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1:34 p.m. April 11, 2013
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The structure of reductive groups
Bill Casselman
University of British Columbia, Vancouver
[email protected]
An algebraic group defined over F is an algebraic variety G with group operations specified in algebraic
terms. For example, the group GLn is the subvariety of (n + 1) × (n + 1) matrices
A 0
0 a
with determinant det(A) a = 1. The matrix entries are well behaved functions on the group, here for
example a = det−1 (A). The formulas for matrix multiplication are certainly algebraic, and the inverse
of a matrix A is its transpose adjoint times the inverse of its determinant, which are both algebraic.
Formally, this means that we are given
(a) an F -rational multiplication map G × G −→ G;
(b) an F -rational inverse map G −→ G;
(c) an identity element—i.e. an F -rational point of G.
I’ll look only at affine algebraic groups (as opposed, say, to elliptic curves, which are projective varieties).
In this case, the variety G is completely characterized by its affine ring AF [G], and the data above are
respectively equivalent to the specification of
(a’) an F -homomorphism AF [G] −→ AF [G] ⊗F AF [G];
(b’) an F -involution AF [G] −→ AF [G];
(c’) a distinguished homomorphism AF [G] −→ F .
The first map expresses a coordinate in the product in terms of the coordinates of its terms. For example,
in the case of GLn it takes
X
xik 7−→
j
xij ⊗ xjk .
In addition, these data are subject to the group axioms. I’ll not say anything about the general theory of
such groups, but I should say that in practice the specification of an algebraic group is often indirect—as
a subgroup or quotient, say, of another simpler one.
Contents
1.
2.
3.
4.
5.
6.
7.
8.
9.
Algebraic tori
Split reductive groups
Automorphisms
The Bruhat decomposition
From root datum to group
Quasi-split reductive groups
Reductive groups over finite fields
Unramified groups over a p-adic field
References
The standard references are [Steinberg:1968], [Humphreys:1975], [Borel:1991], and [Springer:1998].
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1. Algebraic tori
The standard reference for the material in this section is §8 of [Borel:1991].
The multiplicative group over a field F is that of non-zero elements of F . It is isomorphic to the group
of points in F 2 satisfying the equation xy = 1, and is hence an algebraic group Gm . Its affine ring is
the quotient F [x, y]/(xy − 1), which may be expressed as F [x, 1/x]. Similarly, the product (Gm )n of n
±1
copies of Gm has affine ring F [x±1
1 , . . . , xn ].
An algebraic torus defined over a field F is by definition an algebraic group defined over F that is
isomorphic to a product (Gm )n after base extension to an algebraic closure of F . For example, the
equation x2 + y 2 = 1 defines an algebraic group over R, and since
x2 + y 2 = (x + iy)(x − iy)
a change of coordinates identifies the group over C with Gm .
A torus defined over F that is isomorphic to some (Gm )n over the field F itself is called an
F -split torus.
SPLIT TORI.
A character of a split torus T is an algebraic homomorphism from it to Gm , which means a ring
∗
homomorphism from F [x±1 ] to F [x±1
i ] compatible with group operations. Define X (T ) be the group
of all characters of the split torus T .
Suppose we have chosen an identification f of T with (Gm )n . To k = (ki ) in Zn corresponds the
character
Y
xk = (xki i ) 7−→
xki i .
This gives us a group homomorphism f ∗ from Zn to X ∗ (T ). On the other hand, let X∗ (T ) be the group
of algebraic homomorphisms from Gm to T —the covharacters of T . To each ℓ = (ℓi ) in Zn corresponds
the map from Gm to T taking x to (xℓi i ), and this gives us a map f∗ from Zn to X∗ (T ).
[xstarT] 1.1. Proposition.
The maps f∗ and f ∗ defined just above are isomorphisms.
Proof. Since the characters of Gm itself are all of the form x 7→ xk and
(x1 , . . . , xn ) = (x1 , 1, . . . , 1) · · · (1, . . . , 1, xn ) ,
Q
the map f ∗ is surjective. If xki i = 1 for all (xi ) then each xki = 1 for all x, hence every ki = 0. Thus
f ∗ is injective. The argument for X∗ is similar.
[star-lin-ind] 1.2. Proposition.
The characters of T form a linear basis of AF [T ].
×
Proof. After choosing coordinates, embed X ∗ (T ) in Hom(T (F ), F ) and apply the following well
known result:
[lin-ind-chars] 1.3. Lemma.
If G is a group and E a field, any set of distinct homomorphisms
χ: G −→ E ×
are linearly independent over E .
