular Theorem 10.1.1] for a proof and exact statement of this theorem.
With the results of Section 5 at our disposal, we investigate parabolic subgroups
of connected reductive groups in Section 6. These have a nice structure which also
relies crucially on the root system: they are uniquely determined by some subset
of the base of the root system which was obtained from the Borel subgroup they
contain. We prove again that these groups are closed, connected, self-normalising
subgroups of G without appealing to the results in Section 4, and that two distinct
parabolic subgroups containing the same Borel subgroup B are not conjugate. We
carry on investigating the structure of parabolic subgroups by decomposing them
into a product of a unipotent group with a connected reductive group, called its
Levi complement. A Levi subgroup is then defined to be a conjugate of a Levi complement, and we show that Levi subgroups are precisely the centralisers of subtori
of G (Proposition 6.13). We then move on to the Borel-Tits theorem (Theorem
6.15), which asserts that for a closed unipotent subgroup U of G contained in a
Borel subgroup, there exists a parabolic subgroup P of G such that NG (U ) ≤ P
and U is contained in a closed connected unipotent normal subgroup of P . Using
this, we immediately obtain that given a maximal closed subgroup H of G, either
H is reductive or H is parabolic (see Theorem 6.18).
Finally, we discuss G-complete reducibility in Section 7. A closed subgroup H
of G is G-completely reducible if whenever it is included in a parabolic subgroup of
G, it is actually included in a Levi subgroup of it. We also introduce the notion
of a strongly reductive subgroup. A closed subgroup H of G is strongly reductive
if it is not contained in any parabolic subgroup of CG (S), where S is a maximal
torus of CG (H). We then show that a closed subgroup is strongly reductive if and
only if it is G-completely reducible (Theorem 7.7). This requires a few lemmas on
the intersection of two parabolic subgroups, which we state beforehand. Using a
theorem of Martin [M], we deduce that a closed normal subgroup of a G-completely
reducible subgroup is itself G-completely reducible (Theorem 7.9), and using the
Borel-Tits theorem, we show that the converse is not true in general.
In Sections 1-6, we follow the texts by Malle and Testerman [MT] and Humphreys
[H], as well as Springer [S] or Borel [B] for some of the more geometrical results. In
Section 7, our main reference is [BMR].
Acknowledgements
I am grateful to David Stewart for setting this essay and for kindly giving me
advice for a talk, based on this work, that I gave in the Part III Seminars.
5
1. Basic definitions and examples
1.1. Introduction to Linear Algebraic Groups. Let k be an algebraically
closed field. Recall that a subset of k n is called an algebraic set if it is of the
form
Z(I) = {(x1 , . . . , xn ) ∈ k n : f (x1 , . . . , xn ) = 0 for all f ∈ I}
where I is an ideal in the polynomial ring k[T1 , . . . , Tn ]. Taking closed sets to be
algebraic sets defines a topolgy on k n , called the Zariski topology. An affine variety
is an algebraic set together with its induced Zariski topology.
Given an algebraic set X, we can define an ideal in k[T1 , . . . , Tn ] by
I(X) = {f ∈ k[T1 , . . . , Tn ] : f (x1 , . . . , xn ) = 0 for all (x1 , . . . , xn ) ∈ X}
The quotient k[X1 , . . . , Xn ]/I(X) is called the coordinate algebra or algebra of regular functions on X, and is denoted by k[X]
If X ⊂ k n , Y ⊂ k m are affine varieties, then the product X × Y naturally has
the structure of an algebraic set in k n+m and thus is also an affine variety when
equipped with the Zariski topology (which in general is not the same as the product
topology). Note that k[X × Y ] ∼
= k[X] ⊗k k[Y ].
A map ϕ : X → Y is called a morphism of affine varieties if it can be defined
by polynomial functions in the coordinates. Note that such maps are continuous
with respect to the Zariski topology. The morphism ϕ induces a k-algebra homomorphism
ϕ∗ : k[Y ] −→ k[X]
f 7−→ f ◦ ϕ
We can now define our main object of study:
Definition 1.1. A linear algebraic group is an affine variety G equipped with a
group structure such that the group operations
µ : G × G −→ G,
(g, h) 7−→ gh,
ι : G −→ G,
g 7−→ g −1 ,
are morphisms of varieties.
Example 1.2. Let’s first look at several examples of linear algebraic groups:
(1) Ga = (k, +), the additive group of k. It’s clear that it satisfies the definition
(it is the zero set of the zero polynomial), and we have k[Ga ] = k[T ], the
usual polynomial ring.
(2) Gm = (k ∗ , ×), the multiplicative group of k ∗ . We can identify it with
{(x, y) ∈ k 2 : xy = 1}
with componentwise multiplication by mapping x 7→ (x, x−1 ). This set is
clearly a closed subset of k 2 , being the zero set of the polynomial T1 T2 − 1.
Multiplication and inversion are clearly given by polynomials so it is a linear
algebraic group, and k[Gm ] = k[T1 , T2 ]/(T1 T2 − 1) ∼
= k[T, T −1 ].
(3) The general linear group GLn = {A ∈ Mn (k) : det A 6= 0} is also a linear
algebraic group. As for Gm , one way of seeing this is to identify it with
2
{(A, y) ∈ k n × k : det A · y = 1}
6
via A 7→ (A, det A−1 ), with componentwise multiplication where we identify
2
k n with Mn (k) (and so multiplication in the first component is matrix
multiplication).
Since det is given by a polynomial in the matrix entries, this is a closed
2
subset of k n +1 . Multiplication is then clearly given by polynomials, and by
Cramer’s rule, so is inversion. We can also find the ring of regular functions:
k[GLn ] = k[Tij , Y : 1 ≤ i, j ≤ n]/(det (Tij ) · Y − 1)
∼
= k[Tij : 1 ≤ i, j ≤ n]det (T )
ij
the localisation of k[Tij : 1 ≤ i, j ≤ n] at the multiplicatively closed subset
generated by det (Tij ).
(4) The special linear group SLn = {A ∈ GLn : det A = 1} is a closed subgroup
of GLn , as det is given by a polynomial in the matrix entries, and so is
also a linear algebraic group. Its ring of regular functions is then clearly
k[Tij : 1 ≤ i, j ≤ n]/(det (Tij ) − 1).
(5) Similarly, the following are closed subgroups of GL(n) and so linear algebraic groups:
• The group of invertible upper triangular matrices




 ∗ ... ∗
Tn =  . . . ...  ∈ GLn = {(aij ) ∈ GLn : aij = 0 for i > j}.


∗
• The group of upper triangular matrices with 1’s on the diagonal


 1 . . . ∗. 
.. .
= {(aij ) ∈ Tn : aii = 1 for 1 ≤ i ≤ n}.
Un = 
. .


