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- Root datums and reductive algebraic groups The main goal in this chapter is to give a sketch of how the classification of reductive algebraic groups goes. I really just want to give you the precise statement and some examples, but am going to skip very many details. This is a non-trivial theorem that would take a substantial amount of work to develop with full proofs...
6.1. Tori and root datums. Let us start by talking about tori. Recall an n-dimensional torus is an algebraic group isomorphic to Gm × · · · × Gm. For example, the subgroup Dn of GLn consisting of all diagonal matrices is an n dimensional torus. Let T be an n-dimensional torus. The character group
X(T ) = Hom(T, Gm ) ∼= Hom(Gm , Gm )⊕n^ ∼= Zn^.
An important point is that, given any two tori T and T ′^ ,
Hom(T, T ′^ ) ∼= Hom(X(T ′^ ), X(T )).
So any homomorphism f : X(T ′^ ) → X(T ) of abelian groups induces a unique morphism T → T ′^ of algebraic groups, and vice versa. To be fancy, you can view X(?) as a contravariant equivalence of categories between the category of tori and the category of finitely generated free abelian groups. All elements of a torus T are semisimple. So if V is any finite dimensional representation of T , every element of T is diagonalizable in its action on V by the Jordan decomposition. Moreover, they commute, hence we can actually diagonalize
V =
λ∈X(T )
V (^) λ
where V (^) λ = {v ∈ V | tv = λ(t)v for all t ∈ T }.
As before, the V (^) λ ’s are called the weight spaces of V with respect to the torus T. Now let G be an arbitrary connected algebraic group. A maximal torus of G is what you’d think: a closed subgroup T that is maximal subject to being a torus. Now start to assume that G is a reductive algebraic group. Let T be a maximal torus. Let g be the Lie algebra of G. We can view g as a representation of T via the adjoint action. It turns out moreover – using for the first time that G is reductive – that the zero weight space of g with respect to T is exactly the Lie algebra t of T itself. So we can decompose
g = t ⊕
α∈Φ
gα
where Φ is the set of all 0 &= α ∈ X(T ) such that the T -weight space gα &= 0. You can already see the root system emerging... The difference now however is that the set Φ of roots is a subset of the free abelian group X(T ). Now using the assumption that G is reductive again, you show:
(1) Each gα is one dimensional, and α ∈ Φ iff −α ∈ Φ. (2) The group W = N (^) G (T )/T is a finite group that acts naturally on X(T ) and permutes the subset Φ ⊆ X(T ). (3) Let Q be the root lattice, the subgroup of X(T ) generated by Φ, and let E = R ⊗Z Q. Fix a positive definite inner product on E that is invariant under the action of W. Then, (E, Φ) is an abstract root system.
We’ve now built out of G a root system (E, Φ), and realized the Weyl group W explicitly as the quotient group N (^) G (T )/T. Moreover, Φ is a subset of the character group X(T ) of T. I must stress that all these things take work to prove – it is usually harder than in the Lie algebra case. BUT everything works in arbitrary characteristic. If G is semisimple, then G is determined up to isomorphism by its root system (E, Φ) together with the extra information given by the fundamental group X(T )/Q. However this is not the most natural point of view to classify the reductive, not just semisimple, groups. This is harder, since X(T ) will in general be of bigger rank than Q, and so there is much more freedom not captured by the fundamental group alone... Let’s prepare the way to state the classification of reductive algebraic groups in general. Let G be a reductive algebraic group, and let T be a maximal torus. Let Φ ⊂ X(T ) be the root system of G, defined from the decomposition of g as above. Let
X(T ) = Hom(T, Gm )
be the character group of T , and let
Y (T ) = Hom(Gm , T )
be the cocharacter group. This is also a free abelian group of rank dim T. Moreover, there is a pairing
X(T ) × Y (T ) → Z
defined as follows. Given λ ∈ X(T ) and φ ∈ Y (T ), the composite λ ◦ φ is a map Gm → Gm. So since Aut(Gm ) = Z,
(λ ◦ φ)(x) = x〈λ,φ〉
for a unique 〈λ, φ〉 ∈ Z. For each α ∈ Φ, you prove that there is a (unique up to scalars) homo- morphism
xα : Ga → G
such that
txα (c)t −^1 = xα (α(t)c)
for all c ∈ Ga , t ∈ T , such that the tangent map
dxα : L(Ga ) → gα
Theorem 6.1. Two reductive algebraic groups G, G′^ are isomorphic if and only if their root datums (relative to some maximal tori) are isomorphic.
There is also an existence theorem:
Theorem 6.2. For every root datum, there exists a corresponding reductive algebraic group G.
Finally, one intriguing thing: given a root datum (X, Φ, Y, Φ ∨^ ) there is the dual root datum (Y, Φ ∨^ , X, Φ). If G is a reductive algebraic group with root datum (X, Φ, Y, Φ ∨^ ) you see there is a dual group G ∨^ with the corresponding dual root datum. Note the process of going from G to G ∨^ is very clumsy: I don’t think there is any direct way of constructing the dual group out of the original.
Example 6.3. Suppose that G is a semisimple algebraic group. Let Q = ZΦ ⊂ X(T ). Here, Q and X(T ) have the same rank, so Q is a lattice in X(T ), and X(T )/Q is a finite group, the fundamental group. Let P be the dual lattice to Q. Fixing a positive definite W -invariant inner product on E = R ⊗Z Q, we can identify P with the weight lattice of the root system of G, and then everything is determined by the relationship between Q ⊆ X(T ) ⊆ P. You can formulate the classification just of the semisimple algebraic groups in these terms.
