ALGEBRAIC GROUPS: PART I
EYAL Z. GOREN, MCGILL UNIVERSITY
Contents
1. Introduction
2. First Definitions
3. The main examples
3.1. Additive groups
3.2. Tori
3.3. The general linear group GLn
3.3.1. The unitary groups Up,q
3.4. The orthogonal group Oq .
3.4.1. Quadratic forms
3.4.2. Clifford algebras
3.4.3. The Clifford and Spin groups
3.4.4. Reflections
3.5. The symplectic group
2
3
4
4
4
6
8
9
9
10
12
13
14
References:
• Springer, T. A.: Linear algebraic groups. Reprint of the 1998 second edition.
Modern Birkhuser Classics. Birkhuser Boston, Inc., Boston, MA, 2009.
• Borel, Armand: Linear algebraic groups. Second edition. Graduate Texts in
Mathematics, 126. Springer-Verlag, New York, 1991.
• Humphreys, James E.: Linear algebraic groups. Graduate Texts in Mathematics, No. 21. Springer-Verlag, New York-Heidelberg, 1975.
• Goodman, Roe; Wallach, Nolan R.: Symmetry, representations, and invariants. Graduate Texts in Mathematics, 255. Springer, Dordrecht, 2009.
• Shimura, G.: Arithmetic and analytic theories of quadratic forms and
Clifford groups. Mathematical Surveys and Monographs, 109. American Mathematical Society, Providence, RI, 2004.
• Jacobson, Nathan: Basic algebra. I & II. Second edition. W. H. Freeman and
Company, New York, 1989.
Date: Winter 2011.
1
ALGEBRAIC GROUPS: PART I
2
1. Introduction
This course is about linear algebraic groups. These are varieties V over a field k,
equipped with a group structure such that the group operations are morphisms, and, in
addition, V is affine. It turns out that then V is a closed subgroup of GLn (k), for some n,
and for that reason they are called linear.
On the other side of the spectrum are the projective algebraic groups. Such a group, if it’s
connected, is automatically commutative and therefore one call the connected projective
algebraic groups abelian varieties. Their theory is very different from the theory of
linear algebraic groups. Linear algebraic groups have plenty of linear representations, and
their Lie algebra has rich structure. On the other hand, abelian varieties have only trivial
linear representations and their Lie algebras are commutative and so of little interest. On
other hand, the arithmetic of abelian varieties is very deep; abelian varieties produce some
of the most interesting Galois representations, while linear algebraic group have less rich
structure when it comes to Galois representations (which is not to say there aren’t very
deep issues going on there as well).
The two aspects of algebraic groups are connected, but hardly mix in practice, and the
development of their theories is completely different. The theories connect, for example,
in a special class of algebraic groups - the semi-abelian varieties - but, from the point of
view of a general theory, this is a very particular case, and so, by and large, one studies
the two classes of algebraic groups separately.
Notation. We shall use the letters k or F to denote fields and k, k sep , F , F sep to denote their algebraic and separable closures. Γk and ΓF will denote then the Galois groups
of k sep /k and F sep /F , respectively.
ALGEBRAIC GROUPS: PART I
3
2. First Definitions
Let k be a field. A linear, or affine, algebraic group G over k, is an affine variety1 G
over k, equipped with a k-rational point e ∈ G and k-morphisms
µ : G × G → G,
i : G → G,
such that G(k) becomes a group with identity e, multiplication xy = µ(x, y) and inverse x−1 = i(x).
Write k[G] for the ring of regular functions on G then µ and i corresponds to a homomorphisms of k-algebras
∆ : k[G] → k[G] ⊗k k[G],
ι : k[G] → k[G],
(called comultiplication and coninverse, respectively) and e is defined by a k-homomorphism
(the counit)
: k[G] → k.
One expresses the group axioms as properties of the morphisms ∆, ι, (for example, associativity is (∆ ⊗ 1) ◦ ∆ = (1 ⊗ ∆) ◦ ∆ , and so on.) In return, given an affine variety G and
given morphisms ∆, ι, with these properties, one gets an algebraic group structure on G.
A closed subgroup H of G defined over k is a closed subset H of G defined over k,
thus an affine variety, which is also closed under the group law. It then follows that ∆, ι
and induce k-algebra homomorphisms k[H] → k[H] ⊗k k[H],
ι : k[H] → k[H] and :
k[H] → k. A homomorphism of algebraic groups over k is defined in the obvious manner.
Let G, G1 be linear algebraic groups over k. We say that G1 is a form of G if G and G1
are isomorphic over k.
1An
affine variety over k is defined to be a closed subset of the affine space Ank , defined by an ideal
of k[x1 , . . . , xn ] that has a generating set of polynomials in k[x1 , . . . , xn ].
ALGEBRAIC GROUPS: PART I
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3. The main examples
We would want to carry with us throughout the course the examples of the classical
groups. Besides illustrating the theory, they are also the most useful and frequently occurring algebraic groups.
3.1. Additive groups. The group Ga - the additive group - is simply k with addition,
µ(x, y) = x + y;
more precisely, it is A1 - the affine line over k. The coordinate ring is k[x] and the
homomorphism ∆ is given by
x 7→ x ⊗ 1 + 1 ⊗ y ∈ k[x] ⊗ k[y].
(Under the isomorphism k[x] ⊗k k[y] ∼
= k[x, y], we have x ⊗ 1 + 1 ⊗ y 7→ x + y.) The
2
group Gna is the affine n space An over k, and often we shall identify Gna with Mn2 the n × n matrices.
3.2. Tori. The group k
×
of the non-zero elements of k under multiplication is a linear
algebraic group defined over k. It is isomorphic to the affine variety V defined by {(x, y) ∈
A2 : xy = 1}:
×
k → V,
t 7→ (t, 1/t).
The group law is given by
(x1 , y1 )(x2 , y2 ) 7→ (x1 x2 , y1 y2 ),
and µ is the homomorphism determined by
x 7→ x ⊗ x,
y 7→ y ⊗ y.
(These are viewed as elements of R ⊗k R, where R = k[x, y]/(xy − 1).) Note that it is
well-defined: 1 = xy 7→ (x ⊗ x)(y ⊗ y) = xy ⊗ xy = 1 ⊗ 1 = 1.
We denote this algebraic group by Gm , and call it the multiplicative group; we also
denote it by GL1 . If we need to emphasize the field of definition, we shall write Gm,k
and GL1/k . A torus over k is an algebraic group T over k, which is a form of Gnm,k , for
some n.
Let K be a number field. We consider K ∗ as an algebraic group over Q as follows:
choose a basis α1 , . . . , αn for K as a vector space over Q. Then every element in K can be
written as x1 α1 + · · · + xn αn . The multiplication then has the form
X
X
X
(
xi αi )(
y i αi ) = (
fi αi ),
ALGEBRAIC GROUPS: PART I
5
where the fi are bihomogenous polynomials, with rational coefficients, in the set of variP
ables {xi } and {yi }. The condition that
xi αi has an inverse can be phrased in the
form g(x1 , . . . , xn ) 6= 0 for some rational polynomial g (exercise!) and so we get an algebraic group over Q, which is the open subset of An defined as g(x1 , . . . , xn ) 6= 0 (this set can
be realized as the affine set in An+1 defined by {(x1 , . . . , xn , y) : g(x1 , . . . , xn )y − 1 = 0}).
The map ∆ is nothing else then xi 7→ fi . We denote this algebraic group ResK/Q Gm ; its
Q-points are ResK/Q Gm (Q) = K ∗ .
√
√
Here is a simple example. Let K = Q( 5), with the basis α1 = 1, α2 = 5. Then,
√
√
√
(x1 + x2 5)(y1 + y2 5) = x1 y1 + 5x2 y2 + (x1 y2 + x2 y1 ) 5.
√
We can write the inverse of x1 +x2 5 as
√
x1 −x2 5
.
2
x1 −5x22
The polynomial g(x1 , x2 ) is thus x21 −5x22
and we get the algebraic group
G = {(x1 , x2 ) : x21 − 5x22 6= 0},
with comultiplication
x1 7→ x1 ⊗ y1 + 5x2 ⊗ y2 ,
x2 7→ x1 ⊗ y2 + x2 ⊗ y1 ,
coinverse
x1 7→
x21
x1
,
− 5x22
x2 7→
x21
−x2
,
− 5x22
and counit
x1 7→ 1,
x2 7→ 0.
√
This is a torus, which is a form of G2m,Q . Indeed, already over the field Q( 5), the map
√
√
(x1 , x2 ) 7→ (x1 + x2 5, x1 − x2 5),
is an isomorphism to G2m .
Proposition 3.2.1. We have End(Gnm ) = Mn (Z).
Proof. Because Hom(X × Y, Z) = Hom(X, Z) × Hom(Y, Z) and Hom(X, Y × Z) =
Hom(X, Y ) × Hom(X, Z), it is enough to prove the proposition for n = 1. Given an
integer n, the function
x 7→ xn ,
ALGEBRAIC GROUPS: PART I
6
is a homomorphism of Gm and we get Z ,→ End(Gm ). Let now f : Gm → Gm be a
homomorphism of algebraic groups. This means that the following diagram commutes:
Gm × Gm
f ×f
Gm × Gm
µ
µ
/
Gm
/
f
Gm
Then, we must have
∆ ◦ f ∗ = (f ∗ ⊗ f ∗ ) ◦ ∆.
Abuse notation and write f (x) for the polynomial f ∗ (x), where f ∗ : k[x, x−1 ] → k[x, x−1 ]
is the k-algebra homomorphism corresponding to f . It follows then from this relation that
f (x ⊗ y) = f (x) ⊗ f (y),
P
n
in k[x, x−1 ] ⊗k k[y, y −1 ]. Write f (x) = N
n=−N an x . We then get that
N
X
n=−N
n
n
an x ⊗ y = (
N
X
n
an x ) ⊗ (
n=−N
N
X
an y n ) =
X
an am x n ⊗ y m .
n=−N
Let n0 be such that an0 6= 0. Then, it follows that, for n 6= n0 , an = 0 (since an an0 xn ⊗ y n0
must be zero). Further, since an0 xn0 ⊗ y n0 = a2n0 xn0 ⊗ y n0 , we must have an0 = 1. If follows
that f is the homomorphism x 7→ xn0 .
For a general linear group G, elements f ∈ k[G] such that ∆(f ) = f ⊗ f are called
“group-like” elements. They correspond to homomorphisms G → Gm .
3.3. The general linear group GLn . For a field k, the group GLn over k is the affine
2
variety in An - thought of n × n matrices (xij )- which is the complement of the closed
subvariety det(xij ) = 0. It is an affine variety, as it is isomorphic to the affine variety in
2 +1
An
with coordinates xij and y, defined by the equation det(xij ) · y − 1 = 0. The group
law is, of course, matrix multiplication. It is called the general linear group. The map
GLn → Gm ,
(xij ) 7→ det(xij ),
is a group homomorphism.
Here are some closed subgroups of GLn .
(1) The torus T = {diag(x1 , . . . , xn ) : x1 x2 · · · xn 6= 0}.
(2) The so-called Borel subgroup of upper-triangular matrices
B = {(xij ) ∈ GLn : xij = 0, i > j}.
ALGEBRAIC GROUPS: PART I
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(3) The group of unipotent matrices
U = {(xij ) ∈ B : xii = 1, ∀i}.
Note that
B = T U = UT ∼
= T × U.
Let V be an n-dimensional vector space over k. A flag F of type (d1 , . . . dt ), where 0 ≤
d1 < d2 < · · · < dt ≤ n are integers is a series of subspaces {Vi } of V such that dim(Vi ) = di
and V1 ⊂ V2 ⊂ · · · ⊂ Vd . A maximal flag is a flag of type (0, 1, 2, . . . , n). The collection
of flags of type (d) is an algebraic variety of dimension d(n − d), called the Grassmann
variety G (n, d). The collection of flags of type (d1 , . . . dt ) is a closed subvariety F(d1 ,...,dt )
of G (n, d1 ) × G (n, d2 ) × · · · × G (n, dt ).
We say that an algebraic group G acts on a variety V if we are given a morphism
G × V → V,
(g, v) 7→ gv,
such that
g1 (g2 v) = (g1 g2 )v,
ev = v,
for all v ∈ V, g1 , g2 ∈ G. Let v ∈ V and Stab(v) = {g ∈ G : gv = v} be its stabilizer. It
is a closed subgroup of G.
Let V = k n . The group GLn acts transitively on F(d1 ,...,dt ) . Pick a flag F of type F(d1 ,...,dt ) .
Then Stab(F ) is a closed subgroup of GLn , called a parabolic subgroup of type F(d1 ,...,dt ) .
Choosing a basis {e1 , . . . , en } for V such that the first d1 vectors span V1 , the first d2 span
V2 and so on, the parabolic subgroup consists of matrices of the form


A1 ∗ ∗ . . .
∗

A2 ∗ . . .
∗ 




A
.
.
.
∗
3

,


..


.
At+1
where A1 is a square matrix of size d1 , A2 is a square matrix of size d2 − d1 , A3 is a square
matrix of size d3 − d2 and so on, and At+1 is of size n − dt .
For the maximal flag F = {Span(e1 ), Span(e1 , e2 ), . . . , Span(e1 , e2 , . . . , en )} we get back
the Borel subgroup B.
ALGEBRAIC GROUPS: PART I
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Exercise 1. Calculate the dimension of a parabolic subgroup and, using that, calculate
the dimension of the variety F(d1 ,...,dt ) .
3.3.1. The unitary groups Up,q . Consider the hermitian form on Cn given by
h(x1 , . . . , xn ), (y1 , . . . , yn )i =
p
X
i=1
xi ȳi − (
n
X
xi ȳi ),
i=p+1
where 1 ≤ p ≤ n (and q = n − p). Let U (p, q) be the matrices of Mn (C) that preserve this
form. Seperating real and imaginary parts, we get a collection of algebraic equations with
A B
rational coefficients that define U (p, q). Indeed, writing a matrix as
with A of
C D
size p × p, B of size p × q, etc., and then A = A1 + iA2 , etc., we find the relations
t
A1 + it A2 t C1 + it C2
Ip 0
A1 − iA2 B1 − iB2
Ip 0
=
.
t
B1 + it B2 t D1 + it D2
0 −Iq
C1 − iC2 D1 − iD2
0 −Iq
Multiplying through and separating real and imaginary parts, we find a collection of quadratic equations with integer coefficients. Thus, the group U (p, q) is defined over Q and
its real points, U (p, q)(R) (denoted often just by U (p, q)) are the matrices of Mn (C) that
P
P
preserve the form pi=1 xi ȳi − ( ni=p+1 xi ȳi ).
Exercise 1. Prove that over the complex numbers the group U (p, q) is isomorphic to the
unitary group U (n).
The group U (n), in turn, is isomorphic over the complex numbers to the group GLn .
Indeed, the elements of U (n)(C) are pairs of complex matrices (A, B) such that t AA +
t
BB = In and t AB is symmetric (this implies (t A + it B)(A − iB) = In ).The group law is
given by (A, B)(C, D) = (AC − BD, AD + BC). Define a map
U (n)(C) → GLn (C),
(A, B) 7→ A + iB.
This is a well-defined group homomorphism. It is injective, because, letting M = A + iB,
we have A − iB = t M
−1
and so
1
−1
A = (M + t M ),
2
B=
1
−1
(M − t M ).
2i
This map is also surjective: given any M ∈ GLn (C) define A, B by these formulas. One
then checks that t AA + t BB = In and t AB is symmetric. Thus, the unitary groups U (p, q)
ALGEBRAIC GROUPS: PART I
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are forms of GLn over Q. The calculation above shows that in fact U (p, q) is isomorphic
to GLn over Q(i).
3.4. The orthogonal group Oq . In this section we assume that char(k) 6= 2.
3.4.1. Quadratic forms. Let V be a finite-dimensional vector space over k of dimension n.
A bilinear form on V is a function
B : V × V → k,
such that B(x, y) is a linear map in each variable separately. It is called symmetric if in
addition B(x, y) = B(y, x). We then let
q:V →k
q(x) = B(x, x),
be the associated quadratic form.2 If
q:V →k
is a function satisfying q(λx) = λ2 q(x), such that the function
B : V × V → k,
1
B(x, y) = (q(x + y) − q(x) − q(y)),
2
is a symmetric bilinear form, then we call q a quadratic form. Note that q(x) = B(x, x)
then.
Given a bilinear form B we define the orthogonal group OB = Oq :
OB = {A ∈ GL(V ) : B(Ax, Ay) = B(x, y), ∀x, y ∈ V }
= {A ∈ GL(V ) : q(Ax) = q(x), ∀x ∈ V }.
If one introduces coordinates on V then we can can write
B(x, y) = t [x]M [y],
where [x] are the coordinates of x and M is a symmetric n × n matrix. Then,
OB = {A ∈ GLn (k) : t AM A = M }.
We see that OB is a closed subgroup of GLn , defined by quadratic equations. We let SOB
be OB ∩ SL(V ) (and in coordinates: OB ∩ SLn (k)). We note that B is non-degenerate
(namely, if B(x, y) = 0 for a fixed x and all y, then x = 0) iff det(M ) 6= 0.
The classical orthogonal group, denoted simply O(n), is a special case of this construction, when the matrix M is In (and so the quadratic form is, in coordinates, x21 + x22 +
2In
many texts the normalization is q(x) = 12 B(x, x), and that can matter a lot sometimes.
ALGEBRAIC GROUPS: PART I
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· · · + x2n . Note that if B is non-degenerate then OB is a form of O(n), and SOB is a form
of SO(n)).
Let V, W be vector spaces with bilinear forms BV , BW . An isometry T : V → W is an
isomorphism of vector spaces such that BW (T x, T y) = BV (x, y). For example, OB is the
group of isometries from V to itself.
Theorem 3.4.1 (Witt’s extension theorem). Let V be a vector space equipped with a
bilinear form B. Any isometry T : U → U 0 , between subspaces U, U 0 of V , can be extended
to an isometry of V . (For the proof see, e.g., Jacobson, Part I, p. 369.)
3.4.2. Clifford algebras. Let V be a vector space over k (still char(k) 6= 2) and q a quadratic
form on k. There is a pair (Cliff(V, q), p) consisting of a k-algebra Cliff(V, q) and a k-linear
map
p : V → Cliff(V, q),
with the following properties:
• Cliff(V, q) is generated as k-algebra by p(V ) and 1.
• p(v)2 = q(v) · 1.
• (Cliff(V, q), p) has the following universal property. Given any other k-algebra A
with a k-linear map p1 : V → A such that p1 (v)2 = q(v) · 1A , there is a unique map
f : Cliff(V, q) → A,
of k-algebras such that f ◦ p = p1 .
Clearly these properties determine (Cliff(V, q), p) up to a unique isomorphism. One
further proves the following properties:
• The map p is injective. (This is clear if every non-zero vector v is anisotropic: q(v) 6=
0, but the vector space may have isotropic vectors even when B is non-degenerate.)
We shall therefore write v for p(v) and think of V as a subset of Cliff(V, q), whenever
convenient.
• If e1 , . . . , en are a basis for V over k then the 2n elements ei1 ei2 · · · eit , where 1 ≤
i1 < i2 < · · · < it ≤ n, are a basis of Cliff(V, q) as a k-vector space (the empty
product is understood as 1).
⊗n
• One can construct Cliff(V, q) as T (V )/hx2 − q(x) : x ∈ V i, where T (V ) = ⊕∞
n=0 V
is the tensor algebra (V ⊗0 := k). The natural map V = V ⊗1 → T (V ) induces the
ALGEBRAIC GROUPS: PART I
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map p : V → Cliff(V, q). Note that since the ideal hx2 − q(x) : x ∈ V i is graded,
the Clifford algebra has a Z/2Z-grading.
Example 3.4.2. Since 2B(x, y) = q(x + y) − q(x) − q(y) = (x + y)2 − x2 − y 2 = xy + yx,
we have the relation in Cliff(V, q)
∀x, y ∈ V
xy + yx = 2B(x, y),
In particular, if x, y are orthogonal, namely, if B(x, y) = 0 then xy = −yx.
Suppose that B is identically zero. Then Cliff(V, q) is the exterior algebra of V . This
is easy to see given the statement about a basis of Cliff(V, q) and the relation xy =
−yx holding for all x, y ∈ V , or from the assertion that the Clifford algebra is equal to
T (V )/hx2 : x ∈ V i.
Definition 3.4.3. The canonical automorphism of Cliff(V, q) is the automorphism
induced by the map
p1 : V → Cliff(V, q),
p1 (v) = −p(v).
This induced automorphism, denoted
a 7→ a0 ,
is uniquely determined by the property
x0 = −x,
∀x ∈ V
(where now we identify V with its image under p; alternately, it is the property p(v)0 =
−p(v)).
Note that Cliff(V, q)op , the opposite k-algebra, equal to Cliff(V, q) as a k-vector space, but
with multiplication a × b = ba, where on the right we have the multiplication in Cliff(V, q),
is indeed a k-algbera. The map p : V → Cliff(V, q)op satisfies the properties of a Clifford
algebra and so we get a map
Cliff(V, q) → Cliff(V, q)op ,
extending the identity map on V . Denote this map by a 7→ a∗ . It is a k-linear involution
on Cliff(V, q); it is uniquely determined by the property (ab)∗ = b∗ a∗ (the involution
property) and
x∗ = x,
∀x ∈ V.
The even Clifford algebra Cliff(V, q)+ is defined by
Cliff(V, q)+ = {a ∈ Cliff(V, q) : a0 = a}.
ALGEBRAIC GROUPS: PART I
12
It is indeed a k-algebra. We also define
Cliff(V, q)− = {a ∈ Cliff(V, q) : a0 = −a}.
Then Cliff(V, q)− is a k-vector space, which is a Cliff(V, q)+ -module and
Cliff(V, q) = Cliff(V, q)+ ⊕ Cliff(V, q)− .
Given the description of a basis for Cliff(V, q), it is easy to see that Cliff(V, q)± , are
each 2n−1 -dimensional; a basis of Cliff(V, q)+ (resp. Cliff(V, q)− ) is given by the elements ei1 ei2 · · · ein , for n even (resp., n odd).
3.4.3. The Clifford and Spin groups. From this point on assume that B is non-degenerate.
That is, if x ∈ V and B(x, y) = 0 for all y then x = 0. Note that this doesn’t preclude the
existence of x 6= 0 such that q(x) = 0.
Given a quadratic space (V, q), put
G(V, q) = {a ∈ Cliff(V, q)× : a−1 V a = V }.
Further, let
G+ (V, q) = G(V, q) ∩ Cliff(V, q)+ .
We call G(V, q) the Clifford group and G+ (V, q) the even Clifford group.
Exercise 2. Prove that G(V, q) and G(V, q)+ are algebraic groups over k.
For a ∈ G(V, q) define τ (a) ∈ GL(V ) by
xτ (a) = a−1 xa,
(x ∈ V ).
Linear transformation will act here on the right so that we have τ (ab) = τ (a)τ (b) and we
get a homomorphism
G(V, q) → GL(V ).
In fact, the image of this homomorphism is in the orthogonal group Oq :
q(xτ (a)) = (xτ (a))2
= (a−1 xa)2
= a−1 q(x)a
= q(x).
Therefore, we have a homomorphism of algebraic groups:
G(V, q) → Oq .
ALGEBRAIC GROUPS: PART I
13
Suppose that x ∈ V is invertible in Cliff(V, q), then also x2 = q(x) is invertible and
so q(x) 6= 0. Conversely, if q(x) 6= 0 then x is invertible in Cliff(V, q) and the inverse
is q(x)−1 · x. Thus, the elements of V that are invertible in Cliff(V, q) are precisely the
anisotropic vectors - the vectors x with q(x) 6= 0. Such vectors v are contained in G(V, q)
as for any u ∈ V , v −1 uv = v −1 (2B(u, v) − vu) = 2B(u, v) · q(v)−1 · v − u ∈ V ; Therefore, a
product of an even number of such vectors is in G(V, q)+ . It is a theorem that G(V, q)+ consists of all the products of an even number of elements of V that are invertible in Cliff(V, q),
namely products of an even number of anisotropic vectors (see below).
Define a function
ν : G(V, q)+ → k ∗ ,
ν(a) = aa∗ .
If a = x1 x2 . . . x2r , xi ∈ V, q(xi ) 6= 0, then
ν(a) = q(x1 )q(x2 ) · · · q(x2r ).
Thus, ν is a group homomorphism. We define the spin group of (V, q) by
Spin(V, q) = {a ∈ G(V, q)+ : ν(a) = 1}.
It is an algebraic group and we have a homomorphism
τ : Spin(V, q) → OB .
One can show that the kernel of τ on G(V, q)+ is precisely k ∗ . Since k ∗ ∩Spin(V, q) = {±1},
there is an injection,
Spin(V, q)/{±1} ,→ Oq .
The truth is the following. The image of Spin(V, q) is SOq and so one obtains an exact
sequence:
(3.4.1)
1 → {±1} → Spin(V, q) → SOq → 1.
This is an exact sequence over k̄, and so is also an exact sequence of algebraic groups. In
order to understand why the sequence is exact we need to go deeper into the structure of
the orthogonal group and the spin group.
3.4.4. Reflections. Let x ∈ V with q(x) 6= 0. Let x⊥ = {v ∈ V : B(x, v) = 0}. We have
the orthogonal decomposition
V = k · x ⊕ x⊥ .
There is a unique element of Oq taking x to −x and acting as the identity on x⊥ ; we call
it a reflection in x⊥ . In fact, this linear transformation is −τ (x). Indeed, if v ∈ x⊥
ALGEBRAIC GROUPS: PART I
14
then xv = −vx and so −vτ (x) = −x−1 vx = −x−1 (−x)v = v and, of course −x−1 xx = −x.
We note that det(−τ (x)) = −1.
Every element of Oq is a product of reflections. This can be proved by an elementary
argument using induction on the dimension of q.3 Let us also admit that the intersection
of the center of Cliff(V, q) with Cliff(V, q)+ is k.4 If τ (x) = 1V then x commutes with any
element of V and so x is in the center of Cliff(V, q) and a fortiori in the center of Cliff(V, q)+ ;
that is, x ∈ k × . Now, let a ∈ G(V, q)+ . Then we may write τ (a) = τ (x1 ) · · · τ (xm )
for some x1 , . . . , xm in V . It follows that a = (cx1 ) · · · xm for some c ∈ k ∗ and we see
that every element of G(V, q)+ is a product of elements of V . Because a ∈ G(V, q)+ , it
follows directly from the definition of the canonical automorphism that m is even, m =
Q
2r. Since det(−τ (a)) =
det(−τ (yi )) = −12r = 1, we conclude that Spin(V, q) maps
into SOq .
Let us assume that k is algebraically closed (it suffices that it contains all square roots
of q(V )). Then, further, if ν(a) = 1, we may assume that a = y1 · · · y2r , yi ∈ V , and
that ν(yi ) = 1 for i > 1. It then follows that ν(y1 ) = 1 as well. It follows that every
element a of Spin(V, q) is a product of an even number of elements yi ∈ V with ν(yi ) =
q(yi ) = 1.
On the other hand, since every element in SOq is a product of an even number of
reflections, each of the form −τ (x), where wlog q(x) = 1, it follows that it is the image of
an element in Spin(V, q).
3.5. The symplectic group. Let V be an even dimensional vector space over a field k
of characteristic different from 2. A symplectic form on V is an alternating perfect pairing
h·, ·i : V × V → k.
There’s always a basis {x1 , . . . , xn , y1 , . . . , yn } of V such that in this basis hyi , yj i =
hxi , xj i = 0 and hxi , yj i = δij . (Thus, any two such pairings are equivalent under a
suitable change of basis.) In this basis the form is given by a 2n × 2n matrix of the form
0 In
J :=
.
−In 0
3For the proof, see Jacobson, Part I, p. 371, or Shimura. In fact, every element of the orthogonal group
of q is a product of atmost n reflections. The proof is in Jacobson, Part I, p. 372; this statement is called
the Cartan-Dieudonné theorem.
4Choose an orthonormal basis {e , . . . , e } for V and let δ = e e · · · e . Then:
1
n
1 2
n
(1) If n > 0 is even then the center of Cliff(V, q) is k and the center of Cliff(V, q)+ is k + kδ.
(2) If n is odd then the center of Cliff(V, q) is k + kz and the center of Cliff(V, q)+ is k.
(See Shimura, Theorem 2.8)
ALGEBRAIC GROUPS: PART I
15
Some prefer to reorder the basis elements such that the pairing is given by


