Subgroups of Linear Algebraic Groups

Transcription

1 Subgroups of Linear Algebraic Groups

2 Subgroups of Linear Algebraic Groups Contents Introduction 1 Acknowledgements 4 1. Basic definitions and examples Introduction to Linear Algebraic Groups Connectedness 8 2. Background in algebraic geometry Tori, Unipotent and Connected Solvable Groups Unipotent Groups Tori Connected Solvable Groups Borel Subgroups Actions of algebraic groups The Borel fixed point theorem and consequences Connected reductive groups Reductive and semisimple groups Lie algebras and root systems Bruhat decomposition Parabolic subgroups Standard parabolic subgroups and the Levi decomposition The Borel-Tits Theorem G-complete reducibility 46 References 50 Introduction Let G be a variety over an algebraically closed field k. We call G an algebraic group if it is equipped with a group structure such that the multiplication map G G G and the inversion map G G are morphisms of varieties. Furthermore, when G is an affine variety, it is called a linear algebraic group. For example, the set M n (k) of n n matrices over k can be easily identified with k n2 so is an affine variety. Now, since the determinant of a matrix is a polynomial in its entries, we see that SL n is also an affine variety, and it s easy to show that it is a linear algebraic group. One can also show that GL n is a linear algebraic group. Algebraic groups have applications to several areas of pure mathematics. For instance, they are notably central to the Langlands program in Number Theory. They can also be a good way to construct an important class of finite groups, called finite groups of Lie type. These groups arise as fixed point sets of certain types

3 2 of endomorphisms of some linear algebraic group over k = F q, where q is a prime power. For example, most of the finite simple groups are finite groups of Lie type. We will unfortunately not have time to discuss these applications, and the reader is referred to [MT, Part III] for a detailed introduction to finite groups of Lie type. The combination of the group structure with the variety structure on G forces it to have some nice properties. For instance, a linear algebraic group is irreducible as a variety if and only if it is connected (see Proposition 1.8). Moreover, any linear algebraic group can be embedded as a closed subgroup of GL n for some n (see Corollary 4.8). Thus linear algebraic groups can be viewed as certain groups of matrices. However, the embedding into GL n is not canonical, and in general we have no control over what it is. Therefore, our results will usually be stated and proved in full generality, without assuming that our groups have already been embedded into some GL n. An important ingredient in studying linear algebraic groups is the notion of a Borel subgroup, which is a maximal closed connected solvable subgroup. For example, the group of invertible upper triangular matrices is a Borel subgroup of GL n. This naturally leads to the study of a larger class of subgroups, called parabolic subgroups. These are closed subgroups which contain a Borel subgroup of G. When the group G is connected and reductive, which means it has no non-trivial proper closed connected unipotent normal subgroups, the structure of its parabolic subgroups is well understood. For instance, they can be expressed as a disjoint union of double cosets of the Borel subgroup they contain (see Theorem 6.6). Parabolic subgroups of connected reductive groups can then be used to generalise some familiar concepts from representation theory. In particular, Serre [Se] introduced the notion of a G-completely reducible (G-cr for short) subgroup (see Definition 7.1), which generalises the notion of a group acting completely reducibly on a vector space V. Indeed, in the special case G = GL(V ) (V a finite dimensional k-vector space), a closed subgroup H of G is G-cr if and only if V is a semisimple H-module. A recent theorem of Bate, Martin and Röhrle [BMR] asserts that G-complete reducibility is equivalent to the notion of strong reductivity, due to Richardson [R]. They then used this to show that a closed normal subgroup of a G-cr subgroup is itself G-cr (see [BMR, Theorem 3.10]). In the case G = GL(V ), this reduces to a well known result in Clifford theory: if V is a semisimple H-module and N H, then V is a semisimple N-module. The purpose of this essay is to give an account of the general theory of linear algebraic groups, focusing on their Borel subgroups and their parabolic subgroups, and to prove the aforementioned results of Bate, Martin and Röhrle. In the study of parabolic subgroups, we will restrict ourselves to the case where the group G is connected reductive. We begin in Section 1 by introducing linear algebraic groups, giving several examples. We then define morphisms of algebraic groups, which are just morphisms of varieties that are also group homomorphisms, and we show that their kernels and images are closed. We also investigate the notion of connectedness and give examples of connected linear algebraic groups. In Section 2, we give the background in algebraic geometry that will be needed later on. More specifically, we discuss tangent spaces, differentials, projective varieties, dimension theory and complete varieties, which play an important role in the study of Borel subgroups in Section 4. However, the emphasis of this essay is