1
• The group of invertible diagonal matrices
!
)
(
∗ .
..
∈ GLn = {(aij ) ∈ GL(n) : aij = 0 for i 6= j}.
Dn =
∗
Note that Dn ∼
= Gm × . . . × Gm
|
{z
}
n times
!
0 .1
0
Kn
(6) Let J2n =
where Kn =
. The symplectic group in
..
−Kn 0
1
0
dimension 2n is then defined to be
Sp2n = {A ∈ GL2n : At J2n A = J2n }.
It is a closed subgroup of GL2n , thus a linear algebraic group. It is the
group of linear transformations leaving invariant the non-degenerate skewsymmetric bilinear form given by J2n .
(7) Orthogonal groups are also linear algebraic groups. For simplicity, we assume char(k) 6= 2 (for the general case see [MT, section 1.2]). The orthogonal group in dimension n is given by
GOn = {A ∈ GLn : At Kn A = Kn }.
It is the group of linear transformations leaving invariant the non-degenerate
symmetric bilinear form given by Kn .
7
Having defined linear algebraic groups, we then consider the maps between them.
We require them to preserve both the geometrical structure and the group structure
of algebraic groups.
Definition 1.3. Let G1 , G2 be linear algebraic groups. A map ϕ : G1 → G2 is a
morphism of algebraic groups if it is a group homomorphism and a morphism of
affine varieties.
We would like for such maps to have nice images and kernels. In order to prove
it, we first need a couple of basic results:
Lemma 1.4. Let U, V be two dense open sets of an algebraic group G. Then
G=U ·V
Proof. Pick x ∈ G. From the definition of algebraic groups, inversion is a continous
map with continuous inverse (being its own inverse), and thus is a homeomorphim.
Hence V −1 is also open dense. Similarly, multiplication by x is a homeomorphism,
and thus xV −1 is open dense. Therefore U , being dense, must meet xV −1 and so
x∈U ·V.
Recall that a subset X of a topological space Y is called locally closed if it is
the intersection of an open set with a closed set. Equivalently, X is locally closed
if it is open in X. A subset of Y is called constructible if it is a finite union of
locally closed sets. It is a fact from algebraic geometry that morphisms of varieties
map constructible sets to constructible sets (see [H, Theorem 4.4]). In particular,
the image of a morphism is constructible. Also, it is a standard fact that if X is a
constructible subset of a variety, then it contains an open dense subset of X.
Proposition 1.5. Let H be a subgroup of G, H its closure. Then:
(i) H is a subgroup of G.
(ii) If H is constructible, then H = H.
−1
Proof. (i) Inversion being a homeomorphism, it’s easy to see that H = H −1 = H.
Similarly, for x ∈ H, multiplication by x is a homeomorphism and so xH = xH =
H. Thus H ·H ⊂ H. Hence, for x ∈ H, Hx ⊂ H, and so Hx = Hx ⊂ H. Therefore
H is closed under inverses and multiplication, and so is a subgroup of G.
(ii) If H is constructible, then it contains an open dense subset U of H. Now,
H is a linear algebraic group by (i), and so by Lemma 1.4, we have H = U · U ⊂
H · H = H.
Corollary 1.6. Let ϕ : G1 → G2 be a morphism of algebraic groups. Then ker ϕ
and ϕ(G1 ) are closed, and therefore are linear algebraic groups.
Proof. ϕ is continuous and ker ϕ = ϕ−1 ({1}) is the inverse image of a closed set,
so it is closed. Moreover, ϕ(G1 ) is a constructible subgroup of G2 . By the previous
proposition, it must be closed.
It is clear that closed subgroups of GLn are linear algebraic groups. The following
important result shows that all linear algebraic groups arise in that way. We will
prove it later, in Section 4, when we discuss quotients of algebraic groups.
Theorem 1.7. Any linear algebraic group can be embedded as a closed subgroup
into GLn for some n.
8
1.2. Connectedness. The group structure of a linear algebraic groups allows us
to know more about its geometrical structure. Recall that an affine variety X is
irreducible if it cannot be written as a union U ∪ V ofSnon-empty closed subsets
r
U, V . In general, an affine variety X can be written as i=1 Xi , for some r, where
the Xi are maximal irreducible subsets, called the irreducible components of X. It
is a fact that a morphism of varieties maps irreducible subsets to irreducible subsets
(see [H, Prop 1.3A]). Also, if X and Y are irreducible, so is X × Y (see [H, Prop
1.4]).
Also recall that a topological space X is connected if it cannot be written as a
disjoint union U tV of non-empty closed subsets U, V . It is clear that an irreducible
variety is connected, while the converse is not true in general. However, for algebraic
groups the converse does hold. More specifically:
Proposition 1.8. Let G be a linear algebraic group.
(i) The irreducible components of G are pairwise disjoint, and so are the connected components of G.
(ii) The irreducible component G◦ containing the identity is a closed normal
subgroup of finite index.
(iii) Any closed subgroup of G of finite index contains G◦ .
Proof. (i) Let X, Y be irreducible components of G. Suppose X ∩ Y 6= ∅. Pick
g ∈ X ∩ Y . Since multiplication by g −1 is an isomorphism of varieties, we know
that g −1 X and g −1 Y are irreducible components, and we have 1 ∈ g −1 X ∩ g −1 Y .
So without loss of generality (wlog), we may assume that 1 ∈ X ∩ Y . Now, since
X × Y is irreducible in G × G, it follows that µ(X × Y ) = X · Y is irreducible in G.
Moreover, we have X ⊆ X · Y since 1 ∈ Y . By maximality of X, it must be that
X = X · Y , and similarly, we obtain Y = X · Y = X.
(ii) Since inversion is an isomorphism of varieties, (G◦ )−1 is an irreducible component of G. As it contains 1, it must be G◦ by (i). Similarly, for h ∈ G◦ , multiplication by h is an isomorphism of varieties and so hG◦ is an irreducible component,
and it contains 1 since G◦ is closed under inverses. Therefore hG◦ = G◦ . It follows that for any g, h ∈ G◦ , gh ∈ G◦ . Hence G◦ is a subgroup. Also, for g ∈ G,
conjugation by g is an isomorphism of varieties, so gG◦ g −1 is again an irreducible
component containing 1, and so it equals G◦ . Thus G◦ is a normal subgroup.
For the last part, let X be an irreducible component of G. Pick g ∈ X. We
have that g −1 X is an irreducible component of G containing 1, and so it equals
G◦ . Hence X = gG◦ and so all the irreducible components of G are cosets of G◦ .
It is clear that all cosets of G◦ are irreducible components, so since there are only
finitely many irreducible components of G, G◦ must have finite index.
(iii) Let H ≤ G be a closed subgroup of finite index. Then H ◦ ≤ G◦ ≤ G and
◦
◦
we have
we can write
F [G :◦ H ] = [G : H] · [H : H ], which is finite◦ by (ii). Hence
◦
G = gH , a finite disjoint union of cosets of H . Since G◦ is connected, it
follows that G◦ = H ◦ ≤ H.
We will therefore refer to the irreducible (or connected) components of G as
the components of G. An immediate consequence of the above proposition is that
ϕ(G◦ ) = ϕ(G)◦ for any morphism of algebraic groups ϕ : G → H. Indeed, ϕ(G◦ )
is closed (by Corollary 1.6), connected (since G◦ is connected), contains 1 and has
finite index in ϕ(G) by Proposition 1.8(ii) applied to G. The result follows by
Proposition 1.8(iii).
9
Example 1.9. Let’s see which of our examples are connected. It is a well known
result in algebraic geometry that a variety X is irreducible if and only if its ring
of regular function k[X] is an integral domain. Therefore, we know that Ga , Gm
and GLn are connected algebraic groups since their ring of regular functions were
integral domains. Also, Dn is connected since it is a direct product of connected
algebraic groups (namely n copies of Gm ). On the other hand, it can be shown
that GOn is not connected (assuming char(k) 6= 2), with component at the identity
SOn = GOn ∩ SLn , called the special orthogonal group.
It is also true that SLn , Tn and Un are connected. In order to show this, we
need a geometrical result (see [H, Proposition 7.5]):
Proposition 1.10. Let G be a linear algebraic group and fi : Xi → G, i ∈ I, a
family of morphisms from irreducible varieties Xi , such that 1 ∈ Yi = fi (Xi ) for
all i ∈ I. Then H = hYi : i ∈ Ii is a closed, connected subgroup of G. Moreover,
. In particular, if G is
· · · Yi±1
for some finite sequence i1 , . . . , in in I, H = Yi±1
n
1
generated by some family of closed connected subgroups, then it is connected.
We know from linear algebra that SLn is generated by subgroups
Uij = {(akl ) ∈ GLn : akk = 1, akl = 0 for (k, l) 6= (i, j)}
(i 6= j)
of matrices with 1’s on the diagonal, arbitrary entry in the (i, j) position, and 0
elsewhere. Similarly, Un is generated by the subgroups Uij for i < j. These subgroups are all isomorphic to Ga , which is connected, so the above proposition gives
us that SLn and Un are connected. Similarly, one can show that Tn is connected.
The above proposition has another useful consequence:
Proposition 1.11. Let H, K be subgroups of a linear algebraic group G, with K
closed and connected. Then [H, K] is closed and connected.
Proof. For h ∈ H, define ϕh : K → G by g 7→ [h, g]. It is clearly a morphism, being
a composition of multiplication and inversion. Also, 1 = ϕh (1) for all h. Hence,
we have that [H, K] = hϕh (K) : h ∈ Hi is closed and connected by Proposition
1.10.
Therefore closed connected subgroups behave well under taking commutators.
In particular, if G is a connected linear algebraic group, then its derived subgroup
G0 = [G, G] is a closed connected subgroup. Inductively, we then see that its nth
derived subgroup is a closed, connected subgroup. With this in mind, we recall a
group-theoretic definition:
Definition 1.12. For a group G, define G(0) = G and G(i) = [G(i−1) , G(i−1) ] for
i ≥ 1. We then obtain the derived series of G:
G = G(0) ≥ G(1) ≥ G(2) ≥ . . .
We say G is solvable (or soluble) if G(d) = 1 for some d. The smallest such d is
then called the derived length of G.
Similarly, one can define C 0 G = G and C i G = [C i−1 G, G] for i ≥ 1. We then
define G to be nilpotent if C n G = 1 for some n.
Remark. If G is a nilpotent group and n is the largest integer such that C n G 6= 1,
then C n G must commute with G so in particular Z(G) 6= 1.
10
Example 1.13. Some of the algebraic groups we met are solvable. Indeed, Ga ,
Gm and Dn are obviously solvable since they are abelian. Also, Tn is solvable and
Un is nilpotent. This can be shown by a direct calculation. It is easy to see that
(1)
Tn ⊂ Un and actually, they are equal (one can find generators of Un which are
commutators of elements of Tn ). It’s then not too hard to see that
C m Un = {(aij ) ∈ Un : aij = 0 for 0 < i − j ≤ m}
and so Un is nilpotent, and thus solvable. This then implies Tn is solvable.
Ga , Gm , Dn , Tn and Un are all examples of connected solvable linear algebraic
groups. We will see later that such linear algebraic groups have a nice structure.
2. Background in algebraic geometry
Most of the results in this section will be stated without proof. Proofs can be
found in [H, sections 1-6]. We need to recall a few facts from algebraic geometry
which will be needed later on. In the theory of algebraic groups, we need to know
about tangent spaces, projective varieties, complete varieties or the dimension of
varieties.
Definition 2.1. For an affine variety X, we define the tangent space of X at x ∈ X
by
Tx (X) = {δ : k[X] → k linear : δ(f g) = f (x)δ(g) + δ(f )g(x) for f, g ∈ k[X]}
the k-vector space of point derivations at x.
Having defined the tangent space, we can now define the differential of a morphism:
Definition 2.2. Let ϕ : X → Y be a morphism of affine varieties. The differential
dx ϕ of ϕ at x ∈ X is the map dx ϕ : Tx (X) → Tϕ(x) (Y ) defined by dx ϕ(δ) = δ ◦ ϕ∗
for δ ∈ Tx (X).
Taking differentials behave functorially (see [H, 5.4]):
Proposition 2.3. Let ϕ : X → Y and ψ : Y → Z be morphisms of affine varieties,
and x ∈ C. Then dx (ψ ◦ ϕ)x = dϕ(x) ψ ◦ dx ϕ.
In Section 4, we will consider actions of algebraic groups on varieties other than
just affine varieties. To this end, we recall facts about projective varieties:
Definition 2.4. Projective n-space Pn is defined to be the set of equivalence classes
of k n+1 \ {0, 0, . . . , 0} relative to the equivalence relation
{x0 , x1 , . . . , xn } ∼ {y0 , y1 , . . . , yn } ⇐⇒ ∃λ ∈ k ∗ such that yi = λxi for all i
For a k-vector space V of dimension n + 1, we can identify Pn with the set of all
1-dimensional subspaces of V , usually denoted by P(V ).
We can define a Zariski topology on Pn : define the closed sets to be the sets given
by the vanishing of some collection of homogeneous polynomials in k[T0 , . . . , Tn ]. A
projective variety is a closed subset of some Pn , equipped with the induced topology.
A quasi-projective variety is an open subset of a projective variety. Projective
varieties are clearly quasi-projective.
11
In general, a variety is defined to be a pair (X, OX ), where X is a topological
S
space and OX is a sheaf of functions on X, such that X has a finite open cover Ui
with each (Ui , OX |Ui ) isomorphic to an affine variety. Moreover we also require that
the diagonal ∆ = {(x, x) : x ∈ X} is closed in X × X. Morphisms are then defined
to be continuous maps which preserve the sheaf of functions. For more details see
[H, section 2]. In practice, our varieties will always be affine or quasi-projective.
A useful example of a projective variety is the flag variety of a finite dimensional
vector space V . A flag of V is a chain 0 = V0 ⊂ V1 ⊂ . . . ⊂ Vk = V where all
inclusions are strict. A full flag is one where dim Vi+1 = dim Vi + 1 for all i, i.e one
where k = dim V . The flag variety is defined to be the set of all full flags of V . It
can indeed be given the structure of a projective variety (see [H, 1.8]).
We now recall the notion of dimension:
Definition 2.5. For an irreducible variety X, its ring of regular function k[X] is
an integral domain, so we can take k(X) to be its field of fractions. We define the
dimension of X to be the the transcendence degree of k(X) over k. Equivalently, it
is the maximal length of a chain of prime ideals in k[X]. In general, for a reducible
variety X, we define dim X = max{dim Xi : 1 ≤ i ≤ r} where the Xi are the
irreducible components of X.
Note that for a linear algebraic group G, dim G = dim G◦ since the components
of G are the cosets of G◦ , which are all isomorphic as varieties to G◦ and so all have
the same dimension. In particular dim G = 0 if and only if G is finite: a connected
space of dimension 0 is just a point, so G◦ = 1 and since G is a union of finitely
many cosets of G◦ , it is finite.
The following proposition shows how dimension behaves well with respect to
morphisms (see [H, Theorem 4.3]):
Proposition 2.6. Let ϕ : X → Y be a morphism of irreducible varieties with ϕ(X)
dense in Y . Then there exists a non-empty open subset U ⊆ Y with U ⊆ ϕ(X)
such that
dim ϕ−1 (y) = dim X − dim Y for all y ∈ U
We deduce a ‘rank-nullity’ result for morphisms of algebraic groups:
Corollary 2.7. Let ϕ : G1 → G2 be a morphism of linear algebraic groups. Then
dim ϕ(G1 ) + dim ker ϕ = dim G1
−1
Proof. Every fiber ϕ (y) is a coset of ker ϕ, and thus has the same dimension.
Apply Proposition 2.6 with X = G◦1 and Y = ϕ(G1 )◦ .
Example 2.8. We find the dimension of some algebraic groups:
(1) dim Ga = dim Gm = 1 since clearly k(Ga ) = k(Gm ) = k(T ). An easy
inductive argument using Corollary 2.7 shows that dim Dn = n for all n.
(2) dim GLn = n2 since the field of fractions of k[Tij : 1 ≤ i, j ≤ n]det Tij is
k(Tij : 1 ≤ i, j ≤ n).
(3) dim SLn = n2 −1 using the previous two examples and Corollary 2.7 applied
to the surjective morphism det : GLn → Gm .
We conclude our discussion of dimension with the following result, which will be
useful for inductive arguments:
12
Proposition 2.9. If Y is a proper, closed subset of an irreducible variety X, then
dim Y < dim X.
Proof. Let Y1 ⊆ Y be an irreducible component. The inclusion ϕ : Y1 ,→ X induces
a surjective k-algebra homomorphism ϕ∗ : k[X] → k[Y1 ]. Since Y1 is irreducible,
k[Y1 ] is an integral domain and so ker ϕ∗ is a prime ideal in k[X], non-zero since
Y1 ⊂ X is proper. Then any chain of prime ideals in k[Y1 ] lifts to a chain of prime
ideals in k[X] through ker ϕ∗ , hence of greater length since X is irreducible and so
0 is a prime ideal in the integral domain k[X].
We now move on to complete varieties. These will be useful when we will study
actions of algebraic groups on projective varieties.
Definition 2.10. A variety X is complete if, for any variety Y , the projection
morphism X × Y → Y is a closed map, i.e maps closed sets to closed sets. Clearly,
a closed subvariety of a complete variety is complete.
We summarize all the results we’ll need in the following proposition (see [H,
section 6]):
Proposition 2.11.
(i) A projective variety is complete.
(ii) A complete quasi-projective variety is projective.
(iii) A complete affine variety has dimension 0.
(iv) If ϕ : X → Y is a morphism of varieties and X is complete, then ϕ(X) is
closed in Y , and complete.
3. Tori, Unipotent and Connected Solvable Groups
Having established some basic facts about linear algebraic groups, a first question
we could ask is the following: what do one-dimensional linear algebraic groups look
like? It turns out that any one-dimensional connected linear algebraic group is
isomorphic to either Ga or Gm . This seemingly innocent result is actually quite
difficult to show and we will not prove it here (see [H, section 20] for a proof). This
tells us that one dimensional connected groups are all abelian, and so solvable.
Connected solvable groups are quite important to the general theory, as we shall
see later, and so we study them in this section. We first start by two particular
examples of such groups: unipotent groups and tori.
3.1. Unipotent Groups. Recall the additive Jordan decomposition for endomorphisms: if V is a finite dimensional k-vector space and α ∈ End(V ), then there
exists unique s, n ∈ End(V ) such that s is semisimple, i.e diagonalisable, n is nilpotent, α = s + n and sn = ns. Moreover s and n are both polynomials in α with
constant coefficient equal to zero.
Definition 3.1. An endomorphism u ∈ End(V ) is unipotent if u − 1 is nilpotent.
Equivalently, u is unipotent if the only eigenvalue of u is 1.
There is also a multiplicative version of the Jordan decomposition:
Proposition 3.2. For g ∈ GL(V ), there exists s, u ∈ GL(V ) such that g = su =
us, where u is unipotent and s is semisimple.
Proof. From the additive decomposition, we can write g = s + n with s, n as
described above. Since g is invertible, so is s and so we may define u = 1 + s−1 n.
13
As n is nilpotent and sn = ns, we have that u − 1 = s−1 n is nilpotent. Therefore u
is unipotent and su = s + n = g. If g = su = us is any such decomposition, where
u = 1 + n with n nilpotent and commuting with s, then g = s + sn is the unique
additive Jordan decomposition, and u, s are therefore uniquely determined.
We can then transfer this definition to an arbitrary linear algebraic group:
Theorem 3.3. (Jordan decomposition) Let G be a linear algebraic group.
(i) For any embedding ρ of G into some GL(V ) and for any g ∈ G, there exists
unique gs , gu ∈ G such that g = gu gs = gs gu , where ρ(gs ) is semisimple
and ρ(gu ) is unipotent.
(ii) The decomposition g = gu gs = gs gu is independent of the chosen embedding.
(iii) Let ϕ : G1 → G2 be a morphism of algebraic groups. Then ϕ(gs ) = ϕ(g)s
and ϕ(gu ) = ϕ(g)u .
We won’t give a proof of this result (see [H, Theorem 15.3]), but we give here an
important step which we will need later on. Given a linear algebraic group G, each
x ∈ G defines a morphism G → G given by g 7→ gx, which induces a k[G]-algebra
homomorphism ρx : k[G] → k[G] defined by ρx (f )(g) = f (gx) for f ∈ k[G], g ∈ G.
This defines an action of G on k[G].
Proposition 3.4. Let G be a linear algebraic group and V a finite dimensional
subspace of k[G]. Then there exists a finite dimensional G-invariant subspace X
containing V . In particular, k[G] is a union of finite dimensional G-invariant
subspaces. Moreover, the restriction of any such finite dimensional subspace X
affords a morphism of algebraic groups ρ : G → GL(X).
Proof. It’s enough to prove this for V = hf i, a one-dimensional subspace. Recall
that multiplication gives a morphism µ : G×G
→ G and that k[G×G] P
is isomorphic
P
to k[G]⊗k k[G]. Therefore write µ∗ (f ) = i∈I fi ⊗gi , so that ρx (f ) = i∈I gi (x)fi .
Hence the finite dimensional subspace generated by {fi : i ∈ I} contains ρx (f ) for
all x ∈ G. It follows that the subspace X generated by {ρx (f ) : x ∈ G} is contained
in it and so is finite dimensional. It is clearly G-invariant and it contains V so the
first part follows. For the last part, we see from the above construction that the
coordinates of ρx in X are polynomial functions in x. Therefore the map x 7→ (ρx )|X
affords a morphism of algebraic groups G → GL(X).
An endomorphism x of a vector space V is called locally finite if V is a union
of finite dimensional x-stable subspaces. Proposition 3.4 shows ρx is locally finite.
One can show that locally finite endomorphisms have a Jordan decomposition in
the sense of Proposition 3.2. The idea of the proof of Theorem 3.3 is to use the
unipotent and semisimple parts of ρg , for g ∈ G, to construct gu and gs .
Definition 3.5. Let G be a linear algebraic group. The decomposition g = gu gs =
gs gu in Theorem 3.3 is called the Jordan decomposition of g ∈ G and g is called
semisimple (respectively unipotent) if g = gs (respectively g = gu ). We write
Gu = {g ∈ G : g is unipotent}
Gs = {g ∈ G : g is semisimple}
for the subsets of unipotent and semisimple elements of G. If G = Gu , then we say
G is a unipotent group. Note that Gu is a closed subset since the set of unipotent
elements of GLn is closed, given by the polynomial (T − 1)n .
14
Remarks.
(i) Observe that the subgroup generated by Gu is a characteristic
subgroup of G where by characteristic, we mean here that it is preserved
under all algebraic group automorphisms. Indeed, if α ∈ Aut(G), take any
embedding ρ of G into some GLn . Then ρ ◦ α is another embedding of G
into GLn , and by Theorem 3.3(ii), we have that the image of Gu is still
unipotent, thus Gα
u ⊆ Gu .
(ii) A group G for which G = Gs is not called semisimple. Semisimple groups
have a different definition (see Section 5).
Example 3.6. It’s clear that Un is unipotent
for
any n and that for G = Tn ,
1
∗
∼ U2 =
Gu = Un . Note also that Ga =
is a unipotent group. We also
0 1
met groups where G = Gs , for example Gm or more generally Dn for n ≥ 1.
As said above, Un and more generally subgroups of Un are unipotent subgroups of
GLn . It turns out that all unipotent subgroups of GLn are conjugate to a subgroup
of Un . To get there, we first need the following result:
Theorem 3.7. Let G be a unipotent subgroup of GL(V ) for some non-zero finite
dimensional vector space V . Then G has a common eigenvector in V .
Proof. Identify V with k n where n = dim V . We use induction on n. The result
is obvious if dim V = 1 (every v ∈ V is a common eigenvector of G), so assume
dim V > 1. Suppose V has a proper non-zero subspace W stable under G. Then
by choosing appropriate bases, we may assume that
∗ ∗
G≤
0 ∗
More specifically, every element g ∈ G can be written in the form
∗
ϕ(g)
0
ψ(g)
where ϕ : G → GL(W ) is the canonical restriction morphism, and ψ : G →
GL(V /W ). Now ϕ(G) is also unipotent, so by induction hypothesis there exists a
common eigenvector v ∈ W ⊂ V for G.
Therefore we may assume that V is an irreducible G-module. We need the
following theorem of Burnside (see [L, XVII, section 3]): if R is a subalgebra of
End(V ) which acts irreducibly on V , then R = End(V ).
Now, the assumption that G is unipotent implies Tr(x) = Tr(1) = dim V for all
x ∈ G. Writing x as 1 + n with n nilpotent, we have for all y ∈ G:
Tr(y) = Tr(xy) = Tr(y + ny) = Tr(y) + Tr(ny).
Therefore Tr(ny) = 0. Now, the k-linear combinations of the elements of G must
also satisfy this. These form a subalgebra R of End(V ), which acts irreducibly on
V since G does. Burnside’s theorem then implies that for all y ∈ End(V ) and for
all x = 1 + n ∈ G, Tr(ny) = 0. Taking y to be the standard unit matrices Eij , we
see that we must have n = 0 (by Eij , we mean the matrix whose (i, j)th entry is 1
and all other entries are 0). Hence G = 1 and since V is irreducible, dim V = 1, a
contradiction.
Corollary 3.8. If G ≤ GLn is a unipotent group, then G is conjugate to a subgroup
of Un . Since Un is nilpotent (see Example 1.13), it follows that G is nilpotent, and
so solvable.
15
Proof. By Theorem 3.7, G has a common eigenvector v ∈ V = k n . Let V1 = hvi.
Then G acts on V /V1 , the image of G in GL(V /V1 ) being again unipotent. Induction
on dim V then allows us to construct a basis of V with respect to which elements
of G are represented by upper triangular matrices. Since they are also unipotent,
it follows that these matrices are in Un .
This result can be seen as a generalisation of the fact that p-groups are nilpotent.
Indeed, if char(k) = p > 0, then an endomorphism u is unipotent if and only if
f
f
up = 1 for some f ≥ 1 since for some large enough f , we require that up − 1 =
f
(u − 1)p = 0.
3.2. Tori.
Definition 3.9. A torus is a linear algebraic group isomorphic to Dn for some
n ≥ 0.
It turns out that an important concept in studying tori is their characters.
Definition 3.10. For G a linear algebraic group, a character of G is a morphism
of algebraic groups χ : G → Gm . The set of characters of G is denoted by X(G).
Note that it can be considered as a subset of k[G].
A cocharacter of G is a morphism of algebraic groups γ : Gm → G. The set of
cocharacters is denoted by Y (G).
X(G) is clearly an abelian group with respect to
(χ1 + χ2 )(g) = χ1 (g)χ2 (g)
for χ1 , χ2 ∈ X(G), g ∈ G.
Similarly, if G is commutative then Y (G) is an abelian group with respect to
(γ1 + γ2 )(x) = γ1 (x)γ2 (x)
for γ1 , γ2 ∈ Y (G), x ∈ Gm .
In particular, since we use the additive notation, we will denote by 0 the character
mapping everything to 1, and similarly for cocharacters.
Given χ ∈ X(G) and γ ∈ Y (G), χ ◦ γ ∈ End(Gm ). Now, an element f ∈
End(Gm ) belongs to k[T, T −1 ]. If f is of the form aT n for some a ∈ k ∗ , n ∈ Z,
then since f is a group homomorphism, we must have f = T n as fP
(1) = 1. If f is
m
not of this form then we can find n < m in Z such that f (T ) = i=n ai T i with
ai ∈ k, an , am 6= 0. But since f is a group homomorphism, we must have that
1 = f (T )f (T −1 ). Expanding f (T )f (T −1 ), we see that the coefficient of T m−n is
non-zero, a contradiction.
Therefore, we just proved that End(Gm ) = {t 7→ tj : j ∈ Z} ∼
= Z. In particular,
for χ ∈ X(G) and γ ∈ Y (G), ∃hχ, γi ∈ Z such that χ ◦ γ : t 7→ thχ,γi . This gives us
a map h , i : X(G) × Y (G) → Z.
Note that having established what End(Gm ) is, it’s easy to see what the characters of Dn are. Indeed, writing elements of Dn as g = diag(t1 , . . . , tn ), we can
define a character χi : g 7→ ti . It’s quite easy to see that X(Dn ) is generated by
the χi , a typical element being of the form χa1 1 . . . χann for some a1 , . . . , an ∈ Z. So
X(Dn ) ∼
= Zn . Similarly, we see that for γ ∈ Y (Dn ), composing with the projection
on the ith diagonal element gives an endomorphism of Gm which is therefore of the
form t 7→ tdi . Thus γ is of the form t 7→ diag(td1 , . . . , tdn ).
Using this, it can easily be shown that the map h , i : X × Y → Z is a perfect
pairing, that is, any homomorphism X → Z is of the form χ 7→ hχ, γi for some
16
γ ∈ Y , and any homomorphism Y → Z is of the form γ 7→ hχ, γi for some χ ∈ X,
where X = X(T ) and Y = Y (T ) for a torus T (see [MT, Prop 3.6]).
Now, given any torus T , we can identify T with Dn for some n and so X(T ) ∼
=
n
Z is a finitely generated abelian group. Therefore it makes sense to talk about
characters being linearly independent.
Definition 3.11. Let T be a torus, H ≤ T a subgroup and X1 ≤ X(T ) a subgroup
of the character group. Then we define
H ⊥ = {χ ∈ X(T ) : χ(h) = 1 for all h ∈ H},
a subgroup of X(T ). We also define
X1⊥ = {t ∈ T : χ(t) = 1 for all χ ∈ X1 } =
\
ker χ,
χ∈X1
a closed subgroup of T
It is a fact that given linearly independent characters χ1 , . . . , χn ∈ X(T ) and
elements c1 , . . . , cn ∈ Gm , there exists t ∈ T such that χi (t) = ci for 1 ≤ i ≤ n (see
[H, Lemma 16.2C]). Using this, we can show:
Proposition 3.12. Let T be a torus, H ≤ T a subgroup and X1 ≤ X(T ). Then:
(i) If H1 ≤ H is a subgroup of finite index, then H1⊥ /H ⊥ is finite.
(ii) (X1⊥ )⊥ /X1 is finite.
Proof. (i) By the structure theorem for finitely generated abelian groups, it’s enough
to show that all the elements of H1⊥ /H ⊥ have finite order. Let h1 , . . . , hr be a complete list of cosets representatives for H1 in H. Then every h ∈ H is equal to hi g
for some i and g ∈ H1 . Take χ ∈ H1⊥ . Then χ takes only finitely many values on
H, namely χ(h1 ), . . . , χ(hr ). So χ(H) is a finite subgroup of k ∗ , say of order n ≥ 1.
It follows that nχ ∈ H ⊥ as required.
(ii) If (X1⊥ )⊥ = X1 we’re done. Otherwise pick χ ∈ (X1⊥ )⊥ \ X1 . Again it’s
enough to show nχ ∈ X1 for some n ≥ 1. Suppose not and aim for a contradiction.
Since X1 ≤ X(T ) ∼
= Zr for some r ≤ n and it has a basis
= Zn for some n, X1 ∼
χ1 , . . . , χr . Then we have that χ, χ1 , . . . , χr are linearly independent and therefore
there exists t ∈ T such that χi (t) = 1 (1 ≤ i ≤ r) but χ(t) 6= 1. Hence t ∈ X1⊥ but
t∈
/ ker χ, contradicting the assumption that χ ∈ (X1⊥ )⊥ .
Using the properties of characters, one can show (see [H, 16]):
Proposition 3.13. Any closed subgroup of Dn is a torus.
We finally state a result about the “rigidity” of tori (see [H, Corollary 16.3] for
a proof):
Theorem 3.14. Let G be a linear algebraic group and T ≤ G a torus. Then
NG (T )◦ = CG (T )◦ .
It follows that the quotient NG (T )/CG (T ) is finite since NG (T )◦ ≤ NG (T ) has
finite index and NG (T )◦ ≤ CG (T ) by the theorem.
As an example, take G = GLn and T = Dn . It’s clear that T = CG (T ) and
moreover NG (T ) = M , the set of monomial matrices (i.e matrices with exactly
one non-zero entry on each row). Then since Dn ≤ M has finite index and is
connected, we must have that Dn = M ◦ by Proposition 1.8(c). So indeed we have
NG (T )◦ = CG (T )◦ and here NG (T )/CG (T ) ∼
= Sn is a symmetric group.
17
3.3. Connected Solvable Groups. We wish to get a result similar to Theorem
3.7 for connected solvable groups. This is the analogue of Lie’s theorem for Lie
algebras, except Lie’s theorem only holds in characteristic 0 while here we don’t
make any assumptions on char(k). In the proof we will need the following lemma
(see [H, Prop 15.4] for a proof):
Lemma 3.15. Let M ⊂ GLn be a commuting set of matrices, then M is trigonalisable (i.e we can find a basis with respect to which all elements of M are represented
by upper triangular matrices).
Theorem 3.16. (Lie-Kolchin) Let G be a connected solvable subgroup of GL(V ),
with V 6= 0 finite dimensional. Then G has a common eigenvector in V , i.e V has
a one-dimensional subspace which G stabilises.
Proof. It is a fact that if G is solvable, then so is G (it’s not difficult to see that
(i)
G(i) = G for all i, see [B, 2.4]), so we may assume that G is closed in GL(V ).
Write V = k n for some n ≥ 1. We argue by induction on n and on the derived
length d of G.
If n = 1, the result is trivial. So suppose n > 1. If d = 1, then G is commutative,
and by Lemma 3.15 we have that G has a common eigenvector. So assume d > 1.
Suppose first that there is a proper 0 6= W < V which is stabilised by G. Then
as in the proof of Theorem 3.7, by choosing appropriate bases, we may write any
g ∈ G in the form
ϕ(g)
∗
0
ψ(g)
where ϕ : G → GL(W ) is the canonical restriction morphism, and ψ : G →
GL(V /W ). Now, ϕ(G) is connected solvable and acts on W with dim W < dim V .
So by induction hypothesis there is v ∈ W < V such that v is a common eigenvector
of G.
The only case left to consider is if G acts irreducibly on V . Assume this holds.
We again aim for a contradiction.
Let G0 = [G, G]. It is closed and connected by Proposition 1.11, and obviously
solvable with derived length d−1. Hence by induction hypothesis there is a common
eigenvector v ∈ V for G0 . Note that gv is also a common eigenvector of G0 for any
g ∈ G, since G0 is normal in G: for h ∈ G0 , g −1 hg ∈ G0 , so g −1 hgv = λv for some
λ ∈ k, and therefore h(gv) = λgv.
Let W denote the non-zero subspace of V spanned by the common eigenvectors
of G0 . By the above W is G-invariant. Since G acts irreducibly on V , it follows
that W = V . Hence V has a basis consisting of common eigenvectors of G0 . So the
elements of G0 are diagonal matrices with respect to that basis, and therefore G0 is
commutative.
Now, for fixed h ∈ G0 , all conjugates ghg −1 , for g ∈ G, are in G0 and hence
are diagonal with the same eigenvalues as h. Therefore, there are only finitely
many possibilities for ghg −1 . So let ϕh : G → G0 be the morphism g 7→ ghg −1
(it is a morphism since multiplication and taking inverses are morphisms). The
image ϕh (G) is finite by the above discussion, and connected since G is connected.
Therefore we must have ϕh (G) = {h}, i.e h ∈ Z(G). Hence G0 ≤ Z(G).
Now, every element of Z(G) commutes with G in its action on V , so by Schur’s
lemma they are represented by scalar multiples of the identity. So elements of G0 are
scalar multiples of the identity and they must have determinant 1 since commutators
18
have determinant 1. Therefore there are only finitely many possibilities for elements
of G0 (namely λ.In where λn = 1). As G0 is connected it follows that G0 = 1, and
so G is commutative, contradicting d > 1.
Corollary 3.17. Let G be a connected, solvable subgroup of GLn . Then G is conjugate to a subgroup of Tn , the linear algebraic group of upper triangular matrices.
Proof. Completely similar to Corollary 3.8.
We discuss here one application of this result. There is a natural split exact
sequence
π
1 −→ Un −→ Tn −→ Dn −→ 1
where π is the morphism