I postpone the proof to the end of this section.
The two free Z-modules X ∗ (T ) and X∗ (T ) are canonically dual to each other, which means that
X ∗ (T ) = HomZ X∗ (T ), Z .
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Why is this? Given λ∨ in X∗ (T ) and µ in X ∗ (T ), the composite µ ◦ λ∨ is a character og Gm , hence of the
form x 7→ xn for some unique n, which is defined a hµ, λ∨ i. In short
∨
µ ◦ λ∨ : x 7−→ xhµ,λ i .
A split torus T is completely and simply characterized by X ∗ (T ). For one thing, we may describe the F rational points of the torus as Hom(X ∗ (T ), F × ). For another, if we are given an algebraic homomorphism
of tori ϕ: S → T , there exists a corresponding map ϕ∗ : X ∗ (T ) → X ∗ (S). This is functorial. Conversely,
since AF (T ) is the group algebra of X ∗ (T ), a homomorphism X ∗ (T ) → X ∗ (S) gives rise to an algebraic
homomorphism. Thus the category of split algebraic tori is contravariantly equivalent to that of free
Z-modules of finite rank.
In particular the group of automorphisms of T may be canonically identified with GL X ∗ (T ) , which
may in turn be non-canonically identified with GLn (Z). In other words, automorphisms of a split torus
are of the form
Y
(xi ) 7−→
a
xj i,j
j
where ai,j is an integral matrix of determinant ±1. In particular all automorphisms are invariant under
automorphisms of the field of definition, since the equations of the automorphism do not involve the
field.
The following is essentially a characterization of tori among Zariski-connected algebraic groups.
[tori-reductive] 1.4. Proposition.
Every F -rational algebraic representation of a split torus is a direct sum of characters.
Proof. A representation of T is an algebraic homomorphism from T to GLn (F ). Because it is algebraic
and the affine ring of T is isomorphic to F [x±1
i ], we may write
π(t) =
X
cχ χ(t) ,
χ
where each χ is a character of T and each cχ is an n × n matrix. Since π(1) = I we see that
I=
X
cχ ,
while since χ(st) = χ(s)χ(t) and the χ are linearly independent we see that
cχ cρ =
n
cχ if χ = ρ
0 otherwise.
This implies that F n is the direct sum of the images of the idempotent operators cχ , on which T acts as
χ.
CLASSIFICATION .
How do we classify algebraic tori over an arbitrary field? We know that every torus
becomes split over an algebraically closed extension of the base field. But one can do better:
[split-separable-closure] 1.5. Theorem.
Every torus defined over F splits over some finite Galois extension.
I’ll give the proof at the end of this section.
Say T splits over the a Galois extension E/F , with Galois group G . The endomorphism ring of a
split torus of dimension n is isomorphic to Mn (Z) independently of the field of definition, and all
automorphisms are Galois-invariant. The theory of Galois descent (see [Serre:1959]) therefore asserts
that isomorphism classes of tori defined over
F that split over E are in bijection with conjugacy classes
of homomorphisms from G to GL X ∗ (T ) , each of which then gives rise in turn to an automorphism of
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the E -rational points on the torus. If that associated to σ is ϕσ , then the group of F -rational points on
the new torus is to be identified with the points x of T such that ϕσ (x) = xσ for all σ .
Example. The real group
x2 + y 2 = 1
becomes isomorphic to Gm upon base field extension to C. Its character group may be identified with
Z, with conjugation acting as z 7→ z −1 , since (x + iy)−1 = x − iy .
Example. Suppose E/F to be a separable quadratic extension. The multiplicative group E × may be
identified with the group of F -rational points on a two-dimensional torus over F . This may be seen in
two ways.
• The group Gm over E is the group of (w, z) with wz = 1. Let 1, α be an F -basis of E/F , with
α2 − Bα + C = 0 .
The equations for Gm oevr E can be written as
1 = (u + vα)(x + yα)
= ux + vyα2 + (uy + vx)α
= ux + vy(Bα − C) + (uy + vx)α
= (ux − Cvy) + (Bvy + uy + vx)α ,
which amount to the pair of equations
ux − Cvy = 1
Bvy + uy + vx = 0
defining a two-dimensional group over F .
• Consider the torus G2m over E , and let conjugation swap the factors. This corresponds also to a swap
of factors in Z2 . Its group of E -rational points may be identified with (E × )2 , that of its F -rational points
may be identified with the pairs (x, x) in (E × )2 .
EXERCISE.
