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UNIVERSIT `

A DI PISA

Facolt`a di Scienze Matematiche, Fisiche e Naturali Corso di Laurea Specialistica in Matematica

Tesi di Laurea

ON THE ORBITS OF THE BOREL SUBGROUP

ON AN ABELIAN NILRADICAL

Relatore: Candidato:

Prof. Andrea Maffei Michele Carmassi

Controrelatore: Prof. Jacopo Gandini

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Contents

1 Linear Algebraic Groups 7

1.1 Basic results . . . 8

1.2 G-varieties . . . 9

1.3 Semi-simple and unipotent elements . . . 10

1.4 Commutative Algebraic Groups and Tori . . . 12

1.5 Tangent Spaces and Lie algebras . . . 13

1.6 Quotients . . . 15

1.7 Parabolic subgroups and Borel subgroups . . . 16

1.8 Maximal tori and other connected solvable subgroup . . . 18

1.9 Root systems and Weyl group . . . 19

1.10 Weights . . . 23

1.11 Root Data . . . 25

1.12 Reductive Groups . . . 27

1.13 Bruhat Decomposition . . . 29

1.14 Parabolic Subgroups . . . 30

1.15 The Bruhat Order . . . 31

1.16 The Isomorphism and Existence Theorems . . . 31

2 B-orbits on abelian nilradicals 35 2.1 Involutions and orthogonal roots . . . 37

2.2 B-action on pu and G/L . . . 38

2.3 Minimal parabolic subgroups . . . 39

3 The simply laced case 41 3.1 The Bruhat order on pu . . . 42

4 The type B and the type C cases 51 4.1 The type B case . . . 51

4.2 The type C case . . . 53

4.3 The dimension of the Bv-orbits . . . 59

4.4 The Bruhat order in the type B case . . . 61

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Introduction

Let G be a connected reductive, linear algebraic group on an algebraically closed field K. By fixing a maximal torus T we obtain a root system Φ(G, T ) = Φ. If we now choose a Borel subgroup B ⊇ T it automatically gives a system of positive roots Φ+ and basis ∆ of Φ. Recall that every root α ∈ Φ defines a reflection sα and the subgroup of isometries

generated by such reflections is called Weyl group. We will denote it with W . Note that W is actually generated by the simple reflections sα with α ∈ ∆. Then every element

of W can be written as a word in {sα| α ∈ ∆}. An expression of minimal length will be

called a reduced expression. At last v ∈ W , Φ+(v) will be the set of positive roots that v maps into negative roots. Note that the length of a reduced expression for v coincides with the cardinality of Φ+(v). Moreover, it is isomorphic to the Weyl group of the pair (G, T ) which is the quotient NG(T )/ZG(T ) where NG(T ) is the normalizer of T in G and

ZG(T ) is the centralizer of T in G.

Recall that for every α ∈ Φ there is a one-parameter subgroup uα : K −→ G which is a

morphism of algebraic group with closed image Uα. The Uα are unique and, with T , they

generate the group G. Moreover, B is the product B = TQ

α∈Φ+Uα and the Lie algebra

gof G is

g= t ⊕M

α∈Φ

uα

where t is the Lie algebra of T and uα is the Lie algebra of Uα.

We can consider the parabolic subgroups P ⊇ B. Every one of them corresponds to a subset ∆P ⊆ ∆. Such a subset defines a root system ΦP ⊆ Φ and a subset of roots

Ψ = Φ \ ΦP. We will denote with WP the Weyl group of ΦP and with WP the set

{v ∈ W | v(α) > 0 for every α ∈ ∆P}. Then WP is a set of representatives for the cosets

of WP in W and W = WPWP.

Every parabolic subgroup admits a Levi decomposition P = L Ru(P ) where L is

closed and reductive and Ru(P ) is the unipotent radical of P . We are concerned with

the parabolic subgroups P for which Ru(P ) is abelian. In this case, ∆P = ∆ \ {αP} with

αP that appears with coefficient one in the decomposition of the highest root.

Now, let B act on G/L and on pu = ⊕α∈Ψuα, the Lie algebra of Ru(P ), by

multipli-cation and by the adjoint action respectively. Among the orbits of this action (on pu or

G/L, indifferently) we can define an order called the Bruhat order. Orbit O is greater than orbit O0if and only ifO ⊇ O0, where O is the Zariski closure of O. If we consider the

action of B on the flag variety G/B, Chevalley characterized the Bruhat order in terms of the elements of W . We know by the Bruhat decomposition that the B-orbits are all of the form BwB/B with w ∈ W . Then BwB/B ⊇ BvB/B if and only if w ≥ v by the Bruhat order in W , that is, if for every reduced expression of w, v can be expressed as a subword of that expression. More generally, if P ⊇ B is a parabolic subgroup, the B-orbits in G/P are parametrized by the elements of WP and the Bruhat order between them is again determined by the Bruhat order between the respective elements in WP.

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In 2014, D.I.Panyushev [7] proved that, if the characteristic of the field K is not 2, the B-orbits on pu are parametrized by subsets of orthogonal roots. Note that this implies

that they are only a finite number. If we fix a root vector eα of weight α for every α ∈ Ψ

and for every S ⊆ Ψ we define eS=Pα∈Seα, then every orbit in puis of the form BeS for

some S orthogonal and different subsets give rise to distinct orbits. Note that the action of Ru(P ) is trivial, so the B-orbits coincide with the BL-orbits if BL= B ∩ L. In its paper,

Panyushev also provided two conjectures regarding the Bruhat order between these orbits and their dimension.

In this thesis, we are going to study the Bv-orbits in pu and the Bruhat order among

them under the hypothesis char K = 2.

In the char K 6= 2 case, Panyushev’s conjectures have been proved in 2017 by A.Maffei and J.Gandini [3]. In their paper they introduced, for every v ∈ WP, the subgroups Bv = P ∩ v−1Bv. For a fixed v ∈ WP, the subgroup Bv is the product of BL with

Uv =Qα∈Ψ\Φ+(v)Uα and its action on pu is defined by

1. BL acts through the adjoint action;

2. Uv ∼= ⊕α∈Ψ\Φ+(v)uα acts by sum.

Note that if w0 is the longest root in WP, then the Bw0-orbits coincide with the original

B-orbits on pu. In every characteristic is well defined a exponential map

exp : pu −→ Ru(P )

which is an isomorphism. By composing it with the map G −→ G/L we obtain an isomorphism rP : pu −→ P/L which is Bv-equivariant if we use the action defined above.

If we now consider the projection π : G/L −→ G/P , it is easy to see that the B-orbits on BvP/L correspond to the vBvv−1 = Bv-orbits on vP/L which, in turn, are

in bijection with the Bv-orbits in P/L ∼= pu. We then have an order-preserving bijection

between the Bv-orbits in puand the B-orbits in BvP/L ⊆ G/L. Note that with the known

parametrization of the B-orbits in G/P , this gives a parametrization of all the B-orbits in G/L that is equivalent to the one by R.W.Richardson and T.A.Springer [6].

It is easily proved that the Bv orbits on pu are parametrized by the subsets S ⊆ Φ+(v)

that are orthogonal and the B-orbits on G/L are parametrized by the admissible pairs (v, S) with v ∈ WP and S ⊆ Φ+(v) orthogonal.

The conjecture is then proved by letting the minimal parabolic subgroups Pα = B ∪

BsαB act on the B-orbit in G/L. If BvxS is a B-orbit, then the Pα-orbit PαvxS contains

a finite number of B-orbits (at most three) so there is a open one, whose closure contains the others. This gives relations that can be used inductively to prove the following result: Theorem. If BveS and BveT are Bv-orbits in pu, then

BveS ⊆ BveT ⇔ σv(S)≤ σv(T )

where σv(S) =

Q

α∈v(S)sα and the order on the involutions is the Bruhat order in

W . The dimension formula is proved similarly by induction with the help of the minimal parabolic subgroup.

The B-orbits on the variety G/L have been studied, with the same hypothesis of char K 6= 2, also by R.W.Richardson and T.A.Springer [6] in the greater generality of the symmetric varieties, that is, varieties G/K where K is the fixed point subgroup of an involution θ : G −→ G. Note that if char K 6= 2, then we can find an involution θ with L = Gθ0, but it is not clear if that’s true for the char K = 2 case.

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As we said, we are going to study the same problems in the char K = 2 case. Note that the hypothesis char K 6= 2 is necessary in the results by Panyushev and Maffei/Gandini. For, if we consider SP(4) which has root system C2, it is easy to see that it has 3

or-bits in characteristic different from 2 which correspond to the orthogonal subsets {2e1},

{2e2},{e1+ e2} while it has 4 orbits in characteristic 2, that is, the previous three plus an

orbit corresponding to {2e1, e1+ e2}.

Nevertheless, many results still hold. Among them, the correspondence between the Bv-orbits in pu and the B-orbits in BvP/L as well as the action of the minimal parabolic

subgroups and obviously the results regarding the root systems.

