The Mathematics any Physicist Should Know. Thomas Hjortgaard Danielsen

Transcription

1 The Mathematics any Physicist Should Know Thomas Hjortgaard Danielsen

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3 Contents Preface 5 I Representation Theory of Groups and Lie Algebras 7 1 Peter-Weyl Theory Foundations of Representation Theory The Haar Integral Matrix Coefficients Characters The Peter-Weyl Theorem Structure Theory for Lie Algebras Basic Notions Semisimple Lie Algebras The Universal Enveloping Algebra Basic Representation Theory of Lie Algebras Lie Groups and Lie Algebras Weyl s Theorem Root Systems Weights and Roots Root Systems for Semisimple Lie Algebras Abstract Root Systems The Weyl Group The Highest Weight Theorem Highest Weights Verma Modules The Case sl(3, C) Infinite-dimensional Representations Gårding Subspace Induced Lie Algebra Representations Self-Adjointness Applications to Quantum Mechanics II Geometric Analysis and Spin Geometry Clifford Algebras Elementary Properties Classification of Clifford Algebras

4 4 7.3 Representation Theory Spin Groups The Clifford Group Pin and Spin Groups Double Coverings Spin Group Representations Topological K-Theory The K-Functors The Long Exact Sequence Exterior Products and Bott Periodicity Equivariant K-theory The Thom Isomorphism Characteristic Classes Connections on Vector Bundles Connections on Associated Vector Bundles* Pullback Bundles and Pullback Connections Curvature Metric Connections Characteristic Classes Orientation and the Euler Class Splitting Principle, Multiplicative Sequences The Chern Character Differential Operators Differential Operators on Manifolds The Principal Symbol Dirac Bundles and the Dirac Operator Sobolev Spaces Elliptic Complexes The Atiyah-Singer Index Theorem K-Theoretic Version Cohomological Version A Table of Clifford Algebras 245 B Calculation of Fundamental Groups 247 Bibliography 251 Index 252

5 Preface When following courses given by Ryszard Nest at the Copenhagen University, you can be almost certain that a reference to the Atiyah-Singer Index Theorem will appear at least once during the course. Thus it was an obvious project for me to find out what this, apparently great theorem, was all about. However, from the beginning I was well aware that this was not an easy task and that it was necessary for me to delve into a lot of other subjects involved in its formulation, before the goal could be reached. It has never been my intension to actually prove the theorem (well except for a few moments of utter over ambitiousness) but merely to pave a road for my own understanding. This road leads through as various subjects as K-theory, characteristic classes and elliptic theory. I have tried to treat each subject as thoroughly and self-contained as I could, even though this meant including stuff which wasn t really necessary for the Index Theorem. The starting point is of course my own prerequisites when I began my work half a year ago, that is a solid foundation in Riemannian geometry, algebraic topology (notably homology and cohomology) and pseudodifferential calculus on Euclidean space. From here we develop at first, in a systematic way, topological K-theory. The approach is via vector bundles as it can be found in for instance [Atiyah] or [Hatcher], no C -algebras are involved. In the first two sections the basic theory will be outlined and most proofs will be given. In the third section we present the famous Bott-periodicity Theorem, without giving a proof. The last two sections are dedicated to the Thom Isomorphism. To this end we introduce equivariant K-theory (that is, K-theory involving group actions), a slight generalization of the K-theory treated in the first sections. I follow the outline given in the classical article by [Segal]. One could argue, that equivariant K-theory could have been introduced from the very beginning, however I have chosen not to, in order not to blur the introductory presentation with too many technicalities. The second chapter deals with the Chern-Weil approach to characteristic classes of vector bundles. The first four sections are devoted to the study of the basic theory of connections on vector bundles. From the curvature forms and invariant polynomials we construct characteristic classes, in particular Chern and Pontrjagin classes and their relationships will be discussed. In the following section the Euler class of oriented bundles is defined. I have relied heavily on [Morita] and [Milnor, Stacheff] when working out these sections but also [Madsen, Tornehave] has provided valuable inspiration. The chapter ends with a discussion of certain characteristic classes constructed, not from invariant polynomials but from invariant formal power series. Examples of such classes are the Todd class and the total Ã-class and the Chern character. No effort has been made to include great theorems, in fact there are really no major results in this chapter. It serves as a tool box to be applied to the construction of the topological index. The third chapter revolves around differential operators on manifolds. In the 5