Verify that the algebraic groups one gets in these two ways are isomorphic.
Example. More generally, suppose L/F to be a finite separable extension, say with [L: F ] = r. Suppose
E/F to be a Galois extension containing L, If G(L) is the subgroup fixing elements of L, then G acts
on G/G(L) by left multiplication, effectively becoming a group of permutations of the r cosets. This
gives rise in turn to a representation on Z[G/G(L)], a free module of rank r over Z, or equivalently a
homomorphism into GLr (Z). The associated torus RL/F Gm is the group obtained by restriction of
scalars from L to F of Gm , whose group of F -rational points is L× .
Restriction of scalars is an operation that may be applied to any affine algebraic variety. For example, if
E/F is any finite extension then RE/F Ga has the property that its F -rational points may be canonically
identified with E . As an algebraic group it is just isomorphic to G2a , but there is no canonical isomorphism
with G2a —in other words, there is no canonical basis of E as an F -vector space.
Example. Suppose E/F to be a separable quadratic extension, say with Galois conjugation σ . There
are two homomorphisms from G to Z× = {±1}, one trivial and the other mapping conjugation to
−1. The first just gives us the trivial torus whose group of real points is F × . The other is a torus for
which conjugation takes the complex character into’ its inverse, whose group of F -rational points may
be identified with {z = 1/z} in E × . A third torus is what we get from Gm by restriction of scalars from
E to F . This is associated to the two-dimensional representation of G swapping coordinates. That’s
essentially all there are:
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[quadratic-tori] 1.6. Proposition.
5
Every torus over F that splits over E is a direct product of copies of these three tori.
Proof. This amounts to the claim that if σ is an involution of a lattice L, then L decomposes into a direct
sum of sublattices (a) L+ on which σ = I ; (b) L− on which σ = −I ; (c) L∗ , on which σ is a direct sum of
swaps. Let M+ be the subgroup of L on which σ = I , M− that on which σ = −I . Since
x=
x + σ(x) x − σ(x)
+
2
2
we know that 2L ⊆ M = M+ ⊕ M− . Applying the principal divisor theorem, we get a basis ei of L
such that the vectors ei for i ≤ m and 2ei for i > m are a basis of M . The quotient L/M is a vector
space over F2 on which σ acts. The involution σ has no eigenvalues, so its Jordan form must be a sum
of unipotents. But over F2 a unipotent element a 7→ b, b 7→ a + b is a swap of basis elements a, a + b.
Given an algorithm for implementing the principal divisor theorem, this proof is constructive. This
result may be applied in particular to the case of C/R to classify completely all real algebraic tori. For
details, see [Casselman:2007].
—————
♣ [lin-ind-chars] Proof of Lemma 1.3. I’ll prove by induction on n that if
X
c i χi = 0
χ
with n coefficients ci then all the ci vanish. For n = 1 this is immediate because the values of the
characters are non-zero.
Suppose this to be true for m < n and suppose such a relation
n
X
(1.7 )
[lindepchar]
c i χi = 0
i=1
where we may assume cn = 1. Substituting xg for x in
n−1
X
ci χi (x) + χn (x) = 0
i=1
we get
0=
=
n−1
X
i=1
n−1
X
ci χi (x)χi (g) + χn (x)χn (g)
χn (g)−1 χi (g)ci χi (x) + χn (x)
i=1
♣ [lindepchar] and subtracting this from the original equation (1.7) we get
n−1
X
i=1
ci χn (g)−1 χi (g) − 1 χi (x)
for each x. By induction, each coefficient vanishes. Choose g such that χn (g) 6= χ1 (g). But then by
induction each ci = 0 for i ≤ n − 1, hence also cn = 0.
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—————
[split-separable-closure]
Proof of Theorem 1.5. I follow §X of [Tits:1968]. It suffices to show that every character of T is defined
over a finite Galois extension, and since the group of characters has finite rank, it suffices to show that if
F is separably closed then every character χ of T is defined over F . What we know is that all characters
are defined over the algebraic closure F . Let the ψi be any F -basis of AF [T ]. Thus we have a finite sum
χ=
X
ai ψi
for some ai in F . Since χ is rational over F it is fixed by all automorphisms of F , so the coefficients will
−r
all be contained in some F p . If q = pr then
χq =
X
aqi ψiq
X
ai ψi (tq )
lies in AF [T ]. But χq (t) = χ(tq ), so we also have
χ(tq ) =
for all t. The map t 7→ tq is an isomorphism of T (F ) with itself, so the functions ψi (tq ) are also a basis
of AF [T ], and the left-hand side is in AF [T ], so all the ai lie in F .