Our results will be divided, and different, in function of the type of the root system Φ. We will prove that if Φ is simply laced the parametrization through sets of orthogonal roots still hold as well as the characterization of the Bruhat order given by Maffei and Gandini.

Then we will consider the problem in the cases of Φ of type B and of type C.

In the type B case, the subset of roots Ψ is relatively simple and contains only a single short root α0. We will show that the Bv orbits are parametrized by the sets S ⊆ Φ+(v)

where S is either a singlet, a couple of orthogonal roots or of the form {α0, α} where

α > α0. We will then make explicit the Bruhat order between the roots with a

case-by-case analysis based on the different properties of the three kinds of sets we discussed above.

In the type C case, the parametrization relies on the admissible sets S ⊆ Φ+(v). The admissible sets can be partitioned in two subsets X(S) and Z(S), both of them orthogonal, such that Z(S) contains only long roots and for every β ∈ Z(S) there is a single short root α ∈ X(S) with β − α = γ ∈ ΦP positive.

At last, we will prove that independently from the characteristic and the type of the root system, the dimension of the Bv-orbit BveS is

dim BveS = #Ψ − l(v) + #Y (v, S)

where Y (v, S) =β ∈ Φ+(v) | ∃b ∈ B

L such that Supp(beS∪{β}− eS) > β and by

Supp(beS∪{β}− eS) > β

we mean that every α ∈ Supp(beS∪{β}− eS) verifies α > β. Note that it is true for every subset S that we used to parametrize the relative orbits.

This thesis is organized as follow.

In the first chapter, we will introduce notions, theorems and other basic results to give the theoretical foundations that are needed to understand the problems we are going to study. This introductory part is mostly taken from [4] and [5], while the section about root systems is from [2]. This chapter will be closed by lemma 1.81 which is fundamental to understand the concrete differences between the char K = 2 case and the other cases. In fact, it is known that if char K 6= 2, then the action of the one parameter subgroup uγ(t) on the root vector eα is (note that we are supposing Φ not of type G2)

uγ(t).eα= eα+ ateα+γ+ bt2eα+2γ

with a, b ∈ K non-zero if and only if, respectively, α + γ and α + 2γ are roots. In lemma 1.81 we will show that if char K = 2, then a = 0 also whenever both α + γ and α − γ are

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roots. Note that it’s what happens in the previous example about SP(4) with α = e1+ e2

and γ = e1− e2.

The second chapter is centred on the results from [3] that are still true in characteristic 2 and that we will use in the third and fourth chapters. In particular, we will be interested in the behaviour of the sets of orthogonal roots and in the Bruhat order between involutions. We will also more formally define the bijection between pu and P/L and the bijection

between the Bv-orbits in pu and the B-orbits in BvP/L that we introduced above as

well as the action of the minimal parabolic subgroups. All these results will be of great importance in the last two chapters. While the more technical propositions, such as the fact that the same root can’t be summed to two different orthogonal roots in Ψ (lemma 2.6), will be used more, the most interesting result in this chapter is probably theorem 2.13. It proves that if x, y ∈ pu and Bvx ⊆ Bvy, then Bv exp(x)v−1B ⊆ Bv exp(y)v−1B.

This allows us to use the characterization by Chevalley of the B-orbits in G/B in the context of the Bv-orbits.

In the third chapter, we study the simply laced case. In this case, the action of B is, in a sense, not different from the char K 6= 2 case, because there are no roots α, β such that both α + γ and α − γ are roots. It is not unexpected, then, that our results reflect the char K 6= 2 case. With theorem 3.1 we prove the parametrization of the Bv-orbits and

with theorems 3.3 and 3.11 we prove the characterization of the Bruhat order. To the latter result are paramount theorems 3.5 and 3.7, which let us mirror the inductive proof of Maffei and Gandini in the characteristic 2 case. Note that both theorems are proved in [3] as a corollary of lemma 7.4 of [6] which ultimately derives from lemma 5.1 in [5] and the latter is deeply dependent on the char K 6= 2 hypothesis.

The last chapter is devoted to the cases of root systems of type B or C. The parametrization of the Bv-orbits in the type B case is proved in theorem 4.1 while the

parametrization in the type C cases is given by theorems 4.4 and 4.7. Then we prove the formula on the dimensions with theorem 4.11. The formula will help in the proof of the characterization of the Bruhat order in the type B case, which is theorem 4.14.

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Chapter 1

Linear Algebraic Groups

In this chapter we will introduce the notion of linear algebraic group and many preliminary results regarding them. Linear algebraic groups, and more precisely reductive groups, are the central subject of our study and the propositions in this chapter will be key to the later results. If not otherwise specified, these claims and relative proofs can be found in [5]. Note that we will always work with algebraically closed fields.

Let K be an algebraically closed field. An algebraic group over K is an algebraic variety G over K with a group structure such that the operations

µ : G × G −→ G i : G −→ G µ(x, y) = xy i(x) = x−1

are morphisms of varieties. If the underlying variety is affine, G is a linear algebraic group. Example 1. We will introduce some of the most common examples of linear algebraic groups.

1. G = A1 = K with addition. Denote this group with Ga;

2. G = K∗ with multiplication. Denote this group with Gm;

3. G = GLnthe general linear group of invertible n×n-matrices. Note that GL1= Gm;

4. Any Zariski-closed subgroup of GLn, for example:

(a) G = Dn the group of non-singular diagonal matrices;

(b) G = Tn the group of non-singular upper triangular matrices;

(c) G = Un the group of unipotent upper triangular matrices;

(d) G = SLn the special linear group;

(e) G = On or G = SOn the orthogonal group and the special orthogonal group;

(f ) G = Spn the symplectic group

Spn=X ∈ GLn| XtJ X = J

where J is the anti-symmetric matrix

J = 

0 L

−L 0 

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1.1

Basic results

Let G be an algebraic group.

Proposition 1.1. There is a unique irreducible component G0 that contains the identity element e. It is a closed normal subgroup of finite index and it is the unique connected component of e.

Proof. Let X and Y be irreducible components of G containing e. Then XY = µ(X × Y ) so both XY and its closure XY are irreducible and that implies X = Y = XY = XY . It follows that X is closed under multiplication.

Since the inverse is a homeomorphism X−1 is an irreducible component containing e, hence coincide with X and X is a closed subgroup. Now, for every g ∈ G the map x −→ gxg−1 is an homomorphism, so gXg−1= X and X is normal. The cosets gX must be all the irreducible components of G, which are a finite number. They are also mutually disjoint, hence they are the connected components of G.

This shows that the notions of irreducibility and connectedness coincide for algebraic groups. Moreover, note that all the connected (irreducible) components of G have the same dimension which is, by definition, dim G.

We will now study specifically the linear algebraic groups. Let X be an affine G-variety and a : G × X −→ X the related action. We have K[G × X] = K[G] ⊗KK[X] and a is dual

to a map of algebras a∗: K[X] −→ K[G] ⊗KK[X]. For g ∈ G, x ∈ X and f ∈ K[X] define

s(g) (f ) (x) = f g−1x

Then s is an invertible map and a representation of abstract groups G −→ GL (K[X]). Proposition 1.2. Let X be an affine variety and V a finite dimensional subspace of K[X].

1. There is a finite dimensional subspace W of K[X] which contains V and is stable under all s(g);

2. V is stable under all s(g) if and only if a∗V ⊆ K[G] ⊗ V . In this case, s defines a map sV : G × V −→ V which is a rational representation of G.

Proof. We may assume without loss of generality that V = f K is one dimensional. Then

a∗f = n X i=1 ui⊗ fi with ui∈ K[G] and fi∈ K[X] and s(g)(f )(x) = n X i=1 ui(g−1) ⊗ fi(x)

so all the s(g)(f ) lie in the subspace W0spanned by the fi. The subspace W of W0spanned

by the s(g)(f ) has the properties of (1). It is now clear that if a∗V ⊆ K[G] ⊗ V then V is stable for all the s(g).

Assume V to be s(G)-stable. If {fi} is a basis for V and {fi} t {hj} is a basis for K[X]

then for every f ∈ V we can write

a∗f =Xui⊗ fi+

X

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and

s(g)(f )(x) =Xui(g−1) ⊗ fi(x) +

X

vj(g−1) ⊗ hj(x)

but our assumption implies vj(g−1) = 0 for every g ∈ G so vj ≡ 0 for every j. In this

hypothesis sV : G × V −→ V is well defined and induces a rational representation.

With the theorem above, we can justify the term linear in linear algebraic group. Corollary 1.3. If G is a linear algebraic group, there is an isomorphism of G in a closed subgroup of some GLn

Proof. Consider the action of G on itself given by left multiplication. As before, it induces an action on K[G] given by

λ(g)(f )(x) = f (xg)

The algebra K[G] is finitely generated by elements {x1, . . . , xn}. If we put V = Span(x1, . . . , xn)

where the span is in the sense of vector spaces, then the theorem above gives us a family f1, . . . , fn which generates K[G] as an algebra and a λ(G)-stable subspace as a vector

space. There exist mi,j ∈ K[G] with

λ(g)fi= n

X

i=1

mj,i(g)fj for every g ∈ G

Then the map Φ(g) = (mi,j(g))i,j=1,...,n defines a group homomorphism Φ : G −→ GLn.