t1
∗
t1
0
 ...
 7→  . . .

0
tn
0
tn
Let G ≤ Tn be a closed connected subgroup. The restriction of π to G has kernel
Gu = G∩Un , a closed normal subgroup. The image T = π(G) is a closed connected
subgroup of Dn , hence a torus by Proposition 3.13. So we get an exact sequence
π
1 −→ Gu −→ G −→ T −→ 1
Since T is abelian, it follows that [G, G] ≤ Gu . This proves most of part (i) in the
following structure theorem for connected solvable groups (see [H, Theorem 19.3
and Prop 19.4] for a full proof):
Theorem 3.18. Let G be a connected solvable linear algebraic group. Then:
(i) Gu is a closed, connected, normal subgroup of G and [G, G] ≤ Gu .
(ii) If T is a maximal torus of G, then all maximal tori are conjugate to T and
G = Gu o T . Moreover, NG (T ) = CG (T )
By maximal torus, we mean a subtorus of G which isn’t contained in any other
subtorus. Also, the semidirect product of two algebraic groups G and H is constructed in the same way as for abstract groups, except we require that G acts as a
group of algebraic group automorphisms of H. Analogously, as for abstract groups,
a linear algebraic group G is the semidirect product of the closed subgroups H, K
if H G, H ∩ K = 1 and the product map H o K → G is an isomorphism of linear
algebraic groups. The omitted proof of Theorem 3.18 has the following corollary:
Corollary 3.19. Let G be a connected solvable group. Then any semisimple element of G lies in a maximal torus.
4. Borel Subgroups
Having studied connected solvable linear algebraic groups, we now consider connected solvable subgroups of a linear algebraic group G. This leads to the definition
of a Borel subgroup.
Definition 4.1. A subgroup B ≤ G is called a Borel subgroup if it is a maximal
closed, connected, solvable subgroup.
19
Example 4.2. From Corollary 3.17, we see that a connected solvable subgroup of
GLn is conjugate to a subgroup of Tn , and we saw that Tn is a closed, connected,
solvable subgroup of GLn . Thus, we see that Tn is a Borel subgroup of GLn .
Moreover, if B is a Borel subgroup of GLn , then by the above it is conjugate to a
subgroup of Tn . By maximality, this conjugate must equal the whole of Tn . Thus
all Borel subgroups of GLn are conjugate.
Clearly, Borel subgroups always exist: just take a closed connected solvable
subgroup of maximal dimension. Our first main tool to study Borel subgroups is
the Borel fixed point theorem, a result about the fixed points in the action of a
connected solvable algebraic groups on a projective variety. We therefore start by
considering actions of algebraic groups.
4.1. Actions of algebraic groups.
Definition 4.3. Let G be a linear algebraic group, and let X be a variety. We say
that X is a G-space if there exists a group action
G × X −→ X
(g, x) 7−→ g.x
of G on X which is also a morphism of varieties. If the action of G on X is transitive,
X is said to be homogeneous.
A morphism ϕ : X → Y is called a morphism of G-spaces if ϕ(g.x) = g.ϕ(x) for
all g ∈ G, x ∈ X.
Example 4.4. An easy example of a G-space is G itself. Indeed, the action of
G on itself by conjugation satisfies the above definition. Also, suppose V is a
finite dimensional vector space. A rational representation of G is a morphism
ϕ : G → GL(V ), and V is then called a kG-module. In this situation V is a
G-space (we can identify it with the affine variety k n , where n = dim V ) via the
action (g, v) 7→ ϕ(g)v. The associated projective space P(V ) is also a G-space via
the action (g, hvi) 7→ hϕ(g)vi
Now, we look at some easy consequences of the definition:
Proposition 4.5. Let X be a G-space.
(i) For every x ∈ X the stabiliser Gx = {g ∈ G : g.x = x} is a closed subgroup
of G.
(ii) The set X G = {x ∈ X : g.x = x for all g ∈ G} is closed in X.
(iii) For Y, Z ⊆ X, with Z closed, the transporter
TranG (Y, Z) = {g ∈ G : g · Y ⊆ Z}
is a closed subset of G.
(iv) For a closed subgroup H ≤ G, NG (H) = {g ∈ G : gHg −1 = H} and
CG (H) = {g ∈ G : ghg −1 = h for all h ∈ H} are closed.
Proof. (i) By definition of G-spaces, for x ∈ X the map
ϕx : G −→ X
g 7−→ g.x
is a morphism of varieties, and so Gx = ϕ−1
x ({x}) is closed as {x} is closed.
20
(ii) For g ∈ G, the map
ψ : X −→ X × X
x 7−→ (x, g.x)
is a morphism of varieties by definition of G-spaces, and hence the set
X g = {x ∈ X : g.x = x} = ψ −1 (∆)
G
is
T closedg in X since X is a variety (recall ∆ = {(x, x) : x ∈ X}). Thus X =
g∈G X is closed.
(iii) The orbit map ϕx in the proof of (i) being a morphism, and Z being closed
in X, we have that
\
TranG (Y, Z) =
ϕ−1
y (Z)
y∈Y
is an intersection of closed sets, and therefore is closed.
T
(iv) Let G act on itself by conjugation. Then CG (H) = h∈H Gh is closed by
(i). Moreover, we have NG (H) ⊆ TranG (H, H). Now, if gHg −1 ⊆ H, then we
must have equality since gHg −1 is a closed subgroup of dimension equal to dim H.
Therefore NG (H) = TranG (H, H), which is closed by (iii).
The next obvious step is to consider orbits. However, in general, these are not
necessarily closed. We can still say some things about them. Firstly, we can observe
that if G is irreducible then so is any orbit G.x, x ∈ X, since it is the image of
G under the morphism g 7→ g.x. Secondly, even though orbits are not closed in
general, the orbits of minimal dimension are closed:
Proposition 4.6. Let X be a non-empty G-space. Then
(i) Every orbit G.x is open in its closure.
(ii) Orbits of minimal dimension are closed.
Proof. (i) The orbit map
G −→ X
g 7−→ g.x
is a morphism of varieties with image G.x, which is constructible by the discussion
preceding Proposition 1.5, so contains an open dense subset Y of its closure. As
G.x is the union of G-translates of Y , the result follows.
(ii) Note that for x ∈ X, g ∈ G, g.G.x is closed and contains G.x. Therefore
G.x ⊆ g.G.x. Similarly G.x ⊆ g −1 .G.x. Applying multiplication by g, we see that
G.x = g.G.x. The element g was arbitrary, so we have that G.x is a union of
G-orbits.
Pick x ∈ X such that dim G.x is minimal. Suppose G.x isn’t closed. Then
G.x \ G.x is a (non-empty) union of G-orbits, which is closed in G.x by (i). We aim
to show it has dimension smaller than dim G.x, which would give a contradiction.
If Y ⊆ G.x is an irreducible component intersecting G.x, then (G.x \ G.x) ∩ Y is
a proper closed subset of Y , thus of strictly smaller dimension by Proposition 2.9.
The result then follows from the definition of dimension.
When a group acts transitively on some set, we expect this set to correspond
to some set of cosets G/H for some H ≤ G. In order to translate this into the
context of G-spaces, we first need to give G/H the structure of a variety when G
is an algebraic group. We begin by the following theorem of Chevalley:
21
Theorem 4.7. (Chevalley) Let H ≤ G be a closed subgroup. Then there exists a
rational representation ϕ : G → GL(V ) and a one-dimensional subspace W ≤ V
such that H = {g ∈ G : ϕ(g)W = W }.
Proof. The ideal I k[G] of functions vanishing on H is finitely generated (as k[G]
is Noetherian), say I = (Fj : j ∈ J) for some finite set J. By Proposition 3.4,
there exists a finite dimensional G-invariant subspace X of k[G] containing the Fj ,
and a corresponding morphism ρ : G → GL(X). Then M = X ∩ I is H-invariant.
Conversely, if x ∈ G is such that ρx (M ) = M , then since M generates I, we have
ρx (I) = ρx (M )ρx (k[G]) = M k[G] = I
and it follows that all functions in I vanish at x, and thus x ∈ H. Therefore we
have H = {x ∈ G : ρx (M ) = M }.
If d = dim M , we set V = ∧d X, the dth exterior power of X, with ϕ : G →
GL(V ) the rational representation induced by the natural G-action. Then the onedimensional subspace W = ∧d M is ϕ(H)-invariant. Now assume that ϕ(g)W = W
for some g ∈ G. Let w1 , . . . , wd be a basis for M , and v1 , . . . , vd be a basis for
ρg (M ).
By assumption, we have ϕ(g)(w1 ∧. . .∧wd ) ∈ W , but by the choice of the second
basis it must be a multiple of v1 ∧ . . . ∧ vd . It follows that each vi ∈ M and so
ρg (M ) = M . Hence g ∈ H and W has the required properties.
An immediate consequence of Theorem 4.7 is the embedding theorem in 1.7:
Corollary 4.8. Any linear algebraic group can be embedded as a closed subgroup
of GLn for some n
Proof. Choose H = 1 in Theorem 4.7. We then obtain a faithful (i.e injective)
rational representation ρ : G → GL(V ) ∼
= GLn where n = dim V , since ker ρ ≤ H.
Therefore we have an embedding of G into GLn as a closed subgroup.
We now give the structure of a variety to a set of cosets G/H. Let H ≤ G
be a closed subgroup, and V be as in Theorem 4.7. Let v ∈ P(V ) be the point
corresponding to the line hvi in V stabilised by H. Set X = G.v ⊆ P(V ). Then X
is a homogeneous G-space and the action gives a surjective map ϕ : G → X defined
by g 7→ g.v, with fibers the cosets of H. This induces a bijection ϕ : G/H → X.
Using this bijection, we endow G/H with the structure of a variety, X being a
variety. Indeed, X ⊆ P(V ) is closed so is a projective variety. Also, X being an
orbit in a G-space, it is open in its closure by Proposition 4.6(i). Therefore, X is
a quasi-projective variety. We call G/H, endowed with its structure of a variety,
the quotient space of G by H. By construction, the natural map π : G → G/H
is a morphism of varieties, and it can be shown that the topology on G/H is the
quotient topology obtained from π (see [H,12.1]). Moreover, the variety structure
on G/H satisfies a universal property, and so is independent of the chosen rational
representation (again, see [H, 12.1]). By Proposition 2.6 applied to π, we obtain:
Proposition 4.9. Let H ≤ G be a closed subgroup of a linear algebraic group G.
Then dim G/H = dim G − dim H.
In the case when H G, more can be said about the structure of G/H:
Proposition 4.10. Let H be a closed normal subgroup of G, then we can endow
G/H with the structure of an affine variety and it is then a linear algebraic group
with its usual group structure.
22
The way to prove this is to show that there exists a rational representation
ψ : G → GL(W ) such that H = ker ψ. We can then identify G/H with a closed
subgroup of GL(W ), thus a linear algebraic group. The idea is to use Theorem 4.7
to obtain a rational representation ϕ : G → GL(V ) and a line L in V stabilised
by H. Since H acts on L by scalar multiplication, there is an associated character
χ0 ∈ X(H). Consider then the sum of all non-zero subspaces Vχ (χ ∈ X(H)),
where
Vχ = {v ∈ V : ϕ(h)v = χ(h)v for all h ∈ H}.
It can easily be shown (see [H, 11.4]) that this sum is direct (and so only finitely
many of the Vχ are non-zero), and that G permutes the Vχ , H being normal in G.
Therefore, we may replace V by this sum. Let W ≤ End(V ) be the subspace of
endomorphisms which stabilise each of the Vχ . The fact that G stabilises each Vχ
will produce a rational representation ψ : G → GL(W ) with the required property
(see [H, Theorem 11.5] for the details).
We can now define a new example of a linear algebraic group: taking G = GLn
and H = Z(G) = {tIn : t ∈ k ∗ } ∼
= Gm , the projective general linear group PGLn is
the quotient G/H, a linear algebraic group of dimension n2 − 1.
Being able to take quotients by a normal subgroup is a useful tool. For example,
it is used in the omitted proof of Theorem 3.18(i). As an illustration of its uses,
we show the following elementary result about centers of nilpotent linear algebraic
groups:
Proposition 4.11. Let G be a connected nilpotent linear algebraic group of positive
dimension and H ≤ G a proper closed subgroup. Then Z(G) has positive dimension
and dim H < dim NG (H).
Proof. All C i G are connected by Proposition 1.11, so by the remark following Definition 1.12, Z(G) contains a non-trivial connected C n G and so has positive dimension.
For the last part, use induction on dim G. If dim G = 1 then by Proposition 2.9,
dim H = 0 and so H is finite. But we must also have Z(G) = G since Z(G) is a nontrivial closed subgroup of positive dimension (it is the intersection of the centralisers
of all the elements of G, hence closed). Thus G is abelian and NG (H) = G.
So assume dim G > 1 and let Z = Z(G)◦ . If Z ⊆ H, then G/Z is connected
nilpotent, has smaller dimension than G and is non-trivial, so by induction hypothesis the image H/Z has smaller dimension than its normaliser NG (H)/Z. Thus
dim H < dim NG (H). If Z is not included in H, then HZ is a subgroup of NG (H)
of larger dimension than H.
4.2. The Borel fixed point theorem and consequences. We finally turn to
the fixed point theorem. In order to show it, we need the following geometrical
result on complete G-spaces (see [H, Lemma 21.1]):
Lemma 4.12. Let X, Y be two irreducible, homogeneous G-spaces and let ϕ : X →
Y be a bijective morphism of G-spaces. If Y is complete, then X is complete.
Theorem 4.13. (Borel fixed point theorem) Let G be a connected, solvable linear
algebraic group, acting on a non-empty projective G-space X. Then G has a fixed
point, i.e ∃x ∈ X such that g.x = x for all g ∈ G.
Proof. We argue by induction on dim G. If dim G = 0 then G = 1 as G is connected
and the result is trivial. So suppose dim G ≥ 1.
23
Let G0 = [G, G]; it is a closed, connected, solvable subgroup of G by Proposition
1.11. Since G is solvable, by definition G0 must be a proper subgroup. Therefore
dim G0 < dim G and by induction hypothesis there is x ∈ X such that x is fixed by
G0 . Let Y denote the (non-empty) set of fixed points of G0 .
0
By Proposition 4.5(ii), Y = X G is closed and so also projective. Also, G sta0
bilises Y since G G. Indeed, for y ∈ Y , g ∈ G, h ∈ G0 , we have [h, g]y = y,
i.e (h−1 g −1 hg)y = y, and so h(gy) = g(hy) = gy. Thus gy is fixed by G0 and so
belongs in Y . Therefore, we may replace X by Y .
Now, all stabilisers Gx contain G0 and so are normal subgroups of G. Indeed,
Gx /G0 ≤ G/G0 is normal since G/G0 is abelian.
So G/Gx is an affine variety by Proposition 4.10. Also, there exists x ∈ X with
G.x closed in X by Proposition 4.6(ii), and so G.x is projective. Then the canonical
morphism G/Gx → G.x is bijective, with G/Gx affine and G.x projective. Also
note that since G is irreducible, so are G/Gx and G.x.
Therefore we can apply Lemma 4.12 (recall that G.x is complete by Proposition
2.11(i)) to obtain that G/Gx is complete, and therefore has dimension 0 by Proposition 2.11(iii). It must then be that G/Gx = 1 since it’s connected, i.e Gx = G as
required.
Remarks.
(i) Since a closed subset of a complete variety is also a complete
variety, the same result still holds when X is a complete G-space, with
exactly the same proof. In practice though, we will only use it in the
projective case.
(ii) This gives a short proof of the Lie-Kolchin theorem: if G ≤ GL(V ) then
G naturally acts on P(V ), which then has the structure of a G-space. If G
is connected solvable, we therefore know it must have a fixed point, i.e G
fixes a line in V . In other words, it has an eigenvector in V .
We now discuss the applications of this theorem. Using a similar argument to
the proof of the Borel fixed point theorem, we obtain that all Borel subgroups of a
linear algebraic group are conjugate (something we had already noticed in the case
of GLn ):
Proposition 4.14. Let B be a Borel subgroup of a linear algebraic group G.
(i) If G is connected, then G/B is a projective variety.
(ii) All other Borel subgroups are conjugate to B.
Proof. (i) Let S be a Borel subgroup of maximal dimension. Embed G into some
GL(V ). By Theorem 4.7, there is a one-dimensional subspace V1 whose stabiliser
in G is precisely S. Thanks to the Lie-Kolchin theorem, we know that the induced
action of S on V /V1 is trigonalisable. So there is a full flag 0 = V0 ⊂ V1 ⊂ . . . ⊂
Vn = V stabilised by S, call it f . In fact, by choice of V1 , S is the stabiliser of the
flag f . Therefore the induced morphism from G/S into the orbit of f in the flag
variety is bijective.
But, the stabiliser H of any full flag is solvable. Indeed, if H is the stabiliser
of 0 = V0 ⊂ V1 ⊂ . . . ⊂ Vn = V , then H is conjugate to a subgroup of Tn , and
so is solvable. Therefore, by considering H ◦ we must have that dim H ≤ dim S by
choice of S. Hence f has an orbit of smallest possible dimension. It is therefore
closed by Proposition 4.6, and so projective. The same argument as in the proof of
Theorem 4.13 then gives that G/S is complete. Since it is quasi-projective, it must
then be projective by Proposition 2.11(ii).
24
Now our Borel subgroup B acts on G/S via h : gS 7→ hgS. By Theorem 4.13,
there exists g ∈ G such that BgS = gS, i.e g −1 Bg ⊆ S. By maximality of B,
we must have equality. So B is conjugate to S (similarly any Borel subgroup is
conjugate to S) and the assertion follows.
(ii) As Borel subgroups are connected and contain 1, they all lie in G◦ , and they
are clearly Borel subgroups of G◦ . The result then follows from the proof of (i). Actually, the fact that a quotient G/H is projective, where G is connected, is
a property somehow characterised by Borel subgroups. Indeed, if H ≤ G is a
closed subgroup such that G/H is projective, then taking a Borel subgroup B, we
see that it acts on G/H as in the proof of Proposition 4.14. We then get to the
same conclusion that B g ≤ H for some g ∈ G, and it follows that H contains a
Borel subgroup. Conversely, if H is a closed subgroup containing a Borel subgroup
B, then we have a surjective morphism G/B → G/H with G/B projective and
thus complete, forcing the image G/H to be complete and thus projective (by
Proposition 2.11(ii) and (iv)). We have therefore proved the following:
Proposition 4.15. Let G be a connected linear algebraic group and H ≤ G a closed
subgroup. Then G/H is projective if and only if H contains a Borel subgroup.
Definition 4.16. A closed subgroup P of a connected linear algebraic group G
is called parabolic if G/P is projective. By Proposition 4.15, a closed subgroup is
parabolic if and only if it contains a Borel subgroup.
Example 4.17. Take G = GLn and B = Tn , a Borel subgroup of G. Let v1 , . . . , vn
be the standard basis for k n . The stabiliser of any flag of the form hv1 , . . . , vi(1) i ⊂
hv1 , . . . , vi(2) i ⊂ . . . clearly is a closed subgroup containing B, and so is a parabolic
subgroup. For n = 3, we obtain two parabolic subgroups in this way, consisting of
matrices of one of the forms