0 ··· 0 1

0 · · · 1 0




.
.


.




1
0

.
 0 · · · 0 −1



 0 · · · −1 0





.
.


.
−1
0
The symplectic group Sp(2n, k) are the linear transformations T of V preserving this
pairing. In terms of matrices, these are the matrices M such that
t
M JM = J.
There is an injective homomorphism
GLn → Sp2n ,
A 7→
A
t
A−1
.
B ) (n × n
The image is a closed subgroup of Sp2n , consisting of all symplectic matrices ( CA D
matrices), such that B = C = 0. There is also an injective homomorphism
I B
n(n+1)/2
Ga
→ Sp2n ,
B 7→
,
I
n(n+1)/2
where Ga
is interpreted as symmetric matrices of size n. The symplectic group is
n(n+1)/2
0 I
generated by the the image of Ga
, the image of GLn and the matrix −I
0 .
Let u ∈ V be a non-zero vector and c ∈ k. Then the function
τu,c v 7→ v + chv, ui · u
is a symplectic automorphism of V , called a transvection. One can prove (Jacobson,
Part I, p. 392) that the symplectic group is generated by transvections. Note that if k is
algebraically closed, or just closed under taking square roots, we can simply take c = 1 by
√
the cost of replacing u by c · u. Note the formula
ητu,c η −1 = τηu,c ,
—– —–
η ∈ Sp2n .
ALGEBRAIC GROUPS: PART II
EYAL Z. GOREN, MCGILL UNIVERSITY
Contents
4. Constructible sets
5. Connected components
6. Subgroups
7. Group actions and linearity versus affine
7.1.
8. Jordan Decomposition
8.1. Recall: linear algebra
8.2. Jordan decomposition in linear algebraic groups
9. Commutative algebraic groups
9.1. Basic decomposition
9.2. Diagonalizable groups and tori
9.2.1. Galois action
9.3. Action of tori on affine varieties
9.4. Unipotent groups
Date: Winter 2011.
16
17
18
20
22
24
27
27
30
35
35
35
39
42
44
ALGEBRAIC GROUPS: PART II
17
4. Constructible sets
Let X be a quasi-projective variety, equipped with its Zariski topology. Call a subset C
of X locally closed if C = V ∩ Z, where V is open and Z is closed. We call a subset of X
constructible if it is a finite disjoint union of locally closed sets.
Example 4.0.1. The set T = (A2 − {x = 0}) ∪ {(0, 0)}, being dense in A2 is not locally
closed (Z would have to contain A2 − {x = 0}), hence would be equal to A2 and T is
not open). But T is a constructible set, being the disjoint union of two locally closed
sets A2 − {x = 0} (open) with {(0, 0)} (closed).
Lemma 4.0.2. The following properties hold:
(1) An open set is constructible.
(2) A finite intersection of constructible sets is constructible.
(3) The complement of a constructible set is constructible.
Proof. If C is open then C = C ∩ X is the intersection of an open set with the closed
set X. Similarly if C is closed.
`
`
To show part (2), suppose that Ci and Di are constructible, where each Ci , Dj is
`
`
`
locally closed. Since Ci ∩ Dj = Ci ∩ Dj , it is enough to prove that the intersection
of two locally closed subsets C = V1 ∩ Z1 , D = V2 ∩ Z2 is constructible. In fact, C ∩ D =
(V1 ∩ V2 ) ∩ (Z1 ∩ Z2 ) is even locally closed.
`
For part (3), the formula ( Ci )c = ∩(Ci )c and part (2), show that it is enough to
prove that a complement of a locally closed set is constructible. Indeed, if C = V ∩ Z
`
then C c = V c ∪ Z c = Z c (V c − Z c ), a disjoint union of an open set with a closed set,
both constructible as we have proven above.
Corollary 4.0.3. A finite union of constructible sets is constructible. Thus, one may also
define a constructible set as a union (not necessarily disjoint) of locally closed sets. The
difference of two constructible sets is constructible.
Proof. If Ci constructible, to show ∪Ci is constructible, it is enough to show that (∪Ci )c =
∩Cic is constructible. But that follows (2) and (3) of the Lemma. The proof for the
difference is similar.
Corollary 4.0.4. The collection of constructible sets is the smallest collection F of subsets
of X containing all open sets and closed under finite intersections and complements.
ALGEBRAIC GROUPS: PART II
18
Proof. The lemma tells us that F is also closed under unions and is contained in the
collection of constructible sets. To show the converse, it is enough to show that every
locally closed set belongs to F . This is true because F contains the open sets, hence the
closed sets and hence their intersections.
Exercise 3. Every constructible set contains on open dense set of its closure.
The main relevance of constructible sets to algebraic geometry is the following important
theorem.
Theorem 4.0.5. Let ϕ : X → Y be a morphism of varieties. Then ϕ maps constructible
sets to constructible sets; in particular ϕ(X) is constructible and contains a set open and
dense in its closure.
Corollary 4.0.6. If ϕ is dominant, i.e., if the closure of ϕ(X) is Y , then the image of ϕ
contains and open dense set of Y .
For the proof see Humphreys, p. 33, or, in more generality, Hartshorne, Algebraic Geometry (GTM 52), exercise 3.19 on page 94.
Exercise 4. Find a morphism A2 → A2 whose image is the set T of Example 4.0.1.
5. Connected components
Let G be an algebraic group over an algebraically closed field k. There is no need to
assume in this section that G is linear.
Proposition 5.0.7. There is a unique irreducible component G0 of G that contains the
identity element e. It is a closed subgroup of finite index. G0 is also the unique connected
component of G that contains e and is contained in any closed subgroup of G of finite
index.
Proof. Let X, Y be irreducible components containing e, then XY = µ(X × Y ), XY , X −1
and tXt−1 , for any t ∈ G, are irreducible and contain e. Since an irreducible component is,
by definition, a maximal irreducible closed set, it follows that X = XY = XY , Y = XY
ALGEBRAIC GROUPS: PART II
19
and, so, X = Y . Since X −1 is also an irreducible component containing e, X = X −1 ,
and since tXt−1 is an irreducible component containing e, X = tXt−1 . Putting everything
together, we conclude: (i) there is a unique irreducible component through e; (ii) it is
closed; (iii) it is a subgroup; (iv) it is normal.
An irreducible component is connected. On the other hand, we have G =
`
x
xG0 , the
union being over coset representatives, each being an irreducible component. Thus, there
must be finitely many of them, each is also open, and the union is a disjoint union of
topological spaces. It follows that G0 is a connected component of G, and in fact, the
connected components equal the irreducible components.
Finally, given a subgroup H of G of finite index, we find that H 0 is of finite index in H
and so in G, and is connected. Thus, H 0 is contained in G0 and has finite index in it.
Since G0 is connected, we must have H 0 = G0 .
Example 5.0.8. The group Ga and Gm are connected. Hence, Gna and tori are connected.
The unipotent group of GLn is connected, being isomorphic to Am , for m = n(n − 1)/2,
and so the Borel subgroup is connected as well. The group GLn is connected: it is the
2
closed subset of An × A1 defined by y det(xij ) − 1, which is an irreducible polynomial. One
can also argue that GLn is irreducible, being an open non-empty subset of the irreducible
2
space An . It follows that the unitary groups U (p, q) are connected.
Assume that B is a non-degenerate quadratic form over a field of characteristic different
from 2. Let q be the associated quadratic form. The orthgonal group Oq is not connected,
as it has a surjective homomorphism det : Oq → {±1}.
The spin groups Spin(V, q) are connected and so it follows that the groups SOq are
connected (and that Oq has two connected components). We give the argument under the
assumption that q is not a square. Let Z be the closed subset of isotropic vectors in (V, q).
Its complement U is connected, being an open dense set. In fact, it’s an irreducible affine
variety. Then, also the variety U 0 = {(u, t) : t2 − q(u) = 0} in V × A1 is an irreducible
affine variety. Further, the closed subvariety W = {v ∈ V : q(v) = 1} is also irreducible
because we have a surjective morphism U 0 → W, (v, t) 7→ v/t.
For every r there is a map
W 2r → Spin(V, q),
(w1 , . . . , w2r ) 7→ w1 w2 · · · w2r .
ALGEBRAIC GROUPS: PART II
20
Let the image of these maps be S 2r . Then S 2r is irreducible, S 2r ⊆ S 2r+2 and every
element of Spin(V, q) belongs to S 2r for some r. It follows that for some r 0 we
have S 2r = Spin(V, q) proving Spin(V, q) is irreducible, equivalently, connected.1
6. Subgroups
There is no need to assume in this section that G is linear.
Lemma 6.0.9. Let U, V be two dense open subsets of an algebraic group G then G = U V .
Proof. Let g ∈ G and consider gV −1 . Since V −1 is open (x 7→ x−1 is a homeomorphism)
also gV −1 is open and so gV −1 ∩ U 6= ∅. Thus, for some v ∈ V, u ∈ U we have gv −1 = u,
which implies g ∈ U V .
Proposition 6.0.10. Let H be a subgroup of an algebraic group G. Let H̄ be its closure.
Then H̄ is a closed subgroup of G. If H is constructible then H̄ = H.
Proof. We first need to show that H̄ is a subgroup. We have H̄ −1 = H −1 = H̄ and so
it is closed under inverses. Since HH ⊂ H we have HH ⊂ H̄ and so for every h ∈ H
we have hH = hH ⊂ H̄. That is, H H̄ ⊂ H̄. Now, for every h ∈ H̄ we have Hh ⊂ H̄
and so H̄h ⊂ H̄ and it follows that H̄ · H̄ ⊂ H̄ which shows that H̄ is closed under
multiplication.
Consider H̄ as an algebraic group by itself. Since H is constructible it contains a subset U
which is open and dense in H̄. By the lemma H̄ = U U ⊂ H and so H = H̄.
Proposition 6.0.11. Let ϕ : G → J be a morphism of algebraic groups. Then the kernel
of ϕ and the image of ϕ are closed subgroups. Furthermore, ϕ(G0 ) = ϕ(G)0 .
1Had
we wanted to use the Cartan-Dieudonné theorem we could have been more precise here. It tells
us that every element of SOq is a product of q − p(q) reflections, where p(q) = 0 if the dimension is even
and otherwise p(q) = 1. Thus, up to ±1 we get every element of the spin group from S q−p(q) . Thus, every
element of the spin group is gotten from S q+2−p(q) .
ALGEBRAIC GROUPS: PART II
21
Proof. The kernel is closed being the preimage of a closed set. The image is a constructible
subgroup of J, hence closed. We note that ϕ(G0 ) is connected, closed, and of finite index
in ϕ(G), hence contains ϕ(G)0 . Maximality of the connected component implies that it is
equal to ϕ(G)0 .
Proposition 6.0.12. Let {Xi } be a family of irreducible varieties together with maps φi :
Xi → G. Let H be the minimal closed subgroup of G containing all the images Yi = φi (Xi ).
Assume that all Yi contain the identity element. Then:
(1) H is connected.
(2) We can choose finitely many among the Yi , say Yi1 , . . . , Yin and signs (i) such
(1)
that H = Yi1
(n)
· · · Yin .
Proof. We may assume that each Yi−1 occurs among the Yj . Given a multi-index a =
(a(1), . . . , a(n)) let
Ya = Ya(1) Ya(2) · · · Ya(n) .
We note that each Ya is constructible and irreducible, being the image of Xa and so are Ya .
We also have Ya Yb = Y(a,b) .
Since Yb Yc ⊂ Y(b,c) also Yb Yc ⊂ Y(b,c) . Let u ∈ Ya then uYc ⊂ Y(b,c) and so uYc = u · Yc ⊂
Y(b,c) . It follows that Yb Yc ⊂ Y(b,c) . Repeating the argument, we find
Yb · Yc ⊂ Y(b,c) .
Let us take Ya of maximal dimension. Then Ya ⊂ Ya Yb Y(a,b) . Maximality gives that Ya =
Y(a,b) and Yb ⊂ Ya . Applying this to b = a we conclude that Ya is closed under multiplication
and applying it to b such that Yb = Ya−1 we conclude that it is closed under inverse too and
contains every Yb . Thus, Ya is a closed subgroup and has properties (i) and (ii). Since Ya
is constructible, it contains a sent open and dense in Ya and so, by Lemma 6.0.9, Ya =
Ya Ya = Y(a,a) and so H = Y(a,a) has all the properties stated.
Since a closed connected subgroup is irreducible, the following corollary holds.
Corollary 6.0.13. If the Xi are a family of closed connected subgroups of G then the
subgroup H generated by them is closed and connected. There’s a choice of Gi1 , . . . , Gin
among them (perhaps with repetitions!) such that H = Gi1 · · · Gin .
ALGEBRAIC GROUPS: PART II
22
Exercise 5. If H and K are closed subgroups of G, one of which is connected then (H, K)
- the subgroup of G generated by all the commutators xyx−1 y −1 , x ∈ H, y ∈ K - is closed
and connected.
Exercise 6. Prove that the symplectic group is connected. You may want to use transvections for that.
7. Group actions and linearity versus affine
Recall that a group G acts on a variety V if we are given a morphism
a : G × V → V,
a(g, v) =: gv,
such that 1v = v, ∀v ∈ V and g(hv) = (gh)v, ∀g, h ∈ G, v ∈ V .
The orbit of v ∈ V is Gv and the isotropy group, or stabilizer, of v is
Gv = {g ∈ G : gv = v}.
It is a closed subgroup of G. If there is v such that Gv = V we say that G acts transitively
and that V is a homogeneous space under G.
Example 7.0.14.
(1) The following maps define group actions
G × G → G,
a(x, y) = xy.
This group action is transitive: there is one orbit. In fact, it is simply transitive: the
isotropy groups are trivial. Thus, this is an example of a principle homogenous
space for G.
Another action is:
G × G → G,
a(x, y) = xyx−1 .
Here the orbits Gy are conjugacy classes and the isotropy group of y (or its stabilizer) is its centralizer.
ALGEBRAIC GROUPS: PART II
23
(2) Let V be a vector space over k. A homomorphism
r : G → GL(V )
of algebraic groups over k is called a rational representation over k. It defines
an action
G × V → V,
a(g, v) = r(g)(v).
(3) The group GLn acts on the matrices Mn by
GLn × Mn → Mn ,
(X, Y ) = XY X −1 .
Assume that k is algebraically closed then the orbits corresponds to matrices in
standard Jordan form.
(4) The natural action of GLn on an n-dimensional vector space V induces an action
GLn × P(V ) → P(V ),
where P(V ) is the projective space associated to V . If V = k n then
P(V ) = {(x1 : . . . , xn ) : xi ∈ k, ∃i xi 6= 0},
where, as usual (x1 : . . . , xn ) = (y1 : . . . , yn ) if and only if there’s a λ ∈ k × such
that λxi = yi , ∀i. We can also say that P(V ) is the space of orbits for the action
Gm × V → V,
(λ, (x1 , . . . , xn )) 7→ (λx1 , . . . , λxn ).
This is a special case of the action of GLn on flag spaces we have already discussed.
Proposition 7.0.15. Let G act on V .
(1) An orbit Gv is open in its closure. It is thus a locally closed set of V .
(2) There exist closed orbits.
Proof. Gv is a constructible set, hence there is a set U such that U ⊂ Gv ⊂ Gv and U is
open in Gv. But then Gv = ∪g∈G gU is also open.
It follows that Bdv := Gv − Gv is closed and is a union of orbits of G. That is, if x ∈ Gv
then gx ∈ Gv. This is true because gx ∈ g · Gv = gGv = Gv. In a quasi-projective variety
every family of closed sets contains a minimal one. Thus, there is a minimal element in
the family {Bdv : v ∈ V }. Take that minimal element v(0). If Bdv(0) is not empty, it is
a union of orbits and so there is v(1) such that G · v(1) ⊂ Bdv(0) , but then so is G · v(1)
and so Bdv(1) $ Bdv(0) (because v(1) 6∈ Bdv(1) ). That is a contradiction. Thus, Bdv(0) is
empty, which means that G · v(0) is closed.
ALGEBRAIC GROUPS: PART II
24
Assume from now on that G is an affine algebraic group over k. We shall ultimately prove
that G is linear, namely that G is isomorphic to a closed subgroup of GLn over k. The
converse is clear. In order to do so, we shall analyze the action of G on spaces of functions
on V , and ultimately take V = G itself.
7.1. Assume that G and V are affine. Then, to give a morphism G × V → V corresponds
to giving a k-algebra map
a∗ : k[V ] → k[G] ⊗k k[V = k[G × V ].
The map a∗ has additional properties expressing the axioms of the group action, but, at
any rate, being the pull-back map on functions, we have
a∗ (f )(g, v) = f (gv).
Define now an action of G on functions:
s(g)(f )(x) = f (g −1 x).
This gives a linear representation
G → GL(k[V ]).
Proposition 7.1.1. Let U be a finite dimensional subspace of k[V ].
(1) There is a finite dimensional subspace W of k[V ] such that W contains U and W
is invariant under the action of G via s.
(2) U is stable under G if and only if a∗ U ⊆ k[G] ⊗k U .
Proof. It is enough to prove it when U is one dimensional, say U = kf . Suppose that
X
a∗ f =
ri ⊗ fi ∈ k[G] ⊗ k[V ].
P
Then, s(g)f (v) = a∗ (f )(g −1 , v) =
ri (g −1 ) · fi (v) and so the function s(g)f is equal
P
to
ri (g −1 ) · fi and we see that s(g)f ∈ Span({fi }i ). The subspace W spanned by all
the functions s(g)f is invariant under the action of G and on the other hand, is contained
in k[{fi }], which is finite dimensional.
ALGEBRAIC GROUPS: PART II
25
Let now U be a subspace such that a∗ (U ) ⊂ k[G] ⊗ U . Then, if f ∈ U we have a∗ f =
P
P
ri ⊗ fi ∈ k[G] ⊗ U and so s(g)f =
ri (g −1 )fi ∈ U and so U is invariant under G.
Conversely, suppose that U is invariant under G and let f ∈ U . Let {fi } be a basis of U
and complete it to a basis {fi } ∪ {gj } to k[V ]. For f ∈ U we may write
X
X
a∗ f =
ri ⊗ f i +
tj ⊗ gj .
Then s(g)f = a∗ f (g −1 , ·) =
P
ri (g −1 )fi +
P
tj (g −1 )gj is an element of U and it follows
that tj (g −1 ) = 0 for all g ∈ G and so that tj = 0. Thus, a∗ f ∈ k[G] ⊗ U .
Theorem 7.1.2. Let G be an affine algebraic group then G is linear, namely G is isomorphic to a closed subgroup of GLn for some n.
Proof. Here we choose to work with the action of G given by
ρ(g)(f )(x) = f (xg),
instead of the action considered above of s(g)(f )(x) = f (g −1 x). This is just for notational
convenience. It is clear that the same results we have proved above also hold for this
action.
G is finitely presented k-algebra. Thus,
k[G] = k[f1 , . . . , fn ],
for some fi . The vector space spanned by the fi is contained in a G-stable vector space,
and so we may assume that Spank hf1 , . . . , fn i is G-invariant. Thus,
(7.1.1)
ρ(g)fi =
n
X
mji (g)fj ,
∀g ∈ G,
j=1
where mji are in k[G]. The map
g 7→ φ(g) = (mij (g))1≤i,j≤n
is a linear representation of G into GLn - namely, it is an algebraic homomorphism. We
would like to prove that
G∼
= φ(G).
For that it is enough to show that φ is injective, the image is closed and that the coordinate
ring of the image is isomorphic to k[G].
ALGEBRAIC GROUPS: PART II
26
Note that since G → GLn is a homomorphism of algebraic groups, the image φ(G) is a
closed subgroup of G. The map φ∗ simply takes the coordinate Tij of a matrix and sends
it to mij . Since, by (7.1.1),
fi (·) =
X
mji (·)fj (e),
j
the map φ∗ is surjective. This implies that φ is injective, because given g 6= g 0 in G,
there is some function f of k[G] such that f (g) 6= f (g 0 ). If this f is of the form φ∗ F
then F (φ(g)) 6= F (φ(g 0 )) and so g 6= g 0 .
Now, by definition, Ker(φ∗ ) is the ideal whose radical defines the coordinate ring of the
closure of φ(G) (which is just φ(G)). Since we know k[G] = k[GLn ]/Ker(φ∗ ) it follows
that Ker(φ∗ ) is a radical ideal. Thus, the coordinate ring of k[φ(G)] is k[GLn ]/Ker(φ∗ ) =
k[G]. That means that φ is an isomorphism onto its image.
ALGEBRAIC GROUPS: PART II
27
8. Jordan Decomposition
8.1. Recall: linear algebra. Let k be an algebraically closed field and V a finite dimensional vector space over k. An operator a ∈ End(V ) is called:
• Semi-simple, or diagonalizable, if there is a basis of V consisting of eigenvectors.
• Nilpotent, if aN = 0 for some N > 0.
• Unipotent, if (a − 1)N = 0 for some N > 0.
The Jordan canonical form of a matrix implies that we can write any a as
a = as + an ,