Now, if Φ(g) = Id, then λ(g)fi = fi. But the fi are a basis and λ(g) is an algebra

isomorphism, so it follows that λ(g)f = f for every f ∈ K[X] and g = e. So Φ is injective. We now consider the corresponding algebra homomorphism

Φ∗: K[GLn] −→ K[G]

Recall that K[GLn] = K[Ti,j, det−1] and the homomorphism is defined by Φ∗Ti,j = mi,j

and Φ∗(det−1) = det(mi,j)−1. From

fi(g) =

X

j

mj,i(g)fj(e)

follows that Φ∗ is surjective. Now, the image of an homomorphism of algebraic group is always closed. So Φ(G) is closed and its algebra is isomorphic to K[G]. It follows that Φ is an isomorphism of algebraic groups.

1.2

G-varieties

Definition 1.4 (G-varieties). Let G be an algebraic group. A G-variety (or G-space) is a variety X with an action of G given by a morphism of varieties. More precisely, there is a morphism

φ : G × X −→ X written φ(g, x) = g.x, such that g.(h.x) = gh.x and e.x = x.

Definition 1.5. Given G an algebraic group and X,Y G-varieties, a morphism ψ : X −→ Y is said to be equivariant if for every g ∈ G and x ∈ X we have ψ(g.x) = g.ψ(x).

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• G.x = {g.x | g ∈ G} the orbit of x;

• Gx = {g ∈ G | g.x = x} the isotropy group of x.

In particular, if V is a finite dimensional K-vector space we will say that an homomor-phism of algebraic groups r : G −→ GLn(V ) is a rational representation of G and V is a

G-module. We can view V as a G-variety with the action g.x = r(g)x.

Proposition 1.6. An orbit G.x is open in its closure. The closure of an orbit G.x is a union of orbits and there exist closed orbits.

Proof. Fix x ∈ X and consider the map g 7→ g.x from G to X. It’s a map of varieties, so the image G.x contains a non-empty open subset U of G.x. But G.x =S

g∈GgU so it

is open in its closure. That implies that the set Cx = G.x \ G.x is closed and, being

G-stable, it’s a union of orbits. Now, the family of closed subset {Cx | x ∈ X} must contain

a minimal element (for the inclusion) and this element must be the null set. Therefore, the corresponding orbit G.x is closed.

1.3

Semi-simple and unipotent elements

We begin by recalling some results related to the Jordan decomposition from linear algebra. Let V be a finite dimensional K-vector space. An endomorphism a is semi-simple if there is a basis of eigenvectors for a. So a is semi-simple if and only if there is a basis in which a is represented by a diagonal matrix. An endomorphism a is nilpotent if an = 0

for some integer n > 0 and unipotent if (a − Id) is nilpotent.

By fixing a base of V we may identify the algebra of endomorphisms End(V ) with an algebra of matrices Mn and the group of invertible endomorphisms GL(V ) with GLn

where n = dim V .

Theorem 1.7 (Jordan decomposition). Let a ∈ End(V ).

1. There are unique elements as, an∈ End(V ) such that as is semi-simple, an is

nilpo-tent, as and an commute and a = as+ an;

2. Moreover, if a ∈ GL(V ) there exists au ∈ GL(V ) unipotent such that a = asau.

Now, let V be a non necessarily finite vector space. We say that a ∈ End(V ) is locally finite if V is a union of finite dimensional a-stable subspaces. The endomorphism a is semi-simple if its restriction to every finite dimensional a-stable subspace is semi-simple and it is locally nilpotent if its restriction to every finite dimensional a-stable subspace is nilpotent. If a is locally finite we have the same Jordan decompositions of the finite dimensional case.

Now let G be a linear algebraic group. If g ∈ G, we can define a map ρ(g) as the left multiplication ρ(g)(h) = gh. The induced map on K[G], which we denote with the same symbol, is a locally finite element of GL(K[G]).

Theorem 1.8 (Jordan decomposition in G). 1. There are unique elements gs and gu

such that ρ(gs) and ρ(gu) are, respectively, the semi-simple and nilpotent part of ρ(g)

in the sense described above;

2. g = gsgu. The elements gs and gu are then called, respectively, the semi-simple and

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3. if Φ : G −→ G0 is an homomorphism of algebraic groups then Φ(gs) = Φ(g)s and

Φ(gu) = Φ(g)u;

4. if G = GLnthe semi-simple and nilpotent parts of g are the same elements given by

theorem 1.7.

An element g ∈ G is semi-simple (respectively unipotent ) if ρ(g)s = g (respectively

ρ(g)u = g). Note that g ∈ G is semi-simple if and only if φ(g) is semi-simple for any

isomorphism φ : G −→ GLn and the same is true for g unipotent. We will say that a

linear algebraic group G is unipotent if every element of G is unipotent. We have the following

Proposition 1.9. Let G be a subgroup of GLn of unipotent elements. Then there is an

element x ∈ GLn such that xGx−1⊆ Un where Unis the set of upper triangular matrices

with 1 on the diagonal.

This implies that every unipotent linear algebraic group is isomorphic to a closed subgroup of some Un.

Suppose G to be a group and consider g, h ∈ G. We call commutator of g and h the following element

[g, h] = ghg−1h−1

If M and N are subgroups of G we call commutator of M and N the group generated by the elements [m, n] for every m ∈ M and n ∈ N and we denote it with [M, N ]. Obviously we can compute the commutator in the case M = N = G; that’s G1, the commutator of

G. In general, we can define the following sequence of subgroups 

G1 = [G, G]

Gi+1= [Gi, Gi]

It’s easy to see that Gi+1⊆ Gi. If moreover there is a n ∈ N such that Gn= 1 we will say

that G is solvable.

Corollary 1.10. If G is a unipotent linear algebraic group, then G is solvable.

Consider now a linear algebraic group G and two normal subgroup N and N0 of G. The product subgroup N N0 is also normal. If, moreover, N and N0 where also connected or solvable, then N N0 would have be connected or solvable as well. This means that the following is a good definition.

Definition 1.11 (Radical and Unipotent radical). Let G be a linear algebraic group. Then G has a unique closed connected normal solvable subgroup which is maximal for these properties. This is called the radical of G and denoted R(G). Similarly, the unique closed, connected, normal, unipotent subgroup of G is called the unipotent radical and denoted Ru(G).

Note that the unipotent radical Ru(G) is solvable, so Ru(G) ⊆ R(G). If G is a linear

algebraic group and R(G) = 0 we will say that G is semi-simple. If Ru(G) = 0 we will say

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1.4

Commutative Algebraic Groups and Tori

In this section G is a commutative algebraic group. Define

Gs = {g ∈ G | g is semi-simple}

Gu= {g ∈ G | g is unipotent}

Theorem 1.12. The sets Gs and Gu are closed subgroups. Moreover, the product map

φ : Gs× Gu −→ G is an isomorphism and if G is connected, so are Gs and Gu.

We see that commutative groups have a relatively simple structure. In particular, we can characterize the one-dimensional one.

Theorem 1.13. Let G be a one dimensional connected commutative group. Then G is isomorphic to Ga or Gm.

More precisely, G = Ga if it is unipotent and G = Gm if it is semi-simple.

We saw with theorem 1.3 that every linear algebraic groups is a closed subgroup of some GLn. It should be no surprise that an important class of commutative algebraic

groups is related to Dn, the subgroup of diagonal matrices.

Definition 1.14. A linear algebraic group G is diagonalizable if it is isomorphic to a closed subgroup of some Dn. If, moreover, it is isomorphic to Dn itself, it is an algebraic

torus or simply a torus.

Given a linear algebraic group G, we can consider the set X∗(G) of homomorphisms χ : G −→ Gm. We will call χ a rational character and X∗(G) the set of characters of G.

Dually, we can define the set of cocharacters X∗(G) = {homomorphism λ : Gm −→ G}.

While the former has a natural structure of abelian group (χ1 + χ2)(x) = χ1(x)χ2(x),

the latter has not in general, but we may still define (n.λ)(a) = λ(a)n for every n ∈ N. Notably, if G = Dn is a torus its character group is isomorphic to Zn. For, write x ∈ Dn

as Diag(χ1(x), . . . , χn(x)). The χi are characters and, with their inverse, generate K[Dn].

Hence, the χi are independent and every element of X∗(Dn) is of the form χa11· · · χann.

Moreover, every homomorphism Gm−→ Dnis of the form x 7→ Diag(xa1, . . . , xan).

The character group of G determines if G is diagonalizable and if it is a torus. Theorem 1.15. Let G be a linear algebraic group.