∗ ∗
* 

 *
or  0
*
0 0 ∗
0
We can use Borel subgroups to generalise known facts on connected solvable
groups to arbitrary linear algebraic groups. For instance:
Corollary 4.18. The maximal tori of G are conjugate.
Proof. Let T1 , T2 be maximal tori of G. Since a torus is a connected solvable
subgroup, we can find Borel subgroups B1 , B2 such that Bi contains Ti (i = 1, 2).
By Proposition 4.14(ii), there exists g ∈ G such that B1g = B2 . So T1g and T2 are
maximal tori of B2 , which is a connected solvable group. By Theorem 3.18(ii), they
are conjugate.
The proof essentially shows that the maximal tori of G are the maximal tori of
its Borel subgroups: if T ≤ B is a maximal torus of B, and T1 ≤ G is a maximal
torus of G, then some conjugate of T1 is in B, and therefore conjugate to T , forcing
T to be a maximal torus of G. We can now define the following:
Definition 4.19. The rank of a linear algebraic group G is the dimension of a
maximal torus of G, denoted by rk(G). This is well-defined by Corollary 4.18.
25
Example 4.20.
(i) For G = GLn , we saw that B = Tn is a Borel subgroup,
and T = Dn is obviously a torus. If T1 ≥ T is a maximal torus, then T1
is abelian so centralises Dn . But CG (Dn ) = Dn (take any diagonal matrix
with distinct eigenvalues, its centraliser will be Dn ) so T = T1 is a maximal
torus. Therefore GLn has rank n.
(ii) Similarly, for G = SLn , we have a Borel subgroup B = SLn ∩ Tn and a
maximal torus T = SLn ∩ Dn , so SLn has rank n − 1.
(iii) For G = SO2n , we have a torus



t1








 ...








t
∗
n −1
T = 
∈
D
:
t
∈
k
≤ SO2n

n
i
tn






.




..