where as is semi-simple, an is nilpotent and as an = an as . The key point of this section is
that such a factorization is highly stable under all kind of maps.
Proposition 8.1.1 (Jordan decomposition in End(V )). Let a be an operator on a finite
dimensional vector space V over an algebraically closed field k. Then, we can write
a = as + an ,
where as is semi-simple, an is nilpotent and as an = an as . Furthermore, such a decomposition is unique and a commutes with as , an .
Proof. Let χ(t) be the characteristic polynomial of a. We may write
Y
χ(t) =
(t − αi )ni ,
i
where the αi are the distinct eigenvalues of a. This decomposition induces a decomposition
of V :
V = ⊕Vi ,
Vi = {v ∈ V : (a − αi )ni = 0}.
Using the Chinese Remainder Theorem, we may choose a polynomial P (t) ∈ k[t] such that
P (t) ≡ 0
(mod t),
P (t) ≡ αi
(mod (t − αi )ni ), ∀i.
(Note that this works also if some αi = 0.) Let Q(t) = t − P (t). Consider the linear
operators P (a), Q(a). We have
a = P (a) + Q(a).
P (a) acts as a scalar on each Vi and so P (a) is semi-simple. Q(a) acts as a − αi on each Vi
and so is nilpotent. Furthermore, a commutes with P (a) and Q(a).
ALGEBRAIC GROUPS: PART II
28
Suppose we had another decomposition: a = bs + bn . Then, since bs and bn commute,
they also commute with a and so with P (a), Q(a). That is, bs commutes with as and bn
commutes with an . The following is easy to prove by linear algebra:
• The sum of two commuting semi-simple operators is semi-simple.
• The sum of two commuting nilpotent operators is nilpotent.
Thus, as − bs = bn − an is both semi-simple and nilpotent, hence must be zero.
Corollary 8.1.2. Let W ⊆ V be an a-invariant subspace. Then W is also invariant
under as and an , and so is V /W . Furthermore,
a|W = as |W + an |W ,
is the decomposition of a|W as a sum of a semi-simple operator and a nilpotent operator.
The same holds for V /W .
Proof. We have seen in the proof above that as = P (a), an = Q(a) and that shows that W
is stable under as , an . Note that we can still take P as the polynomial for producing (a|W )s
and similarly for Q and it follows that (a|W )s = as |W and similarly for the nilpotent part.
Similar arguments apply to V /W .
Corollary 8.1.3. Let a be an operator on V and b an operator on another finite dimensional vector space W . Let φ : V → W be a linear map such that
V
a
V
φ
φ
/
/
W
b
W
is a commutative diagram. Then also the following diagrams are commutative:
V
as
V
φ
φ
/
/
W
bs
W,
V
an
V
φ
φ
/
/
W
bn
W.
Proof. We can decompose φ as
V V / ker(φ) ,→ W,
and use the previous corollary.
ALGEBRAIC GROUPS: PART II
29
Corollary 8.1.4 (Jordan decomposition in GL(V )). Let a ∈ GL(V ) then there is a unique
decomposition:
a = as au ,
such that as is semi-simple, au is unipotent and as au = au as . This decomposition if functorial in the sense described in the previous corollaries.
Proof. We have a = as + an . Since a is invertible also as must be invertible. This follows
from the proof constructing as (it acts by a scalar αi on each Vi and, a being invertible, αi 6=
−1
0 for all i). And so, a = as (1 + a−1
s an ). Since as commutes with an , the operator 1 + as an
is unipotent.
Conversely, given a factorization a = as au = as (1 + (au − 1)) it follows that au − 1
is nilpotent and so is as (au − 1) and we get a decomposition a = as + as (au − 1) into a
semi-simple and nilpotent operator that commute. It is easy now to deduce uniqueness
and functoriality.
We now want to generalize the Jordan decomposition to infinite-dimensional vector
spaces under a finiteness assumption. Let V be a vector space over k and let a ∈ End(V ).
We call a locally-finite if V is a union of finite dimensional vector spaces stable under a. a
is then called semi-simple (locally nilpotent) if its restriction of each such subspace
is semi-simple (nilpotent). To avoid un-necessary complications, we shall assume that V
has countable dimension, or, equivalently, that there are a-invariant finite-dimensional
subspaces
V0 ⊂ V1 ⊂ V2 ⊂ . . . ,
V = ∪i Vi .
In this case, we can define as and an as the unique operators whose restrictions to each Vi
is the semi-simple and nilpotent part, respectively, of a restricted to Vi . We have then,
a = as + an ,
as an = an as .
Furthermore, as , an are unique with such properties and preserve each Vi . If a is invertible,
then we have a factorization
a = as au , as au = au as ,
where as is semi-simple and au is unipotent, and, again, this factorization is unique.
ALGEBRAIC GROUPS: PART II
30
8.2. Jordan decomposition in linear algebraic groups. Let G be a linear algebraic
group. The representation
ρ : G → GL(k[G]),
(ρ(g)f )(x) = f (xg),
maps elements of G to locally finite endomorphisms. Thus, for every g ∈ G we have a
decomposition:
ρ(g) = ρ(g)s ρ(g)u .
The idea is to use this to define the semisimple and unipotent part of g itself and moreover
prove that this decomposition is canonical.
Theorem 8.2.1.
(1) There are unique elements gs , gu in G such that gs is semisim-
ple, gu is unipotent, g = gs gu = gu gs and. moreover,
ρ(g)s = ρ(gs ),
ρ(g)u = ρ(gu ).
(2) If φ : G → H is a homomorphism of algebraic groups then φ(gs ) = φ(g)s and φ(gu ) =
φ(g)u .
(3) For G = GLn the decomposition defined here agrees with the Jordan decomposition
defined before.
Proof. Multiplication on k[G] is a k-algebra homomorphism
m : k[G] ⊗k k[G] → k[G].
Since ρ(g) is a k-algebra homomorphism of k[G] it commutes with multiplication in the
sense that
m ◦ (ρ(g) ⊗ ρ(g)) = ρ(g) ◦ m.
If we apply the functoriality of Jordan decomposition to the vector spaces k[G] ⊗k k[G]
and k[G], relative to the map m, we find that
(8.2.1)
m ◦ (ρ(g)s ⊗ ρ(g)s ) = ρ(g)s ◦ m,
m ◦ (ρ(g)u ⊗ ρ(g)u ) = ρ(g)u ◦ m.
We are using here the facts, left as exercises that for two linear maps a, b we have as ⊗ bs
is semi-simple au ⊗ bu is unipotent, which implies (a ⊗ b)s = as ⊗ bs and (a ⊗ b)u = au ⊗ bu .
From now on we just prove the statements for the semi-simple part; the proof for the
unipotent part is the same. The identity (8.2.1) means that ρ(g)s is not just a linear map;
ALGEBRAIC GROUPS: PART II
31
it commutes with multiplication. That is, ρ(g)s is a k-algebra homomorphism of k[G]. We
can thereferore define a k-algebra homomorphism
k[G] → k,
(8.2.2)
f 7→ (ρ(g)s f )(e).
Such a homomorphism is nothing else but a k-point of G, which we call gs . We have an
equality of homomorphisms
f 7→ (ρ(g)s f )(e)
is the homomorphism f 7→ f (gs ).
But, on the other hand,
f (gs ) = (ρ(gs )f )(e).
Thus, for any function f , we have
(ρ(g)s f )(e) = (ρ(gs )f )(e).
Given a function f we can apply this relation to the functions λ(h)f (x) := f (h−1 x), h ∈ G,
to get:
ρ(g)s (λ(h)f )(e) = ρ(gs )(λ(h)f )(e).
Now, since ρ(g) ◦ λ(h) = λ(h) ◦ ρ(g), if we think of λ(h) as a k-linear map on the vector
space k[G], we may conclude by the functoriality of Jordan decomposition that
ρ(g)s ◦ λ(h) = λ(h) ◦ ρ(g)s .
Thus, we have the relation
λ(h)(ρ(g)s f )(e) = ρ(g)s (λ(h)f )(e) = ρ(gs )(λ(h)f )(e).
The left hand side is ρ(g)s f (h−1 ), while the right hand side is f (h−1 gs ) = (ρ(gs )f )(h−1 ).
Thus,
ρ(g)s f = ρ(gs )f,
∀f ∈ k[G].
This means that ρ(g)s = ρ(gs ). The same holds for unipotent parts: ρ(g)u = ρ(gu ). Since ρ
is faithful, that finishes the proof of (1).
We now prove (2). We may factor φ as
G φ(G) ,→ H.
(Note that φ(G) is a closed subgroup of H and so all the groups here are affine.) In the first
case, φ∗ identifies k[φ(G)] with a subspace T of k[G]. The operator ρ(g)|T is nothing else
ALGEBRAIC GROUPS: PART II
32
then ρ(φ(g)) (via φ∗ ). Functoriality of Jordan decomposition for restriction gives ρ(g)s =
ρ(gs ) is equal to ρ(φ(g))s = ρ(φ(g)s ) on T and so, since ρ is faithful, φ(gs ) = φ(g)s .
For φ(G) ,→ H we may view k[G] as k[H]/I where I is the ideal defining the closed
subset φ(G). The ideal I is stable under the action of ρ(φ(G)). Now the result will follow
from functoriality of Jordan decompositions for quotient spaces.
It remains to prove (3). G acts naturally on the vector space V = k n . Fix a non-zero
function f ∈ V ∗ - the dual vector space. For every v ∈ V define a function fv on G by
fv (g) = f (gv).
This gives us an injective linear map
V → k[G],
v 7→ fv .
Now, fgv is the function whose value at h is f (hgv) = (ρ(g)fv )(h). Thus, the map V → k[G]
is equivariant for the natural action of G on V and on k[G] via ρ. Functoriality now gives
us ρ(g)s = ρ(gs ).
Corollary 8.2.2. An element g ∈ G is semi-simple (resp. unipotent) if and only if for
any isomorphism φ from G to a closed subgroup of some GLn , φ(g) is semi-simple (resp.
unipotent).
Proposition 8.2.3. The set of unipotent elements of G is a closed subset.
Proof. Choose an embedding G → GLn . The image is closed and the map takes unipotent
to unipotent. Thus, it is enough to prove the statement for G = GLn . But then the
unipotent elements are defined by the equation (a − 1)n = 0.
Theorem 8.2.4. Let G be a subgroup of GLn consisting of unipotent elements. Then G can
be conjugated to Un , where Un are the upper unipotent matrices. That is, for some x ∈ Gn
we have
xGx−1 ⊂ Un .
Proof. One argues by induction on n. The case n = 1 being clear. Assume n > 1. If V = k n
is a reducible representation of G then there is a proper G-stable subspace W . The action
of G on W and V /W is unipotent and so there are bases B for W and C 0 to V /W relative
to which the action is by upper unipotent matrices. Lift the elements of C 0 in any way
ALGEBRAIC GROUPS: PART II
33
to V obtaining a set C. Then B ∪ C is a basis for V in which the action of G is by upper
unipotent matrices.
If V is an irreducible representation of G then, by a theorem of Burnside (that is not at
all obvious; it follows from Jacobson’s density theorem. See Lang/Algebra, Chapter XVII)
the linear span of G is the whole of End(V ).
Consider the linear functional g 7→ Tr(g). Since g − 1 is nilpotent, Tr(g − 1) = 0 and
so this functional is constant on G, having value n = dim(V ). Let x, y ∈ G and write
x = 1 + N , where N is nilpotent. We have Tr(y) = Tr(xy), because x, xy ∈ G and
Tr(xy) = Tr((1 + N )y) = Tr(y) + Tr(ny). Thus, Tr(N y) = 0 for all y ∈ G. But then this
holds for all y in the k-linear span of G, which is Mn (k). If we choose y to vary over the
elementary matrices (zero everywhere except for 1 in a single place) and calculate Tr(N y)
we find that N = 0 and so G = {1}. This is a contradiction to irreducibility.
The group Un is nilpotent. To see that let U (a) be the matrices M with zero on all the
diagonals mi+x,j+x with 0 ≤ j − i ≤ a − 1. One checks that U (a)U (b) ⊆ U (a + b) and that
1+U (a) is a group. One the proves that if G(0) = Un , G(1) = [Un , Un ], . . . G(`+1) = [Un , G(`) ]
then G(`) ⊆ 1 + U (` + 1). One proves by induction, using that if u is nilpotent then
(1 + u)−1 = 1 − u + u2 − u3 + . . . (this is a finite sum!), and by expanding the expression
[1 + u, 1 + v] = (1 + u)(1 + v)(1 − u + u2 − . . . )(1 − v + v 2 − . . . ).
Corollary 8.2.5. A unipotent algebraic group is nilpotent, hence solvable (both notions in
the sense of group theory).
Proof. We can embed the group into GLn for some n and so into Un . Since Un is nilpotent
so is any subgroup of it. A nilpotent group is solvable.
Corollary 8.2.6. Let G be a unipotent algebraic group and ρ : G → GLn a rational representation. Then there is a non-zero fixed vector for this action.
Proof. This follows directly from the theorem.
Corollary 8.2.7. Let G be a unipotent algebraic group and V an affine variety on which G
acts. Then all the orbits of G in V are closed.
Proof. Let W be a G-orbit. We may assume without loss of generality that W is dense
in V . We want to show that W = V . We know that W is open in V . Let C = V − W and
ALGEBRAIC GROUPS: PART II
34
let I the ideal defining the closed set C. To show C is empty it is enough to show that I
contains a non-zero constant function. The group G acts on I and the representation is
locally finite as I ⊂ k[V ]. Thus, there is a non-zero fixed function f in I. This means
that f is constant on W , hence on V , hence a constant.
ALGEBRAIC GROUPS: PART II
35
9. Commutative algebraic groups
9.1. Basic decomposition.
Lemma 9.1.1. Let V be a finite dimensional vector space over an algebraically closed
field k. Let S ⊂ End(V ) be a set of commuting operators, then there is a basis of V
in which all the semi-simple elements of S are diagonal and all the elements of S are
upper-triangular.
Proposition 9.1.2. Let G be a commutative linear algebraic group over an algebraically
closed field k. The set of semi-simple elements of G, Gs , and the set of unipotent elements
of G, Gu , are both closed subgroups of G. Further,
G∼
= Gs × Gu .
Proof. We may assume that G is a closed subgroup of GL(V ) for some vector space V
over k of dimension n < ∞. A collection of commuting linear maps can be simultaneously
triangularized. Thus, we may assume that G ⊆ Bn , where Bn is the standard Borel
subgroup of GLn . It is easy to see that an element of Bn is semi-simple iff it belongs to the
torus Tn , and unipotent iff it belongs to the unipotent group Un . Thus, Gs = G ∩ Tn , Gu =
G ∩ Un and thus both are closed subgroups.
The existence of Jordan decomposition shows that the map Gs × Gu → G, (gs , gu ) 7→
gs gu is surjective. Since Tn ∩ Un = {In } the map is also injective. The set-theoretic
maps G → Gs , G → Gu taking an element g to gs and gu , respectively, are morphisms
because they are defined by taking a subset of the coordinates of g. Thus, we have an
isomorphism Gs × Gu ∼
= G.
The proposition reduces the study of commutative linear algebraic groups to the study of
semisimple commutative groups and unipotent commutative groups.
9.2. Diagonalizable groups and tori. For an algebraic group G, the group of characters of G is
X ∗ (G) = Homk (G, Gm )
(homomorphism of k-algebraic groups).
It is a commutative (abstract group) which we write additively:
(χ1 + χ2 )(g) := χ1 (g) · χ2 (g).
ALGEBRAIC GROUPS: PART II
36
The cocharacters of G, also called one-parameters subgroups of G, are
X∗ (G) := Homk (Gm , G)
(homomorphism of k-algebraic groups).
In general this is only a set. If G is a commutative group then X∗ (G) is a commutative
group as well, and again the group is written additively. We use the notation nχ for n ∈ Z
to denote the cocharacter
(nχ)(x) = χ(x)n = χ(xn ).
Proposition 9.2.1. Let x ∈ Tn and write x = (χ1 (x), . . . , χn (x)). Then each χi is a
character of Tn . We have
±1
k[Tn ] = k[χ±1
1 , . . . , χn ].
The functions χa11 · · · χann are linearly independent over k and are all characters of Tn .
Furthermore, every character of Tn is of this form. We have X ∗ (Tn ) ∼
= Zn and X∗ (Tn ) ∼
=
Zn .
±1
Proof. The statement k[Tn ] = k[χ±1
1 , . . . , χn ] is equivalent to the statement in case n = 1
(which is clear), together with the statement Tn ∼
= Gn , which is also clear.
m
We have analyzed before End(Tn ) and the arguments made there easily extend to
Hom(Gam , Gbm ) and so give us the statements about the characters. And about the cocharaters.
We say that a linear algebraic group is diagonalizable if it is isomorphic to a closed
subgroup of Tn for some n.
Theorem 9.2.2. The following are equivalent:
(0) G is commutative and G = Gs .
(1) G is diagonalizable.
(2) X ∗ (G) is a finitely generated abelian group and its elements form a k-basis for k[G].
(3) Any rational representation of G is a direct sum of one dimensional representations.
Proof. The equivalence of (0) and (1) is clear from Lemma 9.1.1. Assume that G is diagonalizable, say G ⊆ Tn a closed subgroup. Thus, k[G] = k[Tn ]/I for some ideal I. By
Dedekind’s theorem on independence of characters, the elements of X ∗ (G) are linearly
independent over k and they contain the spanning set obtained as the restriction of the
ALGEBRAIC GROUPS: PART II
37
characters from X ∗ (Tn ) (since k[Tn ] has a basis consisting of the characters χa11 · · · χann , and
the restriction of a character is a character). Thus, X ∗ (G) form a basis for k[G] and
X ∗ (Tn ) → X ∗ (G),
is surjective and, in particular, X ∗ (G) is finitely generated.
Let us assume now that G has the property that X ∗ (G) is a finitely generated abelian
group and its elements form a k-basis for k[G]. Let φ be a rational representation of G, φ :
G → GLn . Consider the function
g 7→ φ(g)ij .
It is a regular function on G, and so there are scalars m(χ)ij , almost all of which are zero,
such that
φ(g)ij =
X
m(χ)ij · χ(g).
χ
Packing this together, we find matrices M (χ), almost all of which are zero, such that
X
χ(g)M (χ).
φ(g) =
χ
We then have
φ(gh) =
X
χ(gh)M (χ) =
χ
X
χ(g)χ(h)M (χ).
χ
On the other hand,
X
X
X
φ(g)φ(h) = (
χ(g)M (χ))(
χ(h)M (χ)) =
χ1 (g)χ2 (h)M (χ1 )M (χ2 ).
χ
χ
χ1 ,χ2
We can view the two formulas for φ(gh) = φ(g)φ(h) as formulas between characters
on G × G (viz., we have characters (g, h) 7→ χ(g)χ(h) and (g, h) 7→ χ1 (g)χ2 (h)). Using independence of characters, we find that
M (χ1 )M (χ2 ) = δχ1 ,χ2 M (χ1 ).
In addition, 1 = φ(1) =
P
χ
M (χ) and so we have decomposed the identity operator into
a sum of (commuting) orthogonal idempotents {M (χ)}. Let
V (χ) = Im(M (χ)).
One easily check that
V = ⊕χ V (χ),
ALGEBRAIC GROUPS: PART II
38
and that G acts on each V (χ) via the character χ. This proves (3).
Assume (3) and choose an embedding G ⊆ GLn , realizing G as a closed subgroup. (3)
implies that the natural representation of G on k n affords a basis in which G is diagonal,
namely, we can conjugate G in GLn into Tn and so G is diagonalizable.
Corollary 9.2.3. If G is diagonalizable then X ∗ (G) is a finitely generated abelian group
with no p-torsion if char(k) = p > 0. The algebra k[G] is isomrophic to the group algebra
of X ∗ (G) and its comultiplication, and coinverse, are given by
χ 7→ χ ⊗ χ,
χ 7→ −χ.
This construction can be reversed. Given a finitely generated abelian group M , with
no p-torsion if char(k) = p > 0, we can define a k-algebra - the group algebra of M - k[M ].
In order not confuse the operations, we write M additively, but we write e(m) for the
corresponding function of k[M ] so that k[M ] has a basis {e(m) : m ∈ M } and
e(m1 )e(m2 ) = e(m1 + m2 ).
We endow k[M ] with comultiplication, coinverse and counit by, respectively,
∆(e(m)) = e(m) ⊗ e(m),
ι(e(m)) = e(−m),
e(m) 7→ 1, ∀m ∈ M.
This construction satisfies
k[M1 ⊕ M2 ] = k[M1 ] ⊗k k[M2 ].
We let G (M ) be the corresponding algebraic group. We have
G (M1 ⊕ M2 ) ∼
= G (M1 ) × G (M2 ).
The group G (M ) is a diagonalizable group. To show that it is enough to consider
the case where M = Z, Z/nZ. It not hard to check that then G is isomorphic to Gm and
to µn , respectively.
There is a canonical isomorphism
M∼
= X ∗ (G (M )).
Indeed, each e(m) is a character of G (M ) (as follows from ∆(e(m)) = e(m) ⊗ e(m)). Any
character of G (M ) is a linear combination of the characters e(m) and so, as before, it must
be one of the characters e(m).
ALGEBRAIC GROUPS: PART II
39
If G is a diagonalizable group then
G∼
= G (X ∗ (G)),
as we have seen.
From this constructions one deduces the following:
Corollary 9.2.4. Let G be a diagonalizable group.
(1) G is isomorphic to a direct product of a torus and a finite abelian group of order