4 on algebraic results, and not so much on the geometry of linear algebraic groups. So the treatment here is mostly expository, and contains almost no proofs. Section 3 is split into three parts. In the first part, we discuss unipotent groups. These are linear algebraic groups which can be embedded into GL n as a group of matrices whose only eigenvalue is 1. In order to study them, we introduce a multiplicative version of the well-known Jordan decomposition for endomorphisms of a finite dimensional vector space. This can be translated back to a decomposition of elements of an arbitrary linear algebraic group into so-called unipotent and semisimple parts. This leads to the definition of a unipotent group, and we then prove that such a subgroup of GL n is conjugate to a group of upper triangular matrices with 1 s along the diagonal (see Theorem 3.7 and Corollary 3.8). In the second part, we introduce tori and outline some of their basic properties. A torus is a linear algebraic group isomorphic to the group of n n diagonal matrices for some n. An important concept needed to understand tori is the notion of a character. For an algebraic group G, a character of G is a morphism between G and the multiplicative group k. The set X(G) of characters of G is easily seen to be an abelian group, and we show that it is finitely generated when G is a torus. We finally investigate connected solvable groups in the third part. Our main result is the Lie-Kolchin theorem (Theorem 3.16), which asserts that a connected solvable subgroup of GL n has a common eigenvector. As an immediate consequence, we obtain that a connected solvable subgroup of GL n is conjugate to a subgroup of the group of all invertible upper triangular matrices (Corollary 3.17). Moreover, we give without proof the result that a connected solvable group G is the semidirect product G u T of its subgroup G u of unipotent elements with a maximal torus T. In Section 4, we study Borel subgroups of arbitrary linear algebraic groups. To do so, we first study actions of algebraic groups. We also explain how to make a quotient G/H into a linear algebraic group when H is a closed normal subgroup of G. Along the way, we will prove that linear algebraic groups can be embedded into some GL n (see Corollary 4.8). Using these tools, we then show the Borel fixed point theorem (Theorem 4.13), which asserts that if a connected solvable group acts on a projective variety in a way so that the action is given by a morphism of varieties, then it must have a fixed point. This has important consequences. For instance, it implies that the Borel subgroups of a linear algebraic group are all conjugate (Proposition 4.14). We also introduce parabolic subgroups and show that they are connected and self-normalising (see Corollary 4.28). Section 5 is devoted to the study of connected reductive linear algebraic groups. The theory there is quite lengthy, and we do not have time for too much details, but we try to explain all the steps required to obtain the structure of connected reductive groups (Theorem 5.13). However, we do not prove the theorem, nor many of the results that build up to it. The structure of these groups relies crucially on their root system, which itself is defined using Lie algebras. Therefore, after introducing reductive groups, we explain how to associate a Lie algebra to a linear algebraic group, and outline basic properties it must satisfy. This allows us to define the root system of G with respect to a maximal torus T. We then work our way to the structure theorem from there. Subsequently, we discuss a few basic facts about abstract root systems and state the result that a Borel subgroup B of G containing T defines a base for the root system. We finally obtain the very important Bruhat decomposition (Theorem 5.26), which states that G can be decomposed as a disjoint 3

5 4 union of double cosets of B. This decomposition is more generally true for any group G with a BN-pair (see Definition 5.24), and it is in that context that we prove it. This result also allows us to deduce that the intersection of two Borel subgroups is connected and contains a maximal torus (Corollary 5.27). Because of lack of time, we do not discuss here the classification theorem of Chevalley, which gives a one-to-one correspondence between isomorphism classes of semisimple linear algebraic groups (these are connected reductive groups with no proper non-trivial closed connected solvable normal subgroups) and isomorphism classes of root data, which are combinatorial objects that can be obtained from the root system. One remarkable feature of this theorem is that it holds in arbitrary characteristic, and not only in characteristic 0 unlike the classification theorem for semisimple Lie algebras. The reader is referred to [S, Chapters 9 and 10, in particular Theorem ] for a proof and exact statement of this theorem. With the results of Section 5 at our disposal, we investigate parabolic subgroups of connected reductive groups in Section 6. These have a nice structure which also relies crucially on the root system: they are uniquely determined by some subset of the base of the root system which was obtained from the Borel subgroup they contain. We prove again that these groups are closed, connected, self-normalising subgroups of G without appealing to the results in Section 4, and that two distinct parabolic subgroups containing the same Borel subgroup B are not conjugate. We carry on investigating the structure of parabolic subgroups by decomposing them into a product of a unipotent group with a connected reductive group, called its Levi complement. A Levi subgroup is then defined to be a conjugate of a Levi complement, and we show that Levi subgroups are precisely the centralisers of subtori of G (Proposition 6.13). We then move on to the Borel-Tits theorem (Theorem 6.15), which asserts that for a closed unipotent subgroup U of G contained in a Borel subgroup, there exists a parabolic subgroup P of G such that N G (U) P and U is contained in a closed connected unipotent normal subgroup of P. Using this, we immediately obtain that given a maximal closed subgroup H of G, either H is reductive or H is parabolic (see Theorem 6.18). Finally, we discuss G-complete reducibility in Section 7. A closed subgroup H of G is G-completely reducible if whenever it is included in a parabolic subgroup of G, it is actually included in a Levi subgroup of it. We also introduce the notion of a strongly reductive subgroup. A closed subgroup H of G is strongly reductive if it is not contained in any parabolic subgroup of C G (S), where S is a maximal torus of C G (H). We then show that a closed subgroup is strongly reductive if and only if it is G-completely reducible (Theorem 7.7). This requires a few lemmas on the intersection of two parabolic subgroups, which we state beforehand. Using a theorem of Martin [M], we deduce that a closed normal subgroup of a G-completely reducible subgroup is itself G-completely reducible (Theorem 7.9), and using the Borel-Tits theorem, we show that the converse is not true in general. In Sections 1-6, we follow the texts by Malle and Testerman [MT] and Humphreys [H], as well as Springer [S] or Borel [B] for some of the more geometrical results. In Section 7, our main reference is [BMR]. Acknowledgements I am grateful to David Stewart for setting this essay and for kindly giving me advice for a talk, based on this work, that I gave in the Part III Seminars.

6 5 1. Basic definitions and examples 1.1. Introduction to Linear Algebraic Groups. Let k be an algebraically closed field. Recall that a subset of k n is called an algebraic set if it is of the form Z(I) = {(x 1,..., x n ) k n : f(x 1,..., x n ) = 0 for all f I} where I is an ideal in the polynomial ring k[t 1,..., T n ]. Taking closed sets to be algebraic sets defines a topolgy on k n, called the Zariski topology. An affine variety is an algebraic set together with its induced Zariski topology. Given an algebraic set X, we can define an ideal in k[t 1,..., T n ] by I(X) = {f k[t 1,..., T n ] : f(x 1,..., x n ) = 0 for all (x 1,..., x n ) X} The quotient k[x 1,..., X n ]/I(X) is called the coordinate algebra or algebra of regular functions on X, and is denoted by k[x] If X k n, Y k m are affine varieties, then the product X Y naturally has the structure of an algebraic set in k n+m and thus is also an affine variety when equipped with the Zariski topology (which in general is not the same as the product topology). Note that k[x Y ] = k[x] k k[y ]. A map ϕ : X Y is called a morphism of affine varieties if it can be defined by polynomial functions in the coordinates. Note that such maps are continuous with respect to the Zariski topology. The morphism ϕ induces a k-algebra homomorphism ϕ : k[y ] k[x] f f ϕ We can now define our main object of study: Definition 1.1. A linear algebraic group is an affine variety G equipped with a group structure such that the group operations are morphisms of varieties. µ : G G G, ι : G G, (g, h) gh, g g 1, Example 1.2. Let s first look at several examples of linear algebraic groups: (1) G a = (k, +), the additive group of k. It s clear that it satisfies the definition (it is the zero set of the zero polynomial), and we have k[g a ] = k[t ], the usual polynomial ring. (2) G m = (k, ), the multiplicative group of k. We can identify it with {(x, y) k 2 : xy = 1} with componentwise multiplication by mapping x (x, x 1 ). This set is clearly a closed subset of k 2, being the zero set of the polynomial T 1 T 2 1. Multiplication and inversion are clearly given by polynomials so it is a linear algebraic group, and k[g m ] = k[t 1, T 2 ]/(T 1 T 2 1) = k[t, T 1 ]. (3) The general linear group GL n = {A M n (k) : det A 0} is also a linear algebraic group. As for G m, one way of seeing this is to identify it with {(A, y) k n2 k : det A y = 1}