t−1
1
isomorphic to Dn , hence of dimension n. So rk(SO2n ) ≥ n. If T1 ≥ T
is a maximal torus, then pick s ∈ T with distinct eigenvalues. Since T1 is
abelian and contains T , we have T1 ≤ CGL2n (s) = D2n . On the other hand,
it can be easily checked from the definition of SO2n that SO2n ∩ D2n = T .
Therefore T = T1 is a maximal torus and rk(SO2n ) = n.
We can deduce some further results about Borel subgroup. Since a Borel subgroup of G is contained in G◦ , and is clearly a Borel subgroup of G◦ , we will assume
for the rest of this section that the group G is connected.
Proposition 4.21. Let B ≤ G be a Borel subgroup. Then:
(i) An automorphism of G which fixes B pointwise must be the identity.
(ii) Z(G)◦ ⊆ Z(B) ⊆ CG (B) = Z(G)
Proof. (i) Let σ be such an automorphism. Then the morphism ϕ : G → G defined
by x 7→ σ(x)x−1 sends B to 1, so factors through G/B. Since G/B is projective,
and so complete, we have that the image of ϕ in G is closed by Proposition 2.11(iv),
and so affine. But by Proposition 2.11(iv) it must also be complete, forcing it to
have dimension 0. But ϕ(G) is connected (as G is), so ϕ(G) = 1 as required.
(ii) Z(G)◦ is a closed, connected, solvable subgroup of G so is contained in some
Borel subgroup S. By Proposition 4.14(ii), there exists g ∈ G such that S g = B.
But conjugation leaves Z(G)◦ invariant, so it must be contained in B. The first
inclusion follows. The second inclusion is obvious, and so is Z(G) ⊆ CG (B). Finally,
for x ∈ CG (B), conjugation by x satisfies the assumptions of (i), and so we must
have x ∈ Z(G).
Corollary 4.22. If G contains a nilpotent Borel subgroup B (so in particular if
B = Bu or Bs ), then G is nilpotent (i.e G = B).
Proof. Argue by induction on dim G. If dim G = 0 the result is trivial. So assume
dim G ≥ 1. If B = 1, then by the previous proposition, Z(G) = CG (1) = G. So
G is abelian and therefore nilpotent. Therefore, we may assume dim B > 0. By
Proposition 4.11, Z(B) has positive dimension. But we know from the previous
proposition that Z(G)◦ ⊆ Z(B) ⊆ Z(G). So we can consider the lower dimensional group G/Z(G)◦ , with a nilpotent Borel subgroup B/Z(G)◦ . By induction
hypothesis, these are equal, which forces G = B.
26
Using this, we obtain a result on centralisers of tori:
Proposition 4.23. Let T ≤ G be a maximal torus, C = CG (T )◦ . Then C is
nilpotent and T is its unique maximal torus. Moreover C = NG (C)◦ .
Proof. First, T is central in C, so by Corollary 4.18, it is the unique maximal torus
of C. Now, let B be a Borel subgroup of C, containing T . Then B/T ∼
= Bu is
unipotent so nilpotent by Theorem 3.18(ii) and Corollary 3.8 (which we apply by
first embedding B in some GLn ). Since T is abelian, it must be that B is nilpotent.
So C is nilpotent by Corollary 4.22. For the last part, the inclusion C ⊆ NG (C)◦ is
obvious since C is connected. Conversely, T is clearly normal in NG (C)◦ (as it’s the
unique maximal torus of C), and so must be central by Theorem 3.14. It follows
that NG (C)◦ ⊆ C.
Corollary 4.22 can be applied more generally to show that maximal tori are nontrivial unless the group G is unipotent. Indeed, if G is solvable, either G = Gu
is unipotent or G has a non-trivial maximal torus by Theorem 3.18. If it’s nonsolvable, then a maximal torus would be contained in a Borel subgroup B, and
would clearly be a maximal torus of B. If it was trivial, then B would be unipotent
by Theorem 3.18, and so nilpotent. Thus G would be nilpotent, contradicting our
assumption that G is non-solvable.
Now, we saw that all Borel subgroups are conjugate. It turns out that given a
Borel subgroup B, its conjugates cover G (see [H, 22.2] for a proof):
Theorem 4.24. LetSG be a connected linear algebraic group, B ≤ G a Borel
subgroup. Then G = g∈G g −1 Bg
Note that for G = GLn , B = Tn , Theorem 4.24 follows from the existence of
Jordan normal form (which follows from k = k). Note also that since k = k, we
have GLn = SLn · Z(GLn ) (given A ∈ GLn , we can pick a scalar matrix whose
determinant is the inverse of det A), and so this also applies to SLn . An immediate
consequence of Theorem 4.24 is the following:
Corollary 4.25. Let G be a connected linear algebraic group.
(i) Every semisimple element of G lies in a maximal torus.
(ii) Every unipotent element of G lies in a closed connected unipotent subgroup.
(iii) The maximal closed, connected, unipotent subgroups of G are all conjugate
and are of the form Bu , where B is a Borel subgroup of G.
Proof. (i) Every g ∈ G lies in some Borel subgroup by Theorem 4.24. In particular,
if g is semisimple, then it lies in some Borel subgroup B, and so in a maximal torus
of it by Corollary 3.19.
(ii) If g is unipotent in G, then g lies in some Borel subgroup B, and more
specifically in Bu , which is closed and connected by Theorem 3.18.
(iii) If U ≤ G is a closed, connected, unipotent subgroup of G, then it must be
solvable by Corollary 3.8. So it is contained in a Borel subgroup B, and hence in
Bu .
We now return to centralisers of tori. Using the Borel fixed point theorem and
Theorem 4.24, we obtain:
Theorem 4.26. Let S be a torus in G. Then CG (S) is connected.
27
Sketch of proof. This result can be shown in the case where G is solvable (see [H,
Prop 19.4]). So we aim to reduce the problem to the solvable case. To do so, given
x ∈ CG (S), we want to find a Borel subgroup containing both S and x. Then this
will force x to lie in CB (S)◦ ⊆ CG (S)◦ as required.
By Theorem 4.24, x lies in a conjugate of a given Borel subgroup B. Moreover,
x acts on G/B via gB 7→ xgB, so the fixed point set
X = {gB ∈ G/B : g −1 xg ∈ B}
is non-empty (as x lies in some conjugate of B) and closed by the proof of Proposition 4.5(ii). Hence X is a projective variety (as G/B is projective) and S acts on
it by left multiplication. By the Borel fixed point theorem, there is a fixed point,
i.e g −1 Sg ≤ B for some gB ∈ X. This gives the required Borel subgroup.
Theorem 4.24 tells us that the conjugates of a Borel subgroup cover G, and so in
particular unless G is solvable, Borel subgroups are not normal subgroups. Indeed,
they are even self-normalising. If N = NG (B), then B is clearly a Borel subgroup
of N ◦ . Since B is normal in N ◦ , it follows that B = N ◦ by Theorem 4.24. The
following stronger assertion requires much more work and won’t be proved here (see
[H, 23.2] for a proof):
Theorem 4.27. Let B be a Borel subgroup of G. Then B = NG (B).
Note again that this is easy to see for G = GLn , with Borel subgroup Tn . This
theorem allows us to uncover some properties of parabolic subgroups of G:
Corollary 4.28. Let P be a parabolic subgroup of G. Then P = NG (P ). In
particular, P is connected.
Proof. By definition, P contains some Borel subgroup B. Actually, B is evidently
a Borel subgroup of P ◦ . Let x ∈ NG (P ). Then both B and xBx−1 are Borel
subgroups of P ◦ , so they are conjugate by some y ∈ P ◦ (by Proposition 4.14).
Hence, xy ∈ NG (B) = B (by Theorem 4.27). But then we have xy ∈ P ◦ . Since
y ∈ P ◦ , we must have x ∈ P ◦ . It follows that P ◦ = P = NG (P ).
Corollary 4.29. Let P , Q be parabolic subgroups of G, both of which include a
Borel subgroup B. If P , Q are conjugate in G, then P = Q.
Proof. Write Q = x−1 P x. Then B and x−1 Bx are both Borel subgroups of Q, and
so are conjugate in Q. Hence there exists y ∈ Q such that B xy = B. Therefore
xy ∈ NG (B) = B ⊆ Q (by the theorem), and again this forces x ∈ Q. It follows
that P = Q.
This implies that the number of conjugacy classes of parabolic subgroups of G
is the number of parabolic subgroups which contain a given Borel subgroup B. We
will prove these results again in section 6 for connected reductive groups without
appealing to Theorem 4.27.
5. Connected reductive groups
In the next section we will investigate the parabolic subgroups of particular kinds
of linear algebraic groups, called connected reductive groups. We therefore start
by discussing the properties of those groups which will be needed to develop the
structure of their parabolic subgroups.
28
5.1. Reductive and semisimple groups. We first observe the following consequence of Proposition 1.10: if N, N 0 are closed connected subgroups of a linear
algebraic group G such that N N 0 is a group, then N N 0 = hN, N 0 i is a closed connected subgroup of G. In particular, if N, N 0 G are closed, connected, solvable
normal subgroups, then N N 0 is also a closed, connected, solvable normal subgroup.
Thus we may define the following:
Definition 5.1. The maximal closed, connected, solvable normal subgroup of a
linear algebraic group G is called the radical R(G) of G.
By Theorem 3.18, we see that R(G)u is a closed, connected, normal unipotent
subgroup of R(G). It actually is a characteristic subgroup of it, and so since
R(G) G, we have that R(G)u G. Since unipotent groups are nilpotent and so
solvable by Corollary 3.8, we have that any closed, connected, unipotent normal
subgroup of G is contained in R(G) and so in R(G)u . Therefore R(G)u is the
maximal closed, connected, unipotent normal subgroup of G, called the unipotent
radical of G, usually denoted by Ru (G).
Definition 5.2. A linear algebraic group G is called reductive if Ru (G) = 1. It is
called semisimple if it is connected and R(G) = 1. Clearly, a semisimple group is
reductive. Also, if G is connected, G/R(G) is semisimple and G/Ru (G) is reductive.
The radical of G is given by its Borel subgroups:
Proposition 5.3. Let G be connected. Then:
T
◦
(i) R(G) = ( B B) .
(ii) if G is reductive, then R(G) = Z(G)◦ .
Proof. (i) R(G) is a connected solvable subgroup, so contained in some Borel subgroup B. If H is another Borel subgroup, then we know B and H are conjugate, say
−1
−1
−1
H = gBg
G. Therefore
T . Then R(G) = gR(G)g ⊆ gBg = H since
T R(G)
◦
R(G) ⊆ B B and as it isTconnected, we have R(G) ⊆ ( B B) .
◦
Conversely, let H = ( B B) . H is clearly a closed, connected subgroup, and
it’s solvable since it’s a subgroup of some Borel subgroup B. Moreover, H G
since any conjugate of H is still contained in all Borel subgroups (as they are all
conjugate), and is connected as conjugation is an automorphism of algebraic group,
so is contained in H. Thus H is a closed, connected, solvable normal subgroup of
G and so by maximality of R(G), we get the other inclusion.
(ii) Clearly, Z(G)◦ ⊆ R(G) since it’s a closed, connected, solvable normal subgroup. Now, R(G) = R(G)u o T for some maximal torus T by Theorem 3.18. Since
G is reductive, R(G)u = Ru (G) = 1 and so R(G) = T is a torus. Therefore, by
Theorem 3.14, the group G/CG (R(G)) = NG (R(G))/CG (R(G)) is finite. Hence,
CG (R(G)) contains G◦ = G and so equals G. Thus R(G) is contained in Z(G) and
since it is connected, we have R(G) ⊆ Z(G)◦ .
This proposition tells us that the only difference between connected reductive
groups and semisimple groups is that connected reductive groups may have a nontrivial centre.
Example 5.4.
(1) If G is connected solvable, then by Theorem 3.18, Ru (G) =
Gu and R(G) = G. So a connected solvable reductive group is a torus and
a semisimple solvable group is trivial.
29
(2) G = GLn is reductive. Indeed, let Tn− be the Borel subgroup consisting of
invertible lower triangular matrices. Then, by Proposition 5.3(i), we have
R(G) ≤ Tn ∩ Tn− = Dn
and thus Ru (G) = R(G)u = 1. However, G is not semisimple. Indeed,
Z(GLn ) = {tIn : t ∈ k ∗ } ∼
= Gm is connected so by Proposition 5.3(ii),
R(GLn ) = Z(GLn ) is non trivial. On the other hand, we see that PGLn is
semisimple.
(3) The same argument as in (2) shows that G = SLn is reductive. Moreover,
R(G) = Z(G)◦ = ({tIn : t ∈ k ∗ } ∩ SLn )◦ = ({tIn : tn = 1})◦ = 1
again by Proposition 5.3(ii). Therefore G is semisimple.
5.2. Lie algebras and root systems. We can associate to connected reductive
groups a root system, which will turn out to be essential to their structure. In order
to define this root system, we first need to associate a Lie algebra to a given linear
algebraic group.
Let G be a connected linear algebraic group, and A = k[G] its ring of regular
functions. A k-linear map D : A → A satisfying D(f g) = f D(g) + D(f )g for
all f, g ∈ A is called a derivation of A. We denote by Derk (A) the set of all
derivations of A. It is an easy calculation to show that if D1 , D2 ∈ Derk (A), then
D1 ◦ D2 − D2 ◦ D1 = [D1 , D2 ] ∈ Derk (A). So we can give Derk (A) the structure of
a Lie algebra.
Now, for x ∈ G, we have an action λx : A → A on A = k[G] given by (λx .f )(g) =
f (x−1 g) for f ∈ A, g ∈ G. We can now give the following definition:
Definition 5.5. The Lie algebra of G is the subspace
Lie(G) = {D ∈ Derk (A) : Dλx = λx D for all x ∈ G}
of left invariant derivations of A = k[G], a Lie subalgebra of Derk (A). In the
general case where G is not necessarily connected, we define Lie(G) to be Lie(G◦ ).
There is a second, more geometric definition. Recall the definition of tangent
spaces from Section 2. Note that since G acts on itself homogeneously by left
translation, the tangent space at any g ∈ G is naturally isomorphic to T1 (G). So
we will consider only T1 (G). We can define the following map:
θ : Lie(G) −→ T1 (G)
D 7−→ (f 7→ D(f )(1))
This is a k-linear map. It can be shown to be an isomorphism (see [H, Theorem
9.1]). So we can use this map to give T1 (G) the structure of a Lie algebra. With
this alternative definition, one can show (see [S, Corollary 4.4.6] and [H, Prop 5.1]):
Theorem 5.6. Let G, G1 , G2 be linear algebraic groups. Then:
(i) dim G = dim G◦ = dim (Lie(G))
(ii) Lie(G1 × G2 ) ∼
= Lie(G1 ) ⊕ Lie(G2 ) as Lie algebras
∼ gl , the algebra of n × n matrices.
Example 5.7. We show that Lie(GLn ) =
n
Indeed, define
f : gln −→ Derk (k[G])
X 7−→ DX
30
Pn
where DX is the derivation defined by DX (Tij ) = l=1 Til Xlj . A simple calculation
gives that
X
−1
λg (DX Tij ) =
gim
Tml Xlj = DX (λg Tij ).
l,m
−1
gij
where
denotes the (i, j)th entry of g −1 . Hence DX is left invariant, and the map
is well defined. It’s easy to see that it is a Lie algebra homomorphism, and that its
kernel is zero, so it’s injective. By Theorem 5.6(i), dim Lie(GLn ) = n2 = dim gln ,
so the map is surjective and we have a Lie algebra isomorphism.
Now, morphisms of algebraic groups give rise to Lie algebra homomorphisms
between their Lie algebras in a natural way (see [H, Theorem 9.1]):
Proposition 5.8. Let ϕ : G1 → G2 be a morphism of algebraic groups. Then
dϕ : Lie(G1 ) → Lie(G2 ) is a Lie algebra homomorphism.
Having obtained the Lie algebra of a linear algebraic group G, we can then
naturally consider the Lie algebras of its closed subgroups as Lie subalgebras of
Lie(G). If H ≤ G is a closed subgroup, defined by an ideal I k[G], then we
have k[H] = k[G]/I. So, any D ∈ Lie(G) with DI ⊆ I defines in a natural way a
derivation of k[H], and similarly any δ ∈ T1 (G) with δI = 0 will define an element
of T1 (H). We then have (see [H, Lemma 9.4, Corollary 10.4A and Theorem 11.5]):
Theorem 5.9. Let H ≤ G be a closed subgroup with ideal I k[G]. Then:
(i) Lie(H) = {D ∈ Lie(G) : DI ⊆ I} and T1 (H) = {δ ∈ T1 (G) : δI = 0}.
(ii) If H G is a closed normal subgroup, then Lie(H) is an ideal of Lie(G).
Moreover, the differential of the projection G → G/H is the canonical projection onto Lie(G)/Lie(H) and this induces an isomorphism Lie(G/H) ∼
=
Lie(G)/Lie(H).
It is a standard fact that Lie(SLn ) = sln , the Lie subalgebra of gln consisting of
traceless n×n matrices over k (see for example [H, 9.4]). The calculation eventually
comes down to showing that differentiating the determinant gives you the trace.
Similarly, the Lie algebra of Dn is the Lie subalgebra of gln consisting of n × n
diagonal matrices.
One of the uses of the Lie algebra of an algebraic group G is that it allows us to
define a rational representation G → GL(Lie(G)) in the following way: for x ∈ G,
define Intx : G → G by Intx (y) = xyx−1 . Let Ad x = dIntx : Lie(G) → Lie(G).
This defines a map Ad : G → GL(Lie(G)) given by x 7→ Ad x, called the adjoint
representation of G.
It can be shown (see [H, Prop 10.3 and Theorem 10.4]) that Ad defines a rational
representation of G and that ad = d Ad satisfies ad(X)(Y ) = [X, Y ] for X, Y ∈
Lie(G). In the special case of G = GLn , Lie(G) = gln , a direct calculation (see
[MT, Example 7.13]) shows that Ad g for GLn is just matrix conjugation by g on
gln . Moreover, it’s not hard to see in general that if H ≤ G is closed, then Lie(H),
as defined in Theorem 5.9, is AdG (H)-invariant and that AdG |H = AdH on Lie(H).
So for any closed subgroup of GLn , Ad is also given by matrix conjugation.
We now apply this to our study of reductive groups. Let G be a non-trivial
reductive group, and T ≤ G be a maximal torus. Since G is not unipotent, dim T ≥
1 by our discussion following Proposition 4.23. Write g for Lie(G). The image of
T under the adjoint representation Ad T ≤ GL(g) is a set of commuting semisimple elements, so can be simultaneously diagonalised (the proof is contained in
31
[H, Lemma 15.4] along with the one of Lemma 3.15). For χ ∈ X(T ), we write
gχ = {v ∈ g : (Ad t)(v) = χ(t)v for all t ∈ T }
L
for the common T -eigenspaces. Then g = χ∈X(T ) gχ . Using this decomposition,
we define
Φ(G) = {χ ∈ X(T ) : χ 6= 0, gχ 6= 0},
the set of roots of G with respect to T , which we will usually simply denote by Φ.
We also define W = NG (T )/CG (T ), the Weyl group of G with respect to T (which
we will sometimes denote by WG (T )). We already saw in Theorem 3.14 that this
group is finite.
If w = nCG (T ) ∈ W , then we can define tw = n−1 tn for t ∈ T (this clearly
doesn’t depend on the choice of n). We can then use this to define actions of the
Weyl group on the character group X(T ) and on the cocharacter group Y (T ) in
the following way: for all w ∈ W, χ ∈ X(T ), γ ∈ Y (T ), t ∈ T and c ∈ Gm , we define
(w.χ)(t) = χ(tw )
(w.γ)(c) = γ(c)w
−1
It’s easy to see that these actions are faithful and are compatible with the pairing
h , i : X(T ) × Y (T ) → Z in the sense that
hw.χ, γi = hχ, w−1 .γi
Moreover, let w ∈ W , say w = nCG (T ), and let α ∈ Φ, v ∈ gα and t ∈ T . We have
(Ad t Ad n)(v) = (Ad n Ad tw )(v)
= Ad n(α(tw )v)
= α(tw ) Ad n(v).
This shows that (Ad n)(gα ) ⊆ gw.α . In particular, if gα 6= 0 then gw.α 6= 0. Thus
we see that Φ is W -stable.
Example 5.10. We illustrate this with the examples of GLn and SLn . Let G =
GLn and T = Dn . We know that the adjoint action of G on gln is by conjugation
and that Lie(T ) is the set of diagonal matrices in gln . We let Eij be the standard
basis elements of gln (its (i, j)th entry is 1 and all other entries are 0). We have for
i 6= j
−1
−1
Ad (diag(t1 , . . . , tn ))(Eij ) = diag(t1 , . . . , tn )Eij diag(t−1
1 , . . . , tn ) = ti tj Eij ,
and so we see that the character χij : T → Gm defined by diag(t1 , . . . , tn ) 7→ ti t−1
j
L
is a root of G. Moreover, gln ∼
= Lie(T ) ⊕ i6=j hEij i so we have
Φ = {χij : i 6= j}.
All gα , α ∈ Φ, are one dimensional and we already saw in the remark following
Theorem 3.14 that the Weyl group W is isomorphic to the symmetric group Sn .
Similarly, for G = SLn , with maximal torus T = Dn ∩ SLn , we obtain roots χij
defined as above for i 6= j. Indeed, sln is generated by the Eij , i 6= j, and Lie(T ),
the set of traceless diagonal matrices. So the above calculations still go through
and we have Φ = {χij : i 6= j}. Note that the gα , α ∈ Φ, are again one-dimensional.
To carry on further, we need the following result (see [H, Theorem 26.1]):
32
Theorem 5.11. Let G be a connected linear algebraic group, and T ≤ G a maximal
torus. Then Ru (G) is the identity component of the intersection of the unipotent
parts of the Borel subgroups containing T .
We also need the fact that for G connected, S a subtorus of G, then all Borel
subgroups of CG (S) are of the form B ∩ CG (S) for a Borel subgroup B of G (see
[H, Corollary 22.4]). We then obtain:
Corollary 5.12. Let G be connected reductive. If S ≤ G is a subtorus, then
CG (S) is also connected reductive. Moreover, if T ≤ G is a maximal torus, then
CG (T ) = T .
Proof. We already know that CG (S) is connected by Theorem 4.26. Let T ≤ G
be maximal torus containing S. Then T ≤ CG (S) since T is abelian, and by the
theorem applied to H = CG (S) and to G, we have
\
\
Ru (H) =
(B ∩ H)u ≤
Bu = Ru (G) = 1
T ≤B
T ≤B
since G is reductive. Thus CG (S) is reductive.
Finally, putting S = T , we see that CG (T ) is connected reductive. But it’s
also nilpotent by Proposition 4.23. Then by Example 5.4(1), we have that T =
CG (T ).
Therefore we see that the Weyl group is equal to NG (T )/T . Now let α ∈ Φ.
Define Tα = (ker α)◦ ≤ T , a subtorus of T of codimension 1 (by rank-nullity applied
to α). Also define Cα = CG (Tα ), a connected reductive group by Corollary 5.12,
and Gα = [Cα , Cα ], which can be shown to be a dimension 3 semisimple group of
rank 1 isomorphic to either SL2 or PGL2 (see [H, Corollary 25.3] and [S, Theorem
7.2.4])
It is a fact that G = hCα | α ∈ Φi, and that the Cα are non-solvable. Moreover
we have that Lie(Cα ) =Lie(T ) ⊕ gα ⊕ g−α with dim gα = 1 (see [MT, Prop 8.15 and
8.16]). Using this, one can obtain the following structure theorem for connected
reductive groups (see [MT, Theorem 8.17 and Corollary 8.22] for a proof):
Theorem 5.13. Let G be connected reductive, T ≤ G a maximal torus, g =Lie(G)
and Φ = Φ(G). Then:
L
(1) g =Lie(T ) ⊕ α∈Φ gα with dim gα = 1 for all α ∈ Φ.
(2) dim G = dim Lie(G) = |Φ| + rk(G).
(3) For all α ∈ Φ, there exists a morphism of algebraic groups uα : Ga → G
inducing an isomorphism onto Uα = im(uα ). Moreover, Lie(Uα ) = gα and
we have tuα (c)t−1 = uα (α(t)c) for all t ∈ T and c ∈ Ga . Then Uα is the
unique one-dimensional connected unipotent subgroup of G normalised by
T such that Lie(Uα ) = gα .
(4) For w = nT ∈ W , we have nUα n−1 = Uw.α .
(5) Gα = hUα , U−α i.
(6) G = hT, U
Tα | α ∈ Φi.
(7) Z(G) = α∈Φ ker α.
(8) G = [G, G]R(G) = [G, G]Z(G)◦
The one-dimensional subgroups Uα are called the root subgroups of G. For each
α ∈ Φ, since Tα lies in the centre of Cα , it is a normal subgroup of it and so we can
form the quotient C α = Cα /Tα . The canonical projection Cα → C α then induces
33
a bijection WCα (T ) ∼
= WC α (T /Tα ). Now C α is a rank 1 non-solvable connected
linear algebraic group, and it can be shown that such groups have Weyl group of
order 2 (see [H, Theorem 25.3]). Denote by sα the unique non-trivial element of
that Weyl group (which has order 2). Since CG (T ) = CCα (T ) = T by Corollary
5.12, we can naturally view WCα (T ) as a subgroup of W , and so sα as an element
of W .
Example 5.14. If we take G = SL2 , then we have
sl2 = (sl2 )0 ⊕ (sl2 )α ⊕ (sl2 )−α
c
7→ c2 following Example 5.10. Its Weyl group has order 2, with
0 1
1 0
sα being represented by the matrix
. Here, Uα = U2 , U−α =
−1 0
∗ 1
and indeed, SL2 is generated by these.
Similarly, if we take G = PGL2 , then G has a maximal torus T = D2 contained
in a Borel subgroup B = T2 , where we write X for the image of some X ⊆ GL2 via
the projection GL2 → PGL2 . It can easily be shown that for char(k) 6= 2, PGL2
has its Lie algebra isomorphic to sl2 . We then get the same decomposition of sl2 as