1. G is diagonalizable ⇔ X∗(G) is an abelian group of finite type ⇔ Any rational representation of G is the sum of one-dimensional rational representations;

2. G is a torus ⇔ X∗(G) is a free abelian group.

In particular every G diagonalizable is the product of a torus with a finite abelian group of order prime to p = char(K). The torus is the identity component of G, so G is a torus if and only if it is connected.

Now suppose T a torus. Given the characterization before, we can define a pairing h·, ·i : X∗(T ) × X∗(T ) −→ Z the following way: for every χ ∈ X∗(T ) and λ ∈ X∗(T )

consider χ ◦ λ : Gm −→ Gm. It has the form χ(λ(a)) = an for some n ∈ Z independent

from a. Put hχ, λi + n.

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We will conclude this section with a result that will be of great importance later. Recall that if H < G is a subgroup, NG(H) =g ∈ G | gHg−1⊆ G is the normalizer of H in

G and ZG(H) = {g ∈ G | gh = hg ∀h ∈ H} is the centralizer of H in G.

Theorem 1.17. If G is a linear algebraic group and H is a diagonalizable subgroup of G, then NG(H)0 = ZG(H)0 and NG(H)/ZG(H) is finite.

1.5

Tangent Spaces and Lie algebras

Let X be an algebraic varieties and K[X] the associated algebra. Having fixed a point x ∈ X we can define the tangent space Tx(X) of X in x.

Fix R a commutative ring, A an R-algebra and M an R-module. An R-derivation of A in M is an R-linear map D : A −→ M such that

D(ab) = a.Db + b.Da for all a, b ∈ A

The set DerR(A, M ) of these derivation is a left A-module with the operations (D +D0)a =

Da + D0a and (b.D)a = b.Da.

For every fixed x ∈ X, Mx = {f ∈ K[X] | f (x) = 0} is a maximal ideal in K[X], so

K[X]/Mx ∼= K and there is a structure of K[X]-module on K given by the projection

π : K[X] −→ K. We will denote this module with Kx.

Definition 1.18. Let X be a affine variety and x ∈ X a fixed point. Then the tangent space of X in x is the K-vector space

Tx(X) = DerK(K[X], Kx)

If Φ : X −→ Y is a morphism of varieties, it induces a morphism of algebras Φ∗ : K[Y ] −→ K[X] which, in turn, induces for every x ∈ X a linear map

dΦx: TxX −→ Tφ(x)Y

If Φ is an isomorphism, so is dΦx.

Definition 1.19. We will say that a point x ∈ X is simple or non-singular if dimKTxX =

dim X. The variety X is non-singular or smooth if every x ∈ X is non-singular.

Generic varieties need not to be smooth, even if they are affine (take as a counterex-ample the curve y2 = x3+ x in R2), but for linear algebraic groups we have the following. Theorem 1.20. If G is a linear algebraic group, then G is smooth.

There is a natural isomorphism between TgG and ThG for every g, h ∈ G given by the

left or right translation. We will see that those tangent spaces, of which we may study only the one at the identity e, have more structure than usual.

Definition 1.21 (Lie Algebra). A Lie algebra is a vector space L with a bracket product [·, ·] : L × L −→ L with the following properties:

1. [·, ·] is bilinear;

2. [x, x] = 0 for every x ∈ L;

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Example 2. 1. if V is a vector space, it can be made a Lie algebra by putting every bracket [x, y] = 0. This way we obtain an abelian Lie algebra;

2. if A is an associative algebra with the operation •, it has a natural structure of Lie algebra by defining [x, y] = x • y − y • x;

3. as an application of the previous case, the space of matrices Mnis a Lie algebra with

bracket

[A, B] = AB − BA

Lie algebras naturally arise from linear algebraic groups in the following way. Let G be a linear algebraic group; the space D = DerK(K[G], K[G]) has a Lie algebra structure given by

[D, D0] = D ◦ D0− D0◦ D Moreover, G acts on D through right (and left) translation

λ(x)D = λ(x) ◦ D ◦ λ(x)

Definition 1.22. The Lie algebra of G is the set L(G) of D ∈ D fixed by the above action. At times, we will denote the Lie algebra of G or H linear algebraic groups with the corresponding gothic letter g and h.

Since left and right translation commute, all ρ(x) stabilize L(G).

Now denote with e the identity element of G and consider the following map α :D −→ TeG

f 7→ (Df )(e)

It is linear and induces a isomorphism of vector spaces L(G) ∼= TxG. We can then define

a Lie algebra structure on TxG by transporting the one on L(G). In fact, we will often

see the Lie algebra L(G) as the tangent space TxG endowed with such structure.

Note that the composition α ◦ ρ(x) ◦ α−1 is the linear automorphism of TeG induced by

the inner automorphism y 7→ xyx−1 of G. We will denote this automorphism with Ad(x). It is a rational representation of G in TeG that is called adjoint representation.

As can be expected by theorem 1.20 we have the following Lemma 1.23. If G is a linear algebraic group, then dimKg= dim G

Moreover, consider H a closed subgroup of a linear algebraic group G. It’s related algebra is K[H] = K[G]/J where J is the ideal of functions vanishing on H. We then have a natural homomorphism Φ between DG,H = {D ∈ DG| DJ ⊆ J } and DH . The

restriction φ : DG,H∪ L(G) −→ L(H) is a isomorphism, so we have an immersion of h in

g.

Example 3. 1. G = Ga. Then K[G] = K[T ], the polynomial algebra on one variable.

The derivations commuting with all λ(x) are the the multiple of Df = dTdf. So g is one-dimensional and abelian;

2. G = GLn. Now K[G] = K[Ti,j, det−1] and there is an injective map gln+ Mn−→ g

given by (xi,j) 7→ DX such that DX = −Pnh=1Ti,hxh,j. This map is surjective

because the two spaces have the same dimension, hence g ∼= gln. The adjoint

repre-sentation is given by Ad(x)M = xM x−1 for every x ∈ GLn and every M ∈ gln;

3. if H < GLn is a closed subgroup, for example if H = SLn, SOn and so on, h can be

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1.6

Quotients

Let G denote a linear algebraic group and H a closed subgroup. We shall establish the existence of a variety G/H, or rather, the existence of a structure of variety on the set of cosets {gH | g ∈ G}.

Lemma 1.24. Let G a linear algebraic group and H a closed subgroup. There is a finite dimensional subspace V of K[G] and a subspace W ⊆ V with the properties:

1. V is stable under the right translation ρ(x) for every x ∈ G; 2.

H = {x ∈ G | ρ(x)W = W } h= {X ∈ g | X.W ⊆ W }

Now let V be a finite dimensional vector space and W ⊆ V a subspace of dimension d. Recall that the exterior product V ∧ V is the tensor product V ⊗ V quotiented by the ideal generated by the element {v ⊗ v | v ∈ V } and the kth exterior power is the exterior product of V with itself k times. If we take the dth exterior power Vd

V it contains the subspace L = Vd

W and this subspace is one-dimensional. Now, if by abuse of notation we denoteVd

V with V , we get the following theorem.

Theorem 1.25. There exists a rational representation φ : G −→ GL(V ) and a non-zero v ∈ V such that

H = {x ∈ G | φ(x)v ∈ Span(v)} h= {X ∈ g | (dφ)(x)v ∈ Span(v)}

We will now define the quotient of G by H through a universal property.

Definition 1.26. A quotient of G by H is a pair (G/H, x) of a variety G/H and a point x ∈ G/H such that:

1. G/H is an homogeneous space for G and a ∈ G/H;

2. (Universal property) for any pair (Y, y) of a G-space Y and y ∈ Y such that the isotropy group of y contains H, there exists a unique equivariant morphism

Φ : G/H −→ Y that maps x in y.

A quotient exists and it’s unique. The uniqueness derives trivially from the universal property, while the existence derives, with some additional work, from the following the-orem. Recall that a quasi-projective variety is an open subvariety of a projective variety. Theorem 1.27. If G is a linear algebraic group and H is a subgroup, then there is a quasi-projective homogeneous space X for G and a point x ∈ X verifying;

1. the isotropy group of x in G is H;

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Proof. Let V and v as in the theorem 1.25 and consider the projective space P(V ). Put x = [v] in P(V ) and denote π : V \ {0} −→ P(V ) the projection.

G acts on P(V ) by g.π(x) = π(g.v). Take X = G.x. It is homogeneous and quasi-projective by 1.6. The properties (1) and (2) follow from theorem 1.25

We then have that the pair (X, x) above is the quotient G/H. If H is normal we have the following stronger properties.

Theorem 1.28. If H is normal in G then 1. G/H is affine;

2. G/H with the usual structure of quotient of group is a linear algebraic group. Moreover, as we can expect, dim G/H = dim G − dim H

To end this section we will introduce a peculiar kind of quotients.

Definition 1.29 ([4], Definition 1.18). Let G be a linear algebraic group that acts on a variety X. We will say that Y is a geometric quotient of X by G if there is a projection π : X −→ Y such that

1. π is surjective and its fibers are exactly the G-orbits in X; 2. U ⊆ Y is open ⇔ π−1(U ) is open;

3. for every U ⊆ Y the algebras K[U ] and K[π−1(U )]G are isomorphic through the comorphism π∗(f (x)) = f (π(x)).