Example 6.4. Let G be a semisimple algebraic group, and suppose that Q ⊆ X(T ) ⊆ P are as in the previous example. If X(T ) = P , then G is called the simply-connected group of type Φ. If X(T ) = Q, then G is called the adjoint group of this type. Now let G (^) sc be the simply-connected one, G (^) ad be the adjoint one. Let G be any other semisimple group of type Φ. Then, there is an inclusion X(T ) ↪→ P = X(T (^) sc ). This induces a map G (^) sc! G. Similarly, there is always a map G! G (^) ad.
Example 6.5. (1) Consider the root datum of GL 2. Here, X(T ) has basis % 1 , % 2 , these being the characters picking out the diagonal en- tries. Moreover, the positive root is α = % 1 − % 2. Also Y (T ) has basis %∨ 1 , % ∨ 2 , the dual basis, mapping Gm into each of the diagonal slots. The coroot is α ∨^ = %∨ 1 − %∨ 2. (2) GL 2 is its own dual group. (3) Consider the root datum of SL 2 × Gm. Here, X(T ) has basis α/ 2 , %, Y (T ) has the dual basis α ∨^ , % ∨^ (here α is the usual positive root of SL 2 ). (4) Consider the root datum of P SL 2 × Gm. Here, X(T ) has basis α, %, Y (T ) has the dual basis α ∨^ / 2 , %. So P SL 2 × Gm is the dual group to SL 2 × Gm.
Exercise 6.6. (7) As an exercise in applying the classification, you can show that (1),(3) and (4) plus one more, the 4 dimensional torus, are all the reductive algebraic groups of dimension 4.
(8) The dual group to SLn is P SLn. The dual group to Sp 2 n is SO 2 n+. The dual group to P Sp 2 n is Spin (^2) n+1. The dual group to SO 2 n is SO 2 n. The dual group to Spin (^2) n is P SO 2 n.
For more explicit constructions of root datums, see Springer, 7.4.7.
6.2. Complete varieties and the Borel fixed point theorem.
Definition 6.7. A variety X is called complete if for all varieties Y the projection pr (^) Y : X × Y → Y
is a closed map.
This is the analogue of compactness in algebraic geometry. Here is an example of a space that is not complete:
Example 6.8. A 1 is not complete. For consider the projection map A 1 × A 1 → A 1 , (x, y) .→ y. It sends the closed subvariety {(x, y) | xy = 1} to A 1 − { 0 } which is not closed.
Theorem 6.9. P n^ is complete.
Proof. We need to show for any variety Y that the projection morphism
π : P n^ × Y → Y
is closed. It suffices to deal with the case that Y is affine and irreducible. Put A = k[Y ], S = A[T 0 ,... , T (^) n ]. We can view S as an algebra of functions on k n+1^ × Y. If I is a proper homogeneous ideal in S put
V (I) = {([x], y) ∈ P n^ × Y | f (x, y) = 0 for all f ∈ I}.
where [x] denotes the point of P n^ defined by x ∈ k n+1^ − { 0 }. You should remember we looked at something like this when we described explicitly the closed sets in P n^ : they were the common zeros of proper homogeneous ideals of k[T 0 ,... , T (^) n ]. In the new situation you show similarly:
- the closed sets in P n^ ×Y are of the form V (I) for proper homogeneous ideals in S;
- V (I) = ∅ if and only if
I = (T 0 ,... , T (^) n );
- V (I) is irreducible if and only if
I is a prime ideal. Now we have to show that π maps closed sets to closed sets. Its enough to show it maps closed irreducible sets to closed irreducible sets (since any closed set is a finite union of irreducible closed sets). Thus we have to show all the irreducible sets πV (I) are closed for all proper prime homogeneous ideals I in S. Let Y 0 be the closure of πV (I), also irreducible and affine. Then V (I) is contained in P n^ ×Y 0 and π : V (I) → Y 0 is dominant. Replacing Y by Y 0 this reduces to showing that πV (I) = Y for all proper prime homogeneous ideals I in S such that π : V (I) → Y is dominant. Note that V (I) ⊆ P n^ × V (A ∩ I) so πV (I) ⊆ V (A ∩ I). So the dominance assumption on π means V (A ∩ I) = Y , i.e. A ∩ I = { 0 }.
through X × Z → Y × Z → Z where both are closed on the way. Hence X is complete. "
Now we can prove the all important Borel’s fixed point theorem:
Theorem 6.12. Let G be a connected solvable algebraic group, and X be a non-empty complete G variety. Then, G has a fixed point on X.
Proof. Proceed by induction on dim G, the case G = { 1 } being trivial. Sup- pose then that dim G > 0 and let H = G ′^ , which is connected solvable of strictly smaller dimension. By induction,
Y = {x ∈ X | Hx = x}
is non-empty. It is closed, hence complete, and G stabilizes Y as H # G. So we may as well replace X by Y to assume that H ⊆ G (^) x for all x ∈ X. Since G/H is abelian, this implies that each G (^) x $ G. Now choose x so that G.x is of minimal dimension. Then, G.x is closed hence complete. The map G/G (^) x → G.x is bijective, so we deduce that G/G (^) x is complete by the preceeding lemma. But G/G (^) x is affine as G (^) x $ G. So in fact G/G (^) x is a point, i.e. G = G (^) x and x is a fixed point. "
Corollary 6.13 (Lie-Kolchin theorem). Let G be a connected solvable sub- group of GL(V ). Then G fixes a flag in V.
Proof. Let G act on the flag variety F (V ). This is projective, so G has a fixed point. "