c
above, with α :
7→ c (which corresponds to our χ12 earlier), Uα = U2 and
1
0 1
sα having preimage
.
1 0
where α :
c−1
Now, furthermore, for each α ∈ Φ, there is a unique α∨ ∈ Y (T ) such that
sα .χ = χ − hχ, α∨ iα for all χ ∈ X(T ). We then have hα, α∨ i = 2 and α∨ is
called the coroot corresponding to α (see [MT, Lemma 8.19]). The elements sα are
reflections on the real vector space X(T ) ⊗Z R, which we usually denote by ER ,
that is to say they have an eigenvector with eigenvalue -1 (namely α) and they
fix a hyperplane of ER pointwise (namely the kernel of χ 7→ hχ, α∨ i). Moreover,
W = hsα | α ∈ Φi (this is proved for example in [S, Theorem 8.2.8]). All these facts
lead to the definition of a root system:
Definition 5.15. A subset Φ of a finite-dimensional real vector space E is called
an (abstract) root system if:
(R1) Φ is finite, 0 ∈
/ Φ, hΦi = E.
(R2) if c ∈ R is such that α, cα ∈ Φ, then c = ±1.
(R3) for each α ∈ Φ there exists a reflection sα ∈ GL(E) along α stabilising Φ.
(R4) for α, β ∈ Φ, sα .β − β is an integral multiple of α.
The group W = hsα |α ∈ Φi is called the Weyl group of Φ. The dimension of E is
called the rank of Φ.
Since Φ is finite, generates E and is stable under W , it follows that the Weyl
group of a root system is always finite. For a connected reductive group G, its set
of roots Φ forms a root system of hΦiR when viewed as a subset of ER , with Weyl
group W . Indeed, (R1) and (R3) are clear from what we’ve already said. (R2)
follows from the fact that the only roots of Cα with respect to T are ±α. For a
proof of (R4), see [MT, Lemma 15.4 and Example 15.5].
We now recall a few standard facts about abstract root systems (see [MT, Appendix A] for proofs). A subset ∆ ⊆ Φ is called a base of Φ if it is a vector space
34
P
basis of E such that any β ∈ Φ can be written as α∈∆ cα α, where cα ∈ Z for all
α ∈ ∆ and either all cα ≥ 0 or all cα ≤ 0. Then the set
X
Φ+ = {β ∈ Φ : β =
cα α with cα ≥ 0}
α∈∆
is called the system of positive roots of Φ with respect to ∆. Also define Φ− = −Φ+ .
Proposition 5.16. Any abstract root system Φ has a base, and the Weyl group W
acts simply-transitively on the set of bases of Φ. Moreover, given a base ∆, we have
W = hsα | α ∈ ∆i and for every α ∈ Φ, there exists w ∈ W such that wα ∈ ∆.
So we see that a root system can be recovered from a base. Therefore a root
system is completely determined by a base. Now, returning to connected reductive
groups, we see that a Borel subgroup B determines a base in the following way:
Theorem 5.17. Let G be connected reductive, T ≤ G a maximal torus, contained
in some Borel subgroup B. Then:
(i) There exists a base ∆ of Φ such that
Y
B=T·
Uα
α∈Φ+
for any fixed order of the factors Uα . Moreover, we have uniqueness of the
expression with respect to the product in the fixed order.
(ii) G = hT, Uα | α ∈ ±∆i
Sketch of proof. For (i), see [S, Prop 8.2.1]. For the proof of (ii), note that Gα =
[Cα , Cα ] = hUα , U−α i contains a preimage of sα . Indeed, Gα is normalised by T
by Theorem 5.13(3) and (5), so Gα T is a group. It is therefore equal to hT, Gα i,
a closed connected subgroup by Proposition 1.10. Since Gα T contains U±α and T ,
its Lie algebra must be equal to Lie(Cα ) by the discussion preceding Theorem 5.13.
Therefore, by Theorem 5.6(i), we have that dim Gα T = dim Cα and so Cα = Gα T .
It follows that if nα ∈ NCα (T ) is a preimage for sα , then we can write it as gt with
g ∈ Gα and t ∈ T , and we see that g is also a preimage of sα .
Thus we have that H = hUα | α ∈ ±∆i contains preimages of all w ∈ W since
W = hsα | α ∈ ∆i. Hence Uβ ≤ H for all β ∈ Φ by Proposition 5.16 and Theorem
5.13(4), and the result follows by Theorem 5.13(6).
Definition 5.18. The base ∆ obtained in Theorem 5.17 is called the set of simple
roots with respect to T ≤ B, and the elements sα , α ∈ ∆, are called simple
reflections.
Example 5.19. Carrying on with our example of SLn , with maximal torus T =
Dn ∩ SLn , we had Φ = {χij : i 6= j} where χij : diag(t1 , . . . , tn ) 7→ ti t−1
j . A base of
Φ is given by
∆ = {χi,i+1 : 1 ≤ i < n}
since we have, in additive notation, χij = χi,i+1 + χi+1,i+2 + . . . χj−1,j for i < j
and χij = −χji for i > j. So Φ+ = {χij : i < j}. Moreover, for i 6= j, we have
Uχij = Uij where
Uij = In + hEij i
since Uij is clearly an isomorphic copy of Ga so is a one-dimensional closed connected unipotent subgroup of SLn , with T acting on it in the way described in
35
TheoremQ15.3(3). Let B = Tn ∩ SLn , a Borel subgroup containing T . Then clearly
B = T · i<j Uij . So we indeed have
Y
B=T ·
Uα
α∈Φ+
as in the theorem. Finally, the algorithm of Gaussian elimination show that SLn is
generated by the Uij , i 6= j, and diagonal matrices.
We now use Theorem 5.17 to understand the structure of B some more. First,
fix isomorphisms uα : Ga → Uα for all α ∈ Φ.
Proposition 5.20. Let G be connected reductive, B ≥ T a Borel subgroup, and let
U ≤ Ru (B) be a T -stable subgroup. Then U is the product of the root subgroups Uα
that it contains, and so is closed and connected. In particular, the Uα , α ∈ Φ+ , are
the minimal non-trivial T -stable subgroups of Ru (B).
Proof. First note that if uα (c) ∈ U for some α ∈ Φ+ and c 6= 0, then tuα (c)t−1 =
uα (α(t)c) ∈ U for all t ∈ T , and so Uα ≤ U since α 6= 0. Also observe that since U is
unipotent, it is nilpotent and so U 0 = [U, U ] is a T -stable closed, connected subgroup
of smaller dimension. Therefore we can argue by induction on the dimension of U .
If U has dimension 0 then the result is trivial.
Suppose U has dimension at least 1. By induction hypothesis, U 0 is the product
of the Uα it contains. Assume
Q the result fails. By Theorem 5.17(i), there exists
1 6= u ∈ U of the form u = α∈M uα (cα ) where M ⊆ Φ+ such that Uα * U for all
α ∈ M . Take u such that |M | is minimal. By the above, we must have |M | > 1.
Take distinct β, γ ∈ M . Then β and γ are linearly independent (since they’re both
in Φ+ ) and so we must have ker β 6= ker γ. Indeed, if we had ker β = ker γ then γ
and β would give two different isomorphisms T / ker β ∼
= Gm , and composing those
would give an endomorphism of Gm , namely the one mapping β(t) 7→ γ(t) for all
t ∈ T . Since we know endomorphisms of Gm are given by raising to some integer
power, this would give a linear dependence between β and γ.
Thus ∃t ∈ T with, say, β(t) = 1 but γ(t) 6= 1. We then have, modulo U 0 ,
Y
Y
Y
tut−1 u−1 ≡
uα (α(t)cα ) ·
uα (−cα ) ≡
uα ((α(t) − 1)cα )
α∈M
α∈M
α∈M
which does not involve uβ , but which does still involve a non-trivial element of
Uγ . So we have a non-trivial product of smaller length, contradicting our choice of
M.
Corollary 5.21. Let G be connected reductive with maximal torus T and H ≤ G
a connected reductive subgroup normalised by T . Then H = hT ∩ H, Uα | Uα ≤ Hi.
Proof. Note that HT is a connected reductive subgroup of G. Let BH be a Borel
subgroup of HT containing T . It is contained
Q in some Borel subgroup B of G. Let
U = Ru (BH ). By Proposition 5.20, U = α∈M Uα for some subset M ⊆ Φ. Then,
by Theorem 5.17, HT = hT, Uα | α ∈ ±M i. Now, by Theorem 5.13(8), we have
U ≤ [HT, HT ] ≤ H and the result follows.
We also have the following result which describes how elements in Ru (B) multiply
(see [H, Lemma 32.5]):
36
Proposition 5.22. (Commutator formula) Given a connected reductive group G
with root system Φ, take a fixed total ordering of Φ compatible with addition. Then
there exists integers cmn
αβ such that the morphisms uα can be chosen so that for all
roots α 6= β we have
Y
m n
[uα (t), uβ (u)] =
umα+nβ (cmn
for all t, u ∈ k
αβ t u )
m,n>0
where the product is taken over all integers m, n > 0 such that mα + nβ ∈ Φ, taken
according to the chosen ordering.
This indeed even allows us to know how elements of B multiply, since we know
how elements of T multiply with elements of Ru (B) thanks to Theorem 5.13(3).
5.3. Bruhat decomposition. We now discuss how to decompose a connected
reductive group
G into double cosets of B. We have from Theorem 5.17 that
Q
Ru (B) = α∈Φ+ Uα . For w ∈ W = NG (T )/T , write ẇ for an arbitrarily fixed
preimage of w in NG (T ). Note that the double coset B ẇB is independent of the
choice of preimage. We first consider how double cosets multiply:
Lemma 5.23. Let α ∈ ∆ be a simple root with corresponding simple reflection s.
Then for all w ∈ W we have
B ẇB · B ṡB ⊆ B wsB
˙ ∪ B ẇB
Proof. It’s easy to check directly that in Gα ∼
= SL2 or PGL2 , we have
ṡ(Uα \ {1})ṡ ⊆ T Uα ṡUα ⊆ B ṡB
by using Examples 5.14 and 5.19, and hence ṡU
Qα ṡ ⊆ B ṡB ∪ B - (∗)
Now, by Theorem 15.7, we have B = T ·Uα · β∈Φ+ \{α} Uβ . Recall from Theorem
5.13(4) that v̇Uβ v̇ = UvβPfor all v ∈ W . Also, observe that sα .β ∈ Φ+ for all α 6= β ∈
Φ+ . Indeed, write β = γ∈∆ cγ γ with cγ ≥ 0 for all γ. Then sα .β = β − hβ, α∨ iα
so all the coefficients of sα .β in its expansion with respect to ∆ are positive, except
maybe for the coefficient of α. If sα .β did belong to Φ− , then this would force
cγ = 0 for all γ 6= α and so β = α, contradicting our assumption on β.
Thus, thanks to these facts, we know that Uβ ṡB = ṡUsβ B = ṡB for all β 6= α
and so we have that B ṡB = Uα ṡB. Now assume first that wα ∈ Φ+ . Then
B ẇB · B ṡB = B ẇB ṡB = B ẇUα ṡB = BUwα wsB
˙
= B wsB.
˙
On the other hand, if wα ∈ Φ− then wsα = w(−α) = −wα ∈ Φ+ . So, writing
v = ws, we have by the previous that B v̇B · B ṡB = B vsB
˙
= B ẇB. Therefore,
putting all these together, we have
B ẇB · B ṡB = B vsB
˙ ṡB ⊆ B v̇(B ṡB ∪ B) = B v̇B ṡB ∪ B v̇B = B wsB
˙ ∪ B ẇB
where the inclusion follows from (∗).
We now define an important combinatorial group theoretical definition which is
useful in the study of connected reductive groups:
Definition 5.24. A pair B, N of subgroups of a group G is called a BN-pair if the
following holds:
(BN1) G is generated by B and N .
(BN2) B ∩ N is a normal subgroup of N .
(BN3) The group W = N/(B ∩ N ) is generated by a set S of involutions.
37
(BN4) If ṡ ∈ N is a preimage of s ∈ S under the natural projection N → W and
n ∈ N , then BnB · B ṡB ⊆ BnṡB ∪ BnB.
(BN5) For ṡ defined as above, ṡB ṡ 6= B.
The group W is called the Weyl group of the pair.
It doesn’t come to much of a surprise that using a Borel subgroup B of a connected reductive group G, we can obtain a BN-pair for G:
Theorem 5.25. Let G be a connected reductive algebraic group with Borel subgroup
B, and let N = NG (T ) for some maximal torus T ≤ B. Then B, N form a BN-pair
in G whose Weyl group is the usual Weyl group of G.
Proof. By Theorem 3.18(ii) and Corollary 5.12, we have B ∩ N = NB (T ) =
CB (T ) = T , which is normal in N . Thus we have (BN2). Then W = N/(B ∩ N )
is the usual Weyl group of G, generated by the set S of simple reflections, giving (BN3). Recall that G = hT, Uα | α ∈ Φi and B = hT, Uα | α ∈ Φ+ i. For
α ∈ Φ+ , we have U−α = s˙α Uα s˙α ≤ hB, N i, and so (BN1) follows. (BN4) is just
Lemma 5.23. Finally, writing α for the root corresponding to a simple reflection
s ∈ S, ṡB ṡ contains ṡUα ṡ = U−α , which is not contained in B, since else we would
have Gα = hUα , U−α i ≤ B, contradicting the fact that B is solvable (as Gα is
non-solvable). So we have (BN5).
Theorem 5.26. (Bruhat decomposition) Let G be a group with a BN-pair. Then
G
G=
B ẇB
w∈W
for any choice of preimages ẇ ∈ N mapping to w ∈ W = N/(B ∩ N ).
Proof. We first show that B v̇B ∩ B ẇB = ∅ if w 6= v. Write v as a product of
minimal length `(v) in the generators from S. Argue by induction on `(v), with
wlog `(v) ≤ `(w). If `(v) = 0, then v = 1. If B ∩ B ẇB 6= ∅ then ẇ ∈ B ∩ N (as
ẇ ∈ N ) and so w = 1 = v. So assume `(v) > 0 and suppose B v̇B ∩B ẇB 6= ∅. Then
clearly B v̇B = B ẇB. Now, we may write v = v 0 s where s ∈ S and `(v 0 ) < `(v).
Then
B v̇ 0 ṡ = B v̇ ⊆ B v̇B = B ẇB
Thus, by (BN4), this gives
B v̇ 0 ⊆ B ẇB ṡ ⊆ B wsB
˙ ∪ B ẇB
So either B v̇ 0 B = B wsB
˙
or B v̇ 0 B = B ẇB. By induction hypothesis this gives
0
0
v = ws or v = w. But `(v) ≤ `(w) so v 0 6= w and we must have v 0 = ws.
Therefore v = v 0 s =Fw.
0
0
Now pick x, y ∈ w∈W B ẇB, say x = b1 v̇b2 and y = b01 ẇb
F2 for bi , bi ∈ B and
v, w ∈ W . Then an easy induction on `(w) shows that xy ∈ w∈W B ẇB. Indeed,
if `(w) = 1 this is given by (BN4). If `(w) > 1, writing w = w0 s with s ∈ S and
F
`(w0 ) < `(w), we obtain by induction hypothesis that b1 v̇b2 b01 ẇ0 ∈ w∈W B ẇB, say
it equals bu̇b0 for some b, b0 ∈ B and u ∈ W . So xy = bu̇b0 ṡb02 , which then is in our
disjoint unionFby (BN4).
Therefore w∈W B ẇB is closed under multiplication, and it is clearly closed
under inverses and contains 1, so it is a subgroup. Since it contains B and N , it
must be all of G by (BN1).
38
Thus we see that connected reductive groups are disjoint unions of double cosets
of one of their Borel subgroups thanks to Theorems 5.25 and 5.26. This result has
an immediate corollary which was not obvious at all until now:
Corollary 5.27. Let G be reductive. Then the intersection of any two Borel subgroups of G is connected and contains a maximal torus.
Proof. Borel subgroups and tori are contained in G◦ , so we may assume G is connected. Let B, B 0 be Borel subgroups. We know they are conjugate by Proposition
4.14, say B 0 = gBg −1 . Choose a maximal torus T ≤ B with Weyl group W . By
Theorem 5.26, g = b1 ẇb2 for some b1 , b2 ∈ B and ẇ ∈ NG (T ). Then
−1 −1
B 0 = gBg −1 = b1 ẇb2 Bb−1
b1 = b1 ẇB ẇ−1 b−1
2 ẇ
1 .
−1
Since ẇ ∈ NG (T ), we have ẇT ẇ−1 = T and so b1 ẇT ẇ−1 b−1
1 = b1 T b1 ≤ B. But
−1
−1
−1
by the above b1 T b1 = b1 ẇT ẇ−1 b1 ≤ B 0 , so we have b1 T b1 ≤ B ∩B 0 as required.
0
Now, write T 0 = b1 T b−1
1 . Then B = T U where U = Ru (B). We then have
0
0
0
B ∩ B = T H where H = B ∩ B ∩ U , a closed T 0 -stable subgroup. By Proposition
5.20, H is generated by the Uα it contains and so is connected by Proposition 1.10.
Hence, B ∩ B 0 is also connected thanks to Proposition 1.10.
Actually, for a connected reductive group G, Theorem 5.26 can be strengthened
in the following way: for any w ∈ W , we can define a subgroup
Y
Uw− =
Uα
α∈Φ+ ,w.α∈Φ−
(it is a subgroup by the commutator formula), and it can be shown that every g ∈ G
can be written uniquely as g = uẇb where b ∈ B, w ∈ W and u ∈ Uw− with respect
to the positive system Φ+ ⊆ Φ determined by T ≤ B (see [H, Thm 28.3 and 28.4]).
Now, since W acts simply-transitively on the set of bases, there exists a unique
w0 ∈ W such that w0 (∆) = −∆ (moreover w02 = 1 since it stabilises ∆). We can
then use the strengthened version of 5.26 to show the following:
Lemma 5.28. B w˙0 ∩ B = T
Proof. Write U = Ru (B). Let g ∈ B w˙0 ∩ B, so g = w˙0 utw˙0 = t0 u0 for some
u, u0 ∈ U , t, t0 ∈ T following Theorem 5.17. Then
w˙0 −1 t0 u0 = uw˙0 tw˙0 .
By definition of w0 , we have that Uw−0 = U , so by the uniqueness in the strengthened
version of Theorem 5.26, we see that u0 = 1 = u. Therefore g = t0 ∈ T . Conversely,
T ⊆ B w˙0 ∩ B as w˙0 normalises T .
B w˙0 is called the Borel subgroup opposite to B, and is denoted by B − .
6. Parabolic subgroups
We now use the results from the previous section to investigate parabolic subgroups of connected reductive groups. It turns out that they can be entirely described using the root system. We will also prove the very beautiful Borel-Tits
theorem which has applications to the study of maximal subgroups.
39
6.1. Standard parabolic subgroups and the Levi decomposition. We keep
using the same notation as in the previous section: let G be a connected reductive
linear algebraic group, T ≤ G a maximal torus contained in a Borel subgroup B,
Φ the root system of G with Weyl group W = N/T where N = NG (T ), ∆ the set
of simple roots with respect to T ≤ B and S = {sα | α ∈ ∆} the set of simple
reflections. Recall S generates W .
For I ⊆ S, let WI = hs ∈ Ii. WI is called a standard parabolic subgroup of W . A
parabolic subgroup
P of W is a subgroup conjugate to some WI . Let ∆I = {α | sα ∈ I}
and ΦI = Φ∩ α∈∆I Zα, the corresponding parabolic subsystem of roots. It is fairly
easy then to show that ΦI is a root system in hΦI iR with base ∆I and Weyl group
WI , by using Proposition 5.16 and the fact that Φ is a root system with base ∆.
We can then translate this to subgroups of G. Indeed, for I ⊆ S, define
G
PI = BWI B =
B ẇB.
w∈WI
Clearly, for I = ∅ and I = S, we get B and G respectively by Theorem 5.26. More
generally, PI is always a subgroup of G containing B. Indeed, it contains 1, and
is clearly closed under inverses. Moreover, by Lemma 5.23, B ẇB · B ṡB ⊆ PI for
w ∈ WI and s ∈ I. An easy induction similar to the one in the proof of Theorem
5.26 then gives that PI is closed under multiplication. So PI is a subgroup, and it
clearly contains B.
Proposition 6.1. For I ⊆ S, PI = hT, Uα | α ∈ Φ+ ∪ ΦI i. In particular, PI is a
closed, connected, self-normalising subgroup of G containing B.
±
Proof. Let H = hT, Uα | α ∈ Φ+ ∪ ΦI i and write Φ±
I = ΦI ∩ Φ . Since B ⊆ PI ,
+
it follows that Uα ⊆ PI for all α ∈ Φ . Also, ΦI is a root system with Weyl
group WI , so WI acts simply-transitively on the set of bases by Proposition 5.16,
and thus there exists a unique w ∈ WI such that w(∆I ) = −∆I . In particular,
−
−
+
w(Φ+
I ) = ΦI . Hence, for all β ∈ ΦI , there exists α ∈ ΦI such that β = wα. But
−1
then Uβ = Uwα = ẇUα ẇ ⊆ PI , thus proving H ⊆ PI .
Conversely, note that for α ∈ ∆ with corresponding simple reflection s ∈ S, we
may choose a preimage ṡ ∈ hUα , U−α i (this is the same argument as in the proof
of Theorem 5.17). So H contains preimages of all s ∈ I, and thus of all w ∈ WI .
Therefore, since B ≤ H, it follows that PI ⊆ H and so we have PI = H as required.
The fact that PI is a closed connected subgroup then follows from Proposition
1.10. Since it contains B, it is a parabolic subgroup, and so is self-normalising by
Corollary 4.28.
Definition 6.2. The PI , I ⊆ S, are called standard parabolic subgroups.
We wish to show that all parabolic subgroups are conjugate to some PI . In
what follows, we will make use of the fact from the proof of Theorem 5.26 that
B v̇B = B ẇB ⇐⇒ v = w for v, w ∈ W . We begin by making the statement of
Lemma 5.23 more precise.
Lemma 6.3.
(i) If s ∈ S, w ∈ W with `(ws) ≥ `(w) then B ẇB · B ṡB ⊆
B wsB
˙
(ii) If s ∈ S, w ∈ W with `(ws) ≤ `(w) then B ẇB · B ṡB ∩ B ẇB 6= ∅
Proof. (i) Use induction on `(w). If `(w) = 0, the result is trivial. So suppose
`(w) ≥ 1 and write w = s0 w0 with s0 ∈ S and w0 ∈ W with `(w0 ) = `(w) − 1.
40
Suppose the result is false, and aim for a contradiction. By Lemma 5.23, B ẇB ·
B ṡB ∩ B ẇB 6= ∅. So in particular ẇB ṡ ∩ B ẇB 6= ∅. Multiplying on the left by ṡ0 ,
we get ẇ0 B ṡ ∩ ṡ0 B ẇB 6= ∅.
But `(w0 s) ≥ `(s0 w0 s) − 1 = `(ws) − 1 ≥ `(w) − 1 = `(w0 ) so by induction
hypothesis, B ẇ0 B · B ṡB ⊆ B w˙0 sB and thus B w˙0 sB ∩ ṡ0 B ẇB 6= ∅. Now, note that
if we take inverses in Lemma 5.23, using the fact that s2 = 1, we obtain that for
all s ∈ S, w ∈ W ,
B ṡB · B ẇB ⊆ B swB
˙ ∪ B ẇB
and applying this, we get that
ṡ0 B ẇB ⊆ B s0˙wB ∪ B ẇB = B ẇ0 B ∪ B ẇB.
It follows that B w˙0 sB intersects one of these double cosets, and hence either w0 s =
w0 (but this would imply s = 1, a contradiction) or w0 s = w. So we must have
w0 s = w and therefore w0 = ws. But this is a contradiction since `(w0 ) < `(w) ≤
`(ws) by assumption.
(ii) By Lemma 5.23, ṡB ṡ ⊆ B ṡB ∪ B. Therefore, by (BN5), ṡB ṡ ∩ B ṡB 6= ∅.
Multiplying on the left by ws,
˙ we get ẇB ṡ ∩ wsB
˙ ṡB 6= ∅. But `(ws2 ) = `(w) ≥
`(ws) so by (i), B wsB
˙
· B ṡB ⊆ B ẇB. Hence, ẇB ṡ ∩ B ẇB 6= ∅ and the result
follows.
Lemma 6.4. Let w = s1 · · · sr , with si ∈ S, be a shortest expression for w, so that
r = `(w) and let I = {s1 , . . . , sr } ⊆ S. Then hB, ẇi = hB, ẇ−1 B ẇi = PI .
Proof. We show by induction on r that s˙1 , . . . , s˙r ∈ hB, ẇ−1 B ẇi. Let w0 = wsr ,
so `(w0 ) < `(w). By Lemma 6.3(ii), B ẇB · B s˙r B ∩ B ẇB 6= ∅. In particular,
s˙r ∈ B ẇ−1 B ẇB, which lies in hB, ẇ−1 B ẇi. So if r = 1, we’re done. If r > 1, by
induction hypothesis s˙1 , . . . , ṡr−1 ∈ hB, ẇ0−1 B ẇ0 i = hB, s˙r ẇ−1 B ẇs˙r i which lies in
hB, ẇ−1 B ẇi since s˙r does. So the group generated by B and ẇ−1 B ẇ contains a
preimage of all elements of I, and hence of all elements of WI . Therefore it contains
PI . But we have hB, ẇ−1 B ẇi ⊆ hB, ẇi ⊆ PI , giving equality throughout.
This lemma has the following consequences:
Lemma 6.5.
(i) S is precisely the set of those w ∈ W for which B ∪ B ẇB is
a group.
(ii) Let w ∈ W and I, J ⊆ S. If ẇ−1 PI ẇ ⊆ PJ then ẇ ∈ PJ . In particular, for
any I ⊆ S, we have NG (PI ) = PI .
Proof. (i) If I = {s}, s ∈ S, then WI = {1, s} and so B ∪ B ṡB = PI is a group.
Conversely, if B ∪ B ẇB is a group and w = s1 · · · sr is a shortest expression, then
{s˙1 , . . . , s˙r } ⊆ hB, ẇi = B ∪ B ẇB by Lemma 6.4. Thus B s˙1 B ⊆ B ∪ B ẇB. Since
s˙1 ∈
/ B, we must have B s˙1 B ∩ B ẇB 6= ∅ and therefore s1 = w. It follows that r = 1
and w ∈ S.
(ii) We have that both B and ẇ−1 B ẇ lie in PJ . Write a shortest expression
w = s1 · · · sr and let I 0 = {s1 , . . . , sr } ⊆ S. By Lemma 6.4, PI 0 ⊆ PJ , and so PJ
contains a preimage of all elements in WI 0 . From the double coset decomposition
of PJ and the fact that B v̇B = B ẇB ⇐⇒ v = w for all v, w ∈ W , it follows
that WI 0 ⊆ WJ and so w ∈ WJ and ẇ ∈ PJ . The fact that NG (PI ) = PI is then
immediate.
41
Remark. We had already proved that NG (PI ) = PI , but this time we didn’t appeal
to Corollary 4.28, which itself depended on Theorem 4.27. Actually, since any Borel
subgroup B is evidently of the form PI , we just proved Theorem 4.27 for connected
reductive groups.
We can now prove that every parabolic subgroup is conjugate to some PI :
Theorem 6.6.
(i) The subgroups of G containing B are all of the form PI
for some I ⊆ S. In particular all parabolic subgroups are conjugate to some
PI .
(ii) The PI are mutually non-conjugate. In particular, PI = PJ implies I = J.
Proof. (i) Let P be a subgroup of G containing B. Any element of P can be written
in the form g = bẇb0 for some b, b0 ∈ B, w ∈ W by Theorem
5.26. Thus ẇ ∈ P and
F
so P can be written as a union of double cosets P = w∈M B ẇB for some M ⊆ W .
Let I be the set of all s ∈ S appearing in the shortest expressions of elements of
M . Clearly P ⊆ PI and by Lemma 6.4, we have PI ⊆ P . For the last part, any
parabolic subgroup P contains a Borel subgroup B 0 , which is conjugate to B by
some g ∈ G. Thus P g contains B and so equals PI for some I ⊆ S.
(ii) Suppose PI is conjugate to PJ , say PI = PJg for some g ∈ G. Write g = bẇb0
for some b, b0 ∈ B, w ∈ W . Then PI = PJẇ and so PI = PJ by Lemma 6.5(ii). Using
the same argument as in the proof of Lemma 6.5(ii), we therefore have WI = WJ .
But then, by Lemma 6.5(i), I and J are both the set of elements w ∈ WI = WJ
such that B ∪ B ẇB is a group, ie I = J.
Remark. More generally, PI ⊆ PJ implies I ⊆ J. Indeed, we then have WI ⊆ WJ ,
and so by Lemma 6.5(i), we have I ⊆ J.
Example 6.7. Let G = SLn . Following Examples 5.14 and 5.19, we have a maximal
torus T = Dn ∩ SLn contained in the Borel subgroup B = Tn ∩ SLn , root system
Φ = {χij : 1 ≤ i, j ≤ n, i 6= j}, base ∆ = {χi,i+1 : 1 ≤ i < n} and set of positive
roots Φ+ = {χij : i < j}. We also have Uij = In + hEij i and