7 6 via A (A, det A 1 ), with componentwise multiplication where we identify k n2 with M n (k) (and so multiplication in the first component is matrix multiplication). Since det is given by a polynomial in the matrix entries, this is a closed subset of k n2 +1. Multiplication is then clearly given by polynomials, and by Cramer s rule, so is inversion. We can also find the ring of regular functions: k[gl n ] = k[t ij, Y : 1 i, j n]/(det (T ij ) Y 1) = k[t ij : 1 i, j n] det (Tij) the localisation of k[t ij : 1 i, j n] at the multiplicatively closed subset generated by det (T ij ). (4) The special linear group SL n = {A GL n : det A = 1} is a closed subgroup of GL n, as det is given by a polynomial in the matrix entries, and so is also a linear algebraic group. Its ring of regular functions is then clearly k[t ij : 1 i, j n]/(det (T ij ) 1). (5) Similarly, the following are closed subgroups of GL(n) and so linear algebraic groups: The group of invertible upper triangular matrices T n = GL n = {(a ij) GL n : a ij = 0 for i > j}. The group of upper triangular matrices with 1 s on the diagonal 1... U n =.... = {(a ij) T n : a ii = 1 for 1 i n}. 1 The group of invertible diagonal matrices {( ) }... D n = GL n = {(a ij ) GL(n) : a ij = 0 for i j}. (6) Let J 2n = Note that D n = Gm... G m }{{} n times ( ) 0 Kn where K K n 0 n = dimension 2n is then defined to be Sp 2n = {A GL 2n : A t J 2n A = J 2n }. ( ) The symplectic group in 0 It is a closed subgroup of GL 2n, thus a linear algebraic group. It is the group of linear transformations leaving invariant the non-degenerate skewsymmetric bilinear form given by J 2n. (7) Orthogonal groups are also linear algebraic groups. For simplicity, we assume char(k) 2 (for the general case see [MT, section 1.2]). The orthogonal group in dimension n is given by GO n = {A GL n : A t K n A = K n }. It is the group of linear transformations leaving invariant the non-degenerate symmetric bilinear form given by K n.

8 7 Having defined linear algebraic groups, we then consider the maps between them. We require them to preserve both the geometrical structure and the group structure of algebraic groups. Definition 1.3. Let G 1, G 2 be linear algebraic groups. A map ϕ : G 1 G 2 is a morphism of algebraic groups if it is a group homomorphism and a morphism of affine varieties. We would like for such maps to have nice images and kernels. In order to prove it, we first need a couple of basic results: Lemma 1.4. Let U, V be two dense open sets of an algebraic group G. G = U V Then Proof. Pick x G. From the definition of algebraic groups, inversion is a continous map with continuous inverse (being its own inverse), and thus is a homeomorphim. Hence V 1 is also open dense. Similarly, multiplication by x is a homeomorphism, and thus xv 1 is open dense. Therefore U, being dense, must meet xv 1 and so x U V. Recall that a subset X of a topological space Y is called locally closed if it is the intersection of an open set with a closed set. Equivalently, X is locally closed if it is open in X. A subset of Y is called constructible if it is a finite union of locally closed sets. It is a fact from algebraic geometry that morphisms of varieties map constructible sets to constructible sets (see [H, Theorem 4.4]). In particular, the image of a morphism is constructible. Also, it is a standard fact that if X is a constructible subset of a variety, then it contains an open dense subset of X. Proposition 1.5. Let H be a subgroup of G, H its closure. Then: (i) H is a subgroup of G. (ii) If H is constructible, then H = H. Proof. (i) Inversion being a homeomorphism, it s easy to see that H 1 = H 1 = H. Similarly, for x H, multiplication by x is a homeomorphism and so xh = xh = H. Thus H H H. Hence, for x H, Hx H, and so Hx = Hx H. Therefore H is closed under inverses and multiplication, and so is a subgroup of G. (ii) If H is constructible, then it contains an open dense subset U of H. Now, H is a linear algebraic group by (i), and so by Lemma 1.4, we have H = U U H H = H. Corollary 1.6. Let ϕ : G 1 G 2 be a morphism of algebraic groups. Then ker ϕ and ϕ(g 1 ) are closed, and therefore are linear algebraic groups. Proof. ϕ is continuous and ker ϕ = ϕ 1 ({1}) is the inverse image of a closed set, so it is closed. Moreover, ϕ(g 1 ) is a constructible subgroup of G 2. By the previous proposition, it must be closed. It is clear that closed subgroups of GL n are linear algebraic groups. The following important result shows that all linear algebraic groups arise in that way. We will prove it later, in Section 4, when we discuss quotients of algebraic groups. Theorem 1.7. Any linear algebraic group can be embedded as a closed subgroup into GL n for some n.