6 6 standard literature on this subject not much care is taken, when transferring the differential operators and principal symbols from Euclidean space to manifolds. I ve tried to remedy this, giving a precise and detailed treatment. To this I have added a lot of examples of classical differential operators, such as the Laplacian, Hodge-de Rham operators, Dirac operators etc. calculating their formal adjoints and principal symbols. To shed some light on the analytic properties we introduce Sobolev spaces. Essentially there are two different definitions: in the first one, Sobolev spaces are defined in terms of connections, and in the second they are defined as the clutching of local Euclidean Sobolev spaces. We prove that the two definitions agree, when the underlying manifold is compact, and we show how to extend differential operators to continuous operators between the Sobolev spaces. The major results such as the Sobolev Embedding Theorem, the Rellich lemma and Elliptic Regularity are given without proofs. We then move on to elliptic complexes, which provides us with a link to the K-theory developed in the first chapter. In the fourth and final chapter the Index Theorem is presented. We construct the so-called topological index map from the K-group K(T M) to the integers and state the index theorem, which says that the index function when evaluated on the specific K-class determined from the symbol of an elliptic differential operator, is in fact equal to the Fredholm index. I give a short sketch of the proof based on the original 1968-article by Atiyah and Singer. Then by introducing the cohomological Thom isomorphism, Thom Defect classes etc. and drawing heavily on the theory developed in the previous chapters we manage to deduce the famous cohomological index formula. To demonstrate the power of the Index Theorem, we prove two corollaries, namely the generalized Gauss-Bonnet Theorem and the fact that any elliptic differential operator on a compact manifold of odd dimension has index 0. I would like to thank Professor Ryszard Nest for his guidance and inspiration, as well as answers to my increasing amount of questions. Copenhagen, March Thomas Hjortgaard Danielsen.

7 Part I Representation Theory of Groups and Lie Algebras 7

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9 Chapter 1 Peter-Weyl Theory 1.1 Foundations of Representation Theory We begin by introducing some basic but fundamental notions and results regarding representation theory of topological groups. Soon, however, we shall restrict our focus to compact groups and later to Lie groups and their Lie algebras. We begin with the basic theory. To define the notion of a representation, let V denote a separable Banach space and equip B(V ), the space of bounded linear maps V V, with the strong operator topology i.e. the topology on B(V ) generated by the seminorms A x = Ax. Let Aut(V ) B(V ) denote the group of invertible linear maps and equip it with the subspace topology, which turns it into a topological group. Definition 1.1 (Representation). By a continuous representation of a topological group G on a separable Banach space V we understand a continuous group homomorphism π : G Aut(V ). We also say that V is given the structure of a G-module. If π is an injective homomorphism the representation is called faithful. By the dimension of the representation we mean the dimension of the vector space on which it is represented. If V is infinite-dimensional the representation is said to be infinite-dimensional as well. In what follows a group without further specification will always denote a locally compact topological group, and by a representation we will always understand a continuous representation. The reason why we demand the groups to be locally compact should be apparent in the next section. We will distinguish between real and complex representations depending on whether V is a real or complex Banach space. Without further qualification, the representations considered will all be complex. The requirement on π to be strongly continuous can be a little hard to handle, so here is an equivalent condition which is more applicable: Proposition 1.2. Let π : G Aut(V ) be a group homomorphism. Then the following conditions are equivalent: 1) π is continuous w.r.t. the the strong operator topology on Aut(V ), i.e. π is a continuous representation. 2) The map G V V given by (g, v) π(g)v is continuous. For a proof see [1] Proposition 18.8.

