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LINE BUNDLES OVER FLAG VARIETIES Contents 1. Introduction LINE BUNDLES OVER FLAG VARIETIES NG HOI HEI JANSON Abstract. In this paper, we will prove the Borel-Weil Theorem, which relates representation theory to line bundles over flag varieties. We will cover the preliminary definitions required to understand and prove the theorem. We give some examples based on the algebraic group SL(2; C). Contents 1. Introduction 1 2. Preliminary definitions & results 1 2.1. Varieties 2 2.2. Lie algebras 3 2.3. Algebraic groups 5 3. Vector Bundles 6 4. Review of Basic Representation Theory 8 5. Borel{Weil Theorem 9 6. Pl¨ucker Embedding 12 Acknowledgments 14 References 14 1. Introduction To simplify presentation, our base field will be the field of complex numbers C. All the results will be true if we work in any algebraically closed field of character- istic 0. The purpose of this paper is to state and prove the Borel{Weil Theorem. Doing so requires us to know about algebraic groups and some basic representation theory, such as the theory of highest weight representations, and induced representations. This material can be found in sections 2 to 4 in this paper, but a more in-depth treatment of section 2 can be found in [3]. Section 5 is devoted to proving the Borel{Weil Theorem. To provide a more complete picture, we discuss the Pl¨ucker embedding in section 6. The Pl¨ucker embedding provides some intuition about the flag variety, since the projective space is relatively well understood. 2. Preliminary definitions & results This section contains some useful definitions and theorems from the theory of algebraic groups and Lie algebras. The proofs of the theorems in this section are Date: August 31, 2015. 1 2 NG HOI HEI JANSON omitted as they are not the main aim of the paper. Readers who are interested in the proofs are referred to [3]. 2.1. Varieties. The idea of a variety is a geometric object that locally looks like the locus cut out by the vanishing of a collection of polynomials. A thorough treatment of varieties can be found in [1, Chapter 1]. We first define 2 classes of varieties, affine and projective varieties. Then we will define varieties in general. An affine algebraic variety is the set of common roots of a collection of polynomials in Cn. We sometimes omit 'algebraic' and call it an affine variety. Before we define projective varieties, we recall that homogeneous polynomials mean that each term has the same degree. For example, x2y + 5xyz + z3 + x3 is a homogeneous polynomial (of degree 3), but x2y +5xyz +z is not (because there are 2 terms of degree 3 and a term of degree 1). The projective n-space, written Pn, is the set of one dimensional subspaces of Cn+1.A projective algebraic variety in Pn is the set of common roots of a collection of homogeneous polynomials in n+1 variables. We sometimes omit 'algebraic' and call it a projective variety. The reader might find it strange why we only allow homogeneous polynomials in the above definition. The reason is because the collection of polynomials should have a well-defined vanishing set in projective n-space, and it makes sense only if we restrict ourselves to only homogeneous polynomials. This is because if f is a homo- m geneous polynomial of degree m, then f(kx0; kx1; : : : ; kxn) = k f(x0; x1; : : : ; xn), so that if (p0; : : : ; pn) is a root of f, then so is (kp0; : : : ; kpn) for any k 2 C. We wish to define a topology on affine and projective varieties. A way to do this is to give them the Zariski topology, which we now define. This captures the relationship of affine and projective varieties with vanishing polynomials. Let X be an affine variety in Cn (resp. a projective variety in Pn). Let S be a collection of polynomials in n variables (resp. collection of homogeneous polynomials in n + 1 variables) over C. Define V (S) to be the vanishing set of S, explicitly fx 2 X j f(x) = 0 for any f 2 S. The Zariski topology of X is the topology on X where V (S) for any collection S of polynomials in n variables (resp. collection S of homogeneous polynomials in n + 1 variables) are all the closed sets of X. An example of a closed set in affine n-space is a singleton set. This is because p = (p1; : : : ; pn) is the vanishing set of fx1 − p1; : : : ; xn − png. By the definition of a topology, this implies that all finite sets in affine