[split-separable-closure] Theorem 1.5 is the same as Proposition 8.11 in [Borel:1991]. The proof there is quite different.
2. Split reductive groups
If F is an algebraically closed field of characteristic 0, a reductive algebraic group may be defined as one
for which all finite-dimensional representations split into a sum of irreducible representations. Over an
arbitrary field of characteristic 0, a reductive group is one that becomes reductive in this sense upon base
♣ [tori-reductive] field extension. Thus, according to Proposition 1.4 a torus is a reductive group. One could take this as
the definition of reductive groups over fields of arbitrary characteristic, but that would turn out to be an
unfruitful idea. The proper definition of a reductive group is slightly technical.
Suppose for the moment that the field F is algebraically closed. A Zariski-connected algebraic group G
defined over F is called solvable if it possesses a filtration
[filtration-solvable] (2.1 )
G0 = {1} ⊂ G1 ⊂ · · · ⊂ Gn = G
with each Gk /Gk−1 isomorphic to either Gm or Ga . It is a torus if the factors here are all copies of Gm ,
and of course every torus is solvable. It is said to be unipotent if all factors are isomorphic to Ga . The
standard example of a solvable group is that of upper triangular matrices in GLn .
Every solvable group fits into a short exact sequence
1 −→ N −→ G −→ T −→ 1
where N is unipotent and T a torus. In particular, every short exact sequence of tori splits. This is
Theorem 10.6 of [Borel:1969/91]. The unipotent group N is called the unipotent radical of G. The torus
T is isomorphic to G/N , so its isomorphism class is determined canonically by G. All maximal tori in G
are conjugate in G(F ).
If F is arbitrary, the F -group G is said to be solvable (or unipotent) if it becomes solvable (or unipotent)
♣ [filtration-solvable] over an algebraic closure of F . A unipotent group is said to be split if it has a filtration (2.1) with each
Gi /Gi−1 isomorphic to Ga over F . Every unipotent group in characteristic 0 is split, but this is not true
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in characteristic p > 0. However, in the theory of reductive groups, which is what we shall be most
interested in, this will not be significant.
The radical of a connected algebraic group is its largest normal solvable subgroup. The group is called
reductive if its radical is a torus, and semi-simple its radical is trivial. In characteristic 0 this turns
out to be equivalent to the reducibility of F -rational finite-dimensional representations, but is not in
characteristic p > 0.
Suppose G to be a connected reductive group defined over F . It is said to be split if it contains a maximal
torus that is split. In these circumstances there exists also a maximal solvable group B defined over
F that contains T , and it is connected. All the maximal solvable F -rational subgroups of G are then
conjugate in G(F ). They are called Borel subgroups. The unipotent radical of a Borel subgroup in a
split reductive group is always split (this is not quite trivial). The maximal torus T is its own centralizer
in G, and B is its own normalizer. Any element of G that conjugates both T and B into themselves will
lie in T .
♣ [tori-reductive] According to Proposition 1.4, the adjoint action of T on the Lie algebra decomposes into a direct sum
of eigenspaces gλ for certain λ ∈ X ∗ (T ). The eigenspace for the trivial character is the Lie algebra t of
T itself. The eigenspaces of the complement are each of dimension one, and the eigencharacters λ that
occur are called the roots of G with respect to T . They form a subset Σ of X ∗ (T ).
Let N be the unipotent radical of B . The adjoint action of T takes the Lie algebra n into itself. The roots
Σ+ occurring in the corresponding eigenspace decomposition are called the positive roots of (G, B, T ).
There exists in Σ+ a unique linearly independent subset ∆ with the property that every other positive
root is an integral sum
X
λ=
cα α
α∈∆
with cα ≥ 0. Every root lies in either Σ+ or Σ− = −Σ+ .
Choose an eigenvector eα 6= 0 in nλ for each α in ∆. I call the triple (B, T, (eα ) a frame for G
(corresponding to épinglage in French). If f is any non-zero element of g−α , then [eα , f ] lies in t. There
exists a unique choice f = e−α such that hα, [eα , e−α ]i = −2. (I follow a useful convention of Tits in
choosing the sign, which is not the most common choice.) In these circumstances, let hα = [eα , e−α ].
There exists now a unique embedding dϕα of the Lie algebra sl2 into g taking
h
=
e+ =
e− =
1
◦
−1
1
◦
◦
◦
◦
◦
◦
−1
◦
7−→ hα
7−→ eα
7−→ e−α .