prime to p = char(k).
(2) G is connected if and only if it is a torus.
(3) G is a torus if and only if X ∗ (G) is a free abeian group.
We have constructed a correspondence
{f.g. abelian groups with no p-torsion} ←→ {diagonalizable groups}.
In fact, this correspondence is an equivalence of categories. Given a morphism of diagonalizable groups G1 → G2 we get a homomorphism of groups X ∗ (G2 ) → X ∗ (G1 ) and
conversely. This is left as an exercise. Note that the categories are not abelian; the quotient
of an abelian group with no p-torsion by a subgroup may well have p-torsion. That complicates the picture a little bit. Note that this equivalence implies that Hom(Ga , Gb ) ∼
=
m
m
Ma,b (Z).
9.2.1. Galois action. The picture can be made richer taking into account the Galois action. Let F be a field and k its separable closure. An F -group G is called diagonalizable if G(k) is. One can show that G(k) is diagonalizable iff G(F alg ) is. Let X ∗ (G) :=
X ∗ (G(k)) = X ∗ (G(F alg )). The Galois group Γ = Gal(k/F ) acts on X ∗ (G) as follows.
Let γ ∈ Γ, χ ∈ X ∗ (G):
γ
χ(x) = γ(χ(γ −1 (x))).
Then γ χ is a group homomorphism. It is also a morphism: If we choose a presentation k[G] = k[x1 , . . . , xn ]/({fj }) and χ is any function G → A1 we can make the same
P
definition for γ χ. Then, letting χ(x1 , . . . , xn ) = I cI · xI we have
γ
X
X
χ(a1 , . . . , an ) = γ(
cI · (γ −1 (a))I ) =
γ(cI ) · aI .
I
I
ALGEBRAIC GROUPS: PART II
40
That is,
χ=
X
cI · x I ⇒
γ
χ=
X
I
γ(cI ) · xI .
I
Finally, we clearly have
α β
( χ) = αβ χ.
This makes X ∗ (G) into a continuous Γ-module. The main theorem is that there is an
antiequivalence of categories
{Γ-modules that are f.g. abelian groups with no p-torsion} ←→ {diagonalizable F -groups}.
Example 9.2.5. Let K/L be a finite separable extension of fields. Let G = ResK/L Gm .
Recall that this is the algebraic group over L associating to an L-algebra R the group
G(R) = (K ⊗L R)× .
Choose an algebraic closure K̄ of K. It is also an algebraic closure of L. We have
G(K̄) = (K ⊗L K̄)× ∼
= ⊕τ ∈HomL (K,K̄) K̄ × .
The isomorphism is given on generators by
α ⊗ λ 7→ (τ (α)λ)τ .
This is the isomorphism showing that G is a torus of dimension [K : L].
Let us write χτ for the character
χτ (α ⊗ λ) = τ (α)λ
P
P
(its values on a general invertible element αi ⊗λi is τ (αi )λi ). To determine the Galois
action it is enough to consider generators. Let σ ∈ Gal(L̄/L) then
σ
χτ (α ⊗ λ) = σ(χτ (σ −1 (α ⊗ λ)))
= σ(χτ (α ⊗ σ −1 (λ)))
= σ(τ (α) · σ −1 (λ))
= (σ ◦ τ )(α) · λ
= χστ (α ⊗ λ).
Therefore,
σ
χτ = χστ .
ALGEBRAIC GROUPS: PART II
41
Since Gal(L̄/L) acts transitively on HomL (K, K̄), and so on the set {χτ : τ ∈ HomL (K, K̄)},
one concludes that as a Galois module
Gal(L̄/L)
X ∗ (ResK/L Gm ) = IndGal(L̄/K) 1.
(The induction here is in a more subtle sense than in the theory of representations on
vector spaces; it is at the level of lattices.)
Here is a particular situation. The Deligne torus is defined as
S = ResC/R Gm .
This is a 2-dimensional torus defined over R. Let us write C = R · 1 + R · i. Then x + iy
is invertible iff x2 + y 2 6= 0. The group S was then constructed as the affine variety
R[x, y, (x2 + y 2 )−1 ].
Since, (x1 + iy1 )(x2 + iy2 ) = (x1 x2 − y1 y2 ) + (x1 y2 + x2 y1 )i, the co-multiplication of S is
given by
x 7→ x ⊗ x − y ⊗ y,
y 7→ x ⊗ y + y ⊗ x.
Over C we have the isomorphism
S∼
= G2m ,
(x, y) 7→ (x + iy, x − iy).
(Note that since 0 6= x2 + y 2 = (x + iy)(x − iy) the right hand side is indeed in G2m .) The
morphism is clearly invertible and one verifies directly that it is a homomorphism. A basis
for the characters of S is thus visibly
χ1
(x, y) 7→ x + iy,
χσ
(x, y) 7→ x − iy.
Here σ is a formal symbol, but now let σ denote complex conjugation. Then
σ
χ1 (x, y) = σ(χ1 (σ −1 (x, y)))
= σ(χ1 (x̄, ȳ))
= σ(x̄ + iȳ)
= x − iy
= χσ (x, y).
Thus,
σ
χ1 = χσ
ALGEBRAIC GROUPS: PART II
42
(justifying our notation) and, necessarily, σ χσ = χ1 . Thus, as a Galois module,
∗
2
X (S) = Z = Zχ1 ⊕ Zχσ ,
σ 7→
1
1
.
We have an exact sequence of Galois modules:
0 → Z · (χ1 , −χσ ) → Zχ1 ⊕ Zχσ → Z → 0,
where Z on the right is given the trivial Galois action. The map to it is simply n1 χ1 +
nσ χσ 7→ n1 +nσ . The module Z·(χ1 , −χσ ) is isomorphic to Z with σ acting as multiplication
by −1; it defines a torus T over R.
This exact sequence of Galois module corresponds to an exact sequence of tori over R:
1 → Gm → S → T → 1.
(9.2.1)
Using the Gm × Gm model for S(C), the complex points of this sequence are
1 → C× → C× × C× → C× → 1,
where the maps are z 7→ (z, z) and (z1 , z2 ) 7→ (z1 /z2 ). (Check this!). The real points of S
are written in the Gm × Gm model as {(z, z̄) : z ∈ C× }. And so the real points of the
sequence (9.2.1) are
1 → R× → C× → T (R) → 1,
where the maps are r 7→ (r, r) and z 7→ z/z̄ ∈ C1 . One finds that T (R) = C1 and the
sequence of real points of (??) is just
1 → R× → C× → C1 → 1.
9.3. Action of tori on affine varieties. Let T be a torus. There is a canonical perfect
pairing
X ∗ (T ) × X∗ (T ) → Z,
(χ, φ) 7→ χ ◦ φ ∈ End(Gm ) = Z.
Let V be an affine variety on which T acts. Then T has a locally finite rational representation on k[V ]; letting k[V ]χ denote the eigenspace corresponding to χ ∈ X ∗ (T ), namely
the functions f satisfying s(t)f (v) = f (t−1 .v) = χ(t)f (v) for all t ∈ T, v ∈ V , we have
k[V ] = ⊕χ∈X ∗ (T ) k[V ]χ ,
k[V ]χ k[V ]ψ = k[V ]χ+ψ .
ALGEBRAIC GROUPS: PART II
43
Theorem 9.3.1. Let V be an affine T -variety. For λ ∈ X∗ (T ), a one-parameter subgroup,
put
V (λ) = {v ∈ V : the morphism λ : Gm → V, g 7→ λ(g).v,
extends to a morphism λ0 : A1 → V }.
We then say limt → 0 λ(t).v exists and define it as λ0 (0). We have that V (−λ) is the set
of v such that limt → ∞ λ(t).v exists. Then:
(1) V (λ) is a closed set.
(2) V (λ) ∩ V (−λ) is the set of fixed points for Imλ.
P
Proof. Let f = χ fχ . Then,
X
X
ahχ,λi fχ .
χ(λ(a))fχ =
s(λ(a))f =
χ
χ
Then lima → 0 λ(a).v exists iff lima → 0 f (λ(a).v) exists for every f , iff lima → 0 ahχ,−λi fχ (v)
exists, iff for every χ such that hχ, −λi < 0, fχ (v) = 0. That is, v ∈ V (λ) iff v is a zero of
the ideal ⊕{χ:hχ,λi>0} k[V ]χ .
It follows that V (λ) ∩ V (−λ) is the set of v that annihilate all k[V ]χ with hχ, λi 6= 0,
which are the set of v such that f (λ(a)v) = f (v) for all a ∈ k ∗ . That is, the fixed points
for λ.
Example 9.3.2. The group SO2 = {
Let a = 21 (x + 1/x), b =
−i
(x
2
a b
−b a
: a2 + b2 = 1} is actually a form of Gm .
− 1/x). This gives a homomorphism Gm → SO2 which is
clearly invertible, x = a + ib. Given a vector v = (v1 , v2 ) we have
1
a b
v1
(x + 1/x)v1 + −i
(x − 1/x)v2
2
2
= i
.
−b a
v2
(x − 1/x)v1 + 21 (x + 1/x)v2
2
This has a limit as x → 0 iff v1 + iv2 = 0. It has a limit as x → ∞ iff v1 − iv2 = 0. And
so we find that the only fixed point under SO2 is the zero vector (0, 0).
Example 9.3.3. Let λ : Gm → G be a one parameter group and define an action of G
on G (playing now the role of the affine variety) by conjugation. So that Gm acts by
a.g = λ(a)gλ(a)−1 . In this case the variety V (λ) is denoted P (λ). One checks that P (λ)
is a subgroup, hence a closed subgroup. In fact, it is a parabolic subgroup (a notion we
discussed for GLn and discuss in general later). We have P (λ) ∩ P (λ−1 ) equal to the fixed
points of λ, which is precisely the centrailizer of λ(Gm ) in G.
ALGEBRAIC GROUPS: PART II
44
9.4. Unipotent groups. We shall be very brief here. More information can be found in
Springer’s book.
Let k = k̄ be an algebraically closed field of characteristic p ≥ 0. A unipotent linear
algebraic group G/k is called elementary if it is abelian and, moreover, if p > 0 then its
elements have order dividing p.
Example 9.4.1. The additive group Ga = (A1 , +) is isomorphic to the subgroup {( 10 ∗1 )}
and hence is unipotent. If p > 0 then a + · · · + a (p times) is pa = 0. Thus, Ga is an
elementary unipotent group.
Theorem 9.4.2. Let G be a connected linear algebraic group of dimension 1. Then, either
G∼
= Gm or G ∼
= Ga .
For the proof see Springer, §3.4. We sketch part of the proof.
• G is commutative. Fix g ∈ G and consider the morphism G → G, x 7→ xgx−1 . The
closure of the image is either G, or e, by dimension considerations and connectedness. Assume it is G. Then, every element, but finitely many, is of the form xgx−1
and thus (viewing G as a subgroup of some GLn ) and has the same characteristic
polynomial as g. Altogether there are finitely many possibilities for the characteristic polynomial of elements of g. Since G is connected, the characteristic polynomial
(whose coefficients are algebraic functions on G) must then be constant. However,
the identity has characteristic polynomial (t − 1)n and, thus, so does every element
of G. Thus, G is unipotent. In that case, it is nilpotent and so its commutator
subgroup must be strictly contained in G (and connected), hence equal to {e}. It
follows that G is commutative and that contradicts our assumption that the image
of x 7→ xgx−1 is G. Thus, the image is always {e} and G is commutative.
• Therefore, G ∼
= Gs × Gu . Both factors must be connected and exactly one of which
1-dimensional (and so the other must be trivial). If Gs is one dimensional, it is a
connected diagonalizable group of dimension one hence isomorphic to Gm . Else,
G∼
= Gu , a commutative unipotent group.
• If G = Gu then G is elementary. One views G as a subgroup of the upper unipotent
group Un in GLn , for a suitable n. If the characteristic is p, writing x = 1+N , where
N is nilpotent, using the binomial formula and divisibility of binomial coefficients
n
by p, one concludes that xp = 1.
ALGEBRAIC GROUPS: PART II
45
Let G(m) the image of the homomorphism x 7→ xm . It is a closed connected
subgroup of G and so is either G or {1}. If G(p) = G then, inductively, G(p
(pn−1 )
(G
)(p) = G and that is a contradiction, because x
pn
n)
=
= 1 for all x ∈ G. Thus,
G(p) = {1} and therefore G is elementary.
• The next (hard) step is to show that an elementary unipotent group of dimension 1
is isomorphic to Ga . For that see Springer’s book.
ALGEBRAIC GROUPS: PART III
EYAL Z. GOREN, MCGILL UNIVERSITY
Contents
10. The Lie algebra of an algebraic group
10.1. Derivations
10.2. The tangent space
10.2.1. An intrinsic algebraic definition
10.2.2. A naive non-intrinsic geometric definition
10.2.3. Via point derivations
10.3. Regular points
10.4. Left invariant derivations
10.5. Subgroups and Lie subalgebras
10.6. Examples
10.6.1. The additive group Ga .
10.6.2. The multiplicative group Gm
10.6.3. The general linear group GLn
10.6.4. Subgroups of GLn
10.6.5. A useful observation
10.6.6. Products
10.6.7. Tori
10.7. The adjoint representation
10.7.1. ad - The differential of Ad.
Date: Winter 2011.
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ALGEBRAIC GROUPS: PART III
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10. The Lie algebra of an algebraic group
10.1. Derivations. Let R be a commutative ring, A an R-algebra and M an A-module.
A typical situation for us would be the case where R is an algebraically closed field, A the
ring of regular functions of an affine k-variety and M is either A itself, or A/M , where M
is a maximal ideal. Returning to the general case, define D, an M -valued R-derivation
of A, to be a function
D : A → M,
such that D is R-linear and
D(ab) = a · D(b) + b · D(a).
We have used the dot here to stress the module operation: a ∈ A and D(b) ∈ M and
a · D(b) denotes the action of an element of the ring A on an element of the module M .
The collection of all such derivations, DerR (A, M ), is an A-module, where we define
(f · D)(a) = f · D(a),
f, a ∈ A.
Example 10.1.1. Let k be an algebraically closed field, X an affine k-variety, OX,x the
local ring of x on X and view k as an OX,x -module via f · α = f (x)α, for f ∈ OX,x , α ∈ k.
Then Derk (OX,x , k) are the k-linear functions
δ : OX,x → k,
such that δ(f g) = f · δ(g) + g · δ(f ). Using the definition of the module structure, these
are the functions δ : OX,x → k such that
δ(f g) = f (x)δ(g) + g(x)δ(f ).
10.2. The tangent space. There are many interpretations for the tangent space. We
bring here three such. Yet another one can be given using the notion of k[] points (2 = 0).
See Springer’s, or Hartshorne’s book.
10.2.1. An intrinsic algebraic definition. Let X be a variety. The tangent space at x ∈ X,
TX,x , is
TX,x := (mx /m2x )∗ ,
namely, TX,x = Homk (mx /m2x , k), where mx is the maximal ideal of the local ring OX,x and
mx /m2x is viewed as a k = OX,x /mx -vector space.
ALGEBRAIC GROUPS: PART III
48
10.2.2. A naive non-intrinsic geometric definition. Assume that X is affine (else, pick a
Zariski open affine negihborhood of x), say X ⊆ An , defined by an ideal I, and let M be
the maximal ideal of k[X] comprising the functions vanishing at x. Then,
OX,x = k[X]mx ,
mx = M OX,x ,
and so
OX,x /mx = k[X]/M,
mx /m2x = M/M 2 .
As a result,
TX,x = (M/M 2 )∗ .
The point is that (M/M 2 )∗ affords a description which is closer to our geometric intuition.
Suppose that T1 , . . . , Tn are the variables on An and f is a function vanishing on X.
Develop f into a Taylor series at x = (x1 , . . . , xn ); the leading term, which we denote dfx ,
is
(10.2.1)
n
X
∂f
(x)(Ti − xi ).
dx f =
∂Ti
i=1
naive
to be the affine-linear variety defined by all the equations (10.2.1) as f
We define TX,x
ranges over I:
dx f = 0,
f ∈ I.
naive
.
We remark that if I = hf1 , . . . , fm i then it is enough to use dx f1 , . . . , dx fm to define TX,x
naive
Thus, TX,x
with origin at x, is the solutions to the homogenous system of equations
∂fi
∂Tj
.
1≤i≤m,1≤j≤n
We can in fact define dx f for any f ∈ k[X], by the same formula (10.2.1), and view it
naive
as a function on TX,x
, which is linear once we make x = (x1 , . . . , xn ) the origin. In this
naive
interpretation dx f , for f ∈ I, is the zero function on TX,x
. Now, since k[X] = k + M and
dx f = 0 if f is constant, we may view dx as a map
naive ∗
dx : M → (TX,x
).
Since dx (f g) = f (x)dx (g) + g(x)dx (f ) we see that dx vanishes on M 2 and so we get a map
naive ∗
dx : M/M 2 → (TX,x
).
ALGEBRAIC GROUPS: PART III
49
One proves this map is an isomorphism (it’s not hard; see Springer, Humphries) and, thus,
dualizing, we have
naive ∼
TX,x
= (M/M 2 )∗ ∼
= (mx /m2x )∗ ,
showing that the naive non-intrinsic geometric definition agrees with the intrinsic algebraic
definition.
10.2.3. Via point derivations. We can also view TX,x , or, rather, (mx /m2x )∗ as point derivations at x. Let δ be a point derivation at x, δ : OX,x → k. Then δ is determined by its
restriction to mx and δ|m2x ≡ 0. Thus, we get a map
δmx /m2x → k,
which is k-linear. This is an element of (mx /m2x )∗ . Conversely, a functional δ ∈ (mx /m2x )∗
defines a derivation δ by
δ(f ) = δ(f − f (x)
(mod m2x )).
10.3. Regular points. (Also called “non-singular”, or “simple”, points). Let X be an
equi-dimensional variety (that is, all the components of X have the same dimension). A
point x ∈ X is called regular if
dimk TX,x = dim(X).
X itself is called regular if all its points are regular. A basic result is
Proposition 10.3.1. The set of regular points in X is a dense Zariski-open set.
Let ϕ : X → Y be a morphismx ∈ X, y = ϕ(x). There is an induced homomorphism of
local rings
ϕ∗ : OY,y → OX,x ,
taking the maximal ideal my into the maximal ideal mx . There is therefore a k-linear map
my /m2y → mx /m2x ,
hence, by dualizing a k-linear map, which we denote dϕx ,
dϕx : TX,x → TY,y .
There is another way to describe it. If δ is a point derivation at x then dϕ(δ) is the point
derivation at y defined by
dϕ(δ)(f ) = δ(f ◦ ϕ).
ALGEBRAIC GROUPS: PART III
50
The matching of these two definitions is an easy exercise. Other properties that follow
easily are
d(ψ ◦ ϕ)x = dψϕ(x) ◦ dϕx ,
d(IdX )x = Id.
From this follows formally that if ϕ is an isomorphism so is dϕx for any x. The following
theorem is useful to know (although we do not use it in the sequel).
Theorem 10.3.2. Let ϕ : X → Y be a morphism. Then ϕ is an isomorphism if and only
if it is bijective and dϕx is an isomorphism for all x ∈ X.
Example 10.3.3. Consider the cuspidal curve Y : y 2 = x3 in A2 . The point (0, 0) is
a singular point (that is, it is not regular) as M = (x, y, y 2 − x3 ) = (x, y), M 2 =
(x2 , y2 , xy, y 2 − x3 ) = (x2 , y 2 , xy) and M/M 2 is two dimensional.
The normalization of Y , X is the affine line A1 . The map
X → Y,
t 7→ (t2 , t3 ),
is a bijection. It cannon be an isomorphism though, because 0 is a regular point of X.
And indeed, at the point 0 the map of tangent spaces can be calculated thus:
The point derivations of X at 0 are the derivations of the form f (t) 7→ α · (∂f /∂t)(0)
where α ∈ k. Let D be the derivation corresponding to α = 1 (a basis for the 1-dimensional
space of derivations). The point derivations of Y at (0, 0) are the derivations f (x, y) 7→
α · (∂f /∂x)(0, 0) + β · (∂f /∂y)(0, 0). The map dϕ0 is the following
dϕ0 (D)(f (x, y)) = D(f (x, y) ◦ ϕ) = D(f (t2 , t3 )) = 0,
as f (t2 , t3 ) − f (0, 0) ∈ (t2 ).
Let us also calculate it the other way. The map OY,(0,0) → OX,0 is f 7→ f (t2 , t3 ), which
belongs to m20 if f ∈ m(0,0) . Once again, the map is the zero map.
Proposition 10.3.4. Let X be an algebraic group. Then X is regular.
Proof. Let x0 ∈ X be a regular point. Let x1 ∈ X an other point. The morphism
X → X,
x 7→ x1 x−1
0 x,
is an isomorphism taking x0 to x1 , hence inducing an isomorphism
TX,x0 → TX,x1 .
It follows that X is regular at x1 as well.
ALGEBRAIC GROUPS: PART III
51
10.4. Left invariant derivations. Let G be an algebraic group over k. Let A = k[G]
and consider Derk (A, A). If D1 , D2 are derivations then so is
[D1 , D2 ] := D1 ◦ D2 − D2 ◦ D1 ,
as a simple calculation shows. This makes Derk (A, A) into a Lie algebra. That means
that apart from the vector space structure, the bracket
[·, ·] : Derk (A, A) × Derk (A, A) → Derk (A, A),
is a k-bilinear alternating pairing, such that the Jacobi identity holds:
[D1 , [D2 , D3 ]] + [D2 , [D3 , D1 ]] + [D3 , [D1 , D2 ]] = 0.
Recall that G acts on A locally finitely by
(λx f )(y) = f (x−1 y).
We define L (G), the left invariants derivations of G, as
L (G) = {D ∈ Derk (A, A) : D ◦ λx = λx ◦ D, ∀x ∈ G}.
It is a Lie algebra under the bracket operation. Let
g = TG,e
be the tangent space to G at the identity element e. We commonly think about it in terms
of point derivations. There is a map
Derk (A, A) → g,
D 7→ {f 7→ (Df )(e)}.
This restricts to a k-linear map
L (G) → g.
In fact, a point of explanation is in order. The derivation D is defined on k[G] and to define
the point derivation we need to extend D to the local ring at e, OG,e = k[G]M , where M is
the maximal ideal corresponding to e. Firstly, if D can be extended to OG,e then it would
have to satisfy D(f ) = D( fg · g) = g · D( fg ) +
f
g
· D(g). That is, D( fg ) =
g·D(f )−f ·D(g)
.
g2
shows that if D extends to OG,e this extension is unique. Secondly, D( fg ) =
This
g·D(f )−f ·D(g)
g2
actually defines the extension of D to OG,e (one has to verify it’s well defined and is indeed
a derivation, but this is just a simple verification).
ALGEBRAIC GROUPS: PART III
52
Theorem 10.4.1. The map
L (G) → g