n-space are closed under the Zariski topology. We proceed to define varieties in general. A quasi-affine algebraic variety is an open subset (under the Zariski topology) of an affine variety. A quasi- projective algebraic variety is an open subset (under the Zariski topology) of a projective variety. An algebraic variety (called a variety for short) is any affine, quasi-affine, projective, or quasi-projective variety. This definition (which agrees with [1]) is sufficient for our purposes. In fact, varieties contain a bit more structure. Let X be a variety. A map f : X ! C1 is a regular function if f is locally a rational function. Explicitly, for any x 2 X, there is an open U containing x such that f, when restricted to U, can be written as f1=f2, where f1; f2 are polynomials and f2 is never 0 on U. LINE BUNDLES OVER FLAG VARIETIES 3 Now that we have varieties, we should define the maps that preserve the structure of the varieties. In other words, we want to define morphisms of varieties. Let X be a variety. A morphism of varieties is a continuous map ': X ! Y such that for any regular function f : U ! C on an open set U ⊂ Y , the function f ◦ ' is regular on the open set '−1(U) ⊂ X. ' '−1(U) U f f◦' C In short, a morphism of varieties is a map such that the pullback of open sets are open, and the pullback of regular functions are regular. This is a good way to remember the definition. An algebraic group is a variety G with group structure such that multiplication and inversion are morphisms of varieties. A morphism of algebraic groups is a morphism of varieties that is also a group homomorphism. A linear algebraic group is an algebraic group where the underlying variety is an affine variety. From now on, when we say `algebraic group', we mean `linear algebraic group', unless stated otherwise. 2.2. Lie algebras. Lie algebras are a useful tool to study representation theory. A standard book to learn about Lie algebras is [2]. The aim of this subsection is to give some basic definitions and results in the theory of Lie algebras, and use it to say something about the representation theory of algebraic groups. But first we need to show how to `linearize' algebraic groups to get Lie algebras. This is the content of the next definition. Given an affine variety X, define P (X) to be the set of polynomials in n vari- ables vanishing on X. Note that P (X) is an ideal in the polynomial ring in n variables. We define the affine algebra of an affine variety X ⊂ Cn to be C[X] = C[x1; : : : ; xn]=P (X). Definition 2.1. For any x 2 G, let λx denote the map defined by −1 (λxf)(g) = f(x g) for all f 2 C[G], the affine algebra of G. The Lie algebra g of an algebraic group G is L (G) = fδ 2 Der(C[G]) j δλx = λxδ for any x 2 Gg where Der(C[G]) denotes the set of derivations of the affine algebra of G. Definition 2.2. A Lie algebra is a vector space g over C with a bilinear form [; ]: g × g ! C, called the Lie bracket, satisfying (1) [x; y] = −[y; x] (2) [x; [y; z]] + [y; [z; x]] + [z; [x; y]] = 0 for any x; y; z 2 g. An ideal of a Lie algebra g is a vector subspace I such that [x; y] 2 I for any x 2 g; y 2 I. In particular, the first property implies that [x; x] = 0. An example of a Lie algebra is gln, the vector space of all n × n complex matrices with Lie bracket [x; y] = xy − yx. Another example is sln, the vector space of all n × n complex matrices of trace 0. 4 NG HOI HEI JANSON The Lie algebra of an algebraic group is an example of a Lie algebra, defined abstractly in Definition 2.2. Geometrically, the Lie algebra of an algebraic group can be visualised as the `tangent space at 1'. This suggests that any morphism between algebraic groups ': G ! G0 can be `linearised' to get a Lie algebra homomorphism which we will define below. This is, in fact, true and we denote the associated Lie algebra homomorphism by d': g ! g0 where g and g0 are the Lie algebras of G and G0 respectively. Before moving on to representations of Lie algebras, we will need some struc- ture theory of Lie algebras. Some notation: [g; g] denotes the vector subspace of g spanned by elements of the form [x; y], where x; y 2 g. A Lie algebra g is simple if [g; g] 6= 0 and the only ideals of g are 0 or g . A Lie algebra g is semisimple if it is a direct sum of simple Lie algebras. In particular, simple Lie algebras are semisimple.
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