There then exists a unique homomorphism ϕα from SL2 to G inducing this as its differential. Define α∨
to be the composition of ϕα with the embedding
Gm → SL2 ,
x 7−→
x
◦
◦
1/x
.
It is an element of X∗ (T ). This is independent of the choice of ealpha .
The quadruple (X ∗ (T ), ∆, X∗ (T ), ∆∨ ) (and, implicitly, the map α 7→ α∨ ) is called the root datum of G.
Because the normalizer of T in B is T itself, and all Borel subgroups are conjugate, any two choices of
B , T give rise to canonically isomorphic root data.
There exists a unique involutory automorphism θ of g taking e±α to e∓α , acting as −I on t. There then
exists for each λ ∈
/ ±∆ a choice of eλ 6= 0 in gλ with θ(eλ ) = e−λ . This choice is unique up to sign but
The structure of reductive groups (1:34 p.m. April 11, 2013)
8
only up to sign, and any basis of t together with the eλ are called a Chevalley basis of g. These data
give rise to homomorphisms ϕλ from SL2 to G for each root λ. Define λ∨ in X∗ (T ) as before. All these
embeddings give rise to embeddings uλ of Ga into G, whose images have Lie algebra gλ .
The group NG (T )/T may be identified with the Weyl group W of the root system. It is a Coxeter group,
generated by the images ṡα of
◦
1
−1
◦
with respect to the ϕα (α ∈ ∆) as distinguished generators. The Weyl group fits into a short exact
sequence
1 −→ T −→ NG (T ) −→ W −→ 1 .
This does not in general split (although it does for GLn ). If w in W has an expression
w = s 1 . . . sk
of minimal length in W then according to a theorem of [Tits:1966]
ẇ = ṡ1 . . . ṡk
depends only on w, not on its reduced expression. The group Wext generated in G by the ṡα has relations
ṡα ẇ =
α∨ (−I)ẋ if x = sα w < w
ẋ if x = sα w > w.
Computation in Wext therefore reduces easily to computation in W . The group Wext encapsulates much
of the information one wants to know about G.
3. Automorphisms
In the rest of this essay I’ll look at several interesting cases in detail, including those where F is finite
and p-adic. This will require knowing something about the group of F -automorphisms of a split group,
which I’ll discuss in the rest of this section. In the next I’ll discuss a large group of reductive groups that
possess Borel subgroups but are not split.
If G is a split reductive group defined over F , say with based root datum (L, ∆, L∨ , ∆∨ ), then so is
the adjoint quotient G/ZG of G by its centre. This too is a split reductive group defined over F . Its
∨
∆
root datum is (Lad , ∆, L∨
ad , ∆ ), where L = Z . Conjugation by elements of G(F ) on itself extends to
an action of Gad (F ), which in general will be a bit larger than G(F )/ZG (F ). (Keep in mind the case
G = SL2 , Gad = PGL2 .) These make up the group of inner automorphisms of G(F ) defined over F .
The group Gad is the Zariski-connected component (but not, where relevant, the topological component)
of Aut(G), and may also be identified with the connected component of the automorphism group of g.
[frame-trans] 3.1. Proposition.
The group Gad (F ) acts simply transitively on the triples (B, T, {eα }) of all frames of
G(F ).
Let’s see roughly how this goes. Suppose given (B, T ) and (B∗ , T∗ ). There exists an inner automorphism
of G taking B to B∗ , so we may assume B = B∗ . There exists an inner automorphism of B taking T
to T∗ . Since the normalizer of T in B is itself, the isomorphism of T with T∗ is canonical, so we may
assume T = T∗ . Let Tad be the image of T in Gad . Given two frames for T , there exists a unique t in Tad
taking one to the other.
If α lies in AutF (G) then there exists g in Gad (F ) such that Int(g)◦α takes B to B , T to T . It is unique
up to left multiplication by an element of T . It therefore determines an automorphism of the root datum
L of G. The kernel is Gad (F ). We get in this way a short exact sequence
1 → Gad (F ) → AutF (G) → Aut(L) → 1 .
This sequence splits. If we are given a frame of G then every automorphism of L lifts to one of G that
leaves the frame stable (but will permute its individual elements as it permutes ∆).
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9
4. The Bruhat decomposition
Suppose (G, B, T, {ϕλ }) to be a reductive group together with a Chevalley frame. Let W = NG (T ),
W = W/T , S the involutions in W associated to the simple roots ∆ ⊂ X ∗ (T ) determined by B .