Note that the quotient G/H defined above is a geometric quotient.

Not every action admits a geometric quotient. For example if we let K∗ act on Kn by scalar multiplication there is no geometric quotient, because 0 lies in every orbit closure. Recall that a section on U of π : G −→ G/H is a morphism σ : U −→ G which verifies π ◦ σ = IdU.

Theorem 1.30. Let G be a linear algebraic group, H a subgroup and π : G −→ G/H the projection. Assume that H acts on a variety X and define an action of H on G × X as h.(g, x) = (gh, h−1.x). Then, if π admits local section, the quotient (G × X)/H exists.

We will denote the quotient above with G ×H X.

1.7

Parabolic subgroups and Borel subgroups

In this section we will introduce two classes of subgroups of linear algebraic group which will be of fundamental importance.

Definition 1.31. An algebraic variety X is complete if for every algebraic variety Y the morphism π : X × Y −→ Y of projection on the second component is closed, that is, it maps closed sets in closed sets.

Example 4. 1. the affine line A1 is not complete. Take the map A1× A1 −→ A1 and

consider the set X = (x, y) ∈ A1× A1| xy = 1 . It is closed, but π(X) = A1\ {0}

which is not closed;

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Complete varieties have many properties.

Theorem 1.32. Let X be a complete variety. Then: 1. every subvariety of X is closed and complete; 2. if Y is also complete so is X × Y ;

3. given a morphism φ : X −→ Y the image φX is closed and complete; 4. if X is affine then X is finite.

We will now give two fundamental definitions.

Definition 1.33 (Borel subgroups). Let G be a linear algebraic group. A Borel subgroup of G is a subgroup B which is closed, connected and solvable and it is maximal for these properties.

It’s important to note that Borel subgroup do exist. It is in fact sufficient to take one of maximal dimension.

Definition 1.34 (Parabolic subgroups). Let G be a linear algebraic group. A closed subgroup P of G is a parabolic subgroup if and only if G/P is a complete variety.

The whole group G is trivially a parabolic subgroup. We have the following criterion. Proposition 1.35. A connected group G contains proper parabolic subgroups if and only if G is non-solvable.

This results will be useful soon.

Theorem 1.36. Let G be a linear algebraic group: 1. if P is parabolic then G/P is projective;

2. if P is parabolic in G and Q is parabolic in P then Q is parabolic in G; 3. P is parabolic in G if and only if P0 is parabolic in G0.

Example 5. We will give example in the case G = SLn:

1. a Borel subgroup is Tn, the set of upper triangular matrices. Having fixed x ∈ G all

the sets xTnx−1 are still Borel subgroup. We will see that every Borel subgroup of

G is of this kind;

2. a parabolic subgroup of G is the set P of matrices of the form J =A B

0 C 

where A is a k × k block and C is a (n − k) × (n − k) block.

Perhaps surprisingly, Borel and Parabolic subgroup are closely related. The ingredient to prove this relation is Borel’s fixed point theorem.

Theorem 1.37 (Borel’s fixed point theorem). Let G be a connected solvable linear alge-braic group and X a complete G-space. Then there is a point x ∈ X such that Gx = G.

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Proof. We know that G has a closed orbit in X. Then the isotropy group Gx of a point x

in that orbit is parabolic. We will show that a connected solvable linear algebraic group contains no trivial parabolic subgroup.

Let P be a maximal parabolic subgroup in G and consider Q = P.[G, G]. Q is a closed subgroup that contains P and it is parabolic (to see this apply 1.32 (3) to π : G/P −→ G/Q). Hence, Q = G or Q = P .

In the first case [G, G]/ ([G, G] ∩ P ) ∼= G/P . Now, [G, G] = 0 or dim[G, G] < dim G. If [G, G] = 0 then G is abelian and P is normal. It follows that G = P because G/P is complete and affine, so it is finite. On the other hand, if dim[G, G] < dim G, by induction we can suppose [G, G] ∩ P = [G, G] ⇒ [G, G] ⊆ P and P is again normal.

In the case Q = P we have again [G, G] ⊆ P .

We will now apply Borel’s theorem to Borel subgroups.

Theorem 1.38. A closed subgroup of G is parabolic if and only if it contains a Borel subgroup and every two Borel subgroups are conjugate.

Proof. We may assume G connected. Let B be a Borel subgroup and P be a parabolic subgroup. By applying Borel’s fixed point theorem to B and G/P we see that B fixes a coset xP . So B0 = x−1Bx ⊆ P and B0 is a Borel subgroup.

We will now show that B itself is parabolic. We can suppose G non-solvable. Suppose B 6= G, then G contains a proper parabolic subgroup which we may assume contains B. If P 6= B, then P is a non-solvable linear algebraic group and B is a Borel subgroup of P . By induction on dimension we obtain that B is parabolic.

For the last part, we obtain from before that if B and B0 are both Borel subgroup, then B contains a conjugate of B0 and vice versa. But this implies that dim B = dim B0 so they are conjugate themselves.

Parabolic and Borel subgroups behave well with morphisms, meaning that if Φ : G −→ G0 is a surjective homomorphism of linear algebraic groups and X is a parabolic (respec-tively Borel) subgroup then Φ(X) is parabolic (respec(respec-tively Borel).

1.8

Maximal tori and other connected solvable subgroup

In the previous section, we used the set Tn⊆ GLn of triangular matrices as an example

of a Borel subgroup. We will now show that every connected solvable algebraic group can be realized as a subgroup of Tn.

Theorem 1.39 (Lie-Kolchin theorem). Let G be a closed subgroup of GLn. Then there

is x ∈ GLn that verifies xGx−1 ⊆ Tn.

Proof. It is sufficient to prove that all the elements of G have a non-zero common vectors. In fact, if that is true, an induction on n gives the thesis.

The existence of a common eigenvector follows from Borel’s fixed point theorem applied to G acting on the projective space Pn−1.

Corollary 1.40. Assume that G is also nilpotent. Then the sets Gs and Gu are closed,

connected subgroups and Gsis a central torus. Moreover, the product map Gs× Gu −→ G

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If G is a linear algebraic group, a maximal torus T of G is a subgroup of G which is a torus and it is not strictly contained in any other torus of G. The identity component of the centralizer of a maximal torus is called Cartan subgroup.

Theorem 1.41. Two maximal tori of G are conjugate. We will now study the properties of the Cartan subgroup.

Theorem 1.42. Fix a maximal torus T of G and C = ZG(T )0 the corresponding Cartan

subgroup.

1. C is nilpotent and T is its maximal torus;

2. There exist elements t ∈ T lying in only finitely many conjugates of C.

1.9

Root systems and Weyl group

In this section we will introduce some combinatorial data and in the next section we will show how they are related to the theory of linear algebraic groups. The results in this section are taken from [2].

Fix an Euclidean vector space V , that is, a finite dimensional vector space over R endowed with a positive definite symmetrical bilinear form (·, ·). An element of GL(V ) is a reflection if it is an involution that pointwise fixes an hyperplane. Every reflection is characterized by the hyperplane it fixes or, symmetrically, by an element of its orthogonal on which the reflection acts as − Id. So, for every α ∈ V we can define σα as the reflection

that fixes {β ∈ V | (β, α) = 0} and maps α to −α. This reflection can be written as σα(β) = β −

2(β, α) (α, α) α

Definition 1.43 (Root system). A root system Φ is a finite set of non-zero vectors called roots in V with the following properties:

1. the roots generate V ;

2. if α ∈ Φ and β = xα ∈ Φ with x ∈ R then x = ±1; 3. for every α ∈ Φ the root system Φ is σα-invariant;

4. if α, β ∈ Φ then hβ, αi + 2(β,α)(α,α) is an integer.

The second axiom is sometimes omitted. In this case, the root system that verify (2) are reduced and the ones that don’t are non-reduced. The axioms restrict the possible angle between roots. In fact if α, β ∈ Φ then hβ, αihα, βi = (α,α)(β,β)4(β,α)2 . Remember that if θ is the angle between the vectors α and β then (α, β) = kαkkβk cos θ. So we obtain

hβ, αihα, βi = 4 cos2θ Analysing all the possibilities we obtain table 1.1.

It follows that in a fixed root system, there are at most two distinct value for the norm of the roots. If there are two, we will call the roots with highest norm the long roots and the other the short roots. From the table we obtain the following criterion.

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Table 1.1: hβ, αi hα, βi θ kαkkβk22 0 0 π/2 undetermined 1 1 π/3 1 −1 −1 2π/3 1 1 2 π/4 2 −1 −2 3π/4 2 1 3 π/6 3 −1 −3 5π/6 3

Proposition 1.44. Let α, β be linearly independent roots. If (α, β) > 0 then α − β is a root. Similarly, if (α, β) < 0 then α + β is a root.