10 10 Chapter 1 Peter-Weyl Theory Example 1.3. The simplest example one can think of is the trivial representation: Let G be a group and V a Banach space, and consider the map G g id V. This is obviously a continuous group homomorphism and hence a representation. Now, let G be a matrix Lie group (i.e. a closed subgroup of GL(n, C)). Choosing a basis for C n we get an isomorphism Aut(C n ) GL(n, C), and we can thus define a representation of G on C n simply by the inclusion map G GL(n, C). This is obviously a continuous representation of G, called the defining representation. We can form new representations out of old ones. If (π 1, V 1 ) and (π 2, V 2 ) are representations of G on Banach spaces we can form their direct sum π 1 π 2 to be the representation of G on V 1 V 2 (which has been given the norm (x, y) = x + y, turning V 1 V 2 into a Banach space) given by (π 1 π 2 )(g)(x, y) = (π 1 (g)x, π 2 (g)y). If we have a countable family (H i ) i I of Hilbert spaces we can form the direct sum Hilbert space i I H i to be the vector space of sequences (x i ), x i H i, satisfying i I x i 2 H i <. Equipped with the inner product (x i ), (y i ) = i I x i, y i this is again a Hilbert space. If we have a countable family (π i, H i ) of representations such that sup i I π i (g) < for each g G, then we can form the direct sum of the representations i I π i on i I H i by ( π i )(g)(x i ) = (π i (g)x i ). i I Finally, if (π 1, H 1 ) and (π 2, H 2 ) are representations on Hilbert spaces, we can form the tensor product, namely equip the tensor product vector space H 1 H 2 with the inner product x 1 x 2, y 1 y 2 = x 1, y 1 x 2, y 2 which turns H 1 H 2 into a Hilbert space, and define the tensor product representation π 1 π 2 by (π 1 π 2 )(g)(x y) = π 1 (g)x π 2 (g)y. Definition 1.4 (Unitary Representation). By a unitary representation of a group G we understand a representation π on a Hilbert space H such that π(g) is a unitary operator for each g G. Obviously the trivial representation is a unitary representation. As is the defining representation of any subgroup of the unitary group U(n). In the next section we show unitarity of some more interesting representations. Definition 1.5 (Intertwiner). Let two representations (π 1, V 1 ) and (π 2, V 2 ) of the same group G be given. By an intertwiner or an intertwining map between π 1 and π 2 we understand a bounded linear map T : V 1 V 2 rendering the following diagram commutative π 1(g) V 1 V 1 T T V 2 V 2 π 2(g) i.e. satisfying T π 1 (g) = π 2 (g) T for all g G. The set of all intertwining maps is denoted Hom G (V 1, V 2 ).