Let N be the unipotent radical of B . The frame gives for each positive root λ an embedding uλ of Ga
into N , and for each negative one an embedding of Ga into its opposite N .
1.
2.
3.
4.
5.
B and W generate G
N is a product
G is union of BW B
BwB = BnNw
Multiplication
Suppose N to have dimension m and (λi ) to be any ordering of the m positive roots. The
product map from (Ga )m to N taking
[nproduct] 4.1. Lemma.
(x1 , . . . , xm ) 7−→ uλ1 (x1 ) . . . uλm (xm )
is an isomorphism of algebraic varieties.
This is Proposition 2.6.3 of [Macdonald:1971]. A constructive form of this reduces to a formula due to
Chevalley for the commutator of certain pairs of elements of N . Since any reordering of the product can
be effected by a number of swaps, this leads to an explicit proof of the Lemma. This process is explained,
for example, in §3 of [Cohen-Murray-Taylor:2005], and I’ll discuss it at the end of this section.
For w in W , the group w−1 N w depends only on w. Let wN = N ∩ wN w−1 .
[bruhat-decomp] 4.2. Theorem. (Bruhat decomposition) Every g
in G may be expressed uniquely as
g = u1 tẇu2
with u1 in N , t in T , u2 in wN .
♣ [nproduct] Because of Lemma 4.1, if one fixes a linear ordering of all positive roots this gives for each element of G a
unique expression. [Cohen-Murray-Taylor:2005] explains how to use this to find explicit algorithms for
multiplication in G.
—————
I’ll discuss here the structure of N —how to represent it and how to compute in it.
5. From root datum to group
A based root datum is specified by a free Z-module L of finite rank), and subsets ∆ of L, ∆∨ of the dual
lattice L∨ , together with a map
∆ −→ ∆∨ ,
α 7−→ α∨
such that the matrix (hα, β ∨ i) is a Cartan matrix. The map from ∆ to ∆∨ is then necessarily a bijection.
Root data are usually specified as the quadruple (L, ∆, L∨ , ∆) for reasons that should become clear later
on. In particular, the map from ∆ to ∆∨ is almost always implicit.
The point is that isomorphism classes of root data classify isomorphism classes of split reductive groups.
Given a split reductive group, deriving its root datum is relatively straightforward. I want to sketch here
how for any field F the group G(F ) is constructed from a datum.
If ∆ = ∅, the associated group will be a torus T . The lattice L is taken to be its character group X ∗ (T ),
which means that T (F ) may be identified with Hom(L, F × ). So this case is very simple. For the rest
The structure of reductive groups (1:34 p.m. April 11, 2013)
10
of the cases I have to recall briefly something about the structure of reductive groups, or in other words
about how the group determines a datum.
Suppose given a split reductive group G. Because it is split, it contains a split torus which is its own
centralizer, hence a maximal torus. Because it is split, the adjoint action of T on the Lie algebra of G
decomposes into the Lie algebra of T plus a sum of one-dimensional eigenspaces. The corresponding
characters of T make up the set Σ roots of (G, T ). Let B be a Borel subgroup containing T . It factors
as B = T N , where N is a unipotent normal subgroup of B . Let Σ+ be the roots corresponding to the
adjoint action on the Lir algebra of N . It contains a subset ∆ called the basis of Σ+ such that every root
in Σ+ is a non-negative linear combination of roots in ∆.
For α in ∆, pick eα in the one-dimensional subgroup of N associated to gα . This gives rise to a
homomorphism ια from SL2 to G taking
1 x
7 → eα
−
0 1
and the diagonal matrices into T . For each α in ∆ let α∨ be the composite map
x 7−→ ια
x 0
0 1/x
.
With L = X ∗ (T ), this gives the quadruple (L, ∆, L∨ , ∆∨ ). Let W be the group N (T )/T . It turns out to
be generated by the reflections
sα = ια
0 −1
1 0
.
General theorems about groups generated by reflections imply that (hα, β ∨ i) is a Cartan matrix.
Now go backwards, starting with a root datum (L, ∆, L∨ , ∆∨ >. The maps
sα : λ 7−→ λ − hλ, α∨ iα
is a reflection, orthogonal with respect to the Euclidean metric that the propertioes of a Cartan matrix
guarantee. Let W be the group these generate, which must be finite. Let Σ be the finite set of transforms
of ∆ by elements of W , Σ+ the non-negative integral combinations of ∆.
Let T be the torus defined so that L = X ∗ (T ).