9 Connectedness. The group structure of a linear algebraic groups allows us to know more about its geometrical structure. Recall that an affine variety X is irreducible if it cannot be written as a union U V of non-empty closed subsets U, V. In general, an affine variety X can be written as r i=1 X i, for some r, where the X i are maximal irreducible subsets, called the irreducible components of X. It is a fact that a morphism of varieties maps irreducible subsets to irreducible subsets (see [H, Prop 1.3A]). Also, if X and Y are irreducible, so is X Y (see [H, Prop 1.4]). Also recall that a topological space X is connected if it cannot be written as a disjoint union U V of non-empty closed subsets U, V. It is clear that an irreducible variety is connected, while the converse is not true in general. However, for algebraic groups the converse does hold. More specifically: Proposition 1.8. Let G be a linear algebraic group. (i) The irreducible components of G are pairwise disjoint, and so are the connected components of G. (ii) The irreducible component G containing the identity is a closed normal subgroup of finite index. (iii) Any closed subgroup of G of finite index contains G. Proof. (i) Let X, Y be irreducible components of G. Suppose X Y. Pick g X Y. Since multiplication by g 1 is an isomorphism of varieties, we know that g 1 X and g 1 Y are irreducible components, and we have 1 g 1 X g 1 Y. So without loss of generality (wlog), we may assume that 1 X Y. Now, since X Y is irreducible in G G, it follows that µ(x Y ) = X Y is irreducible in G. Moreover, we have X X Y since 1 Y. By maximality of X, it must be that X = X Y, and similarly, we obtain Y = X Y = X. (ii) Since inversion is an isomorphism of varieties, (G ) 1 is an irreducible component of G. As it contains 1, it must be G by (i). Similarly, for h G, multiplication by h is an isomorphism of varieties and so hg is an irreducible component, and it contains 1 since G is closed under inverses. Therefore hg = G. It follows that for any g, h G, gh G. Hence G is a subgroup. Also, for g G, conjugation by g is an isomorphism of varieties, so gg g 1 is again an irreducible component containing 1, and so it equals G. Thus G is a normal subgroup. For the last part, let X be an irreducible component of G. Pick g X. We have that g 1 X is an irreducible component of G containing 1, and so it equals G. Hence X = gg and so all the irreducible components of G are cosets of G. It is clear that all cosets of G are irreducible components, so since there are only finitely many irreducible components of G, G must have finite index. (iii) Let H G be a closed subgroup of finite index. Then H G G and we have [G : H ] = [G : H] [H : H ], which is finite by (ii). Hence we can write G = gh, a finite disjoint union of cosets of H. Since G is connected, it follows that G = H H. We will therefore refer to the irreducible (or connected) components of G as the components of G. An immediate consequence of the above proposition is that ϕ(g ) = ϕ(g) for any morphism of algebraic groups ϕ : G H. Indeed, ϕ(g ) is closed (by Corollary 1.6), connected (since G is connected), contains 1 and has finite index in ϕ(g) by Proposition 1.8(ii) applied to G. The result follows by Proposition 1.8(iii).

10 9 Example 1.9. Let s see which of our examples are connected. It is a well known result in algebraic geometry that a variety X is irreducible if and only if its ring of regular function k[x] is an integral domain. Therefore, we know that G a, G m and GL n are connected algebraic groups since their ring of regular functions were integral domains. Also, D n is connected since it is a direct product of connected algebraic groups (namely n copies of G m ). On the other hand, it can be shown that GO n is not connected (assuming char(k) 2), with component at the identity SO n = GO n SL n, called the special orthogonal group. It is also true that SL n, T n and U n are connected. In order to show this, we need a geometrical result (see [H, Proposition 7.5]): Proposition Let G be a linear algebraic group and f i : X i G, i I, a family of morphisms from irreducible varieties X i, such that 1 Y i = f i (X i ) for all i I. Then H = Y i : i I is a closed, connected subgroup of G. Moreover, for some finite sequence i 1,..., i n in I, H = Y ±1 i 1 Y ±1 i n. In particular, if G is generated by some family of closed connected subgroups, then it is connected. We know from linear algebra that SL n is generated by subgroups U ij = {(a kl ) GL n : a kk = 1, a kl = 0 for (k, l) (i, j)} (i j) of matrices with 1 s on the diagonal, arbitrary entry in the (i, j) position, and 0 elsewhere. Similarly, U n is generated by the subgroups U ij for i < j. These subgroups are all isomorphic to G a, which is connected, so the above proposition gives us that SL n and U n are connected. Similarly, one can show that T n is connected. The above proposition has another useful consequence: Proposition Let H, K be subgroups of a linear algebraic group G, with K closed and connected. Then [H, K] is closed and connected. Proof. For h H, define ϕ h : K G by g [h, g]. It is clearly a morphism, being a composition of multiplication and inversion. Also, 1 = ϕ h (1) for all h. Hence, we have that [H, K] = ϕ h (K) : h H is closed and connected by Proposition Therefore closed connected subgroups behave well under taking commutators. In particular, if G is a connected linear algebraic group, then its derived subgroup G = [G, G] is a closed connected subgroup. Inductively, we then see that its nth derived subgroup is a closed, connected subgroup. With this in mind, we recall a group-theoretic definition: Definition For a group G, define G (0) = G and G (i) = [G (i 1), G (i 1) ] for i 1. We then obtain the derived series of G: G = G (0) G (1) G (2)... We say G is solvable (or soluble) if G (d) = 1 for some d. The smallest such d is then called the derived length of G. Similarly, one can define C 0 G = G and C i G = [C i 1 G, G] for i 1. We then define G to be nilpotent if C n G = 1 for some n. Remark. If G is a nilpotent group and n is the largest integer such that C n G 1, then C n G must commute with G so in particular Z(G) 1.