11 1.1 Foundations of Representation Theory 11 A bijective intertwiner with bounded inverse between two representations is called an equivalence of representations and the two representations are said to be equivalent. This is denoted π 1 = π2. It s easy to see that Hom G (V 1, V 2 ) is a vector space, and that Hom G (V, V ) is an algebra. The dimension of Hom G (V 1, V 2 ) is called the intertwining number of the two representations. If π 1 = π2 via an intertwiner T, then we have π 2 (g) = T 1 π 1 (g) T. Since we thus can express the one in terms of the other, for almost any purpose the two representations can be regarded as the same. Proposition 1.6. Hom G respects direct sum in the sense that Hom G (V 1 V 2, W ) = Hom G (V 1, W ) Hom G (V 2, W ) and (1.1) Hom G (V, W 1 W 2 ) = Hom G (V, W 1 ) Hom G (V, W 2 ). (1.2) Proof. For the first isomorphism we define Φ : Hom G (V 1 V 2, W ) Hom G (V 1, W ) Hom G (V 2, W ) by Φ(T ) := (T V1, T V2 ). It is easy to check that this is indeed an element of the latter space. It has an inverse Φ 1 given by Φ 1 (T 1, T 2 )(v 1, v 2 ) := T 1 (v 1 ) + T 2 (v 2 ), and this proves the first isomorphism. The latter can be proved in the same way. Definition 1.7. Given a representation (π, V ) of a group G, we say that a linear subspace U V is π-invariant or just invariant if π(g)u U for all g G. If U is a closed invariant subspace for a representation π of G on V, we automatically get a representation of G on U, simply by restricting all the π(g) s to U (U should be a Banach space, and therefore we need U to be closed). This is clearly a representation, and we will denote it π U (although we are restricting the π(g) s to U and not π). Here is a simple condition to check invariance of a given subspace, at least in the case of a unitary representation Lemma 1.8. Let (π, H) be a representation of G, let H = U U be a decomposition of H and denote by P : H U the orthogonal projection onto U. If U is π-invariant then so is U. Furthermore U is π-invariant if and only if P π(g) = π(g) P for all g G. Proof. Assume that U is invariant. To show that U is invariant let v U. We need to show that π(g)v U, i.e. u U : π(g)v, u = 0. But that s easy, exploiting unitarity of π(g): π(g)v, u = π(g 1 )(π(g)v), π(g 1 )u = v, π(g 1 )u which is 0 since π(g 1 )u U and v U. Thus U is invariant Assume U to be invariant. Then also U is invariant by the above. We split x H into x = P x + (1 P )x and calculate P π(g)x = P (π(g)(p x + (1 P )x)) = P π(g)p x + P π(g)(1 P )x. The first term is π(g)p x, since π(g)p x U, and the second term is zero, since π(g)(1 P )x U. Thus we have the desired formula.

12 12 Chapter 1 Peter-Weyl Theory Conversely, assume that P π(g) = π(g) P. Every vector u U is of the form P x for some x H. Since U is an invariant subspace. π(g)u = π(g)(p x) = P (π(g)x) U, For any representation (π, V ) it is easy to see two obvious invariant subspaces, namely V itself and {0}. We shall focus a lot on representations having no invariant subspaces except these two: Definition 1.9. A representation is called irreducible if it has no closed invariant subspaces except the trivial ones. The set of equivalence classes of finitedimensional irreducible representations of a group G is denoted Ĝ. A representation is called completely reducible if it is equivalent to a direct sum of finite-dimensional irreducible representations. Any 1-dimensional representation is obviously irreducible, and if the group is abelian the converse is actually true. We prove this in Proposition 1.14 If (π 1, V 1 ) and (π 2, V 2 ) are irreducible representations then the direct sum π 1 π 2 is not irreducible, since V 1 is an π 1 π 2 -invariant subspace of V 1 V 2 : (π 1 π 2 )(g)(v, 0) = (π 1 (g)v, 0). The question is more subtle when considering tensor products of irreducible representations. Whether or not the tensor product of two irreducible representations is irreducible and if not, to write is as a direct sum of irreducible representations is a branch of representation theory known as Clebsch-Gordan theory. Lemma Let (π 1, V 1 ) and (π 2, V 2 ) be equivalent representations. Then π 1 is irreducible if and only if π 2 is irreducible. Proof. Given the symmetry of the problem, it is sufficient to verify that irreducibility of π 1 implies irreducibility of π 2. Let T : V 1 V 2 denote the intertwiner, which by the Open Mapping Theorem is a linear homeomorphism. Assume that U V 2 is a closed invariant subspace. Then T 1 U V 1 is closed and π 1 -invariant: π 1 (g)t 1 U = T 1 π 2 (g)u T 1 U But this means that T 1 U is either 0 or V 1, i.e. U is either 0 or V 2. Example Consider the group SL(2, C) viewed as a real (hence 6-dimensional) Lie group. We consider the following 4 complex representations of the real Lie group SL(2, C) on C 2 : ρ(a)ψ := Aψ, ρ(a)ψ := Aψ, ρ(a)ψ := (A T ) 1 ψ, ρ(a)ψ := (A ) 1 ψ, where A simply means complex conjugation of all the entries. All four are clearly irreducible. They are important in physics where they are called spinorial representations. The physicists have a habit of writing everything in coordinates, thus ψ will usually be written ψ α, where α = 1, 2 but the exact notation will vary according to which representation we have imposed on C 2 (i.e. according to how ψ transforms as the physicists say). In other words they view C 2 not as a vector space but rather as a SL(2, C)-module. The notations are ψ α C 2, ψ α C 2, ψ α C 2, ψ α C 2.