is an isomorphism of k-vector spaces. In particular, dimk (L (G)) = dim(G).
Let ϕ : G → G0 be a homomorphism of algebraic groups then the induced map,
dϕe : g → g0 ,
is a homomorphism of Lie algebras (where g, g0 are given the bracket operation via the
isomorphism to L (G), L (G0 ), respectively).
Remark 10.4.2. Although we do not need it in the proof, let’s see that the map L (G) → g
is injective. Suppose that the point derivation
f 7→ (Df )(e)
is identically zero. Then, for every f ∈ k[G], and every x ∈ G,
0 = [D(λx f )](e)
= [λx (Df )](e)
= (Df )(x−1 ).
Thus, Df = 0 for all f ∈ G and so D is the zero derivation.
Proof. Given a point derivation δ at e and f ∈ A, define the convolution of f with δ,
(f ∗ δ)(x) := δ(λx−1 f ).
Then, f 7→ f ∗ δ is a derivation:
((f g) ∗ δ)(x) = δ(λx−1 (f g))
= δ(λx−1 f · λx−1 g)
= δ(λx−1 f ) · (λx−1 g)(e) + δ(λx−1 g) · (λx−1 f )(e)
= δ(λx−1 f ) · g(x) + δ(λx−1 g) · f (x)
= (f ∗ δ)(x) · g(x) + (g ∗ δ)(x) · f (x)
= ((f ∗ δ) · g + (g ∗ δ) · f )(x).
ALGEBRAIC GROUPS: PART III
53
It is left-invariant:
(λy (f ∗ δ))(x) = (f ∗ δ)(y −1 x)
= δ(λx−1 y f )
= δ(λx−1 λy f )
= ((λy f ) ∗ δ)(x).
We claim that this is an inverse to the map L (G) → g. Indeed, given D ∈ L (G), let δ
be the derivation f 7→ (Df )(e). Then,
(f ∗ δ)(x) = δ(λx−1 f )
= (Dλx−1 f )(e)
= (λx−1 Df )(e)
= (Df )(x).
Conversely, let f ∈ k[G] and δ a point derivation at e. Then the derivation f 7→ (f ∗ δ)(e)
is just δ as (f ∗ δ)(e) = δ(λe f ) = δ(f ).
Now let ϕ : G → G0 be a homomorphism. The map
dϕe : g → g0
is the map
dϕe (δ)(f 0 ) := δ(ϕ∗ f 0 ) = δ(f 0 ◦ ϕ),
f 0 ∈ OG0 ,e0 .
Let f = ϕ∗ f 0 . Let δ1 , δ2 ∈ g (point derivations) and let δ10 = dϕe (δ1 ), δ20 = dϕe (δ2 ). Then,
on the one hand,
[δ10 , δ20 ](f 0 ) = [∗δ10 , ∗δ20 ](f 0 )(e0 )
= ((f 0 ∗ δ20 ) ∗ δ10 )(e0 ) − ((f 0 ∗ δ10 ) ∗ δ20 )(e0 )
(sic!)
= δ10 (f 0 ∗ δ20 ) − δ20 (f 0 ∗ δ10 )
= δ1 (ϕ∗ (f 0 ∗ δ20 )) − δ2 (ϕ∗ (f 0 ∗ δ10 )).
On the other hand,
dϕe ([δ1 , δ2 ])(f 0 ) = [δ1 , δ2 ](ϕ∗ f 0 )
= (((ϕ∗ f 0 ) ∗ δ2 ) ∗ δ1 )(e) − (((ϕ∗ f 0 ) ∗ δ1 ) ∗ δ2 )(e)
= δ1 ((ϕ∗ f 0 ) ∗ δ2 ) − δ2 ((ϕ∗ f 0 ) ∗ δ1 ).
ALGEBRAIC GROUPS: PART III
54
It is therefore enough to prove the following identity,
(ϕ∗ f 0 ) ∗ δ2 = ϕ∗ (f 0 ∗ dϕe (δ2 )).
We do that by calculating the values of these functions on G at every x ∈ G. On the one
hand,
∗ 0
((ϕ∗ f 0 ) ∗ δ2 )(x) = δ2 (λ−1
x ϕ f )
= δ2 (ϕ∗ (λϕ(x)−1 f 0 ))
= dϕe (δ2 )(λϕ(x)−1 f 0 ).
on the other hand,
ϕ∗ (f 0 ∗ dϕe (δ2 ))(x) = (f 0 ∗ dϕe (δ2 ))(ϕ(x))
= dϕe (δ2 )(λϕ(x)−1 f 0 ).
10.5. Subgroups and Lie subalgebras. Let H be a closed subgroup of G defined by an
ideal J. The natural inclusion
H → G,
induces a map on tangent spaces
TH,e → TG,e .
The relations between local rings is OH,e = OG,e /J and mH,e = mG,e /J. Thus
Theorem 10.5.1. The image of TH,e → TG,e is the subspace of TG,e consisting of derivations δ such that δ(f ) = 0, ∀f ∈ J.
This simple observation is very useful in calculating Lie algebras, as we shall see below.
More theoretically, we have:
Theorem 10.5.2. Let g be the Lie algebra of G and h the Lie algebra of H. Then h is a
subalgebra of g and
h = {δ ∈ g : f ∗ δ ∈ J, ∀f ∈ J}.
Proof. Let δ ∈ h and f ∈ J. For h ∈ H we have
(f ∗ δ)(h) = δ(λh−1 f ) = 0,
because λh−1 f ∈ J and δ, being a derivation on (the localization of) k[G]/J, vanishes on J.
Thus, f ∗ δ ∈ J.
ALGEBRAIC GROUPS: PART III
55
Conversely, let δ ∈ g be a derivation such that f ∗ δ ∈ J, ∀f ∈ J. Then, 0 = (f ∗ δ)(e) =
δ(λe−1 f ) = δ(f ). Thus, δ ∈ h.
10.6. Examples.
10.6.1. The additive group Ga . Let the coordinate be t. The tangent space at zero is
k·
∂(·)
(0).
∂t
Since the bracket operation is alternating, the bracket is trivial and this is the
full story. What is the invariant derivation corresponding to δ :=
∂(·)
(0)?
∂t
Let f be a
polynomial, x ∈ Ga and g the function g(t) = f (t + x), then
(f ∗ δ)(x) = δ(λ−x f )
∂g
(0)
∂t
∂f
=
(x),
∂t
=
by the chain rule. That is, ∗δ is just the derivation f 7→
∂f
.
∂t
To check, in general, that a
derivation is invariant on A = k[G], it is enough to test it on algebra generators, because,
if Dλx f = λx Df and Dλx g = λx Dg then, Dλx (f g) = D(λx f · λx g) = λx f D(λx g) +
λx gD(λx f ) = λx f λx (Dg) + λx gλx (Df ) = λx D(f g), etc..
In the particular case at hand t is a generator and
∂(t + x)
∂t
∂λ−x t
=
= 1 = λ−x .
∂t
∂t
∂t
10.6.2. The multiplicative group Gm . Again the tangent space is one dimensional and so
the bracket is identically zero. We let δ be the point derivation at 1 given by δ(f ) =
∂f
(1).
∂t
For a fixed x, letting g(t) = f (xt), the invariant derivation corresponding to δ is
(f ∗ δ)(x) = δ(λx−1 f )
= δ(g(t))
∂g
(1)
∂t
∂f
=x·
(x).
∂t
=
We conclude that up to multiplication by a scalar every invariant derivation on Gm is the
derivation
f 7→ t ·
∂f
.
∂t
ALGEBRAIC GROUPS: PART III
56
We can again check. Call that derivation D. As an algebra k[Gm ] is generated by t±1 . We
have
D(λx−1 t) = t ·
λx−1 Dt = λx−1 t = xt,
∂(tx)
= xt,
t
and
λx−1 Dt−1 = λx−1 (t · (−t−2 )) = −1/(xt),
D(λx−1 t−1 ) = t · D(1/(xt)) = t · (−x/(xt)2 ) = −1/(xt).
10.6.3. The general linear group GLn . A basis of the point derivations at e are the derivations
f 7→
∂f
(1).
∂tij
2
(The inclusion GLn ,→ An identifies the tangent spaces.) Let us calculate the corresponding invariant derivation. Let δ =
∂f
(1).
∂tij
Consider the function f ((tij )) = tk` , where k, `
are some fixed indices. Then, the function g((tab )) = λ(xab )−1 f ((tab )) is simply the function
P
(tab ) 7→ r xkr tr` and
(f ∗ δ)((xab )) = δ(g((tab )))
P
∂( r xkr tr` )
=
(1)
∂tij
= δj` · xki
(here δj` is, unfortunately, the Kronecker delta symbol). Consider the derivations
Dij f :=
X
a
tai
∂f
.
∂taj
Since f 7→ ∂f /∂tab is a derivation, and the derivations are a module over the coordinate
ring, these are indeed derivations. The value of Dij on the function f ((tij )) = tk` is δj` · tki .
Since any derivation is determined by its values on the functions tk` it follows that D must
be the derivation ∗δ and, in particular, left invariant.
P
∂(·)
If we write an element of TGLn ,1 as (mij ), corresponding to the derivation
mij ∂t
(1)
ij
P
and so to the left-invariant derivation ij mij Dij we can calculate the Lie bracket. It will
be determined uniquely by taking the elementary matrices Ek` , Eij , corresponding to Dk`
ALGEBRAIC GROUPS: PART III
57
and Dij . One calculates (it is enough to check on the basic functions we used before) that
[Dkl , Dij ] = δ`i Dkj − δjk Dil ,
which is the derivation associated with [Ek` , Eij ]. Thus, we conclude that gln is canonically
identified with the k-vector space of all n × n matrices with Lie bracket
[X, Y ] = XY − Y X.
10.6.4. Subgroups of GLn . Let H be a subgroup of GLn defined by the vanishing of the
2
2
ideal J, where we view GLn ⊂ An . Then, the tangent space of GLn is An , identified as a
Lie algebra with Mn (k) with the bracket XY − Y X (see above), and the tangent space of
H is defined as the subspace of Mn (k) determined by the vanishing of the linear equations
dfe =
X ∂f
(Idn ),
∂t
ij
ij
f ∈ J.
P
∂f
We note that if we develop such f as f (Idn + (tij )) =
ij tij ∂tij (0n ) + h.o.t. then h is
P
defined by the equations ij tij ∂t∂fij (0n ), that is, by the equations
X
ij
tij
∂f
∂tij
(mod (tij tk` )i,j,k,` ).
Consider for example H = SLn . Then H is defined by the equation f (Idn + (tij )) − 1 =
det(Idn + (tij )) − 1 = 0. Modulo squares of variables this is the equation
X
tii = 0,
and we conclude that
sln = {M ∈ Mn (k) : Tr(M ) = 0}.
Consider the case of a bilinear form represented by a symmetric matrix B = (bij ) (so that
hx, yi = t xBy. The orthogonal group associated to it is
OB = {M ∈ GLn : t M BM = B}.
Write M = Idn + (tij ) then, modulo squares, we have
(Idn + t (tij ))B(Idn + (tij ) − B = B + t (tij )B + B(tij ) + (tij )B(tij ) − B
= t (tij )B + B(tij ).
ALGEBRAIC GROUPS: PART III
58
That is, the Lie algebra are the “B-skew-symmetric matrices”,
oB = {M ∈ Mn (k) : t M B = −BM }.
Now Tr(M ) = Tr(BM B −1 ) = Tr(−t M BB −1 ) = −Tr(M t ) = −Tr(M ) and so Tr(M ) = 0.
It follows that
soB = oB .
This of course can be proven with no calculation. SOB is of index 2 in OB and is equal in
fact to the identity component of OB , hence they have the same Lie algebra.
P
In particular, for B = Idn (corresponding to the quadratic form q(x) = x2i ), we have
on = son = {M ∈ Mn (k) : t M = −M }.
10.6.5. A useful observation. One consequence of the fact that an algebraic group G oveer
k is non-singular and that
dimk L (G) = dimk g = dim G,
is that we can calculate the dimension of G by calculating the dimension of its Lie algebra.
Thus, we easily find that dim(GLn ) = n2 and dim SLn = n2 − 1, which we knew of course,
but also that dim OB = dim SOB = 21 n(n − 1).
10.6.6. Products. Let G1 , G2 be algebraic groups. Then
L (G1 × G2 ) ∼
= L (G1 ) ⊕ L (G2 ).
We leave that as an exercise.
10.6.7. Tori. Let T be a torus. There is a canonical isomorphism
L (T ) = k ⊗Z X∗ (T ).
Further, the bracket operation is trivial. We leave that as an exercise.
10.7. The adjoint representation. Let A = k[G]. We are going to define two actions
of G on g that will be shown to be equal. First, we note the action of G on L (G),
D 7→ ρx ◦ D ◦ ρ−1
x ,
where x ∈ G and (ρx f )(y) = f (yx). It is easy to check this is an action, using the
λy ρx = ρx λy .
ALGEBRAIC GROUPS: PART III
59
This induces an action on g via the isomorphism g → L (G); the action of x takes a
derivation δ to a derivation µ such that
∗µ = ρx ◦ ∗δ ◦ ρ−1
x .
On the other hand, consider the action of G on TG,e coming from conjugation. Let Int(x)
be the automorphism of G given by
Int(x)(y) = xyx−1 .
We denote its differential at the identity by Ad(x). Thus,
Ad(x) = d Int(x)e .
Note that, by definition,
Ad(x)(δ)(f ) = δ(f ◦ Int(x)) = δ(λx−1 ρx−1 f ).
We claim that µ = Ad(x)(δ). We may pass to g. Then µ(f )(e) = [ρx ((ρ−1
x f ) ∗ δ)](e),
which is equal to [(ρ−1
x f ) ∗ δ](x) = δ(λx−1 ρx−1 f ) = Ad(x)(δ)(f ). Thus, we have proven the
following lemma.
Lemma 10.7.1. Ad(x)(δ) = ρx ◦ ∗δ ◦ ρ−1
x .
2
Let us consider now the case G = GLn . The map Int(x) is in fact a linear map on An
and, as is well-known, the differential of a linear map is equal to the map. Thus:
Lemma 10.7.2. Let δ ∈ gln = Mn (k) then
Ad(x)(δ) = xδx−1 .
Corollary 10.7.3. The adjoint representation Ad : GLn → GL(gln ) ∼
= GLn2 is an algebraic representation.
For a general algebraic group we cannot multiply elements of the group with elements
of the Lie algebra, so there is no intrinsic formula like in the lemma. Yet, every algebraic
group is isomorphic to a closed subgroup of GLn and for those we have the following lemma.
Lemma 10.7.4. Let H be a closed subgroup of GLn and view h as a Lie subalgebra of gln .
Let h ∈ H then Ad(h)(δ) = hδh−1 .
ALGEBRAIC GROUPS: PART III
60
Proof. This is just the statement that the differential of conjugation by h on H is the
restriction of the differential of conjugation by h on GLn . This is clear for example from
the interpretation of the tangent space usingthe m/m2 description.
Corollary 10.7.5. Let H be an algebraic group. The adjoint representation Ad : H → GL(h) ∼
=
GLn is an algebraic representation (n = dim(H)).
This corollary is very important as it gives us a way to associate a canonical representation to an algebraic group.
10.7.1. ad - The differential of Ad. We have constructed a linear representation
Ad : G → GL(g) ∼
= GLd (k),
where d = dim(G). Note though that Ad(x) has the additional property that it respects
the Lie bracket on g. At any rate, we have an induced map of Lie algebras - the differential
of Ad at the identity:
ad : g → gl(g).
The calculation for GLn is given by the following theorem.
Proposition 10.7.6. The homomorphism of Lie algebras
ad : gln → gl(gln ) ∼
= Mn2 ,
is given by
ad(X)(Y ) = [X, Y ] = XY − Y X.
The proof is a calculation that can either be done directly, or by more sophisticated
arguments. In any case we omit it. It can be found in Springer’s or Humphreys books.
By embedding an algebraic group G into GLn we conclude:
Corollary 10.7.7. Let G be an algebraic group. The homomorphism
ad : g → gl(g)
is given by
ad(X)(Y ) = [X, Y ] = X ◦ Y − Y ◦ X.
ALGEBRAIC GROUPS: PART IV
EYAL Z. GOREN, MCGILL UNIVERSITY
Contents
11. Quotients
11.1. Some general comments
11.2. The quotient of a linear group by a subgroup
12. Parabolic subgroups, Borel subgroups and solvable subgroups
12.1. Complete varieties
12.2. Parabolic subgroups
12.3. Borel subgroups
13. Connected solvable groups
13.1. Lie-Kolchin
13.2. Nilpotent groups
13.3. Solvable groups
Date: Winter 2011.
59
60
60
61
65
65
65
69
72
72
72
74
ALGEBRAIC GROUPS: PART IV
60
11. Quotients
Very few proofs will be given in this section. The missing proofs can be found in Springer,
Humphreys and Borel (and other references that we give below). This is not because there
is anything very difficult about them; it is done so that we still have time to discuss other
topics.
11.1. Some general comments. Let X be a quasi-projective variety and H an algebraic
group acting on X. Ideally, we want a quotient X/H. One would expect:
• X/H is a quasi-projective variety.
• There is a morphism X → X/H that gives a bijection between points of X/H and
orbits of H in X.
• Additional properties: one may expect the morphism π to be open, that every
morphism X → Y constant on orbits of H factors through X/H and so on.
Unfortunately, already in very simple cases such a quotient doesn’t exists. For example,
let H = Gm act on X = A1 . There are two orbits {0} and A1 − {0}. If A1 /Gm existed
it should thus have two points, corresponding to the two orbits. Since X/H is a variety,
each such point is closed and so both orbits {0} and A1 − {0} are closed, which is not the
case.
On the other ase, if X is an affine variety and H is a finite group then a quotient exists
and moreover, X/H is an affine variety with coordinate ring k[X/H] = k[X]H . So, for
example, although it is not obvious, PSL2 = SL2 /{±1} is an affine variety. Moreover, the
coordinate ring of SL2 is
k[SL2 ] = k[x11 , x12 , x21 , x22 , x11 x22 − x12 x21 − 1],
and so the coordinate ring of PSL2 is
k[xij xkl , x11 x22 − x12 x21 − 1 : i, j, k, l ∈ {±1}].
Note that it is presented as a closed subvariety of A10 !
The following theorem is very useful. There is a proof in Mumford/Abelian varieties.
Theorem 11.1.1. Let X be a quasi-projective variety and G a finite group acting on X,
then there is a quotient X → X/G with all the properties above (including the additional
properties).
ALGEBRAIC GROUPS: PART IV
61
11.2. The quotient of a linear group by a subgroup. Let G be an affine linear group
over an algebraically closed field k. Let H be a closed subgroup of G. The main theorem
- a proof can be found in Springer’s book - is the following:
Theorem 11.2.1. There is a quotient G/H in the following sense: There is a homogenous G-variety, denoted G/H, and a point a ∈ G/H such that:
for any pair (Y, b) comprising a homogenous G-variety Y and a point b ∈ Y such that
StabG (b) ⊇ H, exists a unique morphism
G/H → Y,
a 7→ b.
Sketch of proof.
Step 0. Recall that a variety for Springer is a ringed space (X, OX ) that is quasi-compact
and locally isomorphic to an affine variety. This means the following: X is a topological
space and OX is a sheaf of rings on X (see, e.g., Hartshorne for the concept of “sheaf”).
Quasi-compact means that every open cover contains a finite open subcover. (Some people
call that “compact”; here we follow the convention that a compact space is quasi-compact
and Hausdorff). Every quasi-projective variety is an example of a quasi-compact ringed
space. If (X, OX ) is a ringed space and U ⊂ X is an open set then (U, OX |U ) is a ringed
space. The assumption made on (X, OX ) that it can be covered by open sets U such that
each (U, OX |U ) is isomorphic as a ringed space to one obtained from a quasi-projective
variety (we could have required “from an affine variety” and that wouldn’t matter).
This category of variety is larger and more flexible then the category of quasi-projective
varieties.
Step 1.There exists a G-homogenous quasi-projective variety X for G and x ∈ X such
that:
• StabG (x) = H.
• the morphism ψ : G → X, g 7→ gx gives a separable morphism G0 → G0 x.
• the fibers of ψ are the cosets gH.
[[We recall that a dominant morphism of quasi-projective varieties φ : A → B is called
separable, if for every a ∈ A the induced map TA,a → TB,φ(a) is surjective. Let k be a field
of characteristic p > 0. Here is a typical example of a non-separable morphism:
φ : Ga → Ga ,
φ(x) = xp .
ALGEBRAIC GROUPS: PART IV
62
This morphism is called the Frobenius morphism. Now, let Ga act on A1 via φ. That is
x ? a = xp + a.
This action makes A1 into a homogenous Ga -variety. The stabilizer of a = 0 is the subgroup
{0}. But it is not true that Ga /{0} is isomorphic to A1 via the map φ. (It happens to be
isomorphic to A1 via a different map - the identity - but this is “a coincidence”. One thus
sees that if we are looking for a model for G/H using a homogeneous G-variety we need
to be careful about issues of separability. ]]
Step 2. In this step one constructs G/H as a ringed space. As a set G/H is the set of
cosets G/H and a = eH. The function π : G → G/H is just π(x) = x̄ = xH. One gives
G/H the quotient topology: U ⊂ G/H is open iff π −1 (U ) is open in G. Thus, the function
π is continuous and open.
Finally, for an open set U ⊂ G/H define
O(U ) = {f : U → k : f ◦ π is regular on π −1 (U )
(11.2.1)
= OG (U )H .
This is a sheaf of rings on G/H.
Step 3. Show that the map of ringed spaces
(G/H, a) → (X, x),
gH 7→ gx,
where (X, x) is from step 1 is an isomorphism. It thus shows G/H is a variety, in fact a
quasi-projective variety. We can see some consequences of the proof:
• G → G/H is open, surjective and its fibers are the cosets of H. dim(G/H) =
dim(G) − dim(H).
• G/H is quasi-projective.
• if H1 ⊇ H there is a natural surjective open morphism G/H → G/H1 .
Proposition 11.2.2. A stronger universal property holds for the quotient G/H: Let φ :
G → Y be a morphism such that ∀g ∈ G, h ∈ H, φ(gh) = φ(g). Then there exists a unique
ALGEBRAIC GROUPS: PART IV
63
morphism φ̄ : G/H → Y making the following diagram commutative
φ
G DD
DD
DD
π DD
!
/Y
z=
z
zz
zz
z
z φ̄
G/H
Proof. Uniqueness is clear, as two morphism agreeing as functions are equal and φ̄ is
determined as a function by the commutativity of the diagram. Define
φ̄ : G/H → Y,
φ̄(x̄) = φ(x).
This is a well-defined function rendering the diagram commutative. It is a continuous
function: let U ⊂ Y be open, then φ̄−1 (U ) = π(φ−1 (U )). Since φ−1 (U ) is open and π
is open, also φ̄−1 (U ) is open. Thus, the only property we still need to show is that if
f ∈ k[U ] is regular then φ̄∗ f ∈ k[φ̄−1 (U )] is regular. To show that we need to show that
π ∗ φ̄∗ f ∈ k[φ−1 (U )] is regular and H invariant. Now, π ∗ φ̄∗ = φ∗ and φ∗ f (xh) = f (φ(xh)) =
f (φ(x)) = φ∗ f (x).
Example 11.2.3. Recall that GLn acts transitively on the flag variety F(d1 ,...,dt ) parameterizing a flag of subspaces V0 = {0} ⊂ V1 ⊂ V2 ⊂ · · · ⊂ Vt ⊂ V = k n , where dim(Vi ) = di .
Let d0 = 0 and dt+1 = n. The stabilizer of the standard flag
{Vi } :
Vi = Spank {e1 , . . . , edi }
is the subgroup