So far, all is fairly straightforward, and explicit constructions are practical. But now something subtle is
necessary in order to define the Borel group B . First one constructs the Lie algebra n after calculating
explicitly a special basis, called a Chevalley basis, as well as the structure constants of n with respect
to this basis. For this, see [Cohen-Murray-Taylor:2005] or [Casselman:2012]. From this, formulas due
to Chevalley for multiplication on N represented as a direct product of copies of Ga will define N as a
unipotent group. From this to the construction of B is immediate.
Now comes a second subtle step in which the group N (T ), an extension of W by T , is defined according
to recipes of Tits. For this, see also [Cohen-Murray-Taylor:2005]. Finally, the group G is taken to be
the union of double cosets BwNw as w ranges over a special set of representatives of W in N (T ) and
Nw is a particular subgroup of N , the product of the Nλ with λ > 0, w−1 λ < 0. Explicit formulas for
multiplication in G can also be found in [Cohen-Murray-Taylor:.] In other words, G(F ) is defined as a
group by its Bruhat decomposition.
The group G can be covered by translates of an open Bruhat cell, and this can be used to define G as an
algebraic group. I am not quite sure how to prove it is an affine group, but it is—perhaps as the group
preserving certain tensor invariants.
The structure of reductive groups (1:34 p.m. April 11, 2013)
11
The classification of split reductive groups by root data can be made very precise. We have seen that from
a root datum one can construct the group G, a torus T , and Borel subgroup B , and certain embeddings
eα of Ga into B . These data make up a frame of G.
If one is given two framed reductive groups then any isomorphism of their root data
can be extended to a unique isomorphism of framed groups.
[frame-aut] 5.1. Proposition.
As I have said, the standard notation for a root datum is (L, ∆, L∨ , ∆∨ ). It is convenient if not precie,
because the dual root datum is specified as (L∨ , ∆∨ , L, ∆). Here The map from ∆∨ to ∆ is the inverse
of the original one. The Cartan matrix is the transpose tC of the original, and I leave it s an exercise to
construct a diagonal matrix D with positive entries such that that tCD is symmetric and positive definite.
b of G which can in particular be defined over C.
This dual root datum gives rise to the Langlands dual G
6. Quasi-split reductive groups
A reductive group defined over F is called quasi-split if it possesses a Borel subgroup defined over F ,
which is to say a maximal solvable subgroup. It will be of the form B = T N where T is a maximal torus
and N a split unipotent subgroup. The torus will not in general be split.
One way to construct quasi-sploit groups is by Galois descent from a split group, upon twisting by
a homomorphism from G into the image of Aut(L) determined by a frame. These are classified by
conjugacy classes of such homomorphisms. Since such automorphisms take B and T to themselves, a
reductive group constructed in this way possesses a Borel subgroup defined over F as well.
[quasi-split-borel] 6.1. Proposition.
Any quasi-split reductive group G defined over F can be obtained in this way.
Proof. Because the descent data for G are also descent data for a Borel subgroup, and it can be adjusted
by inner automorphisms to fix any given frame.
Quasi-split groups are important for two reasons, only one of which I shall discuss in detail.
(1) One group is said to be an inner form of another if it obtained from it by means of descent through
Gad . Every group is an inner form of a unique quasi-split one. The quasi-split form is obtained by
projection from AutE (G) to Aut(L). Thus quasi-split groups are important in the classification of
arbitrary reductive groups.
(2) If G is a reductive group defined over a global field F then at almost all completions v of F the group
G(Fv ) is quasi-split and, as I shall explain in the next section, unramified. Langlands’ conjectures about
functoriality and L-functions cannot be understood without understanding what happens for arbitrary
unramified groups. This point does not seem to be well known.
Example. I’ll not discuss quasi-split groups in general, but look now at one basic example. Let E/F be
a quadratic extension, let


0 0 1
H =  0 −1 0  ,
1 0 0
and let G = U (H) be the unitary group of H over E —the X in GL3 (E) such that tXHX = H . This
becomes an algebraic group over F upon reading off the equations satisfied by the coefficients of X in
terms of a basis of E/F . The group G splits over E , where it becomes isomorphic to GL3 (E). The
equation defining G can be rewritten as
X = H −1 tX −1 H .
This means that it is the group over F descended from GL3 (E) by the involution X 7→ σ(X) =
H −1 tX −1 H .
The structure of reductive groups (1:34 p.m. April 11, 2013)
12
Define the subgroup T to be that of all diagonal matrices in G, N that of upper triangular unipotent
matrices in G, B = T N .
[unitaryB] 6.2. Proposition.