13 1.1 Foundations of Representation Theory 13 The representations are not all ( mutually ) inequivalent, actually the map ϕ : 0 1 C 2 C 2 given by the matrix intertwines ρ with ρ and intertwines ρ 1 0 with ρ. On the other hand ρ and ρ are actually inequivalent as we will se in Section 1.4. These two representations are called the fundamental representations of SL(2, C). In short, representation theory has two goals: 1) given a group: find all the irreducible representations and 2) given a representation of this group: split it (if possible) into a direct sum of irreducibles. The rest of this chapter deals with the second problem (at least for compact groups) and in the end we will achieve some powerful results (Schur Orthogonality and the Peter-Weyl Theorem). Chapter 5 revolves around the first problem of finding irreducible representations. But already at this stage we are able to state and prove two quite interesting results. The first result is known as Schur s Lemma. We prove a slightly more general version than is usually seen, allowing the representations to be infinitedimensional. Theorem 1.12 (Schur s Lemma). Let (π 1, H 1 ) and (π 2, H 2 ) be two irreducible unitary representations of a group G, and suppose that F : H 1 H 2 is an intertwiner. Then either F is an equivalence of representations or F is the zero map. If (π, H) is an irreducible unitary representation of G and F B(H) is a linear map which commutes with all π(g), then F = λ id H. Proof. The proof utilizes a neat result from Gelfand theory: suppose that A is a commutative unital C*-algebra which is also an integral domain (i.e. ab = 0 implies a = 0 or b = 0), then A = Ce. The proof is rather simple. Gelfand s Theorem states that there exists a compact Hausdorff space X such that A = C(X). To reach a contradiction, assume that X is not a one-point set, and pick two distinct points x and y. Then since X is a normal topological space, we can find disjoint open neighborhoods U and V around x and y, and the Urysohn Lemma gives us two nonzero continuous functions f and g on X, the first one supported in U and the second in V, the product, thus, being zero. This contradicts the assumption that A = C(X) was an integral domain. Therefore X can contain only one point and thus C(X) = C. With this result in mente we return to Schur s lemma. F being an intertwiner means that F π 1 (g) = π 2 (g) F, and using unitarity of π 1 (g) and π 2 (g) we get that F π 2 (g) = π 1 (g) F where F is the hermitian adjoint of F. This yields (F F ) π 2 (g) = F π 1 (g) F = π 2 (g) (F F ). In the last equality we also used that F intertwines the two representations. Consider the C -algebra A = C (id H2, F F ), the C -algebra generated by id H2 and F F. It s a commutative unital C -algebra, and all the elements are of the form n=0 a n(f F ) n. They commute with π 2 (g): ( a n (F F ) n) π 2 (g) = (a n (F F ) n π 2 (g)) = a n (π 2 (g)(f F ) n ) n=1 n=1 = π 2 (g) a n (F F ) n. n=1 We only need to show that A is an integral domain. Assume ST = 0. Since π 2 (g)s = Sπ 2 (g) it s easy to see that ker S is π 2 -invariant. π 2 is irreducible n=1