11 10 Example Some of the algebraic groups we met are solvable. Indeed, G a, G m and D n are obviously solvable since they are abelian. Also, T n is solvable and U n is nilpotent. This can be shown by a direct calculation. It is easy to see that T (1) n U n and actually, they are equal (one can find generators of U n which are commutators of elements of T n ). It s then not too hard to see that C m U n = {(a ij ) U n : a ij = 0 for 0 < i j m} and so U n is nilpotent, and thus solvable. This then implies T n is solvable. G a, G m, D n, T n and U n are all examples of connected solvable linear algebraic groups. We will see later that such linear algebraic groups have a nice structure. 2. Background in algebraic geometry Most of the results in this section will be stated without proof. Proofs can be found in [H, sections 1-6]. We need to recall a few facts from algebraic geometry which will be needed later on. In the theory of algebraic groups, we need to know about tangent spaces, projective varieties, complete varieties or the dimension of varieties. Definition 2.1. For an affine variety X, we define the tangent space of X at x X by T x (X) = {δ : k[x] k linear : δ(fg) = f(x)δ(g) + δ(f)g(x) for f, g k[x]} the k-vector space of point derivations at x. Having defined the tangent space, we can now define the differential of a morphism: Definition 2.2. Let ϕ : X Y be a morphism of affine varieties. The differential d x ϕ of ϕ at x X is the map d x ϕ : T x (X) T ϕ(x) (Y ) defined by d x ϕ(δ) = δ ϕ for δ T x (X). Taking differentials behave functorially (see [H, 5.4]): Proposition 2.3. Let ϕ : X Y and ψ : Y Z be morphisms of affine varieties, and x C. Then d x (ψ ϕ) x = d ϕ(x) ψ d x ϕ. In Section 4, we will consider actions of algebraic groups on varieties other than just affine varieties. To this end, we recall facts about projective varieties: Definition 2.4. Projective n-space P n is defined to be the set of equivalence classes of k n+1 \ {0, 0,..., 0} relative to the equivalence relation {x 0, x 1,..., x n } {y 0, y 1,..., y n } λ k such that y i = λx i for all i For a k-vector space V of dimension n + 1, we can identify P n with the set of all 1-dimensional subspaces of V, usually denoted by P(V ). We can define a Zariski topology on P n : define the closed sets to be the sets given by the vanishing of some collection of homogeneous polynomials in k[t 0,..., T n ]. A projective variety is a closed subset of some P n, equipped with the induced topology. A quasi-projective variety is an open subset of a projective variety. Projective varieties are clearly quasi-projective.

12 11 In general, a variety is defined to be a pair (X, O X ), where X is a topological space and O X is a sheaf of functions on X, such that X has a finite open cover U i with each (U i, O X Ui ) isomorphic to an affine variety. Moreover we also require that the diagonal = {(x, x) : x X} is closed in X X. Morphisms are then defined to be continuous maps which preserve the sheaf of functions. For more details see [H, section 2]. In practice, our varieties will always be affine or quasi-projective. A useful example of a projective variety is the flag variety of a finite dimensional vector space V. A flag of V is a chain 0 = V 0 V 1... V k = V where all inclusions are strict. A full flag is one where dim V i+1 = dim V i + 1 for all i, i.e one where k = dim V. The flag variety is defined to be the set of all full flags of V. It can indeed be given the structure of a projective variety (see [H, 1.8]). We now recall the notion of dimension: Definition 2.5. For an irreducible variety X, its ring of regular function k[x] is an integral domain, so we can take k(x) to be its field of fractions. We define the dimension of X to be the the transcendence degree of k(x) over k. Equivalently, it is the maximal length of a chain of prime ideals in k[x]. In general, for a reducible variety X, we define dim X = max{dim X i : 1 i r} where the X i are the irreducible components of X. Note that for a linear algebraic group G, dim G = dim G since the components of G are the cosets of G, which are all isomorphic as varieties to G and so all have the same dimension. In particular dim G = 0 if and only if G is finite: a connected space of dimension 0 is just a point, so G = 1 and since G is a union of finitely many cosets of G, it is finite. The following proposition shows how dimension behaves well with respect to morphisms (see [H, Theorem 4.3]): Proposition 2.6. Let ϕ : X Y be a morphism of irreducible varieties with ϕ(x) dense in Y. Then there exists a non-empty open subset U Y with U ϕ(x) such that dim ϕ 1 (y) = dim X dim Y for all y U We deduce a rank-nullity result for morphisms of algebraic groups: Corollary 2.7. Let ϕ : G 1 G 2 be a morphism of linear algebraic groups. Then dim ϕ(g 1 ) + dim ker ϕ = dim G 1 Proof. Every fiber ϕ 1 (y) is a coset of ker ϕ, and thus has the same dimension. Apply Proposition 2.6 with X = G 1 and Y = ϕ(g 1 ). Example 2.8. We find the dimension of some algebraic groups: (1) dim G a = dim G m = 1 since clearly k(g a ) = k(g m ) = k(t ). An easy inductive argument using Corollary 2.7 shows that dim D n = n for all n. (2) dim GL n = n 2 since the field of fractions of k[t ij : 1 i, j n] det Tij is k(t ij : 1 i, j n). (3) dim SL n = n 2 1 using the previous two examples and Corollary 2.7 applied to the surjective morphism det : GL n G m. We conclude our discussion of dimension with the following result, which will be useful for inductive arguments:

13 12 Proposition 2.9. If Y is a proper, closed subset of an irreducible variety X, then dim Y < dim X. Proof. Let Y 1 Y be an irreducible component. The inclusion ϕ : Y 1 X induces a surjective k-algebra homomorphism ϕ : k[x] k[y 1 ]. Since Y 1 is irreducible, k[y 1 ] is an integral domain and so ker ϕ is a prime ideal in k[x], non-zero since Y 1 X is proper. Then any chain of prime ideals in k[y 1 ] lifts to a chain of prime ideals in k[x] through ker ϕ, hence of greater length since X is irreducible and so 0 is a prime ideal in the integral domain k[x]. We now move on to complete varieties. These will be useful when we will study actions of algebraic groups on projective varieties. Definition A variety X is complete if, for any variety Y, the projection morphism X Y Y is a closed map, i.e maps closed sets to closed sets. Clearly, a closed subvariety of a complete variety is complete. We summarize all the results we ll need in the following proposition (see [H, section 6]): Proposition (i) A projective variety is complete. (ii) A complete quasi-projective variety is projective. (iii) A complete affine variety has dimension 0. (iv) If ϕ : X Y is a morphism of varieties and X is complete, then ϕ(x) is closed in Y, and complete. 3. Tori, Unipotent and Connected Solvable Groups Having established some basic facts about linear algebraic groups, a first question we could ask is the following: what do one-dimensional linear algebraic groups look like? It turns out that any one-dimensional connected linear algebraic group is isomorphic to either G a or G m. This seemingly innocent result is actually quite difficult to show and we will not prove it here (see [H, section 20] for a proof). This tells us that one dimensional connected groups are all abelian, and so solvable. Connected solvable groups are quite important to the general theory, as we shall see later, and so we study them in this section. We first start by two particular examples of such groups: unipotent groups and tori Unipotent Groups. Recall the additive Jordan decomposition for endomorphisms: if V is a finite dimensional k-vector space and α End(V ), then there exists unique s, n End(V ) such that s is semisimple, i.e diagonalisable, n is nilpotent, α = s + n and sn = ns. Moreover s and n are both polynomials in α with constant coefficient equal to zero. Definition 3.1. An endomorphism u End(V ) is unipotent if u 1 is nilpotent. Equivalently, u is unipotent if the only eigenvalue of u is 1. There is also a multiplicative version of the Jordan decomposition: Proposition 3.2. For g GL(V ), there exists s, u GL(V ) such that g = su = us, where u is unipotent and s is semisimple. Proof. From the additive decomposition, we can write g = s + n with s, n as described above. Since g is invertible, so is s and so we may define u = 1 + s 1 n.

14 13 As n is nilpotent and sn = ns, we have that u 1 = s 1 n is nilpotent. Therefore u is unipotent and su = s + n = g. If g = su = us is any such decomposition, where u = 1 + n with n nilpotent and commuting with s, then g = s + sn is the unique additive Jordan decomposition, and u, s are therefore uniquely determined. We can then transfer this definition to an arbitrary linear algebraic group: Theorem 3.3. (Jordan decomposition) Let G be a linear algebraic group. (i) For any embedding ρ of G into some GL(V ) and for any g G, there exists unique g s, g u G such that g = g u g s = g s g u, where ρ(g s ) is semisimple and ρ(g u ) is unipotent. (ii) The decomposition g = g u g s = g s g u is independent of the chosen embedding. (iii) Let ϕ : G 1 G 2 be a morphism of algebraic groups. Then ϕ(g s ) = ϕ(g) s and ϕ(g u ) = ϕ(g) u. We won t give a proof of this result (see [H, Theorem 15.3]), but we give here an important step which we will need later on. Given a linear algebraic group G, each x G defines a morphism G G given by g gx, which induces a k[g]-algebra homomorphism ρ x : k[g] k[g] defined by ρ x (f)(g) = f(gx) for f k[g], g G. This defines an action of G on k[g]. Proposition 3.4. Let G be a linear algebraic group and V a finite dimensional subspace of k[g]. Then there exists a finite dimensional G-invariant subspace X containing V. In particular, k[g] is a union of finite dimensional G-invariant subspaces. Moreover, the restriction of any such finite dimensional subspace X affords a morphism of algebraic groups ρ : G GL(X). Proof. It s enough to prove this for V = f, a one-dimensional subspace. Recall that multiplication gives a morphism µ : G G G and that k[g G] is isomorphic to k[g] k k[g]. Therefore write µ (f) = i I f i g i, so that ρ x (f) = i I g i(x)f i. Hence the finite dimensional subspace generated by {f i : i I} contains ρ x (f) for all x G. It follows that the subspace X generated by {ρ x (f) : x G} is contained in it and so is finite dimensional. It is clearly G-invariant and it contains V so the first part follows. For the last part, we see from the above construction that the coordinates of ρ x in X are polynomial functions in x. Therefore the map x (ρ x ) X affords a morphism of algebraic groups G GL(X). An endomorphism x of a vector space V is called locally finite if V is a union of finite dimensional x-stable subspaces. Proposition 3.4 shows ρ x is locally finite. One can show that locally finite endomorphisms have a Jordan decomposition in the sense of Proposition 3.2. The idea of the proof of Theorem 3.3 is to use the unipotent and semisimple parts of ρ g, for g G, to construct g u and g s. Definition 3.5. Let G be a linear algebraic group. The decomposition g = g u g s = g s g u in Theorem 3.3 is called the Jordan decomposition of g G and g is called semisimple (respectively unipotent) if g = g s (respectively g = g u ). We write G u = {g G : g is unipotent} G s = {g G : g is semisimple} for the subsets of unipotent and semisimple elements of G. If G = G u, then we say G is a unipotent group. Note that G u is a closed subset since the set of unipotent elements of GL n is closed, given by the polynomial (T 1) n.

15 14 Remarks. (i) Observe that the subgroup generated by G u is a characteristic subgroup of G where by characteristic, we mean here that it is preserved under all algebraic group automorphisms. Indeed, if α Aut(G), take any embedding ρ of G into some GL n. Then ρ α is another embedding of G into GL n, and by Theorem 3.3(ii), we have that the image of G u is still unipotent, thus G α u G u. (ii) A group G for which G = G s is not called semisimple. Semisimple groups have a different definition (see Section 5). Example 3.6. It s clear that U n is unipotent { ( ) for } any n and that for G = T n, G u = U n. Note also that G a 1 = U2 = is a unipotent group. We also 0 1 met groups where G = G s, for example G m or more generally D n for n 1. As said above, U n and more generally subgroups of U n are unipotent subgroups of GL n. It turns out that all unipotent subgroups of GL n are conjugate to a subgroup of U n. To get there, we first need the following result: Theorem 3.7. Let G be a unipotent subgroup of GL(V ) for some non-zero finite dimensional vector space V. Then G has a common eigenvector in V. Proof. Identify V with k n where n = dim V. We use induction on n. The result is obvious if dim V = 1 (every v V is a common eigenvector of G), so assume dim V > 1. Suppose V has a proper non-zero subspace W stable under G. Then by choosing appropriate bases, we may assume that { ( ) } G 0 More specifically, every element g G can be written in the form ( ) ϕ(g) 0 ψ(g) where ϕ : G GL(W ) is the canonical restriction morphism, and ψ : G GL(V/W ). Now ϕ(g) is also unipotent, so by induction hypothesis there exists a common eigenvector v W V for G. Therefore we may assume that V is an irreducible G-module. We need the following theorem of Burnside (see [L, XVII, section 3]): if R is a subalgebra of End(V ) which acts irreducibly on V, then R = End(V ). Now, the assumption that G is unipotent implies Tr(x) = Tr(1) = dim V for all x G. Writing x as 1 + n with n nilpotent, we have for all y G: Tr(y) = Tr(xy) = Tr(y + ny) = Tr(y) + Tr(ny). Therefore Tr(ny) = 0. Now, the k-linear combinations of the elements of G must also satisfy this. These form a subalgebra R of End(V ), which acts irreducibly on V since G does. Burnside s theorem then implies that for all y End(V ) and for all x = 1 + n G, Tr(ny) = 0. Taking y to be the standard unit matrices E ij, we see that we must have n = 0 (by E ij, we mean the matrix whose (i, j)th entry is 1 and all other entries are 0). Hence G = 1 and since V is irreducible, dim V = 1, a contradiction. Corollary 3.8. If G GL n is a unipotent group, then G is conjugate to a subgroup of U n. Since U n is nilpotent (see Example 1.13), it follows that G is nilpotent, and so solvable.