Given a root system Φ, we can find a subset ∆ of Φ with the following properties: 1. ∆ is a basis of V ;

2. each root can be written as P aiαi with αi ∈ ∆ and ai ∈ Z. Moreover, the ai are

all negative or all positive.

Such a subset is called a base of Φ. The roots in ∆ are called simple. The roots which can be written as P aiαi with ai > 0 are called positive, while the ones with ai < 0 are

negative. We will sometime say that a root α is positive (respectively negative) by writing α > 0 (respectively α < 0). Having fixed a base we get the following order on the roots of Φ

α < β ⇔ β − α is a sum of roots in ∆ with positive coefficients

Consider now the subgroup W of GL(V ) generated by the reflection σα with α ∈ Φ. It

is called the Weyl group of Φ. We see that Φ is W-invariant and that W can be identified with a subgroup of the permutation group of Φ. Hence W is finite. Note that (·, ·) is W-invariant.

Theorem 1.45. The Weyl group W is generated by the simple reflection, that is, by the σα with α ∈ ∆.

It is obvious that if ∆ is a base, then for w ∈ W the set w(∆) is another base. In fact, for every two bases ∆ and ∆0 there exists an element w ∈ W such that ∆0 = w(∆).

Note that if Φ is a root system in V and Φ0 is a root system in V0, then (Φ ⊕ {0}) ∪ ({0} ⊕ Φ0) is naturally a root system in V ⊕ V0. We will say that a root system Φ is irreducible if it can’t be written as the sum of two root system in the sense above, that is, it can’t be partitioned in two subsets that are mutually orthogonal.

Having fixed a base ∆ consider the Weyl group W and the corresponding generators S = {sα1, . . . , sαn}. The length function l(w) is the smallest integer l > 0 such that w can

be written as the product of h elements in S. We will say that a sequence sαi1, . . . , sαil is

a reduced decomposition for w if w = sαi1· · · sαil and l(w) = h.

Now, if v ∈ W , denote with Φ+(v) the set of positive roots that v maps into negative roots.

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Proposition 1.46. The length of v ∈ W coincides with the cardinality of Φ+(v).

More-over, if α is simple, then Φ+(sα) = {α} and

l(sαv) = l(v) + 1 ⇔ v(α) > 0

Similar properties hold for vsα. Note that if α ∈ ∆ and l(sαw) = l(w) − 1 then w has

a reduced decomposition with first element sα.

If we choose a subset S ⊆ ∆, then the set of roots generated by S is still a root system and it has S as a base. We will denote such a root system with ΦS.

The choice of such a subset fixes a subgroup in W which is the subgroup generated by the reflections sα with α ∈ S. This subgroup is denoted by WS and is the Weyl subgroup

of ΦS.

The cosets of WS in W have a set of privileged representatives. It is the set

WS =v ∈ W | v(α) ∈ Φ+ for every α ∈ ∆

Suppose to have fixed a base ∆ and a subset of positive roots Φ+. Then every root β ∈ Φ can be written as β = P

α∈∆aαα in a unique way. For every α ∈ ∆ denote

[β : α] = aα.

Definition 1.47 (height of a root). Let β =P

α∈∆aαα be a root in Φ. Then the height

of α is the integerP

α∈∆aα=

P

α∈∆[β : α].

Fixing a base ∆ gives also an order on the Weyl group W.

Definition 1.48 (Bruhat order). Let u, v ∈ W be elements of the Weyl group. Then u ≤ v with the Bruhat order if having fixed a reduced decomposition v = sα1· · · sαn of

v, u can be written as a subword of v, that is, there exist indices i1 < . . . < ih for which

u = sαi1· · · sαih

We will now show the classification of irreducible root systems. To do this, we introduce the Dynkin diagrams. Consider a root system Φ and ∆ a base of Φ. Label the roots in ∆ as α1, . . . , αn. Now we can write a graph with the following rules:

1. the graph has n vertices;

2. the ithvertex is joined to the jthvertex by a number of edges equal to hαi, αjihαj, αii;

3. whenever a double or triple edge occurs in the graph, we add an arrow pointing to the vertex associated with the shorter root.

Note that the Dynkin diagram doesn’t depend on the specific choice of a base because every base is equivalent under the action of W.

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5π/6

α β

G2

The roots labelled α and β form a base. The associated Dynkin diagram is:

G2

α β

It is, in fact, the only irreducible Dynkin diagrams where the ratio of the length of the roots is√3.

The following proposition implies that irreducible root systems correspond to con-nected Dyinkin diagrams.

Proposition 1.49. Let Φ be a root system and ∆ a base for Φ. Then Φ is irreducible if and only if ∆ can’t be partitioned in two mutually orthogonal subsets.

Moreover, to every connected component of the Dynkin diagram correspond an irre-ducible component of the root system. We then have that every root system Φ decomposes uniquely as the union of irreducible root systems Φi each having Dynkin diagram a

con-nected component of the Dynkin diagram of Φ.

It follows that to classify the irreducible root system is sufficient to classify the con-nected Dynkin diagrams.

Theorem 1.50 (Classification of irreducible root systems). Let Φ be an irreducible root system of rank n. Then its Dynkin diagram is one of the following.

An: (n ≥ 1) 1 2 3 n − 2 n − 1 n Bn: (n ≥ 2) 1 2 3 n − 2 n − 1 n Cn: (n ≥ 3) 1 2 3 n − 2 n − 1 n Dn: (n ≥ 4) 1 2 3 n − 3 n − 2 n − 1 n

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E6: 1 3 4 5 6 2 E7: 1 3 4 5 6 7 2 E8: 1 3 4 5 6 7 8 2 F4 : 1 2 3 4 G2 : 1 2

An irreducible root system has only a single highest root, that is, a root θ for which P

α∈∆[θ : α] is maximal.

Proposition 1.51. Let Φ be a irreducible root system and θ the highest root in Φ. Then 1. for every α ∈ ∆ [θ : α] > 0;

2. if γ ∈ Φ, for every α ∈ ∆ we have

[γ : α] ≤ [θ : α]

If we label the simple roots as the vertices of the Dynkin diagrams above, table 1.2 gives us the highest root of every type of root systems.

1.10

Weights

We will now show how linear algebraic groups are related to root systems.

Let T be a torus e consider a rational representation r : T −→ GL(V ). By theorem 1.15 V is the direct sum of one-dimensional representation. To be more precise, V is the sum of the non-zero Vχ with χ ∈ X∗(T ) where

Vχ= {v ∈ V | r(t)v = χ(t)v for all t ∈ T }

The character χ such that Vχ 6= 0 are called weight of T in V and the respective Vχ are

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Table 1.2: Type Highest root

An α1+ α2+ · · · + αn−1+ αn Bn α1+ 2α2+ · · · + 2αn−1+ 2αn Cn 2α1+ 2α2+ · · · + 2αn−1+ αn Dn α1+ 2α2+ · · · + 2αn−2+ αn−1+ αn E6 α1+ 2α2+ 2α3+ 3α4+ 2α5+ α6 E7 2α1+ 2α2+ 3α3+ 4α4+ 3α5+ 2α6+ α7 E8 2α1+ 3α2+ 4α3+ 6α4+ 5α5+ 4α6+ 3α7+ 2α8 F4 2α1+ 3α2+ 4α3+ 2α4 G2 3α1+ 2α2

In particular, consider the case T ⊆ G, V = g and r = Ad the adjoint representation. Then

g= t ⊕M

α

and the subspace t = gT is the Lie algebra L(T ).

Denote with P the set of non-zero weights of such action. For every α ∈ P denote with Gα the centralizer of (ker α)0.

Proposition 1.52. The Gα generate G. Moreover, if all Gα are solvable, then G is also

solvable.

Suppose G non-solvable and denote with P0 the subset of non-zero weights α such that Gα is non-solvable. Now consider the group W = W (G, T ) = NG(T )/ZG(T ). It is called

the Weyl group of (G, T ). It is finite and it acts faithfully on X∗(T ) as n.χ(t) = χ(ntn−1). If S lies in the center of G there is an isomorphism W (G, T ) ∼= W (G/S, T /S).

It follows that if α ∈ P0 and S = (ker α)0, the pair (Gα, T ) is such that W (Gα, T ) ∼=

W (Gα/S, T /S) and the latter has order 2 (Proposition 7.1.5, [5]). Moreover, W (Gα, T ) ⊆

W (G, T ), so we can define sα∈ W (G, T ) as the non-zero element of W (Gα, T ) and nα as

one representative of sα.

The Weyl group acts on the character group X and on its dual X∨ which we see as the group of cocharacters of T .

Denote V = R ⊗ X. One can define a symmetric bilinear form on V that is invariant under the induced action of W ; it is sufficient to take any positive symmetric bilinear form f and define

(x, y) = X

w∈W

f (w.x, w.y) Then the sα are reflection for the metric defined by (·, ·) and

sα(x) = x −

2(x, α) (α, α)α Theorem 1.53. W is generated by the sα.