A1 ∗ ∗ . . .
∗










A
∗
.
.
.
∗


2





A
.
.
.
∗
3
P = 
 : Ai ∈ GLdi −di−1 , i = 1, . . . , t + 1 ,




..






.






At+1
where A1 is a square matrix of size d1 , A2 is a square matrix of size d2 − d1 , A3 is a square
matrix of size d3 − d2 and so on, and At+1 is of size n − dt . The flag variety is projective
and we have a natural bijective morphism
GLn /P → F(d1 ,...,dt ) .
ALGEBRAIC GROUPS: PART IV
64
In fact, this morphism is an isomorphism, but we shall not prove it here. It requires an
understanding of the tangent space of F(d1 ,...,dt ) at each of its point. The following Lemma,
whose proof can be found in Springer, suffices to conclude many properties of GLn /P .
Lemma 11.2.4. Let φ : Y1 → Y2 be a bijective morphism of G-varieties, where G is a
linear algebraic group. Then, for every variety Y , the morphism φ × 1Y : Y1 × Y → Y2 × Y
is (topologically) a homeomorphism.
We mention one last important result about quotients.
Proposition 11.2.5. Let G be a linear algebraic group and H a closed normal subgroup
of G. Then the quotient G/H has a natural structure of an algebraic group. Further, G/H
is an affine algebraic group whose coordinate ring is k[G]H .
Thus, for example, PGLn = GLn /Gm is an affine algebraic group.
ALGEBRAIC GROUPS: PART IV
65
12. Parabolic subgroups, Borel subgroups and solvable subgroups
12.1. Complete varieties. A variety X is called complete if for every variety Y the
projection map X × Y → Y is closed. Here are some fundamental facts about complete
varieties.
• A quasi-projective variety is complete if and only if it is projective.
• A closed subvariety of a complete variety is complete.
• If X1 , X2 are complete varieties so is X1 × X2 .
• If X is complete and irreducible then any regular function on X is constant.
• X is complete and affine if and only if X is finite.
• If φ : X → Y is a morphism then φ(X) is closed and complete.
Of these facts the hardest to prove is the first one. The rest are not too hard to prove from
the definition, or deduce from the first fact. According to these facts A1 is not complete
and indeed we can show that directly. Consider the closed set Z = {(x, y) : xy = 1} in
A1 × A1 . Its projection to A1 is A1 − {0}, which is not closed.
12.2. Parabolic subgroups. A closed subgroup P of G is called parabolic if G/P is a
complete variety.
Lemma 12.2.1. If P is parabolic then G/P is a projective variety.
Proof. We know already that G/P is a quasi-projective variety. Since it is also complete,
it must be projective.
Lemma 12.2.2. Let G act transitively on a projective variety V and let v0 ∈ V , P =
StabG (v0 ). Then P is parabolic.
Proof. The natural map
G/P → V,
gP 7→ gv0 ,
is a bijective map of homogeneous G-varieties. By Lemma 11.2.4, for every Y the map,
G/P × Y → V × Y,
is a homeomorphism and it commutes with the projection to Y . Thus, G/P is complete if
and only if V is complete, which, being projective, it is.
ALGEBRAIC GROUPS: PART IV
66
Corollary 12.2.3. The subgroup P of GLn given by



A1 ∗ ∗ . . .
∗










A
∗
.
.
.
∗


2





A
.
.
.
∗
3
:
A
∈
GL
,
i
=
1,
.
.
.
,
t
+
1
,
P = 

i
di −di−1




.


..










At+1
is a parabolic subgroup.
Corollary 12.2.4. Let {e1 , . . . , en , δ1 , . . . , δn } be the standard symplectic basis of k 2n , rel
ative to the pairing hx, yi =t xJy, where J = −I0 n I0n . Thus, hei , ej i = hδi , δj i = 0 for all
i, j and hei , δj i = −hδj , ei i = δij (where δij is Kronecker’s delta).
be the projective variety
Let d0 = 0 < d1 < · · · < dt = n be integers and let Fdiso
1 ,...,dt
parameterizing flags {0} ⊂ V1 ⊂ V2 ⊂ · · · ⊂ Vt , where each Vi is isotropic and of dimen. The
sion di . By Witt’s theorem, the symplectic group Sp2n acts transitively on Fdiso
1 ,...,dt
stabilizer of the standard flag Vi = Span{e1 , . . . , edi } is thus a parabolic subgroup. This
stabilizer has the shape



A1 ∗ ∗ . . .
∗










A
∗
.
.
.
∗


2





A
.
.
.
∗
3
Sp2n ∩ 
 : Ai ∈ GLdi −di−1 , i = 1, . . . , t ,




..






.






At+1
B ) is
where dt+1 = 2n and At+1 is in GLn . Now, it is easy to verify that a matrix ( A0 D
symplectic iff A ∈ GLn , D = t A−1 and t BD is symmetric. Note then that D has the shape

 −1

A1
0
0
0
0






−1




∗
A
0
0
0


2



−1

 ∗
∗
A
0
0
3
 : Ai ∈ GLdi −di−1 , i = 1, . . . , t .


 ..

..
..
.. 
..





 .
.
.
.
.