In these circumstances:
(a) the subgroup N consists of all matrices


1 x y
X = ◦ 1 z
◦
◦
1
with TRACE(y) = NE/F (x);
(b) the matrices in T are those of the form

a
◦
◦
◦
◦
b
◦
◦
1/a


with NE/F (b) = 1.
Proof. These are both straightforward computations.
1
As an algebraic torus over F , T is isomorphic to the direct product of RE/F Gm and NE/F
. The group
N has an an obvious filtration with associated graded pieces isomorphic, although not canonically, to
Ga and RE/F Ga .
7. Reductive groups over finite fields
The main result is this:
[finie-sq] 7.1. Theorem.
Every reductive group defined over a finite field is quasi-split.
Proof. By a well known theorem of Serge Lang, every reductive group defined over a finite field possesses
a Borel subgroup.
The Galois groups of finite extensions of the finite field Fq are cyclic and generated by the Frobenius automorphism x 7→ xq . Therefore reductive groups over Fq are classified by root data L = (L, ∆, L∨ , ∆∨ )
together with a single automorphism Φ of L, representing the descent datum that determines how the
Frobenius acts on X ∗ (T ) for T a maximal (split) torus defined over an algebraic closure of F .
8. Unramified groups over a p-adic field
Now suppose k to be a p-adic field, with integers o, prime ideal p. A reductive group defined over k
is said to be unramified if it splits over an unramified extension of k. These are the groups defined by
base extension from a smooth reductive group scheme over o. These groups are in bijection with the
reductive groups defined over o/p.
The structure of reductive groups (1:34 p.m. April 11, 2013)
13
9. References
1. Armand Borel, ‘Linear algebraic groups’, pp. 3–19 in Algebraic groups and discontinuous subgroups, Proceedings of Symposia in Pure Mathematics IX, AMS, 1966.
2. ——, Linear algebraic groups, W. A. Benjamin, 1969. Reissued in an expanded second edition by
Springer-Verlag in 1991.
3. —— and Tonny Springer, ‘Rationality properties of linear algebraic groups’, pp. 26–32 in Algebraic
groups and discontinuous subgroups, Proceedings of Symposia in Pure Mathematics IX, AMS, 1966.
4. —— and Jacques Tits, ‘Groupes réductifs’, Publications Mathématiques de l’Institut des Hautes
Études Scientifiques 27 (1965), 55-150.
5. N. Bourbaki, Chapitres IV–VI of Groupes et Algèbres de Lie, Hermann, 1968.
6. Bill Casselman, ‘Computations in real tori’, Representation theory of real groups, Contemporary
Mathematics, 472, A.M.S., 2007.
7. ——, ‘Structure constants’, preprint, 2012.
8. Claude Chevalley, The theory of Lie groups, Princeton, 1955.
9. ——, ‘Sur certains groupes simples’, Tohoku Journal of Mathematics 48 (1955), 14–66.
10. Arjeh M. Cohen and Scott H. Murray, ‘Algorithm for Lang’s theorem’, preprint.
11. ——, ——, and Don Taylor, ‘Computing in groups of Lie type’, Mathematics of Computation 73
(2004), 1477–1498.
12. Brian Conrad, ‘The structure of solvable groups over general fields’, preprint, 2011.
13. ——, Reductive group schemes, notes from the SGA Summer School of 2011.
14. Michel Demazure and Alexandre Grothendieck, Schemas en groupes I, II, III, Lecture Notes in
Mathematics 151–153, Springer, 1970.
15. James Humphreys, Linear Algebraic Groups, Springer, 1975.
16. Ian G.Macdonald, Spherical functions on a group of p-adic type, University of Madras, 1971.
17. P. Müller, ‘Algebraic groups over finite fields, a quick proof of Lang’s theorem’, Proceedings of the
American Mathematical Society 131(2) (2003), 369–370.
18. Joseph Oesterlé, ‘Nombres de Tamagawa et groupes unipotents en caractéristique p’, Inventiones
Mathematicae 78 (1984), 13–88.
19. Jean-Pierre Serre, Groupes algébriques et corps de classes, Hermann, 1959.
20. Tonny Springer, Linear algebraic groups (second edition), Birkhäuser, 1998.
21. Robert Steinberg, Lectures on Chevalley groups, Tale Lecture Notes, 1967.
22. ——, ‘Normalisateurs de tores I. Groupes de Coxeter étendus’, Journal of Algebra 4 (1966), 96–116.
23. Jacques Tits, Lectures on algebraic groups I., Yale Lecture Notes, 1968.