If T is a maximal torus of G, the integer dim T is the rank of G. The semisimple rank of G is the rank of G/ R(G). Consider now a linear algebraic group G of rank one and non-solvable. Then its Weyl group is cyclic of order two and if we fix a Borel subgroup B containing a maximal torus T and put U = Ru(B) the following holds.

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Theorem 1.54. G is the disjoint union of B and U nB where n is a representative of the non-zero element of W . Moreover:

1. dim U = 1 and ZG(T ) = T ;

2. there is a unique weight α of T in g such that g= t ⊕ gα⊕ g−α

where t is the Lie algebra of T and both gα = L(U ) and g−α = L(nU n−1) are

one-dimensional weight spaces;

3. the product map U × B −→ U nB is a isomorphism of variety.

We already defined the linear algebraic groups of the family SLn, e.g. the group of n×n

matrices with determinant 1. It contains a subgroup Z that acts as scalar multiplication; it is the subgroup of matrices λ Id where λn = 1. The quotient SLn/Z is the projective

special linear group and it is denoted PSLn.

Theorem 1.55. If G is a linear algebraic group which is connected, semi-simple and of rank one, then G is isomorphic to SL2 or PSL2.

As a generalization of rank one semi-simple groups we can consider the reductive groups of semi-simple rank one, that is, the reductive groups G such that G/ R(G) has rank one. In this case, we have seen that G/ R(G) is isomorphic to SL2 or PSL2. It follows that

g= t ⊕ gα⊕ g−α

with g±α one dimensional.

Theorem 1.56. There exists an homomorphism of algebraic groups uα : Ga −→ G such

that

1. tuα(x)t−1= uα(α(t)x) for every t ∈ T and x ∈ Ga;

2. Im duα= gα;

3. T and Im uα generate a Borel subgroup of G.

Moreover, if u0α is another homomorphism with the same properties, there is a unique a ∈ K∗ with u0α(x) = uα(ax).

Note that the subgroup generated by T and Im u±α has Lie algebra equal to g, so G

is generated by T and the Im u±α.

1.11

Root Data

Definition 1.57 (Root Datum). A root datum is a quadruple Ψ = (X, R, X∨, R∨) where: 1. X and X∨ are free abelian group of finite rank with a pairing h·, ·i : X × X∨ −→ Z; 2. R and R∨ are finite subsets of X and X∨ and there is a bijection α 7→ α∨ of R into

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For every α ∈ R, we define endomorphisms sα and s∨α of X and X∨ respectively

sα(x) = x − hx, α∨iα

s∨α(y) = y − hy, α∨iα and, for every α ∈ R we impose the following:

1. hα, α∨i = 2;

2. sαR = R, s∨α(R∨) = R∨.

The Weyl group of Ψ is the group of automorphisms generated by sα.

The first thing to note is that if we denote by Q the subgroup of X generated by R and V = Q ⊗ R, then R is a (non-reduced) root system in V and the Weyl group W of R as a root system is isomorphic to the Weyl group of Ψ.

Now let G be a connected linear algebraic group. With the notations of the previous section, consider β ∈ P0 and the group Gβ. The group Gβ/ Ru(Gβ) is reductive, so there

are two weights ±α of T Ru(Gβ)/ Ru(Gβ) in Gβ/ Ru(Gβ). But T Ru(Gβ)/ Ru(Gβ) ∼= T so

these weights correspond to characters ±α of T . We have (ker α)0 = (ker β)0 so α = λβ

and α ∈ P0. The characters so obtained are the roots of G relative to T . The set of roots is denoted Φ or Φ(G, T ) if we want to specify group and torus.

Now suppose Φ is a root system obtained as above and put X = X∗(T ) the character group of T and X∨= X∗(T ) the cocharacter group of T . There is a bijective map α −→ α∨

that maps Φ onto a subset Φ∨ of X∨. It is defined by the relation hx, α∨i = 2(x,α)(α,α) for every x ∈ V . Note that hα, α∨i = 2. It is bijective because if α∨ = βthen

sαsβ(x) = sα x − hx, β∨iβ



= x − hx, β∨iβ − hx, α∨iα + hx, β∨ihβ, α∨iα = x + hx, α∨i(α − β)

But since hα − β, β∨i = 0 all eigenvalues of sαsβ are 1 so sα= sβ and α = β. The elements

of Φ∨ are the coroots of G relative to T . To sum, given a connected linear algebraic group G and a maximal torus T ⊆ G we can define a quadruple (X = X∗(T ), Φ, X∨ = X∗(T ), Φ∨) that is, indeed, a root datum. Since maximal tori are conjugate the root

datum is determined by G, up to isomorphism, and the same holds for the root system Φ. Lemma 1.58. If α ∈ Φ and λα ∈ Φ then λ = ±1.

Proof. We have Gα= Gλα. But the pair of roots ±α is determined by Gα, so it must be

c = ±1.

Hence, The root system Φ of a linear algebraic group G is reduced and verifies all axioms of definition 1.43.

Now we can replicate the theory of root systems in the specific case of Φ(G, T ). We obtain that there can be defined a system of positive roots and hence a base. In particular, if B is a Borel subgroup T ⊆ B and α ∈ Φ, then Gα∩ B is a Borel subgroup of Gα. By

applying the theory of semisimple groups to Gα ⊇ Gα ∩ B it follows that L(Gα ∩ B)

contains exactly one of the weight spaces corresponding to ±α in Gα. So, B picks exactly

one root out of each pair ±α. Denote the set of roots so obtained with Φ+(B). The following holds.

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Suppose again that G is a connected linear algebraic group and T a maximal torus. We now characterize the unipotent radical Ru(G) of G.

Denote by C the identity component of the intersection of the unipotent parts Bu

of the Borel subgroups B that contain T . For every α ∈ Φ denote by Cα the identity

component of the intersection of the Bu such that B contains T and α ∈ Φ+(B). Then C

is a closed subgroup of Cα.

Theorem 1.60. Ru(G) = C

Proof. It is obvious that Ru(G) ⊆ C.

To prove the other inclusion it is sufficient to show that C is normal. Consider Gγ for

some γ ∈ P . If Gγ is non-solvable, then Gγ = Gα for some α ∈ Φ. Then Gα is generated

by its Borel subgroup that contain T and every such Borel subgroup lies in every Borel subgroup B of G such that α ∈ Φ+(B) or −α ∈ Φ+(B), so it is contained in some T.Cα.

If Gγ is solvable, then Gγ is contained in every Borel subgroup that contains T , so it

is contained in T.C.

In both cases, it follows that C is normal in Gγ because Gγ is generated by its Borel

subgroup that contain T , so C is normal in G.

Corollary 1.61. Let G be reductive and T a maximal torus of G: 1. If S is a subtorus of G, then ZG(S) is connected and reductive;

2. ZG(T ) = T ;

3. the center of G lies in T .

1.12

Reductive Groups

Let G denote a connected, reductive, linear algebraic group and T a maximal torus of G. Let (X, Φ, X∨, Φ∨) be a the root datum of (G, T ).

Theorem 1.62. 1. For every α ∈ Φ there exists an isomorphism uα: Ga−→ Uαwhere

Uα is a closed subgroup of G such that

(a) tuα(x)t−1= uα(α(t)x);

(b) the subgroup Uα is uniquely determined;

(c) Im duα= gα the weight space for the weight α.

2. T and the Uα generate G.

Corollary 1.63. The roots of Φ are the non-zero weights of T in g. For each α we have dim gα = 1.

Note that Uα ⊆ (Gα, Gα) and the latter is isomorphic to SL2 or PSL2. In both cases,

with some computation the following can be deduced. Proposition 1.64. The maps uα can be chosen such that

nα= uα(1)uα(−1)uα(1)

where nα∈ NG(T ) is a representative of sα∈ W . For every x ∈ K∗ we then have:

uα(x)uα(−x−1)uα(x) = α∨(x)nα

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1. n2

α = α∨(1);

2. n−α = n−1α ;

3. if u0α is another family with the same properties then there exist cα∈ K∗ such that:

u0α(x) = uα(cαx)

cαc−α= 1

Now fix a total order on Φ and for x, y ∈ G recall the commutator [x, y] = xyx−1y−1

Proposition 1.65. Let α, β ∈ Φ with α 6= ±β. Then there exist constants cα,β,i,j ∈ K

such that [uα(x), uβ(y)] = Y iα+jβ∈Φ;i,j>0 uiα+jβ cα,β,i,jxiyj 

where the product is computed in the order prescribed by the ordering of Φ.

Let B be a Borel subgroup of G containing T . We denote the unipotent radical of B with Bu.

Lemma 1.66. Let H be a connected, solvable linear algebraic group with S a maximal torus. If there is a family of isomorphisms νi : Ga −→ H i ∈ {1, . . . , n} that for each i

verify:

1. Im νi is a closed subgroup of B;

2. there exist a non-trivial character βi with sνi(x)s−1 = νi(βi(s)x) for every x ∈ K

and s ∈ S;

3. the previous character are two by two linearly independent; 4. the weight spaces hβi are one-dimensional and span L(Hu).

then the morphism

Ψ : Gna −→ Hu

(x1, . . . , xn) 7−→ ν1(x1) · · · νn(xn)

is an isomorphism of varieties.