−1
∗
∗
∗ . . . At
Proposition 12.2.5. Let Q ⊂ P ⊂ G be closed subgroups. If Q is parabolic in G then it
is parabolic in P .
Proof. We have an injective map P/Q → G/Q. The image is closed, as its complement is
the image of the open set G − P under the projection map G → G/Q. Thus, the image is
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complete and in bijection by a map of homogeneous P -spaces with P/Q. It follows that
P/Q is complete and so Q is parabolic in P .
Proposition 12.2.6. If P is parabolic in G and Q is parabolic in P then Q is parabolic
in G.
Proof. We need to show that for every Y the morphism,
G/Q × Y → Y,
is closed.
Let A ⊆ G × Y be a closed set such that
(a, y) ∈ A ⇒ (aq, y) ∈ A,
∀q ∈ Q.
We shall call such a set Q-closed (or Q-closed in G×Y ). Then, to show that the morphism
G × Y → Y is closed is equivalent to showing that the morphism G × Y → Y takes a Qclosed set in G × Y to a closed set in Y .
Let A be a Q-closed set in G × Y . Consider the morphism
P × G × Y → G × Y,
(p, g, y) 7→ (gp, y).
The preimage of A is
A+ = {(p, g, y) : (gp, y) ∈ A}.
It is a Q-closed set in P × (G × Y ). Since Q is parabolic in P , the projection of A+ to
G × Y is thus closed. This projection is equal to the set
à = {(gp, y) : (g, y) ∈ A, p ∈ P }.
(Indeed, if (g, y) ∈ A, p ∈ P then (p−1 , gp, y) ∈ A+ ; conversely, if (p, g, y) ∈ A+ then
(gp, y) ∈ A and so (g, y) = ((gp)p−1 , y) ∈ Ã.) The set à is P -closed in G × Y and so its
projection to Y is closed. But this projection is equal to the projection of A.
Proposition 12.2.7. Let P ⊂ Q ⊂ G be closed subgroups. If P is parabolic in G so is Q.
Proof. There is a surjective morphism G/P → G/Q. Since G/P is complete, so is its
image, namely G/Q.
Proposition 12.2.8. Let P ⊂ G be a closed subgroup. Then P is parabolic in G if and
only if P 0 is parabolic in G0 .
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Proof. First, G0 is parabolic in G as G/G0 is a finite group.
If P 0 is parabolic in G0 then, because G0 is parabolic in G, P 0 is parabolic in G by
Proposition 12.2.6. Suppose that P is parabolic in G. Since P 0 is parabolic in P , it is
parabolic in G, again by Proposition 12.2.6. Now G → G/P 0 is an open map. Thus,
the open set G − G0 maps to an open set in G/P 0 , whose complement is the closed set
G0 /P 0 . Since G0 /P 0 is a closed subset of a complete variety it is complete too and so P 0
is parabolic in G0 .
Theorem 12.2.9. A connected linear group G contains a proper parabolic subgroup (that
is, a parabolic subgroup different from G itself ) if and only if G is non-solvable.
Proof. Assume, without loss of generality, that G is a closed subgroup of GLn . Recall also
that the subgroup of upper triangular matrices in GLn is solvable. (We have seen that
the upper triangular unipotent matrices are a nilpotent, hence solvable group, and the
quotient is Gnm which is also solvable.)
The group GLn , hence also G, acts on the projective space Pn−1 . Let X be a closed orbit
for the action of G. Let x ∈ X and P = StabG (x). Then the morphism of homogeneous
G-varieties G/P → X, gP 7→ gx is bijective. Since X is complete, so is G/P (we have made
this argument before). Thus, P is parabolic. If P = G, it follows kx̃ ⊂ k n , where x̃ is a
lift of x to k n , is stable under G. Consider then the action of G on P(k n /kx). Continuing
this way, either we get a proper parabolic subgroup of G at some stage, or we arrive to a
basis of k n in which G is upper-triangular, hence solvable. Thus, if G is non-solvable, it
has a proper parabolic subgroup.
Suppose now that G is solvable (and connected, by our original assumption). We need to
show that it doesn’t have any proper parabolic subgroups. We argue by induction on the
dimension of G. The base case is dimension 0, where the statement is trivial as G = {1}.
Now, if G has proper parabolic group we may choose one of maximal dimension, say P .
We may assume, using Proposition 12.2.8 that P is connected.
Recall that the commutator subgroup (G, G) of G is closed and connected (by an exercise). Consider the group
Q = P · (G, G).
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Since (G, G) is normal in G, P · (G, G) is a subgroup of G, equal to the subgroup generated
by P and (G, G). Since both P and (G, G) are connected Q is connected. Since Q ⊇ P ,
Q is also parabolic. It follows that either (1) Q = G, or (2) Q = P .
Suppose that Q = G, then the homomorphism (G, G) → G/P is surjective and so induces a bijective morphism
(G, G)
G
−→
.
(G, G) ∩ P
P
Since G/P is complete so is (G, G)/((G, G) ∩ P ) and so (G, G) ∩ P is a parabolic subgroup
of (G, G). Since G is solvable, (G, G) is a proper subgroup of G and so has strictly small
dimension than G (G is irreducible) . Thus, by induction, (G, G) ∩ P is equal to (G, G).
Then (G, G) ⊂ P and G = P · (G, G) = P , which is a contradiction.
The other case is Q = P . In this case (G, G) ⊂ P . In this case, because G/(G, G) is
commutative, P is a normal subgroup of G and so G/P is affine and complete, hence finite.
But G/P is also connected. If follows that G/P is trivial and so that G = P , which is a
contradiction.
Theorem 12.2.10 (Borel’s fixed point theorem). Let G be a connected linear solvable
algebraic group and let X be a complete G-variety. Then G has a fixed point on X.
Proof. Let Y ⊂ X be a closed orbit and y ∈ Y . By the usual argument, the stabilizer of y
is a parabolic subgroup, but cannot be a proper parabolic; hence it must be equal to G.
That is, y is a fixed point for G.
12.3. Borel subgroups. A closed subgroup B of a linear algebraic group G is called a
Borel subgroup is it is closed, connected and solvable subgroup of G and is maximal
relative to these properties.
Example 12.3.1. Let G = GLn and B the upper triangular matrices. Then B is a
Borel subgroup. Indeed, it is certainly a closed subgroup. It is solvable, as we had already
remarked. It is also connected, being isomorphic, as a variety only, to Gnm ×An(n−1)/2 . Why
is it maximal? Suppose that B $ P ⊂ G and P is connected. Then, since B is parabolic
in G, B is parabolic in P and so P has a proper parabolic subgroup which implies that P
is non-solvable.
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Remark 12.3.2. The same argument shows that whenever we find a connected solvable
closed subgroup of G which is parabolic, then it is a Borel subgroup. Thus, for example, the
B ) such that A is an invertible n × n upper-triangular, D = t A−1 ,
matrices of the form ( A0 D
and B is a matrix such that t BD is symmetric, form a Borel subgroup of Sp2n .
Theorem 12.3.3. The following assertions hold true:
(1) A closed subgroup of G is parabolic if and only if it contains a Borel subgroup.
(2) A Borel subgroup is parabolic.
(3) Any two Borel subgroups of G are conjugate.
Corollary 12.3.4. A closed subgroup of G is Borel if and only if it is a minimal parabolic
subgroup.
Proof of Theorem. Let P be a subgroup of G. Since P is parabolic in G if and only if P 0
is parabolic in G0 , and since any Borel subgroup is contained in G0 , we may assume that
G is connected.
Let B be a Borel subgroup and P a parabolic subgroup. Then B acts on the complete
variety G/P by (b, gP ) 7→ bgP . Since B is solvable and G/P complete, by Borel’s fixed
point theorem, B has a fixed point, say gP . Then g −1 Bg ⊂ P . Since g −1 Bg is also a
Borel subgroup, it follows that any parabolic subgroup contains a Borel subgroup. That
prove the “only if” part of (1). If we prove (2) then we get that every Borel subgroup is
parabolic and if B ⊂ P ⊂ G also P is parabolic by Proposition 12.2.7, hence the “if” part
of (1).
We now prove (2). Let B be a Borel. If G is solvable, then G = B (we assumed G is
connected) and a Borel is parabolic trivially. If G is not solvable, then there is a proper
parabolic subgroup P of G. By what we had proven above, we may assume that P ⊃ B.
Since the dimension of P is smaller than G’s, by induction B is a parabolic subgroup of
P . Thus, by Proposition 12.2.6, B is parabolic in G.
It remains to prove (3). Given two Borel subgroups B1 , B2 , since both are parabolic, we
may conjugate B1 into B2 and B2 into B1 . Thus, dim(B1 ) = dim(B2 ). It follows that if
g −1 B1 g ⊂ B2 then, in fact, g −1 B1 g = B2 .
Proposition 12.3.5. Let φ : G → H be a surjective homomorphism of algebraic groups
and let P be a parabolic, resp. Borel, subgroup of G. Then φ(P ) is parabolic, resp. Borel.
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Proof. Since any parabolic contains a Borel and every subgroup containing a parabolic is
parabolic, it is enough to proof the assertion for Borel subgroups P ⊂ G. The subgroup
φ(P ) is closed, connected and solvable. Further, the morphism G/P → H/φ(P ) induced by
φ is surjective and so H/φ(P ) is complete. Thus, φ(P ) is parabolic. As we have remarked
above, this implies that φ(P ) is Borel.
Proposition 12.3.6. Let B be a Borel subgroup of a connected linear group G. Denote
the centre of G and B by C(G), C(B), respectively. Then
C(G)0 ⊂ C(B) ⊂ C(G).
Proof. The subgroup C(G)0 is closed, connected and commutative, hence contained in
some Borel subgroup. Since all Borel subgroups are conjugate, it is contained in every
Borel subgroup.
Let g ∈ C(B) and consider the morphism G/B → G given by xB 7→ gxg −1 x−1 . Since
G/B is projective (and so is its image) and G is affine, the image must consists of finitely
many points. Since the image is connected, it must be just the identity element. That
implies that g ∈ C(G).
Remark 12.3.7. We shall later see that in fact C(B) = C(G). This simplifies in practice
the calculation of the centre of a group.
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13. Connected solvable groups
We have skipped over many results in Chapter 5 of Springer, in order to be able to cover
some deeper aspects of algebraic groups. In this chapter we will occasionally need such
results. Those that have short proofs, will be proven; the proofs for the others can be
found in the book.
13.1. Lie-Kolchin.
Theorem 13.1.1 (Lie-Kolchin). Let G be a closed connected solvable subgroup of GLn .
Then G can be conjugated into Tn - the upper triangular matrices.
Proof. We use Borel’s fixed point theorem. GLn acts on Pn−1 . Thus G has a fixed point on
Pn−1 , corresponding to a line kx1 . Consider now the action of G on P(k n /kx1 ). There is a
fixed point, corresponding to a line in k n /kx1 , hence to a plane kx1 + kx2 in k n . And so on.
This way we obtain a basis x1 , . . . , xn in which G consists of upper triangular matrices.
Note that this proves that Tn is a Borel subgroup of GLn . Conversely, supposing that
Tn is a Borel of GLn , one can also argue differently. Since G is connected and solvable,
it is contained in a Borel subgroup of GLn . Since all Borel subgroups are conjugate, it
follows that G can be conjugated into Tn .
13.2. Nilpotent groups.
Lemma 13.2.1 (Springer’s Corollary 5.4.8). Assume G to be a connected, nilpotent, linear
algebraic group G. The set Gs of semi-simple elements is a subgroup of the centre of G.
−1
Proof. A commutator of length 1 is denoted (x1 , x2 ) = x1 x2 x−1
1 x2 ; a commutator of length
two is either (x1 , (x2 , x3 )) or ((x1 , x2 ), x3 ), etc.. Since G is nilpotent, there exists an n so
that all iterated commutators of length n, (x1 , (. . . (xn , xn+1 ) . . . )), are trivial.
Let s ∈ G be a semisimple element, σ = Int(s) the automorphism of conjugation by s
and χ : G → G the morphism x 7→ σ(x)x−1 (and so χ(x) = (s, x)). The commutators of
length n being trivial implies that χn G = {1}. One the other hand, one has the following
identity on tangent spaces
dχe : g → g,
dχe = Ad(s) − 1.
Since χn is the trivial map Ad(s) − 1 is nilpotent. On the other hand, Ad(s) is semisimple
(any representation takes a semisimple element to a semisimple element) and the difference
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of two commuting semisimple operators is semisimple. Thus, Ad(s) − 1 is at the same time
semisimple. It follows that Ad(s) − 1 = 0. Thus, Ad(s) is the identity map. We need to
use the following
Lemma 13.2.2 (Springer’s Corollary 5.4.5). Let G be a connected algebraic group. Let
s ∈ G be a semisimple element. Then:
(1) The conjugacy class C = {xsx−1 : x ∈ G} is closed. The morphism x 7→ xsx−1 is
separable.
(2) Let Z = ZG (s) = {x ∈ G : xsx−1 = s} be the centralizer of s. Then, with the
notation Z = TZ,e ,
g = z ⊕ (Ad(s) − 1)g.
It follows that g = z in our situation and so dim(G) = dim(Z). Since Z is a closed
subgroup of G and G is connected (hence irreducible), we must have Z = G. That is, s
belongs to the centre of G.
Since the product of commuting semisimple elements is semisimple, we can now deduce
that Gs is a group.
The following proposition improves on Lemma 13.2.1.
Proposition 13.2.3. Assume that G is a nilpotent connected algebraic group.
(1) The set Gs , resp. Gu , of semi-simple, resp. unipotent, elements is a closed, connected subgroup.
(2) Gs is a central torus of G.
(3) The product map Gs × Gu → G is an isomorphism of algebraic groups.
Proof. We have just seen that Gs is a subgroup of the centre of G. Let us assume that
G is contained in GLn . We may simultaneously diagonalize the elements of Gs , obtaining
a decomposition of V = V1 ⊕ · · · ⊕ Vd into subspaces according to characters of Gs ; In
particular, on each Vi each element of Gs acts by a scalar and because Gs is contained
in the centre of G, each Vi is G-invariant. Let Gi = G|Vi ; the group G embeds into the
product G1 × · · · × Gd . Note that each Gi is nilpotent and (Gi )s = (Gs )i and so the
semisimple elements of Gi are precisely the scalar matrices in Gi .
Each Gi is nilpotent, hence solvable. Thus, by Lie-Kolchin, we may assume that each
Gi consists of upper-triangular matrices. The unipotent elements of Gi are precisely those
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with 1 on the diagonal, and thus, the Jordan decomposition associates to a matrix M in Gi
a semisimple element Ms of Gi , which is therefore a diagonal matrix with the same diagonal
entries as in M , as they have the same eigenvalues, and a unipotent part determined by
M = Ms Mu . In fact, that semisimple part is a scalar matrix. One concludes that in this
basis of V
Gs = G ∩ Dn ,
Gu = G ∩ Un ,
where we are denoting by Dn the diagonal matrices of GLn and by Un its upper triangular
unipotent matrices. It follows that both Gs and Gu are closed subgroup. The map
Gs × Gu → G,
(gs , gu ) 7→ gs gu ,
is a homomorphism because Gs is in the centre and it is bijective since it exhibits the
Jordan decomposition for G. The inverse map is also a morphism, since, as we have seen,
Ms is just the morphism M 7→ diag(m11 , . . . , mnn ) and Mu = Ms−1 M .
Finally, since G is connected this isomorphism shows that both Gs and Gu are connected.
13.3. Solvable groups.
Proposition 13.3.1. Let G be a connected solvable algebraic group.
(1) The commutator subgroup (G, G) (also called the derived subgroup) of G is a closed
connected unipotent normal subgroup.
(2) The set Gu of unipotent elements is a closed connected nilpotent normal subgroup
of G. It contains (G, G). The quotient G/Gu is a torus.
Example 13.3.2. The group (G, G) is contained in Gu but may be smaller than Gu . This
is so in the case G is unipotent (thus, under a suitable embedding to GLn , consists entirely
of upper triangular unipotent matrices) since it is then nilpotent and the derived series
must descend to the identity. For example, look at G = Un .
To illustrate the proposition, let Tn be the upper triangular matrices of GLn , which is
a Borel subgroup of GLn . The commutator subgroup is Un , which is equal to the set of
unipotent elements of Tn (but, as said, this need not be the case in general). The quotient
Tn /Un is a torus. In fact, it is isomorphic to Dn . As we’ll see, the fact that there is a
subgroup in Tn isomorphic to the quotient Tn /Un is not a coincidence.
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Proof. We already know that (G, G) is a closed connected normal subgroup. To see it is
unipotent, embed G in Tn , for some n, by Lie-Kolchin. Then it is easy to check that (G, G)
is a closed subgroup of Un , hence consists of unipotent elements.
As for (2), still by Lie-Kolchin, we view G as a closed subgroup of Tn . Thus, Gu = G∩Un
and so is a closed normal subgroup of G. We have an injective homomorphism of algebraic
groups,
G/Gu ,→ Tn /Un ∼
= Dn .
Thus G/Gu is commutative and all its elements are semisimple. As we have seen, G/Gu
can then be diagonalized and being connected must be a torus. (Here we are avoiding
arguing that G/Gu is isomorphic to its image, which is true in fact.)
To show Gu is connected, consider its identity component G0u . Gu is normal in G and
G0u is a characteristic subgroup of Gu so it is normal in G. Consider then the group G/G0u .
The group homomorphism G → G/G0u takes unipotent elements to unipotent elements.
Thus Gu /G0u are unipotent elements of G/G0u . Since G/Gu is a torus, these are all the
unipotent elements. That is
(G/G0u )u = Gu /G0u .
Thus, the algebraic group H = G/G0u has the property that Hu is finite and H/Hu is
commutative.
Lemma 13.3.3. Let J be a connected algebraic group and N a finite normal subgroup of J
then N is contained in the center of J.
Proof of Lemma. Let n ∈ N . Consider the map J → N, g 7→ gng −1 . The image is connected and contains n. Thus, the image is {n} and so n is a central element.
Coming back to the main proof, we conclude that Hu is central and so we conclude that
H is nilpotent (because it is so modulo a central subgroup) and connected. By Proposition 13.2.3, Hu is connected, hence trivial. Thus, Gu = G0u .
Definition 13.3.4. Let G be a connected solvable algebraic group. We call a torus T in
G maximal if its dimension is dim(G/Gu ).
It is clear that such a torus is indeed maximal in the sense that it is not properly
contained in any other torus, because any torus of G maps injectively into G/Gu . The
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converse is also true: every torus is contained in a maximal torus in the sense that its
dimension is dim(G/Gu ). We will prove that later.
Theorem 13.3.5. Let G be a connected solvable algebraic group.
(1) Let s ∈ G be a semisimple element. Then s lies in a maximal torus. In particular,
maximal tori exist (take s = 1, if you must).
(2) The centralizer ZG (s) of a semisimple element s ∈ G is connected.
(3) Any two maximal tori of G are conjugate.
(4) If T is a maximal torus then the morphism T × Gu → G is an isomorphism of
varieties.
Proof. We first prove (4). Since G/Gu is a torus and the homomorphism T → G/Gu is
injective (as T ∩ Gu = {1}, the identity being the only element that is both unipotent and
semisimple), it is also surjective, by dimension considerations. Thus, the morphism
T × Gu → G
is bijective as well. It is a morphism of T × Gu -homogeneous varieties, where we give G
the action (t, u) ∗ x = txu−1 . Since a morphism of homogeneous varieties is separable if
and only if it is separable at one point (Springer, Theorem 5.3.2), it is enough to check
that it is separable at the identity elements.
We have
(13.3.1)
L (T × Gu ) = L (T ) ⊕ L (Gu ) → L (G),
(X, Y ) 7→ X − Y.
(This is so because we are considering the morphism of T × Gu -varieties taking (t, u) to
teu−1 . For the natural multiplication morphism T × Gu → G the map is (X, Y ) 7→ X + Y .)
Lemma 13.3.6. L (T ) ∩ L (Gu ) = {0} (interesection inside L (G)).
Proof of Lemma. We show that the Lie algebra of a torus consists of semisimple elements
and the Lie algebra of a unipotent group consists of nilpotent elements (viewed as derivations they are linear operators). Hence their intersection is just {0}.
For a torus, we may reduce to Gnm , whose Lie algebra, thought of as left-invariant
derivations, is spanned by the invariant derivations {t1 ∂/∂t1 , . . . , tn ∂/∂tn }. As we have
proven, the characters are a basis for k[Gnm ]. The effect of ti ∂/∂ti on a character ta11 · · · tann
ALGEBRAIC GROUPS: PART IV
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is multiplication by ai . That means that in the basis of characters each invariant derivation
is already diagonalized and so is a semisimple operator.
For a unipotent group, using Lie-Kolchin, we may reduce to the case of Un . It is
isomorphic as a variety (not as a group!) to An(n−1)/2 and so k[Un ] = k[tij : 1 ≤ i < j ≤ n].
A basis for the Lie algebra consists of the functions ∂/∂tij , for i < j. In this way we view
the tangent space of Un as a subspace gln . The operation δ 7→ ∗δ clearly agrees with this
identification. Now, recall that for GLn we have
Dij = ∗∂/∂tij =
n
X
tai ∂(·)/∂taj .
a=1
Since on Un the functions tij = 0 for i > j and are equal to 1 for i = j, the invariant
derivation in L (Un ) corresponding to ∂/∂tij , for i < j, is
Kij = ∂(·)/∂tij +
n
X
tai ∂(·)/∂taj .
1≤a<i
To show each linear combination of such operators is nilpotent it is enough to show each
Kij is nilpotent and for that it is enough to show that it eventually reduces the degree of
every monomial. It is enough to consider
kij =
n
X
tai ∂(·)/∂taj .
1≤a<i
Now kij takes a homogeneous polynomial (e.g., a monomial) to either zero, or a homogeneous polynomial of the same degree. However, if we put a lexicographic order on the
monomial where the letters are ordered t12 < t13 < · · · < t1n < t23 < t24 < . . . then it
decreases the order because
tai
∂
(
tαk`k` ) =
∂taj 1≤k<`≤n
Y
(
0
αaj tai
αaj = 0
Q
α
k`
1≤k<`≤n tk`
taj
αaj > 0.
(So, either the monomial is killed, or the weight is shifted to a preceding letter.)
Thus, the map (13.3.1) on tangent spaces is injective. Since both source and target
groups are non-singular and of the same dimension, the map on tangent spaces is also
surjective. And so we have a bijective separable morphism T × Gu → G. By Springer, loc.
cit., it is an isomorphism. This finishes the proof of (4).
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The rest of the theorem is proven by induction on the dimension of Gu .
Case 1. dim(Gu ) = 0. In this case, since Gu is connected, Gu = {0} and Proposition 13.3.1
gives that G = G/Gu is a torus and the theorem holds trivially.
Case 2. dim(Gu ) = 1. This case is non-trivial and will take a while to prove. Using again
the classification of connected algebraic groups of dimension 1, we know that Gu ∼
= Ga .
Fix an isomorphism,
φ : Ga → Gu ,
and use the notation ψ for the quotient map
ψ : G → S := G/Gu .
The group S is a torus of dimension dim(G) − 1. Consider the map
Ga → Gu ,
a 7→ gφ(a)g −1 .
Since Aut(Ga ) = Gm , there is a unique scalar α(g) such that
gφ(a)g −1 = φ(α(g) · a).
It is easy to see that α is a character of G that factors through G/Gu . Thus, abusing
notation, there is a character α of S such that
gφ(a)g −1 = φ(α(ψ(g)) · a).
Claim. If α is trivial then G is commutative.
Proof of Claim. We first remark that this doesn’t hold for abstract groups. For example,
{±1} is a normal subgroup of the quaternion group Q8 of 8 letters. The action of G on
{±1} by conjugation is trivial and Q8 /{±1} ∼
= Z/2Z2 is commutative. Still Q8 is not
commutative. However, in our case, consider the commutator map
G × G → G,
(x, y) 7→ [x, y].
Since α is trivial, it means that Gu is in the centre (for every g ∈ G, a ∈ Ga , gφ(a) =
φ(α(ψ(g)) · a) · g = φ(a)g). Thus, the pairing factors through G/Gu × G/Gu → G. In fact,
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this pairing has image in Gu because taking the values mod Gu we get the commutator
pairing on a torus. Thus, we have a pairing
G/Gu × G/Gu → Gu ,
(ḡ, h̄) 7→ ghg −1 h−1 .
Now, this pairing is in fact a homomorphism in, say, the first argument: [ḡ1 ḡ2 , h̄] =
g1 g2 hg2−1 g1−1 h−1 = g1 [g2 , h]hg1−1 h−1 = [g1 , h][g2 , h]. Thus, since G/Gu is a torus and Gu is
unipotent, the map ḡ 7→ [ḡ, h] must be trivial for any g ∈ G, h ∈ G and that means G is
commutative.
Thus, we have established that if α is trivial then G is commutative. In this case we have
proven that G = Gs × Gu and the assertions of the theorem are easy to check. (There is a
unique maximal torus, equal to Gs .). Thus, we assume henceforth that α is not trivial.
Let s ∈ G be a semisimple element and let Z = ZG (S) be its centralizer. By Lemma 13.2.2,
g = z ⊕ (Ad(s) − 1)g,
z := L (Z).
The relation,
ψ(sxs−1 x−1 ) = 1,
(that follows from the commutativity of G/Gu ) gives on the tangent spaces the relation
dψ ◦ (Ad(s) − 1) = 0.
Now, the map
dψ : TG,e → TG/Gu ,e ,
is not the zero map. In fact, from the construction of ψ : G → G/H in general, it follows
that dψ is surjective with kernel L (H). In particular, ker(dψ) = L (Gu ), which is one
dimensional and contains (Ad(s) − 1)g. Thus, dim(z) = dim(Z 0 ) ≥ dim(G) − 1.
Consider first the case where α(ψs) 6= 1. It is important later that this case occurs.
Indeed, if g is such that α(ψg) 6= 1 then also α(ψgs ) 6= 1. In this case, using gφ(a)g −1 =
φ(α(ψ(g)) · a), we see that
ZG (s) ∩ Gu = {1}.
This shows that Z 0 must be a proper subgroup of G and hence it is a closed connected
subgroup of G of dimension dim(G) − 1 with Zu0 = {1}. We proved that the quotient of a
ALGEBRAIC GROUPS: PART IV
80
connected solvable group by its unipotent elements is a torus. Thus, Z 0 , having dimension
dim(G/Gu ) is a maximal torus and by (4) we have
G = Z 0 Gu .
We claim that in fact Z = Z 0 . Let g ∈ G. Then g = g0 gu with g0 ∈ Z 0 and gu ∈ Gu .
Then, s commutes with g0 and so commutes with g if and only if it commutes with gu . But
that would imply α(ψs) = 1, unless gu = 1 (meaning gu = φ(0)). It follows that Z = Z 0 .
We have therefore shown (1) and (2) provided α(ψs) 6= 1.
Suppose then that α(ψs) = 1. This means that Gu ⊆ Z = ZG (s) and so L (Gu ) ⊆ z.
But, the decomposition g = z ⊕ (Ad(s) − 1)g and (Ad(s) − 1)g ⊂ L (Gu ) is only possible
if Ad(s) − 1 = 0. That means that Z = G and so that s is an element of the centre of
G. In this case (2) is trivial. To see (1), namely that s lies in a maximal torus, take any
element s0 such that α(ψs0 ) 6= 1. We have then seen that ZG (s0 ), which surely contains s,
is a maximal torus.
We still need to prove (iii) (still in the case dim(Gu ) = 1). Let T be a maximal torus. If
T 0 is a maximal torus, each of its elements is semi-simple, and from our discussion above
it follows that there must be an element t0 ∈ T 0 such that
α(ψt0 ) 6= 1,
T 0 = ZG (t0 ).
Let us write
t0 = tφ(a),
t ∈ T, a ∈ Ga .
If a = 0 then t0 ∈ T and T 0 = ZG (t0 ) ⊇ T , whence T = T 0 . Assume a 6= 0. Let
b ∈ Ga , b 6= 0. Then
t · φ(a + ([α(ψt0 )]−1 − 1)b) = t · φ(a) · φ([α(ψt0 )]−1 b) · φ(b−1 )
= t · φ(a) · (t0−1 φ(b)t0 ) · φ(b−1 )
= φ(b)t0 φ(b)−1 .
The point is that we can choose b 6= 0 so that the left hand side is t. We have shown that
t0 can be conjugated to get t. Thus, the centralizer of t0 , which is T 0 , can be conjugated
to contain the centralizer of t, which, in turn, contains T . That is, φ(b)T 0 φ(b)−1 ⊇ T .
Dimensions being the same, and T and T 0 being connected, we must have φ(b)T 0 φ(b)−1 = T .
ALGEBRAIC GROUPS: PART IV
81
This finishes the proof in the case dim(Gu ) = 1. For the induction step we shall need the
following lemma (that we use without proof; its proof is independent of our proof of the
theorem).
Lemma 13.3.7 (Springer’s Lemma 6.3.4). Under the assumptions of the theorem, assume
that G is not a torus. There exists a closed normal subgroup N of G, contained in Gu and
isomorphic to Ga .
We now assume that dim(Gu ) > 1 and choose such a subgroup N . Let Ḡ = G/N , a
connected solvable group, and let Ḡu = Gu /N . Note that dim(Ḡu ) = dim(Gu ) − 1 and
Ḡu = (Ḡ)u . We can apply induction to Ḡ then.
Let s be a semisimple element of G and s̄ its image in Ḡ. It is a semisimple element and so
lies in a maximal torus T̄ of Ḡ. Recall that this means that dim(T̄ ) = dim(Ḡ)−dim(Ḡu ) =
dim G − dim Gu . Consider the preimage T̃ of T̄ in G. Since T̃u = N is one dimensional,
we may apply induction and conclude that s lies in a maximal torus of T̃ . This torus has
dimension dim T̃ − dim N = dim T̄ = dim G − dim Gu and so is a maximal torus of G as
well. Hence, (1).
Let T, T 0 be maximal tori of G. Their projections to Ḡ are maximal tori and thus for
some b̄, b̄T̄ b̄−1 = T̄ 0 . It follows that bT b−1 is a maximal torus of T̃ 0 , the preimage of T̄ 0 .
Thus, we reduced the assertion to maximal tori of T̃ 0 . Since T̃ 0 is connected (being fibered
over T̄ 0 with fiber Ga ) and T̃u0 is one-dimensional, the assertion follows from the case we
have already proven. This proves (3).
Consider now assertion (2). We know that ZḠ (s̄) is connected. Let Z̃ be the preimage
in G. It is the subgroup of elements g of G such that [s, g] ∈ N . Z̃ is a connected
variety(being a fibration of ZḠ (s̄) with fiber Ga ). It is thus a connected solvable group.
The centralizer of s in G is contained in Z̃ and is equal to the centralizer of s in Z̃. If s
is central everything is easy. If s is not central and dim Z̃ < dim G we have by induction
on dim G that the centralizer is connected. If s is not central and dim Z̃ = dim G then
G = Z 0 · N (because Z 0 Ḡ implies Z 0 Ḡ). Further, for dimension reasons, Z 0 ∩ N is
finite. But a unipotent group doesn’t have finite subgroups. Thus Z 0 ∩N = {1}. As before
`
(cf. page 80), one concludes that Z = Z 0 . Alternately, one can argue that G = zZ 0 · N
over coset representatives for Z/Z 0 and use that G is connected.
ALGEBRAIC GROUPS: PART IV
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Corollary 13.3.8. Let G be a connected solvable group and H ⊂ G a closed subgroup all
whose elements are semisimple.
(1) H is contained in a maximal torus of G. In particular, a subtorus of G is contained
in a maximal torus (and so the maximal tori are maximal with respect to inclusion
as well).
(2) The centralizer ZG (H) is connected and coincides with the normalizer NG (H).
Proof. Since H ∩Gu = {1}, the morphism H → G/Gu is injective and so H is commutative.
In fact, a torus. If H is contained in the centre of G the assertions about the centralizer
and normalizer are obvious. Let s ∈ H. Then s is contained in a maximal torus T . We
claim that T ⊃ H, otherwise the homomorphism T ×H → G shows that T is not maximal.
(In particular, we proved that every maximal torus contains the centre of the group.)
Otherwise, let s ∈ H be an element which is not in the centre of G. Then H ⊆ ZG (s),
which is a connected subgroup of G of smaller dimension. By induction on the dimension
of G we conclude that H is contained in a maximal torus T of ZG (s). It follows from
(1) of the theorem that ZG (s) contains a maximal torus of G, hence the dimension of the
maximal tori of G and of ZG (s) are the same. Thus, H is contained in a maximal torus
of G.
It remains to prove the assertion about NG (H). Let x ∈ NG (H) and h ∈ H. Then
xhx−1 h−1 ∈ H ∩ (G, G) ⊆ H ∩ Gu = {1},
and so x ∈ ZG (H).
Example 13.3.9. All Borel subgroups are conjugate. Thus, to study Borel subgroups
of GLn it is enough to consider Tn . Also, every maximal torus of GLn is contained in a
maximal connected closed subgroup, that is, in a Borel. Thus, every maximal torus of
GLn is conjugate to a torus of Tn . All maximal tori in a Borel are conjugate. Thus, every
maximal torus of GLn is conjugate to Dn .
The unipotent elements of Tn are Un , the upper unipotent matrices. The map
Dn × Un ,
(M, N ) 7→ M N,
is an isomorphism of varieties, but not of groups (unless n = 1) as Dn and Un do not
commute in Tn for n > 1. It is important to note that there are plenty of elements
in Tn that are semi-simple besides Dn . For example, any matrix whose diagonal entries
ALGEBRAIC GROUPS: PART IV
83
are distinct is semisimple. Every semisimple element is contained in a maximal torus, in
particular, every such element can be conjugated in Tn to a diagonal matrix, and vice-versa.
That is, the semisimple elements of Tn are the conjugates of the diagonal matrices.
Let us take a diagonal matrix and consider its centralizer in Tn . If the matrix is a scalar
then the centralizer is obviously Tn . Consider, as a sample case, a matrix of the form
diag(a, 1, . . . , 1), where a 6∈ {0, 1}. One calculates that the centralizer are the matrices
(tij ) in Tn such that t1j = 0 for j > 1. Thus, the centralizer is isomorphic to Gm × Tn−1 .
Example 13.3.10. Let us now look at Borel subgroups in Sp2n . It will be convenient to
change basis so that the pairing is given by
0 K
t
hx, yi = x
y,
−K 0
1
K :=
..
.
1
!
.
1
(K = K −1 .) As we have done before, the symplectic group acts transitively on the flag
variety of maximal isotropic flats. The stabilizer of the standard flag are the following
matrices:
A B
0 D
:
t
0 K
A B
0 K
A B
, A ∈ Tn .
=
−K 0
0 D
−K 0
0 D
One finds that this amounts to the following conditions:
D = K −1 · t A−1 · K,
A ∈ Tn ,
t
BKD is symmetric.
Note that if M = (mij ) and σ is permutation of {1, 2, . . . , n},
σ = (1n)(2 n − 2)(3 n − 2) · · · ,
then
K −1 M K = (mσ(i)σ(j) ).
Let
τ
A = K ·t A−1 · K.
Then A 7→ τ A is an automorphism that preserves Tn and furthermore, if A = (aij ) ∈ Tn
−1
then the diagonal entries of τ A are a−1
nn , . . . , a11 . This means that we can write the Borel
we have found as
B :=
A
0
B
τ
A
: A ∈ Tn , BKD symmetric .
t
ALGEBRAIC GROUPS: PART IV
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Note that these are upper triangular matrices lying also in GL2n and thus
A B
Bu =
∈ B : A ∈ Un .
0 τA
We find that the torus
−1
×
diag(t1 , . . . , tn , t−1
n , . . . , t1 ) : ti ∈ k
is a maximal torus since it maps isomorphically onto B/Bu .
As before, every maximal torus of Sp2n is conjugate to this torus and every maximal
torus of B is conjugate to this torus in B. Every Borel of Sp2n is conjugate to B. Every
semisimple element of B is conjugate to an element of the torus we found. As an example
of how a centralizer in B of such an element may look like, consider a diagonal matrix
diag(a, 1, . . . , 1, a−1 ). One calculates that the stablizer are the matrices of the form