16 15 Proof. By Theorem 3.7, G has a common eigenvector v V = k n. Let V 1 = v. Then G acts on V/V 1, the image of G in GL(V/V 1 ) being again unipotent. Induction on dim V then allows us to construct a basis of V with respect to which elements of G are represented by upper triangular matrices. Since they are also unipotent, it follows that these matrices are in U n. This result can be seen as a generalisation of the fact that p-groups are nilpotent. Indeed, if char(k) = p > 0, then an endomorphism u is unipotent if and only if u pf = 1 for some f 1 since for some large enough f, we require that u pf 1 = (u 1) pf = Tori. Definition 3.9. A torus is a linear algebraic group isomorphic to D n for some n 0. It turns out that an important concept in studying tori is their characters. Definition For G a linear algebraic group, a character of G is a morphism of algebraic groups χ : G G m. The set of characters of G is denoted by X(G). Note that it can be considered as a subset of k[g]. A cocharacter of G is a morphism of algebraic groups γ : G m G. The set of cocharacters is denoted by Y (G). X(G) is clearly an abelian group with respect to (χ 1 + χ 2 )(g) = χ 1 (g)χ 2 (g) for χ 1, χ 2 X(G), g G. Similarly, if G is commutative then Y (G) is an abelian group with respect to (γ 1 + γ 2 )(x) = γ 1 (x)γ 2 (x) for γ 1, γ 2 Y (G), x G m. In particular, since we use the additive notation, we will denote by 0 the character mapping everything to 1, and similarly for cocharacters. Given χ X(G) and γ Y (G), χ γ End(G m ). Now, an element f End(G m ) belongs to k[t, T 1 ]. If f is of the form at n for some a k, n Z, then since f is a group homomorphism, we must have f = T n as f(1) = 1. If f is not of this form then we can find n < m in Z such that f(t ) = m i=n a it i with a i k, a n, a m 0. But since f is a group homomorphism, we must have that 1 = f(t )f(t 1 ). Expanding f(t )f(t 1 ), we see that the coefficient of T m n is non-zero, a contradiction. Therefore, we just proved that End(G m ) = {t t j : j Z} = Z. In particular, for χ X(G) and γ Y (G), χ, γ Z such that χ γ : t t χ,γ. This gives us a map, : X(G) Y (G) Z. Note that having established what End(G m ) is, it s easy to see what the characters of D n are. Indeed, writing elements of D n as g = diag(t 1,..., t n ), we can define a character χ i : g t i. It s quite easy to see that X(D n ) is generated by the χ i, a typical element being of the form χ a χan n for some a 1,..., a n Z. So X(D n ) = Z n. Similarly, we see that for γ Y (D n ), composing with the projection on the ith diagonal element gives an endomorphism of G m which is therefore of the form t t di. Thus γ is of the form t diag(t d1,..., t dn ). Using this, it can easily be shown that the map, : X Y Z is a perfect pairing, that is, any homomorphism X Z is of the form χ χ, γ for some

17 16 γ Y, and any homomorphism Y Z is of the form γ χ, γ for some χ X, where X = X(T ) and Y = Y (T ) for a torus T (see [MT, Prop 3.6]). Now, given any torus T, we can identify T with D n for some n and so X(T ) = Z n is a finitely generated abelian group. Therefore it makes sense to talk about characters being linearly independent. Definition Let T be a torus, H T a subgroup and X 1 X(T ) a subgroup of the character group. Then we define H = {χ X(T ) : χ(h) = 1 for all h H}, a subgroup of X(T ). We also define X 1 = {t T : χ(t) = 1 for all χ X 1 } = a closed subgroup of T χ X 1 ker χ, It is a fact that given linearly independent characters χ 1,..., χ n X(T ) and elements c 1,..., c n G m, there exists t T such that χ i (t) = c i for 1 i n (see [H, Lemma 16.2C]). Using this, we can show: Proposition Let T be a torus, H T a subgroup and X 1 X(T ). Then: (i) If H 1 H is a subgroup of finite index, then H 1 /H is finite. (ii) (X 1 ) /X 1 is finite. Proof. (i) By the structure theorem for finitely generated abelian groups, it s enough to show that all the elements of H1 /H have finite order. Let h 1,..., h r be a complete list of cosets representatives for H 1 in H. Then every h H is equal to h i g for some i and g H 1. Take χ H1. Then χ takes only finitely many values on H, namely χ(h 1 ),..., χ(h r ). So χ(h) is a finite subgroup of k, say of order n 1. It follows that nχ H as required. (ii) If (X1 ) = X 1 we re done. Otherwise pick χ (X1 ) \ X 1. Again it s enough to show nχ X 1 for some n 1. Suppose not and aim for a contradiction. Since X 1 X(T ) = Z n for some n, X 1 = Z r for some r n and it has a basis χ 1,..., χ r. Then we have that χ, χ 1,..., χ r are linearly independent and therefore there exists t T such that χ i (t) = 1 (1 i r) but χ(t) 1. Hence t X1 but t / ker χ, contradicting the assumption that χ (X1 ). Using the properties of characters, one can show (see [H, 16]): Proposition Any closed subgroup of D n is a torus. We finally state a result about the rigidity of tori (see [H, Corollary 16.3] for a proof): Theorem Let G be a linear algebraic group and T G a torus. N G (T ) = C G (T ). Then It follows that the quotient N G (T )/C G (T ) is finite since N G (T ) N G (T ) has finite index and N G (T ) C G (T ) by the theorem. As an example, take G = GL n and T = D n. It s clear that T = C G (T ) and moreover N G (T ) = M, the set of monomial matrices (i.e matrices with exactly one non-zero entry on each row). Then since D n M has finite index and is connected, we must have that D n = M by Proposition 1.8(c). So indeed we have N G (T ) = C G (T ) and here N G (T )/C G (T ) = S n is a symmetric group.