14 14 Chapter 1 Peter-Weyl Theory so ker S is either H 2 or {0}. In the first case S = 0, and we are done, in the second case, S is injective, and so T must be the zero map. This means that A = C id H2, in particular, there exists a λ C so that F F = λ id H2. Likewise, one shows that F F = λ id H1. Thus, we see λf = F (F F ) = (F F )F = λ F which implies F = 0 or λ = λ. In the second case if λ = λ = 0 then F F v = 0 for all v H 1, and hence 0 = v, F F v = F v, F v, i.e. F = 0. If λ = λ and λ 0 then it is not hard to see that λ 1 2 F is unitary, and that F therefore is an isomorphism. The second claims is an immediate consequence of the proof of the first. The content of this can be summed up to the following: If π 1 and π 2 are irreducible unitary representations of G on H 1 and H 2, then Hom G (H 1, H 2 ) = C if π 1 and π 2 are equivalent and Hom G (H 1, H 2 ) = {0} if π 1 and π 2 are inequivalent. Corollary Let (π, H 1 ) and (ρ, H 2 ) be finite-dimensional unitary representations which decompose into irreducibles π = i I m i δ i and ρ = i I n i δ i. Then dim Hom G (H 1, H 2 ) = i I n im i. Proof. Denoting the representations spaces of the irreducible representations by V i we get from (1.1) and (1.2) that Hom G (H 1, H 2 ) = i I n i m j Hom(V i, V j ), and by Schur s Lemma the dimension formula now follows. j I Now for the promised result on abelian groups Proposition Let G be an abelian group and (π, H) be a unitary representation of G. If π is irreducible then π is 1-dimensional. Proof. Since G is abelian we have π(g)π(h) = π(h)π(g) i.e. π(h) is an intertwiner. Since π is irreducible, Schur s Lemma says that π(h) = λ(h) id H. Thus, each 1-dimensional subspace of H is invariant, and by irreducibility H is 1-dimensional. Example With the previous lemma we are in a position to determine the set of irreducible complex representations of the circle group T = R/Z. Since this is an abelian group, we have found all the irreducible representations when we know all the 1-dimensional representations. A 1-dimensional representation is just a homomorphism R/Z C, so let s find them: It is well-known that the only continuous homomorphisms R C are those of the form x e 2πiax for some a R. But since we also want it to be periodic with periodicity 1, only integer values of a are allowed. Thus, T consists of the homomorphisms ρ n (x) = e 2πinx for n Z. Proposition Every finite-dimensional unitary representation is completely reducible.

15 1.2 The Haar Integral 15 Proof. If the representation is irreducible then we are done, so assume we have a unitary representation π : G Aut(H) and let {0} U H be an invariant subspace. The point is that U is invariant as well cf. Lemma 1.8. If both π U and π U are irreducible we are done. If one of them is not, we find an invariant subspace and perform the above argument once again. Since the representation is finite-dimensional and since 1-dimensional representations are irreducible, the argument must stop at some point. 1.2 The Haar Integral In the representation theory of locally compact groups (also known as harmonic analysis) the notions of Haar integral and Haar measure play a key role. Some preliminary definitions: Let X be a locally compact Hausdorff space and C c (X) the space of complex valued functions on X with compact support. By a positive integral on X is understood a linear functional I : C c (X) C such that I(f) 0 if f 0. The Riesz Representation Theorem tells us that to each such positive integral there exists a unique Radon measure µ on the Borel algebra B(X) such that I(f) = fdµ. We say that this measure µ is associated with the positive integral. Now, let G be a group. For each g 0 G we have two maps L g0 and R g0, left and right translation, on the set of complex-valued functions on G, given by (L g0 f)(g) = f(g 1 0 g), (R g 0 f)(g) = f(gg 0 ). These obviously satisfy L g1g 2 = L g1 L g2 and R g1g 2 = R g1 R g2. Definition 1.17 (Haar Measure). Let G be a locally compact group. A nonzero positive integral I on G is called a left Haar integral if I(L g f) = I(f) for all g G and f C c (X). Similarly a nonzero positive integral is called a right Haar integral if I(R g f) = I(f) for all g G and f C c (X). An integral which is both a left and a right Haar integral is called a Haar integral. The measures associated with left and right Haar integrals are called left and right Haar measures. The measure associated with a Haar integral is called a Haar measure. Example On (R n, +) the Lebesgue integral is a Haar integral: it is obviously positive, and it is well-known that the Lebesgue integral is translation invariant: f(x + a)dx = R n f( a + x)dx = R n f(x)dx. R n The associated Haar measure is of course the Lebesgue measure m n. On the circle group (T, ) we define an integral I by X C(T) f 1 2π 2π 0 f(e it )dt. As before this is obviously a positive integral and since I(L e iaf) = 1 2π = 1 2π 2π 0 2π 0 f(e ia e it )dt = 1 2π f(e it )dt 2π 0 f(e i( a+t) )dt