a 0 ... 0
0 ... 0
0

A
B

.
τ

A 0 
a−1
The middle matrix here is the Borel of Sp2n−2 .
ALGEBRAIC GROUPS: PART V
EYAL Z. GOREN, MCGILL UNIVERSITY
Contents
14. General structure theorems for connected algebraic groups
15. Summary of some results so far
15.1. Some definitions
Date: Winter 2011.
84
85
93
94
ALGEBRAIC GROUPS: PART V
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14. General structure theorems for connected algebraic groups
Let G be a connected linear algebraic group. By a maximal torus of G we mean a
torus of G not properly contained in any other torus.
Theorem 14.0.1. Let G be a connected linear algebraic group. Any two maximal tori in
G are conjugate.
Proof. Every maximal torus, being connected and solvable, is contained in a Borel subgroup. We proved that all Borel subgroups are conjugate and all the maximal tori of a
Borel subgroup are conjugate (in that Borel subgroup).
Definition 14.0.2. Let G be a connected linear algebraic group and let T be a maximal
torus in G. Then dim(T ) is called the rank of G. It is independent of the choice of T .
Example 14.0.3. rk(GLn ) = n, rk(SLn ) = n − 1, rk(Sp2n = n, rk(Tn ) = n.
Proposition 14.0.4 (Rigidity of Tori). Let G and H be two diagonalizable groups and let
V be a connected affine variety. Assume given a morphism
φ : V × G → H,
such that for any v ∈ V the map x 7→ φ(v, x) is a homomorphism of algebraic groups
G → H. Then φ(v, x) is independent of v. (Colloquially, a family of homomorphisms
G → H indexed by a connected variety V is constant.)
The proposition states that one cannot continuously deform homomorphisms of diagonalizable groups. This makes sense heuristically as morphism of diagonalizable groups are
determined by their effect on characters groups, which are discrete objects.
Proof. Let ψ ∈ X ∗ (H) be a character of H. It is in particular a regular function on H and
so
φ∗ ψ(v, x) =
X
fχ,ψ (v)χ(x),
χ∈X ∗ (G)
with fχ,ψ ∈ k[V ]. Here we have used that k[V × G] = k[V ] ⊗k k[G] and that X ∗ (G) is a
basis for k[G] over k.
By our assumption, for a fixed v the sum
P
χ∈X ∗ (G)
fχ,ψ (v)χ(x) is a character of G,
because x 7→ φ(v, x) is a homomorphism of algebraic groups. By Dedekind’s independence
ALGEBRAIC GROUPS: PART V
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of characters, we must have fχ,ψ (v) = 1 for exactly one χ and zero for all the other χ. For
a fixed χ, the conditions fχ,ψ (v) = 1, or fχ,ψ (v) = 0 are closed conditions and as v varies
over an irreducible component of V they exhibit the component as a disjoint union of two
closed sets, thus one of them must be empty. For every irreducible component choose v
and χ such that fχ,ψ (v) = 1. Then fχ,ψ (v) = 1 for any v in that irreducible component
and fχ0 ,ψ (v) = 0 for any v in that component and any χ0 6= χ. The connectedness of V
now implies that for that χ, fχ,ψ ≡ 1 and for any χ0 6= χ, fχ0 ,ψ ≡ 0.
Proposition 14.0.5. Let G be a linear algebraic group and H a diagonalizable subgroup
of G. Then (i) NG (H)0 = ZG (H)0 , (ii) ZG (H) and ZG (H)0 are normal in NG (H), and
(iii) the groups NG (H)/ZG (H), NG (H)/ZG (H)0 , are finite.
NG (H)
ZG (H)
finite, normal
NG (H)0 = ZG (H)0
Proof. Let z ∈ ZG (H), n ∈ NG (H) and h ∈ H. Then (nzn−1 )h = nz(n−1 hn)n−1 =
n(n−1 hn)zn−1 = h(nzn−1 ) which proves that ZG (H) is a normal subgroup of NG (H).
Since ZG (H)0 is a characteristic subgroup of ZG (H) it is normal in NG (H) too (and this
will also follow from NG (H)0 = ZG (H)0 ).
Apply the previous proposition with
NG (H)0 × H → H,
(n, h) 7→ nhn−1 .
This map is therefore independent of n. For n = 1 it is the identity. Thus, NG (H)0 ⊂
ZG (H) and so NG (H)0 ⊂ ZG (H)0 ; the other inclusion being obvious, we get NG (H)0 =
ZG (H)0 .
The group NG (H)/NG (H)0 is finite and NG (H)/ZG (H) is a quotient of it.
The following Lemma is of independent interest. It is also useful in practice in calculating
centralizers of tori, as it reduces the calculation to calculating the centralizer of a single
element that the proof explains how to choose.
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Lemma 14.0.6. Let G be a linear algebraic group and S a torus of G. There exists s ∈ S
such that ZG (s) = ZG (S).
Proof. Embed G in GLn so that S are diagonal matrices. We may then assume, without
loss of generality, that G = GLn . S being diagonal, its diagonal entries are characters
s ∈ S 7→ sii . Let χ1 , . . . , χm be the distinct characters obtained this way. Let s0 ∈ S be
an element such that χi (s0 ) 6= χj (s0 ) for i 6= j. Such s0 exists, because each of the finitely
many characters χi /χj is equal to 1 on a codimension 1 subtorus of S.
The inclusion ZG (s) ⊃ ZG (S) is obvious. We may arrange the coordinates so that
S = {diag(χ1 (s), . . . χ1 (s), χ2 (s), . . . , χ2 (s), · · · , χm (s), . . . χm (s)) : s ∈ S)}.
|
{z
} |
{z
}
|
{z
}
a1
a2
am
Then one can easily check that ZG (s) ∼
= GLa1 × GLa2 × · · · × GLam and that this group
centralizes S as well. (Remember that this is the centralizer in GLn . If one wants to
apply that to the original G, it is the intersection with the image G under an embedding
diagonalizing S in a very specific manner.)
Example 14.0.7. Consider the torus {diag(t1 , . . . , tm , 1, . . . , 1)} in GLn . Its centralizer is
therefore {diag(t1 , . . . , tm )} × GLn−m .
Let G be a connected linear algebraic group. A Cartan subgroup of G is the identity
component of the centralizer of a maximal torus of G. (We shall see later that in fact the
centralizer of a maximal torus is already connected.)
Proposition 14.0.8. Let G be a connected algebraic group. Let T be a maximal torus and
C = ZG (T )0 the corresponding Cartan subgroup. Then:
(1) C is nilpotent and T is its unique maximal torus. In particular, C is contained in
some Borel subgroup.
(2) There exists elements t ∈ T lying in only finitely many conjugates of C.
Proof. We will need a lemma.
Lemma 14.0.9. Let G be an algebraic group and B a Borel subgroup of G. If B is nilpotent
then B = G0 .
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Proof of Lemma. The proof is by induction on dim(G). The case dim(G) = 0 is obvious.
In general, since B is nilpotent, B has a non-trivial closed connected central subgroup J,
for example that generated by commutators of maximal length that are not yet trivial.
Since C(B) ⊆ C(G), J is normal in G. We may then pass to G/J and B/J (the image of
Borel is Borel, as we have proven) and conclude by induction.
Now, C contains T as a central subgroup; choose a Borel subgroup B of C containing T .
Then T is a central subgroup of B. In this case the isomorphism of varieties T × Bu → B
is also an isomorphism of algebraic groups and so, since Bu is nilpotent, B is nilpotent.
By the lemma B = C. In particular, C is nilpotent and T is its central torus, C = T × Cu .
To prove (2) we choose an element t ∈ T such that ZG (t) = ZG (T ). If t ∈ gCg −1 then
g −1 tg ∈ C = ZG (T )0 and so T ⊂ ZG (g −1 tg) = ZG (g −1 T g) and must be the maximal torus
of ZG (g −1 T g). But, ZG (g −1 T g) also contains g −1 T g as a maximal torus. Thus, T = g −1 T g
and hence g ∈ NG (T ). As NG (T )/ZG (T ) is finite, there are only finitely many conjugates
gCg −1 containing t (note that gCg −1 depends only on the coset gZG (T ) 1).
The next theorem is a very important theorem. It explains the special role played by tori
and Borel subgroups in the study of linear algebraic groups.
Theorem 14.0.10. Let G be a connected linear algebraic group.
(1) Every element of G lies in some Borel subgroup.
(2) Every semisimple element of G lies in a maximal torus.
(3) The union of the Cartan subgroups of G contains a dense open subset of G. (So
“almost” every element of G lies in a Cartan subgroup.)
Before the proof we need a lemma.
Lemma 14.0.11. Let H be a closed subgroup of a connected linear group G. Let X =
∪g∈G gHg −1 and X̄ the Zariski closure of X.
(1) X contains a non-empty open subset of X̄. If H is parabolic then X = X̄.
(2) Assume that H has a finite index in its normalizer and that there exist elements of
H lying in only finitely many conjugates of H. Then X̄ = G.
1We
are not claiming though that distinct cosets give distinct conjugates. This would be the case of
ZG (T ) = NG (T ), but this is almost never the case.
ALGEBRAIC GROUPS: PART V
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Proof of lemma. We may assume H is connected. We can view X as the image of G × H
under the morphism G × H → G, (g, h) 7→ ghg −1 . It follows that X is constructible, hence
contains a non-empty open subset of X̄.
It is also useful to view X as the image of a different morphism. Consider the isomorphism φ : G × G → G × G given by
φ(x, y) = (x, x−1 yx).
Let Y = {(x, y) : x−1 yx ∈ H}. Then Y = φ−1 (G × H). We have a commutative diagram
Y
φ
/
G×H .
w
ww
(x,y)7→y
ww
w
{www (x,y)7→xyx−1
G
Y is a closed subset of G × G and it is H-closed; that is, if h ∈ H, (x, y) ∈ Y then
(xh, y) ∈ Y . Indeed, φ(xh, y) = h−1 (x−1 yx)h ∈ H. Therefore, if H is parabolic, then the
image of Y under the projection p2 : G × G → G, namely X, is closed in G.
We may now let Ȳ be the image of Y in G/H × G. Since Y is closed and equal to the
pre-image of Ȳ , we can conclude from G → G/H being an open map that Ȳ is a closed set
of G/H × G, hence a variety. Ȳ is irreducible, being the image of Y ∼
= G × H. We have
dim(Ȳ ) = dim Y − dim H = dim(G).
We have an induced map
p̄2 : Ȳ → G,
(xH, y) 7→ y.
−1
Let y be an element of H that belongs to finitely many cosets of H, say t1 Ht−1
1 , . . . , tm Htm .
The preimages of y in Y are the elements (x, y) such that x−1 yx ∈ H, or, y ∈ xHx−1 . Such
x is therefore of the form x = ti n for some n ∈ NG (H). It follows that the preimages of y in
Ȳ are the elements (ti αj H, y) where the αj are the finite number of cosets representatives
for NG (H)/H. In particular, the fiber of Ȳ → X over y is finite, hence zero-dimensional.
Thus, the fibre is zero-dimensional over almost any point of X. It follows that
dim(X) = dim(Ȳ ) = dim(G).
Thus, X̄ = G.
Proof of the theorem. Let T be a maximal torus and C = ZG (T )0 the Cartan subgroup.
Suppose that x ∈ NG (C) then x must also conjugate the only maximal torus of C
ALGEBRAIC GROUPS: PART V
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(Proposition 14.0.8) to itself. Thus, x ∈ NG (T ). On the other hand, NG (T ) normalizes
ZG (T )0 = NG (T )0 ,as we have shown above (Proposition 14.0.5). Thus,
NG (C) = NG (T )
It follows that C = ZG (T )0 has finite index in NG (C) and so if we apply the Lemma
with H = C, the conditions of part (ii) hold, because we had also proven that there is
an element of T (hence of C) that lies in only finitely many conjugates of C. Thus, we
conclude that
G = Zariski closure(∪x∈G xCx−1 ),
and (3) follows.
Now, we know C is connected and nilpotent, hence solvable. Thus C is contained in
some Borel subgroup B. Since a Borel is parabolic, the Lemma gives that ∪x∈G xBx−1 is
a closed set. It contains the dense set ∪x∈G xCx−1 . Thus,
G = ∪x∈G xBx−1 .
Finally, let s be a semisimple element of G. Then s lies in some Borel B and we can apply
Theorem 13.3.5 to deduce that x lies in a maximal torus of B (so of G).
Corollary 14.0.12. Let B be a Borel subgroup of a connected linear algebraic group G.
Then
C(B) = C(G).
Proof. We had already proven that C(B) ⊆ C(G). Let g ∈ C(G); g belongs to some Borel
subgroup. Since all Borel subgroups are conjugate and g is fixed by conjugation, g belongs
to all Borel subgroups. Thus, g ∈ C(B).
Theorem 14.0.13. Let S be a subtorus of a connected linear algebraic group G.
(1) The centralizer ZG (S) is connected.
(2) If B is a Borel subgroup containing S then ZG (S) ∩ B is a Borel subgroup of ZG (S).
Every Borel subgroup of ZG (S) is obtained this way.
Proof. We will only prove here the first part. The proof of the second part is completely
within our means now, but in the interest of time we don’t give it. See Springer’s theorem
6.4.7 for the proof.
ALGEBRAIC GROUPS: PART V
91
Let g ∈ ZG (S) and let B be a Borel subgroup containing g. Let
X = {xB ∈ G/B : x−1 gx ∈ B}.
Note that X = (p̄2 )−1 (g) in the notation of Lemma 14.0.11 (used for H = B), and so
X is a closed subset of G/B. Thus, X is a complete variety. The torus S acts on X
by (s, xB) 7→ sxB (since g centralizes S). Using Borel’s fixed point theorem, there is an
element xB of X that is fixed by the action of S: sxB = xB, ∀s ∈ S. That is, x−1 Sx ⊂ B.
Since B is connected solvable, we can apply Corollary 13.3.8 to H = x−1 Sx and conclude
that x−1 gx, which is in the centralizer of x−1 Sx and also in B (by definition of X) belongs
to the identity component of ZB (x−1 Sx). Thus, g belongs to the identity component
of ZxBx−1 (S) and hence to the identity component of ZG (S). We proved that ZG (S) is
connected.
Corollary 14.0.14. Let S be a torus of a connected linear algebraic group G. Then
ZG (S) = ZG (S)0 = NG (S)0 and is of finite index in NG (S). In particular, this holds for a
maximal torus T . Thus, a Cartan subgroup is the centralizer of a maximal torus (no need
to add anymore that it is the connected component of that centralizer). The finite group
NG (T )/NG (T )0 = NG (T )/ZG (T ) =: W (G, T )
is called the Weyl group of G relative to T . Any two Weyl groups of G are conjugate
(because maximal tori are), hence isomorphic, but not canonically.2
Corollary 14.0.15. Let B be a Borel subgroup containing a maximal torus T then B
contains its Cartan subgroup C.
Proof. The theorem tells us that B ∩ C is a Borel subgroup. However, if T is a maximal
torus, its centralizer C is a nilpotent subgroup (Proposition 14.0.8), hence solvable. Thus,
every Borel of C is equal to C. That is, B ⊇ C.
We come now to a very important theorem in the structure of algebraic groups. Unfortunately, its proof employs techniques appearing in parts of Springer’s book that we didn’t
cover. Thus, we omit the proof. But see the remark following the theorem for a weaker
statement.
2One could say canonically up to inner automorphism. However such groups often contain a large
symmetric group and so the distinction between “automorphism” and “inner automorphism” is not so
important.
ALGEBRAIC GROUPS: PART V
92
Theorem 14.0.16. Let G be a connected linear algebraic group and B a Borel subgroup
of G. Then,
NG (B) = B.
Remark 14.0.17. We can at least show that B = NG (B)0 and thus see why it is of finite
index in NG (B).
The group NG (B)0 is connected and B is its Borel subgroup; in fact, its only Borel
subgroup, since all Borel subgroups are conjugate. But, every element of NG (B)0 belongs
to some Borel subgroup. Thus, NG (B)0 = B.
Corollary 14.0.18. Let G be a connected linear algebraic group. Let P be a parabolic
subgroup of G. Then P is connected and NG (P ) = P .
Proof. P contains a Borel subgroup B and, in fact, B ⊂ P 0 . Let x ∈ NG (P ) then
xBx−1 is another Borel subgroup of the connected linear algebraic group P 0 and so since all Borels are conjugate in P 0 , there is a y ∈ P 0 such that xBx−1 = yBy −1 . Then,
y −1 x ∈ NG (B) = B and so x ∈ yB ⊂ P 0 . That is, P 0 ⊃ NG (P ) ⊃ P ⊃ P 0 and everything
follows.
Corollary 14.0.19. Let P be a parabolic subgroup of a connected linear algebraic group
G. Then,
ZG (P ) = C(P ) = C(G).
Proof. Let P be a parabolic subgroup. Then, since ZG (P ) ⊂ NG (P ) = P , we find that
ZG (P ) = C(P ) is the centre of P . Let B be a Borel subgroup contained in P . Then,
as we proved before, C(B) = C(G) and so C(B) ⊆ C(P ). On the other hand, certainly
C(P ) = ZG (P ) ⊂ ZG (B) = C(B). Thus, C(P ) = C(B) = C(G).
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15. Summary of some results so far
Let G be a connected algebraic group. We have 3 notions that are prominent so far: (i)
a maximal torus; (ii) a Borel subgroup; (iii) a parabolic subgroup.
We also have three “operations”: (i) take the centralizer; (ii) take the normalizer; (iii)
conjugate.
It is interesting to examine the knowledge we have so far in light of these concepts and
operations. We know the following.
(1) Every element belongs to some Borel subgroup.
(2) Every semisimple element belongs to some maximal torus.
As to conjugation:
(1) A conjugate of a maximal torus is a maximal torus and any two maximal tori are
conjugate. In particular, they all have the same dimension. Every maximal torus
is contained in some Borel B, hence the dimension of maximal tori is dim(B/Bu ).
(2) A conjugate of a Borel is a Borel and all Borel are conjugate.
(3) A conjugate of a parabolic is parabolic.
As to inclusions:
(1) Every parabolic contains a Borel. Every Borel is parabolic.
(2) Every Borel contains a maximal torus, every maximal torus is contained in some
Borel.
(3) The centralizer of a maximal torus T - a Cartan subgroup C = C(T ) - is connected,
nilpotent, and is equal to NG (T )0 and is contained in every Borel containing T . We
have NG (C) = NG (T ).
(4) The normalizer of a parabolic is equal to the parabolic. The centralizer of a parabolic is equal to its centre and is equal to the centre of the ambient group G.
It is now interesting to look at collections.
(1) The collection of Borel subgroups of G is in bijection with the projective variety
G/B, where B is some Borel. More generally, the collection of Borel subgroups
contained in a given parabolic subgroup P is in bijection with the projective variety
P/B, where B is some Borel of P .
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(2) The collection of maximal tori of G is in bijection with G/NG (T ), which has a finite
cover by G/ZG (T ) (often T = ZG (T )). The collection of maximal tori contained in
a fixed Borel subgroup is in bijection with B/NB (T ) = B/ZB (T ).
(3) The collection of Borel subgroups containing a given maximal torus T is in bijection
with NG (T )/ZG (T ) = W (G, T ) - the Weyl group (which is a finite group).
(4) Let B be a Borel subgroup. Out of every conjugacy class of parabolic subgroups
there’s exactly one element containing B. In particular, the parabolic subgroups
containing B are in bijection with conjugacy classes of parabolic subgroups. These
can be classified by means of roots.
We have more or less proven all these assertions. The assertions we didn’t prove can be
deduced quite easily from those we have proved. For example, one needs that if P, Q are
conjugate parabolic containing a Borel B then P = Q. Indeed, if P = xQx−1 then the
Borels B and xBx−1 are contained in P and so conjugate in P . Thus, for some y ∈ P ,
B = yxBx−1 y −1 . Then yx ∈ NP (B) = B and so x ∈ y −1 B ⊆ P and so P = Q.
All this, while beautiful, doesn’t explain completely why the notions we were occupied
with are so central. It is the role the play in the classification of algebraic groups and their
representations that makes them central to the whole theory.
15.1. Some definitions. Let A, B be closed normal subgroups of a connected algebraic
group G. Then AB is also a closed normal subgroup, connected if A and B are, solvable
if A and B are. Thus, there is a maximal closed connected solvable and normal subgroup
of G. This group is called the radical of G and will be denoted here R(G). A group is
called semi-simple if R(G) = {1}.
Similarly, if A and B are normal and unipotent then so is AB. Thus, there is a maximal
closed connected unipotent normal subgroup of G. This group is called the unipotent
radical of G and is denoted here Ru (G).
How to calculate these? Note that the radical being connected and solvable is contained
in a Borel subgroup. Thus, being normal, it is contained in all Borel subgroups and being
connected, R(G) ⊆ (∩B Borel B)0 . On the other (∩B Borel B)0 is a closed connected normal
solvable subgroup of G, hence contained in the radical. Thus:
R(G) = (∩B Borel B)0 .
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In the same way,
Ru (G) = (∩B Borel Bu )0 .
Since for every Borel B, R(G)u ⊆ Bu , and being connected (as R(G) is connected solvable)
R(G)u ⊆ Ru (G). On the other hand, Ru (G) is contained in R(G) and consists of unipotent
elements, hence Ru (G) ⊆ R(G)u . Thus,
Ru (G) = R(G)u .
A connected linear algebraic group G is called semisimple if R(G) = {1} and reductive
if Ru (G) = {1}.
Example 15.1.1. Suppose that the centre of the group is positive dimensional, equivalently, C(G)0 is not trivial. It is easy to see that C(G)0 ⊆ R(G). Then G cannot be
semisimple. For example, we see that GLn is not semisimple.
On the other hand, GLn is reductive. We know that Gm = C(GLn ) ⊂ R(GLn ). The two
Borel subgroups Tn and t Tn show the radical is contained in Dn = Tn ∩ t Tn . On the other
hand, any diagonal matrix having two non-equal entries is conjugate to a non-diagonal
matrix. For example, diag(2, 1, a, b, c, . . . ) is conjugate to diag(( 2 11 ) , a, b, c, . . . ). That
implies that the largest normal subgroup of GLn contained in Dn are the scalar matrices
Gm . Thus,
R(GLn ) = k × · In ∼
= Gm ,
Ru (GLn ) = {1}.
On the other hand, SLn is semisimple. If G is a closed normal subgroup of H then it is
not hard to check that R(G) is a normal subgroup of H (in fact, R(G) is a characteristic
subgroup of G and in particular preserved under the automorphisms induced by conjugation by elements of H). Thus, R(G) ⊆ R(H). Thus, R(SLn ) ⊆ R(GLn ) = k × · In . But
the only elements of determinant one in that group is the finite subgroup of n-th roots of
unity. Since R(SLn ) is connected, it must be trivial.
Example 15.1.2. Let P be a parabolic subgroup in GLn then P is not reductive. Indeed,
if we write P as the matrices




M =


A1
∗
A2
∗ ...
∗ ...
A3 . . .
..
.

∗
∗

∗
,


At
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(Ai of size ai ) the the subgroup of P where all the Ai are identity matrices is normal,
connected, unipotent, hence contained in Ru (P ). Indeed, this subgroup is the kernel of
the homomorphism P → GLa1 × · · · × GLat , M 7→ diag(A1 , . . . , At ).
Proposition 15.1.3. Let G be a connected linear algebraic group. Then G/R(G) is
semisimple. Its rank is called the semisimple rank of G. Similarly, G/Ru (G) is reductive.
Proof. Let H be a normal connected closed and solvable subgroup of G/R(G). Then, it
preimage in G, say H̃ is a closed connected normal subgroup. It is also solvable, because it
sits in an exact sequence 1 → R(G) → H̃ → H → 1, where both R(G) and H are solvable.
Thus, H̃ ⊂ R(G) and so H = {1}.
The proof for G/Ru (G) is the same, where now one needs that if 1 → Ru (G) → H̃ → H → 1
is an exact sequence and H is unipotent so is H̃. Indeed, if it had a semisimple element h its
projection to H would be both semisimple and unipotent and so trivial. Thus, h ∈ Ru (G)
and again semisimple and unipotent, hence trivial.
We have the following additional useful results:
• If G is reductive then R(G) is a maximal torus. G = R(G)·(G, G) and R(G)∩(G, G)
is finite. Thus, R(G) × (G, G) → G is a surjective homomorphism with a finite
kernel (isogeny). Further, (G, G) is semisimple.
• If G is semisimple then G = (G, G) and it has a finite centre.
• If G is reductive and T is a maximal torus of G then C(T ) = T , that is, Cartan
subgroups are tori, and C(G) ⊆ T . Further, if S is any torus then ZG (S) is
connected and reductive.
• If G is connected, semi-simple, of rank 1 then G ∼
= SL2 or PSL2 .