18 Connected Solvable Groups. We wish to get a result similar to Theorem 3.7 for connected solvable groups. This is the analogue of Lie s theorem for Lie algebras, except Lie s theorem only holds in characteristic 0 while here we don t make any assumptions on char(k). In the proof we will need the following lemma (see [H, Prop 15.4] for a proof): Lemma Let M GL n be a commuting set of matrices, then M is trigonalisable (i.e we can find a basis with respect to which all elements of M are represented by upper triangular matrices). Theorem (Lie-Kolchin) Let G be a connected solvable subgroup of GL(V ), with V 0 finite dimensional. Then G has a common eigenvector in V, i.e V has a one-dimensional subspace which G stabilises. Proof. It is a fact that if G is solvable, then so is G (it s not difficult to see that G (i) = G (i) for all i, see [B, 2.4]), so we may assume that G is closed in GL(V ). Write V = k n for some n 1. We argue by induction on n and on the derived length d of G. If n = 1, the result is trivial. So suppose n > 1. If d = 1, then G is commutative, and by Lemma 3.15 we have that G has a common eigenvector. So assume d > 1. Suppose first that there is a proper 0 W < V which is stabilised by G. Then as in the proof of Theorem 3.7, by choosing appropriate bases, we may write any g G in the form ( ) ϕ(g) 0 ψ(g) where ϕ : G GL(W ) is the canonical restriction morphism, and ψ : G GL(V/W ). Now, ϕ(g) is connected solvable and acts on W with dim W < dim V. So by induction hypothesis there is v W < V such that v is a common eigenvector of G. The only case left to consider is if G acts irreducibly on V. Assume this holds. We again aim for a contradiction. Let G = [G, G]. It is closed and connected by Proposition 1.11, and obviously solvable with derived length d 1. Hence by induction hypothesis there is a common eigenvector v V for G. Note that gv is also a common eigenvector of G for any g G, since G is normal in G: for h G, g 1 hg G, so g 1 hgv = λv for some λ k, and therefore h(gv) = λgv. Let W denote the non-zero subspace of V spanned by the common eigenvectors of G. By the above W is G-invariant. Since G acts irreducibly on V, it follows that W = V. Hence V has a basis consisting of common eigenvectors of G. So the elements of G are diagonal matrices with respect to that basis, and therefore G is commutative. Now, for fixed h G, all conjugates ghg 1, for g G, are in G and hence are diagonal with the same eigenvalues as h. Therefore, there are only finitely many possibilities for ghg 1. So let ϕ h : G G be the morphism g ghg 1 (it is a morphism since multiplication and taking inverses are morphisms). The image ϕ h (G) is finite by the above discussion, and connected since G is connected. Therefore we must have ϕ h (G) = {h}, i.e h Z(G). Hence G Z(G). Now, every element of Z(G) commutes with G in its action on V, so by Schur s lemma they are represented by scalar multiples of the identity. So elements of G are scalar multiples of the identity and they must have determinant 1 since commutators

19 18 have determinant 1. Therefore there are only finitely many possibilities for elements of G (namely λ.i n where λ n = 1). As G is connected it follows that G = 1, and so G is commutative, contradicting d > 1. Corollary Let G be a connected, solvable subgroup of GL n. Then G is conjugate to a subgroup of T n, the linear algebraic group of upper triangular matrices. Proof. Completely similar to Corollary 3.8. We discuss here one application of this result. There is a natural split exact sequence π 1 U n T n D n 1 where π is the morphism t 1... t t n 0 t n Let G T n be a closed connected subgroup. The restriction of π to G has kernel G u = G U n, a closed normal subgroup. The image T = π(g) is a closed connected subgroup of D n, hence a torus by Proposition So we get an exact sequence 1 G u G π T 1 Since T is abelian, it follows that [G, G] G u. This proves most of part (i) in the following structure theorem for connected solvable groups (see [H, Theorem 19.3 and Prop 19.4] for a full proof): Theorem Let G be a connected solvable linear algebraic group. Then: (i) G u is a closed, connected, normal subgroup of G and [G, G] G u. (ii) If T is a maximal torus of G, then all maximal tori are conjugate to T and G = G u T. Moreover, N G (T ) = C G (T ) By maximal torus, we mean a subtorus of G which isn t contained in any other subtorus. Also, the semidirect product of two algebraic groups G and H is constructed in the same way as for abstract groups, except we require that G acts as a group of algebraic group automorphisms of H. Analogously, as for abstract groups, a linear algebraic group G is the semidirect product of the closed subgroups H, K if H G, H K = 1 and the product map H K G is an isomorphism of linear algebraic groups. The omitted proof of Theorem 3.18 has the following corollary: Corollary Let G be a connected solvable group. Then any semisimple element of G lies in a maximal torus. 4. Borel Subgroups Having studied connected solvable linear algebraic groups, we now consider connected solvable subgroups of a linear algebraic group G. This leads to the definition of a Borel subgroup. Definition 4.1. A subgroup B G is called a Borel subgroup if it is a maximal closed, connected, solvable subgroup.