Ii−1


0 1
 T ∈ NG (T )
si = 


−1 0
In−i−1
∼ Sn under the isois the simple reflection corresponding to χi,i+1 . We have W =
morphism given by si 7→ (i i + 1). So we may identify S = {si : i < n} with
{(1 2), . . . , (n − 1 n)}. Let I = S \ {sa } for some a < n. Then
∼ Sa × Sn−a
WI = hsi | a 6= i < ni =
and ΦI = {χ
≤ a or i, j ≥ a}. Then by Proposition 6.1 it’s not hard to see
ij : i, j A ∗
that PI =
: A ∈ GLa , B ∈ GLn−a ∩ SLn . This parabolic subgroup is
0 B
the stabiliser of the subspace of k n given by its first a basis vectors. In general, as
pointed out in Example 4.17, stabilisers of flags are parabolic subgroups.
Now that we know what parabolic subgroups of G look like, we can decompose
them furthermore in the following way: for I ⊆ S, define
Y
UI =
Uα = hUα | α ∈ Φ+ \ ΦI i
α∈Φ+ \ΦI
42
(the product is a subgroup thanks to the commutator formula) and let
LI = hT, Uα | α ∈ ΦI i.
We can then show that LI is a complement to UI in PI :
Proposition 6.8. Let I ⊆ S. Then Ru (PI ) = UI and LI is complement to UI , so
PI = UI o LI . In particular, LI is reductive with root system ΦI . Moreover, all
closed complements to UI are conjugate to LI in PI and LI = CG (Z(LI )◦ ).
Proof. Thanks to the commutator formula (Proposition
T 5.22), we have that UI PI
and PI = hLI , UI i by Proposition 6.1. Now let Z = ( α∈ΦI ker α)◦ ≤ T , a subtorus
of T , and let L = CG (Z). It’s clear from Theorem 5.13(3) that LI ≤ L.
Conversely, L is connected reductive by Corollary 5.12 so by Corollary 5.21 it
is generated by T and the Uβ it contains. Suppose Uβ ≤ L. Then β is trivial
⊥
on
T Z again thanks ⊥to Theorem 5.13(3) so β ∈ Z . Now, Z has finite index in
ker
α
=
hΦ
i
,
so
thanks
to
Proposition
3.12(i)
we know that some multiple
I
α∈ΦI
⊥ ⊥
of β lies in (hΦI i ) . Furthermore, some multiple of β lies in hΦI i by Proposition
3.12(ii). So, by (R2), β ∈ ΦI and thus L = LI .
Therefore LI ∩UI is a normal unipotent subgroup of the reductive group LI = L,
hence trivial. So PI = UI o LI . As LI is reductive, we also have Z(LI )◦ = Z (by
Theorem 5.13(7)), so LI = CG (Z(LI )◦ ) and Z is a maximal torus in R(PI ) = ZUI .
Now, suppose L0 is another closed complement to UI in PI . Then L0 ∼
= LI as
abstract groups (since they’re both isomorphic to PI /UI as abstract groups), and
so if we let Z 0 = Z(L0 )◦ , we have Z 0 ∼
= Z(LI )◦ = Z. As Ru (PI ) = UI , we have
0 ∼
Ru (L ) = Ru (PI /UI ) = 1 and so Ru (Z 0 ) = 1. It follows that Z 0 is a connected
solvable reductive group and so a torus (by Example 5.4). Now, Dn ∼
6 Dm as
=
abstract groups for n 6= m (by counting the number of elements of fixed finite
order), so Z 0 and Z must have the same dimension. Hence Z 0 is a maximal torus
of R(PI ). Maximal tori of an algebraic group are conjugate (by Corollary 4.18), so
Z and Z 0 are conjugate, and thus L0 = CG (Z 0 ) (which holds by the first part) is
conjugate to L = CG (Z).
Definition 6.9. The decomposition PI = UI o LI is called the Levi decomposition
of PI , and LI is called the standard Levi complement or Levi factor of PI . A Levi
subgroup of G is a subgroup conjugate to some LI .
Remark. The proof of Proposition 6.8 shows that for any Levi subgroup L, we have
L = CG (Z(L)◦ ).
Corollary 6.10. For any I ⊆ S, we have PI = NG (UI ).
Proof. Let N = NG (UI ). Clearly PI ≤ N . Thus N is a parabolic subgroup of G,
so equal to PJ for some J ⊇ I in S. Suppose J 6= I. Then let s = sα ∈ J \ I. We
have ṡ ∈ N and Uα ≤ UI (as α ∈ Φ+ \ ΦI ). Therefore U−α = ṡUα ṡ−1 ≤ UI , which
is a contradiction. So J = I and PI = N .
Example 6.11. If we take G = SLn as in Example 6.7, we had a parabolic subgroup
PI with I = S \ {sa }. Then since ΦI = {χij : i, j < a or i, j > a}, it follows that
its standard Levi complement is
A 0
LI =
: A ∈ GLa , B ∈ GLn−a ∩ SLn
0 B
43
and so we see LI is conjugate to LJ where J = S \ {sn−a }. Thus, unlike for
parabolic subgroups, Levi subgroups for different subsets I of S can be conjugate.
In a more general context, for a flag 0 = V0 ⊆ V1 ⊆ . . . ⊆ Vr = k n with
dim Vi /Vi−1 = ni , its stabiliser in GLn is a parabolic subgroup whose Levi complement is the subgroup of block diagonal matrices isomorphic to GLn1 × . . . × GLnr
(where we identify GLni with GL(Vi /Vi−1 )). For example if r = 3 we get a parabolic
subgroup