If we apply the previous lemma to the case H = B and νi = uαi where (α1, . . . , αn) is

any numbering of the roots in Φ+(B) we obtain that any Borel subgroup can be written as:

B = T · Y

α∈Φ+(B)

Given a root datum (X, Φ, X∨, Φ∨), we know that Φ is a root system. Moreover, if we fix a W -invariant positive definite symmetric bilinear form and an x ∈ X, we can define a system of positive root Φ+ as

Φ+= {α ∈ Φ | (α, x) > 0} we have the following

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Theorem 1.67. Let Φ+ be an arbitrary system of positive roots in Φ. Then T and the

Uα generate a Borel subgroup of G.

Given a system of positive root Φ+, we can define a base ∆ as the roots α ∈ Φ+ such that #(Φ+∩ sα.Φ+) = #Φ+− 1. It follows that to every Borel subgroup is associated a base for Φ.

Corollary 1.68. Let B be a Borel subgroup of G and ∆ the associated base. G is generated by T and by the groups U±α with α ∈ ∆.

1.13

Bruhat Decomposition

G is still a connected reductive linear algebraic group, T a maximal torus and B a Borel subgroup of G. We have seen that to G is associated a root datum (X, Φ, X∨, Φ∨) and a Weyl group W .

Lemma 1.69. Let w ∈ W . Then Uw =

Q

α∈Φ+(w)Uα is a closed connected subgroup and

if w0 is the longest element of W , then the product

Uw× Uw0w −→ U = Bu

is an isomorphism.

Fix a set of representatives { ˙w}w∈W in NG(T ) of the elements of W and consider the

double cosets B ˙wB. For every ˙w, B ˙wB is a locally closed subvariety of G.

Lemma 1.70. Let w = s1. . . sn be a reduced expression of w with si = sαi. Then the

morphism

φ : Gna× B −→ G

(x1, . . . , xn, b) 7−→ uα1(x1) ˙s1uα2(x2) ˙s2· · · uαn˙snb

is an isomorphism. It follows that B ˙wB ∼= Uw−1× B

Now denote C(w) = B ˙wB. A simple application of the previous lemma gives Lemma 1.71. Let w ∈ W and α ∈ ∆. Then

C(sα)C(w) =



C(sαw) if sαw > w

C(w) ∪ C(sαw) if sαw < w

Theorem 1.72 (Bruhat Lemma). G is the disjoint union of the C(w) for w ∈ W . Proof. Let G1=Sw∈WC(w). Then by the preceding lemma, G1 is stable under

multipli-cation by C(sα) for α ∈ ∆ and so is stable under multiplication by T , Uα and U−α. As

these groups generate G, we have G = G1.

Suppose C(v) ∩ C(w) 6= ∅ and so C(v) = C(w). By lemma 1.70, we have dim C(w) = l(w) + dim B, so it must be l(w) = l(v).

Now, if l(v) = 0 is clear that v = w.

Suppose l(v) > 0 and α ∈ ∆ such that sαv < v. Then by the previous lemma

C(sαv) ⊆ C(sα)C(w) ⊆ C(w) ∪ C(sαw)

Now, is either C(sαv) = C(w) or C(sαv) = C(sαw). The former case is impossible

because l(sαv) 6= l(w), so it must be C(sαw) = C(sαv). By induction on l(v) we get

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1.14

Parabolic Subgroups

Suppose Φ a root system and ∆ a basis. Recall that if I = {α1, . . . , αn} ⊆ ∆, then the

set {a1α1+ · · · + anαn}ai∈Z∩ Φ is a root system with basis I. Denote such a root system

with ΦI and the subset in W generated by the sα, α ∈ I with WI.

Now suppose G reductive, B a Borel subgroup and T a maximal torus such that T ⊆ B. If Φ = Φ(G, T ) and ∆ is the basis associated to B denote SI= Tα∈Iker α

0 and LI = ZG(SI). Then LI is a connected and reductive subgroup of G with maximal torus

T and Borel subgroup BI= B ∩ LI.

Lemma 1.73. The root system Φ(LI, T ) is ΦI, the Weyl group W (LI, T ) is WI, the

system of positive roots Φ+I(BI) is Φ+(B) ∩ ΦI and the corresponding basis is I.

Theorem 1.74. With the notation above: 1. PI =

S

w∈WIC(w) is a parabolic subgroup of G containing B and LI;

2. the unipotent radical Ru(PI) is generated by Uα with α ∈ Φ+\ Φ+I;

3. the product LI× Ru(PI) −→ PI is an isomorphism;

4. if P is a parabolic subgroup containing B then there is a unique I ⊆ ∆ for which P = PI.

Proof. We will prove every point one by one:

1. by lemma 1.71 PI is the subgroup generated by C(Id) = B and C(sα) with α ∈ I.

Given that LI is generated by T and the Uα with α ∈ I, it follows that LI⊆ PI.

If we consider C(w0) with w0 the longest element of WI, then C(w0) has greater

dimension than any other B × B orbit in PI. It follows that it is open in the closure

of PI and so PI itself is closed;

2. note that if w0 is the longest element of I, then Φ+(w0) = Φ+I and w0 stabilizes

Φ+\ Φ+

I. Then by lemma 1.69 Bu = U1U2where U1 is generated by Uα with α ∈ Φ + I

and U2 is generated by Uα with α ∈ Φ+\ Φ+I. According to 1.60, Ru(PI) is the

intersection C of the unipotent parts of Borel subgroups that contain T , that is, of wBw−1 for w ∈ WI. Hence Ru(PI) = \ w∈WI wU w−1=   \ w∈WI wU1w−1  U2

because U2is normalized by WI. But LIis reductive, so

T

w∈WIwU1w

−1 = R

u(LI) =

0 and the claim is proved;

3. if C0(w) = BIwBI, then C(w) = C0(w) Ru(PI) so PI = LIRu(PI). More precisely,

the product map LI× Ru(PI) −→ PI is bijective and it is an isomorphism because

the differential map between the tangent spaces is also bijective;

4. let P be a closed subgroup of G that contains B. The set of roots of (P, T ) is a subset of Φ and by intersecting it with ∆ we obtain a subset I. It is clear that PI ⊆ P because, if α ∈ I then U−α⊆ P and Ru(PI) ⊆ B ⊆ P . But the root system

of P and LI are isomorphic, so dim LI = dim P/Pu and

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We will say that L is a Levi subgroup of P if L is a closed subgroup of P and the product map L × Ru(P ) −→ P is bijective.

1.15

The Bruhat Order

Let G be a reductive group and B a Borel subgroup. Consider the homogeneous space B = G/B which is called flag variety. Denote X(W ) = π (C(w)), where π is the projection π : G −→ G/B. Following Bruhat lemma we can prove:

Proposition 1.75. B is the disjoint union of the locally closed subvarieties X(w). They are the B-orbits of B. Moreover X(w) is isomorphic to Gl(w)a .

The X(w) are called the Bruhat cells while their closures S(w) = X(w) are called Schubert varieties.

Definition 1.76 (Bruhat order). The Bruhat order on the B × B orbits C(w) is defined by

C(v) < C(w) ⇔ C(v) ⊆ C(w)

Recall that in 1.9 we defined the Bruhat order between elements in the Weyl group W . For v, w ∈ W , we had v < w if and only if for any reduced expression of w, v could be expressed as a subexpression. The two definitions are strictly related.

Theorem 1.77. Let C(v) and C(w) with v, w ∈ W . Then C(v) < C(w) ⇔ v < w

Proof. Suppose w = sα1. . . sα1 and, for every α ∈ ∆, the subset Pα = B ∪ BsαB. Then

Pα1. . . Pαh is irreducible and closed. By Lemma 1.71, Pα1. . . Pαh is a union of double

cosets C(v) with v < w. Then C(w) ⊆ Pα1. . . Pαh and they have the same dimension, so

C(w) = Pα1. . . Pαh.

1.16

The Isomorphism and Existence Theorems

In this section we will briefly discuss two important theorems regarding reductive groups and their root systems. Detailed proofs can be found in [5].

The first result is called the isomorphism theorem. It states that the root datum is an invariant for the family of connected, reductive, linear algebraic group.

Definition 1.78. Let Ψ = (X, Φ, X∨, Φ∨) and Ψ1 = (X1, Φ1, X1∨, Φ∨1) be two root data.

An isomorphism of root data f : Ψ −→ Ψ1 is an isomorphism of groups f : X −→ X1

that maps Φ in Φ1 and such that its dual maps Φ∨1 into Φ∨.

If G and G1 are two connected reductive linear algebraic groups and ψ : G −→ G1 is

an isomorphism it defines a map between the character groups f : X1 −→ X that induces

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