∗
 A1 ∗

P =  0 A2 ∗  : Ai ∈ GLni


0
0 Ar
with Levi complement



0
 A1 0

L =  0 A2 0  : Ai ∈ GLni


0
0 A3
We see that stabilisers of flags give typical examples of parabolic subgroups
of GLn or SLn . Using the Levi decomposition, one can show that all parabolic
subgroups of SLn are of that form (see [MT, Prop 12.13]).
From Proposition 6.8, we see that any Levi subgroup is equal to CG (Z) for some
torus Z. We want to show that the converse holds. Recall that a parabolic subgroup
of W is a subgroup conjugate to some WI . We have the following basic result about
abstract root systems (see [MT, Corollary A.29]):
Lemma 6.12. Let Φ be an abstract root system of E with Weyl group W , and let
M ⊆ E. Then the pointwise stabiliser CW (M ) of M in W is a parabolic subgroup.
Conversely, any parabolic subgroup H ≤ W is the centraliser of its fixed point space,
ie H = CW (E H ).
Proposition 6.13. Let G be connected reductive, Z ≤ G a torus. Then CG (Z) is
a Levi subgroup of G.
Proof. Let C = CG (Z). By Corollary 5.12, C is connected reductive. A Borel
subgroup BC of C is contained in a Borel subgroup B of G. Since Z ≤ Z(C)◦ ≤
BC ≤ B (where the second inclusion follows from Proposition 5.3(i)), Z is contained
in a maximal torus T of B (and so of G). Clearly T ≤ C so by Theorem 5.13,
C = hT, Vα | α ∈ ΦC i where ΦC is the root system of C with respect to T and the
Vα are the root subgroups of C.
For α ∈ ΦC , 0 6= Lie(Vα ) ⊆ gα and so α ∈ Φ where Φ is the root system of G
with respect to T . Since all the gα have dimension 1, we have Lie(Vα ) = gα and so
by Theorem 5.13(3), Vα = Uα , the root subgroup of G with respect to α.
By Theorem 5.13(3), Uα ≤ C if and only if Z ≤ ker α and so C = hT, Uα | Z ≤
ker αi. Therefore ΦC = Φ ∩ hα : Z ≤ ker αi, the intersection of Φ with a subspace
of hΦiR . By Lemma 6.12, ΦC is then a parabolic subsystem of Φ, and therefore C
is the Levi complement of the corresponding parabolic subgroup of G.
We introduce one further concept related to Levi subgroups. Recall from Lemma
5.28 that given a Borel subgroup B of G, containing a maximal torus T , there exists
a unique Borel subgroup B − , called the Borel subgroup opposite to B, such that
B ∩ B − = T . This is an example of a more general construction. Indeed, given a
parabolic subgroup P , and a Levi subgroup L of P , there exists a unique parabolic
subgroup P − of G such that L is a Levi subgroup of P − and P ∩ P − = L. The
44
subgroup P − is called the opposite parabolic subgroup to P with respect to L. Since
any parabolic subgroup is conjugate to a standard parabolic subgroup PI for some
I ⊆ S and any Levi subgroup of PI is conjugate to LI , it’s enough to show that P −
exists for P = PI and L = LI . Then just define PI− = hT, Uα | − α ∈ ΦI ∪ Φ+ i (or
in the double coset notation, PI− = B − WI B − ). It is clear that PI− is a parabolic
subgroup, and its intersection with P is LI . Finally, if P − is an opposite
Q parabolic
subgroup to PI with respect to LI , then its intersection with Ru (P ) = α∈Φ+ \ΦI Uα
is trivial and so it must equal PI− .
Note that then we have Ru (P ) ∩ Ru (P − ) = 1 for any parabolic subgroup P with
Levi L, where P − is the opposite to P with respect to L, since it holds for P = PI .
6.2. The Borel-Tits Theorem. We now aim to show a fundamental result on
normaliser of unipotent subgroups. We still assume G is a connected reductive
linear algebraic group. A key step in the proof is the following result:
Proposition 6.14. Let V be a closed unipotent subgroup of G, and N = NG (V ).
Assume that V lies in some Borel subgroup of G, and that Ru (N ) ⊆ V (equivalently
V ◦ = Ru (N )). Then N is a parabolic subgroup of G, with V = Ru (N ).
Assume for now that Proposition 6.14 is true. Then we get:
Theorem 6.15. (Borel-Tits) Let U ≤ G be a closed unipotent subgroup which
lies in a Borel subgroup of G. Then there exists a parabolic subgroup P of G with
U ≤ Ru (P ) and NG (U ) ≤ P .
Proof. Set U0 = U and define inductively Ni+1 = NG (Ui ) and Ui+1 = Ui ·Ru (Ni+1 )
for i ≥ 0. Then U = U0 ≤ U1 ≤ . . . and N1 ≤ N2 ≤ . . . are chains of closed
subgroups. We claim each Ui is contained in a Borel subgroup of G. This holds for
i = 0, so assume Ui ≤ B for i ≥ 0 and some Borel subgroup B of G.
We let Ui act on the projective variety G/B by left multiplication. Then, since
Ui ≤ B, the set X of fixed points is a non-empty closed subset, thus also projective.
X is stabilised by the connected solvable group Ru (Ni+1 ) ≤ NG (Ui ), so by the Borel
fixed point theorem (Theorem 4.13), there exists gB, for some g ∈ G, which is fixed
by Ru (Ni+1 ) and by Ui . So Ui+1 = Ui · Ru (Ni+1 ) fixes gB and hence Ui+1 is
contained in gBg −1 as claimed.
Now,
Ui+1 /Ui = Ui · Ru (Ni+1 )/Ui ∼
= Ru (Ni+1 )/(Ui ∩ Ru (Ni+1 )),
so since Ru (Ni+1 ) is connected, we have that Ui /Ui+1 is connected. Thus either
dim (Ui+1 /Ui ) ≥ 1 or Ui+1 = Ui . Since G has finite dimension, the sequence
U = U0 ≤ U1 ≤ . . . must terminate, ie there is some l ≥ 0 such that for all k ≥ 0,
Ul = Uk+l . Therefore we also have Nl+1 = Nl+k+1 for all k ≥ 0.
Let V = Ul and P = Nl+1 = NG (V ). Then we know V lies in some Borel
subgroup of G, and Ru (P ) ⊆ V . By Proposition 6.14, P is a parabolic subgroup
and V = Ru (P ), so U ≤ V = Ru (P ) and NG (U ) = N1 ≤ Nl+1 = P .
Since a closed connected unipotent subgroup is a solvable, it lies in some Borel
subgroup. Thus we have the following immediate corollary:
Corollary 6.16. Let U ≤ G be a closed connected unipotent subgroup. Then there
exists a parabolic subgroup P of G with U ≤ Ru (P ) and NG (U ) ≤ P .
Now, if we consider inclusions of connected reductive groups, we get the following:
45
Corollary 6.17. Let H be a closed connected reductive subgroup. Let PH be a
parabolic subgroup of H. Then there exists a parabolic subgroup PG of G with
Ru (PH ) ≤ Ru (PG ) and PH ≤ PG .
Proof. Ru (PH ) is a closed connected unipotent subgroup, so by Corollary 6.16
there exists a parabolic subgroup PG of G such that PH ≤ NG (Ru (PH )) ≤ PG and
Ru (PH ) ≤ Ru (PG ).
We now complete the proof of Theorem 6.15:
Proof of Proposition 6.14. Let B be a Borel subgroup containing V , and S =
(B ∩ N )◦ . Then S lies in some Borel subgroup B1 of N , and choose a subgroup
nB1 n−1 , n ∈ N , such that Ru (B1 ∩ nB1 n−1 ) = Ru (N ) (we can do this by taking
an opposite subgroup, as defined after Lemma 5.28, of the image of B1 in the
connected reductive group N ◦ /Ru (N )). Let B 0 = nBn−1 .
By choice of B 0 , we have Ru (N ∩ B ∩ B 0 ) ⊆ Ru (N ) = V ◦ . On the other hand,
n normalises V so V lies in the unipotent part of the solvable group N ∩ B ∩ B 0
and therefore V ◦ = Ru (N ∩ B ∩ B 0 ). Now, we know from Corollary 5.27 that
B ∩ B 0 is connected solvable, so by Theorem 3.18 V ⊆ Ru (B ∩ B 0 ). If V had
smaller dimension than Ru (B ∩ B 0 ), then by Proposition 4.11 the normaliser of
V in that group would have larger dimension than V . But that normaliser is just
N ∩Ru (B ∩B 0 ) (as N = NG (V )), whose connected component is a closed connected
solvable normal subgroup of N ∩ B ∩ B 0 , hence contained in Ru (N ∩ B ∩ B 0 ) = V ◦ ,
giving a contradiction. Therefore V = Ru (B ∩ B 0 ) and so V is connected and
B ∩ B 0 ⊆ N . Since B ∩ B 0 contains a maximal torus T of G by Corollary 5.27, it
follows that N has maximal rank in G.
We now use the root system of G with respect to T to show that N is parabolic.
We let ∆ be the base of Φ determined by B as in Theorem 5.17 and S be the set
of simple reflection. Also define Ψ to be the set of roots of N with respect to T .
These are the roots α ∈ Φ such that Uα ⊆ V and the roots ±β corresponding to
the roots of the connected reductive group N ◦ /V with respect to the image of T .
Take α ∈ ∆. If Uα ⊆ V , then α ∈ Ψ. Otherwise, Uα ⊆ B but Uα * V , so since
V = Ru (B ∩ B 0 ), it follows that Uα * B 0 and thus U−α ⊆ B 0 . Let B 00 = s˙α B s˙α −1 .
Recall from the proof of Lemma 5.23 that sα .β ∈ Φ+ for all α 6= β ∈ Φ+ , and
hence the root subgroups in B 00 are the root subgroups in B except for Uα , which
is replaced by U−α . Therefore U−α normalises B 0 and Ru (B ∩ B 00 ), and hence also
Ru (B ∩ B 0 ∩ B 00 ) = Ru (B ∩ B 0 ) = V . Thus −α ∈ Ψ and since U−α * V , we must
have that ±α ∈ Ψ.
So we proved that ∆ ⊆ Ψ. So B ⊆ N and N is a parabolic subgroup, more
specifically the standard parabolic subgroup PI where I = {sα ∈ S : Uα * V }.
Using the Borel-Tits theorem, we get the following striking result:
Theorem 6.18. Let G be connected reductive.
(i) Suppose H ≤ G is a maximal proper closed subgroup. Then either H ◦ is
reductive or H is a parabolic subgroup.
(ii) Let U ≤ G be a unipotent subgroup. Then U lies in some Borel subgroup
of G.
Proof. (i) Suppose H ◦ is not reductive. Then Ru (H ◦ ) is a non-trivial connected
unipotent subgroup of G. By Corollary 6.16, there exists a parabolic subgroup P of
46
G such that Ru (H ◦ ) ≤ Ru (P ) and NG (Ru (H ◦ )) ≤ P . Thus H ≤ NG (Ru (H ◦ )) ≤
P . By maximality of H, we must have H = P .
(ii) We prove this by induction on dim G. If dim G = 1, then G ∼
= Ga or
Gm and so G is solvable. Thus G is a Borel subgroup containing U . So assume
dim G > 1. Since U is unipotent, it is nilpotent by Corollary 3.8 and so Z(U ) 6= 1
(see Definition 1.12). Then pick 1 6= u ∈ Z(U ) and let U1 = hui, the closure of the
subgroup generated by u. We have U ≤ NG (hui) ≤ NG (U1 ).
By Theorem 4.24, u belongs to some Borel subgroup B of G. So hui ≤ B and
it follows that U1 ≤ B. More specifically, U1 ≤ Ru (B) = Bu by Theorem 3.18.
By the Borel-Tits theorem, there exists a parabolic subgroup P of G such that
U ≤ NG (U1 ) ≤ P and U1 ≤ Ru (P ).
But 1 6= U1 , so Ru (P ) 6= 1 and hence P 6= G. Also, the group P/Ru (P ) is
connected reductive and has smaller dimension than dim G. The image of U in
this quotient, namely U · Ru (P )/Ru (P ), is unipotent by Theorem 3.3(iii), so by
induction hypothesis, it is contained in a Borel subgroup of P/Ru (P ). This Borel
subgroup is of the form H/Ru (P ) where H is a closed connected solvable of G with
Ru (P ) ≤ H. Let B1 be a Borel subgroup of G with B1 ≤ P and Ru (P ) ≤ B1 .
Then B1 /Ru (P ) is a closed connected solvable subgroup of P/Ru (P ) and so lies in
a conjugate of H/Ru (P ). Then B1 ≤ H g for some g ∈ G and by maximality of B1 ,
we have B1 = H g . So H is a Borel subgroup containing U as required.
Therefore we see that Theorem 6.15 holds for arbitrary closed, unipotent subgroups of G. Theorem 6.18(ii) also has the consequence that maximal unipotent
subgroups of G are precisely unipotent radicals of Borel subgroups.
The Borel-Tits theorem can also be used to investigate the representations of
the Levi complement of a parabolic subgroup of G, but we don’t discuss this here,
see [MT, sections 15, 16 and 17].
7. G-complete reducibility
In this section, we follow the work of Bate, Martin and Röhrle [BMR]. In representation theory, a finite dimensional representation V of a group H is said to
be completely reducible if whenever 0 6= W < V is a proper H-invariant subspace,
then it has an H-invariant complement W 0 so that V = W ⊕ W 0 . If we identify V
with k n for some n, W with k m for some m < n, and H with some subgroup of
GLn , the above definition gives that whenever, after changing basis, we can write
elements of H as elements of
A ∗
P =
: A ∈ GLm , B ∈ GLn−m ,
0 B
then in fact they are, again up to a change of basis, elements of
A 0
L=
: A ∈ GLm , B ∈ GLn−m .
0 B
Note that P is a parabolic subgroup of GLn and L is a Levi subgroup of it. This
motivates the following definition due to Serre [Se]:
Definition 7.1. Let G be a connected reductive linear algebraic group, and H ≤ G
a closed subgroup. H is said to be G-completely reducible (G-cr) if whenever it is
contained in a parabolic subgroup of G, it is actually contained in a Levi subgroup
of it.
47
The idea here was to try to extend results about representations of algebraic
groups ρ : H → GL(V ) to the more general case of a morphism H → G where G is
an arbitrary connected reductive group (see [Se]). Now, G itself is obviously G-cr
since it is a parabolic subgroup which is its own Levi subgroup. More generally, any
closed subgroup H which isn’t contained in any proper parabolic subgroup of G is
G-cr. Following Serre [Se], H is said to be G-irreducible (G-irr) in that case. As for
G-complete reducibility, this reduces to a familiar concept in representation theory
for the special case G = GL(V ), namely here that V is an irreducible representation
of H. Serre also defined a closed subgroup to be G-indecomposable (G-ind) if it
is not contained in any Levi subgroup of a proper parabolic subgroup of G. This
unsurprisingly also reduces to a well-known notion from representation theory in
the case G = GL(V ). Indeed, in that case, H being G-ind just says that V is an
indecomposable H-module (i.e it is a representation of H which cannot be written
as a direct sum of two proper non-trivial subrepresentations).
As in the previous two sections, we assume throughout that our linear algebraic group G is connected reductive. Recall from Corollary 5.12 that centralisers
of subtori of G are also connected reductive. We make the following important
definition due to Richardson [R]:
Definition 7.2. A closed subgroup H of G is said to be strongly reductive if it is
not contained in any proper parabolic subgroup of CG (S), where S is a maximal
torus of CG (H). Equivalently, H is strongly reductive if it is CG (S)-irr.
Remark. This is independent of the choice of S, since all maximal tori of CG (H)
are conjugate.
Maximal tori of G, for instance, are strongly reductive since they equal their
own centraliser by Corollary 5.12. Richardson introduced the notion of strong
reductivity in order to study the action of G on Gn by simultaneous conjugation,
and in particular to study the closed orbits of that action (we shall not discuss this
here, see [R], in particular Theorem 16.4). One basic property of strong reductivity
is the following:
Lemma 7.3. If H is a strongly reductive subgroup of G, then H ◦ is reductive.
Proof. Let U = Ru (H). Suppose for a contradiction that U 6= 1. Let S be a
maximal torus of CG (H). Recall that CG (S) is connected reductive. Then by the
Borel-Tits theorem (Theorem 6.15) applied to U in CG (S), there exists a parabolic
subgroup of CG (S) containing H, contradicting strong reductivity.
It turns out that the concepts of strong reductivity and G-complete reducibility
coincide. In order to show this, we need a few results on parabolic subgroups. The
first one is due to Borel and Tits, and determines what the parabolic subgroups of
G contained in a fixed parabolic subgroup P of G are (see [BT, Prop 4.4]):
Proposition 7.4. Let P, P 0 be parabolic subgroups of G, and L ≤ P a Levi subgroup.
(i) (P ∩ P 0 ) · Ru (P ) is a parabolic subgroup of G. It equals P if and only if P 0
contains a Levi subgroup of P . It is a Borel subgroup of G if P 0 is a Borel
subgroup.
(ii) The parabolic subgroups of G contained in P are the semidirect products
of the parabolic subgroups of L with Ru (P ). Two parabolic subgroups of G
48
contained in P are conjugate if and only if their intersections with L are
conjugate in L.
The following lemma tells us some more about the intersection of two parabolic
subgroups (see [BMR, Lemma 6.2(iii)]):
Lemma 7.5. Let P and Q be two parabolic subgroups of G, with Levi subgroups
LP and LQ respectively, such that LP ∩ LQ contains a maximal torus T of G. Then
P ∩ Q = (LP ∩ LQ ) · (LP ∩ Ru (Q)) · (Ru (P ) ∩ LQ ) · (Ru (P ) ∩ Ru (Q))
where Ru (P ∩ Q) is the product of the last three factors.
We finally require the following result about parabolic subgroups of a connected
reductive subgroup of G, which strengthens Corollary 6.17 (see [BMR, Corollary
2.5]):
Lemma 7.6. Let H be a connected reductive subgroup of G. If Q is a parabolic
subgroup of H with Levi subgroup LH , then there exists a parabolic subgroup P
of G, and a Levi subgroup L of P such that P ∩ H = Q, L ∩ H = LH and
Ru (P ) ∩ H = Ru (Q).
Note that we already knew this in the special case of Borel subgroups of centralisers of tori (see the discussion preceding Corollary 5.12). We can now prove
the main theorem of this section:
Theorem 7.7. (Bate, Martin, Röhrle) Let H ≤ G be a closed subgroup. Then H
is G-completely reducible if and only if H is strongly reductive.
Proof. Suppose H is G-cr, and let S be a maximal torus of CG (H). Assume that
H is contained in some proper parabolic subgroup Q of CG (S), and aim for a
contradiction. By Lemma 7.6 and Corollary 5.12, there exists a parabolic subgroup
P of G such that Q = CG (S) ∩ P . Since S lies in Z(CG (S))◦ , it must lie in all
Borel subgroups of CG (S) by Proposition 5.3 and so it lies in Q ⊆ P . Now, as H
is G-cr, we have H ⊆ L for some Levi subgroup L of P . Moreover, T = Z(L)◦ is
a torus of CP (H) (it is a subtorus of L by Theorem 5.13(7), which applies since
L is connected reductive). Recall from Proposition 6.8 that L = CG (T ). Now,
S is a maximal torus of CP (H) (as S ⊆ P ), so some conjugate of T lies in S by
Corollary 4.18, say gT g −1 ⊆ S for some g ∈ CP (H). Thus, CG (S) ⊆ CG (gT g −1 ) =
gCG (T )g −1 = gLg −1 ⊆ P , and so Q = CG (S), contradicting the assumption that
Q was a proper subgroup. Therefore, H is strongly reductive.
Conversely, suppose H is strongly reductive. Let S be a maximal torus of CG (H),
and L = CG (S). By Proposition 6.13, L is a Levi subgroup of some parabolic
subgroup Q of G. Since H is strongly reductive, it is not contained in any proper
parabolic subgroup of L, and thus by Proposition 7.4(ii) we have that Q is a minimal
parabolic subgroup of G containing H. Now, let P be a parabolic subgroup of G
containing H. We want to show that H is contained in a Levi subgroup of P . We
have H ⊆ P ∩ Q. If P 0 ⊆ P is a parabolic subgroup of G, and if M 0 is a Levi
subgroup of P 0 , then there exists a Levi subgroup M of P such that M 0 ⊆ M (this
is clear for standard parabolic subgroups, and the general case then follows easily).
Hence, we may assume wlog that P is also minimal among parabolic subgroups
containing H. Now, by Proposition 7.4(i), (P ∩ Q) · Ru (P ) is a parabolic subgroup
of G containing H which is contained in P , so by minimality of P we must have
P = (P ∩ Q) · Ru (P ). Similarly, we have Q = (P ∩ Q) · Ru (Q). It follows from the
49
same proposition that P contains a Levi subgroup MQ of Q, and Q contains a Levi
subgroup MP of P .
Claim. P ∩ Q contains a common Levi subgroup of both P and Q.
Indeed, fix Levi subgroups LP and LQ of P and Q respectively, such that LP ∩LQ
contains a maximal torus of G. Since P ∩ Q contains a maximal torus T , say, of
G by Corollary 5.27, we can take LP and LQ to be Levi subgroups containing T .
Then by Lemma 7.5, we have
P ∩ Q = (LP ∩ LQ ) · (LP ∩ Ru (Q)) · (Ru (P ) ∩ LQ ) · (Ru (P ) ∩ Ru (Q))
where Ru (P ∩ Q) is the product of the last three factors. Now, MP is connected
reductive so MP ∩ Ru (P ∩ Q) is trivial. Therefore there is a bijective morphism
from MP onto a subgroup of LP ∩ LQ . Now, dim LP = dim MP since they are both
Levi subgroups of P , so this forces LP ≤ LQ . By the same argument applied to
MQ , we get LQ ≤ LP . Hence P ∩ Q contains a common Levi subgroup M of both
P and Q as claimed, namely M = LP = LQ .
Let P − be the unique opposite parabolic subgroup to P in G with respect to
M , so that P ∩ P − = M . The set of roots of G with respect to T decomposes as
the disjoint union Ψ(M ) ∪ Ψ(Ru (P ) ∪ Ψ(Ru (P − )) (this is easy to see when P is a
standard parabolic subgroup). Since Ru (Q) ∩ M is trivial, it follows that we have
Ru (Q) = (Ru (Q) ∩ Ru (P − )) · (Ru (Q) ∩ Ru (P )).
Since L and M are Levi subgroups of Q, there exists x ∈ Q such that xM x−1 = L.
Actually, x can be chosen to be in Ru (Q) since Q = Ru (Q) · M . So we can
write x = yz where y ∈ Ru (Q) ∩ Ru (P − ) and z ∈ Ru (Q) ∩ Ru (P ) by the above
decomposition. Since z ∈ P ∩ Q, we have zM z −1 ⊆ P ∩ Q. Replacing M by zM z −1
if necessary, we may therefore assume that z = 1. But then, as y ∈ P − , we have
that L = yM y −1 ⊆ P − and so H lies in P − . Therefore H ⊆ P ∩ P − = M as
required, and so H is G-cr.
Corollary 7.8. Let H be a closed subgroup of G. Then the following are equivalent:
(i)
(ii)
(iii)
(iv)
H is strongly reductive.
H is G-cr.
H is CG (S)-irr for a maximal torus S of CG (H).
for every parabolic subgroup P of G which is minimal among parabolic subgroups containing H, the subgroup H is L-irr for some Levi subgroup L of
P.
(v) there exists a parabolic subgroup P of G which is minimal among parabolic
subgroups containing H such that H is L-irr for some Levi subgroup L of
P.
Proof. By definition of strong reductivity, (i) ⇐⇒ (iii). The equivalence of (i) and
(ii) is Theorem 7.7, and (iv) ⇒ (v) is obvious. Now, given a maximal torus S of
CG (H), CG (S) is a Levi subgroup of some parabolic subgroup P of G by Proposition
6.13. If H is CG (S)-irr, then P is a minimal parabolic subgroup with respect to
containing H by Proposition 7.4(ii), giving (iii) ⇒ (v). The proof of Theorem 7.7
shows that (v) ⇒ (ii). Finally, if H is G-cr, then H is contained in a Levi subgroup
L of any parabolic subgroup P of G, minimal with respect to containing H. So H
is L-irr by Proposition 7.4(ii), giving (ii) ⇒ (iv).
50
Therefore we see that the study of G-cr subgroups reduces to the study of L-irr
subgroups for a Levi subgroup L of G. We now use these results to extend Clifford
theory to the context of G-complete reducibility. A famous result in Clifford theory
asserts that if V is a completely reducible representation of a group H, and if N H,
then V is a completely reducible representation of N . Equivalently, a semisimple
H-module is also semisimple as an N -module. Now, Martin showed that if H ≤ G
is a strongly reductive subgroup, and N H is a closed normal subgroup, then N
is also strongly reductive (see [M, Theorem 2]). The proof uses geometrical ideas
along the line of the theory developed by Richardson [R]. Using this result, and
Theorem 7.7, we immediately get:
Theorem 7.9. Let H be a closed subgroup of G, and N a closed normal subgroup
of H. If H is G-cr, then so is N . In particular, H ◦ is G-cr.
Therefore we can indeed extend Clifford’s theorem to the context of G-complete
reducibility, since Theorem 7.9 reduces to it in the case G = GL(V ). Note that the
converse of Theorem 7.9 is not true in general. Indeed, take H to be a non-trivial
finite unipotent subgroup of G. Then by the Borel-Tits theorem, there exists a
parabolic subgroup P of G such that H ⊆ Ru (P ). Therefore H is not G-cr, while
on the other hand H ◦ = 1 trivially is. However, it can be shown that the converse
does hold with the further assumption that CG (N ) is contained in H (see [BMR,
Theorem 3.14]).
References
[B] A. Borel, Linear Algebraic Groups. Graduate Texts in Mathematics, 126. Springer, 1991.
[BMR] M.Bate, B. Martin, G. Röhrle, A geometric approach to complete reducibility. Invent.
math. 161 (2005), 177-218.
[BT] A. Borel, J. Tits, Groupes Réductifs. Publ. Math.. Inst. Hautes Étud. Sci. 27 (1965), 55-150.
[H] J. Humphreys, Linear Algebraic Groups. Graduate Texts in Mathematics, 21. SpringerVerlag, New York, 1975.
[L] S.Lang, Algebra. Addison-Wesley, Reading, Mass., 1965.
[M] B. Martin, A normal subgroup of a strongly reductive group is strongly reductive. J. Algebra
265 (2003), 669-674.
[MT] G. Malle, D. Testerman, Linear Algebraic Groups and Finite Groups of Lie Type. Cambridge
Studies in Advanced Mathematics, 133. Cambridge University Press, Cambridge, 2011.
[R] R. W. Richardson, Conjugacy classes of n-tuples in Lie algebras and algebraic groups. Duke
Math. J. 57 (1988), 1-35.
[S] T. A. Springer, Linear Algebraic Groups. Second Edition. Progress in Mathematics, 9.
Birkhäuser, Boston, 1998.
[Se] J.-P. Serre, Complète Réducibilité. Séminaire Bourbaki, 56ème année, 2003-2004, no 932.