16 16 Chapter 1 Peter-Weyl Theory again by exploiting translation invariance of the Lebesgue measure, I is a left Haar integral on T. Likewise one can show that it is a right Haar integral as well, and hence a Haar integral. The associated Haar measure on T is also called the arc measure. In both cases the groups were abelian and in both cases the left Haar integrals were also right Haar integrals. This is no mere coincidence for if G is an abelian group we have L g0 = R g 1 and thus a positive integral is a left Haar integral if 0 and only if it is a right Haar integral. The following central theorem attributed to Alfred Haar and acclaimed as one of the most important mathematical discoveries in the 20th century states existence and uniqueness of left and right Haar integrals on locally compact groups. Theorem Every locally compact group G possesses a left Haar integral and a right Haar integral, and these are unique up to multiplication by a positive constant. If G is compact then the two integrals coincide, and the corresponding Haar measure is finite. It would be far beyond the scope of this thesis to delve into the proof of this. The existence part of the proof is a hard job so we just send some acknowledging thoughts to Alfred Haar and accept it as a fact of life. Now we restrict focus to compact groups on which, as we have just seen, we have a finite Haar measure. The importance of this finiteness is manifested in the following result: Theorem 1.20 (Unitarization). Let G be a compact group and (π, H) are representation on a Hilbert space (H,, ). Then there exists an inner product, G on H equivalent to, which makes π a unitary representation. Proof. Since the measure is finite, we can integrate all bounded measurable functions over G. Let us assume the measure to be normalized, i.e. that µ(g) = 1. For x 1, x 2 H the map g π(g)x 1, π(g)x 2 is continuous (by Proposition 1.2), hence bounded and measurable, i.e. integrable. Now define a new inner product by x 1, x 2 G := π(g)x 1, π(g)x 2 dg. (1.3) G That this is a genuine inner product is not hard to see: it is obviously sesquilinear by the properties of the integral, it is conjugate-symmetric, as the original inner product is conjugate-symmetric. Finally, if x 0 then π(g)x 0 (π(g) is invertible) and thus π(g)x > 0 for all g G. Since the map g π(g)x 2 is continuous we have x, x G = π(g)x dg > 0. G By the translation of the Haar measure we get π(h)x 1, π(h)x 2 G = π(gh)x 1, π(gh)x 2 dg G = π(g)x 1, π(g)x 2 dg G = x 1, x 2 G. Thus, π is unitary w.r.t. this new inner product. We just need to show that the two norms and G corresponding to the two inner products are equivalent, i.e. that there exists a constant C so that C G and G C. To this end, consider the map g π(g)x 2 for some x H. It s a continuous map, hence sup g G π(g)x 2 < for all x, and

17 1.2 The Haar Integral 17 the Uniform Boundedness Principle now says that C := sup g G π(g) <. Therefore x 2 = x 2 dg = π(g 1 )π(g)x 2 dg C 2 π(g)x 2 = C 2 x 2 G. G Conversely we see x 2 G = π(g)x 2 G This proves the claim. G G π(g) 2 x 2 dg C 2 If we combine this result with Proposition 1.16 we get G G x 2 dg = C 2 x 2. Corollary Every finite-dimensional representation of a compact group is completely reducible. The Peter-Weyl Theorem which we prove later in this chapter provides a strong generalization of this result in that it states that every Hilbert space representation of a compact group is completely reducible. We end this section by introducing the so-called modular function which is a function that provides a link between left and right Haar integrals. Let G be a topological group and I : f f(g)dg a left Haar integral. G Let h G and consider the integral Ĩh : f G f(gh 1 )dg. This is positive and satisfies Ĩ h (L g0 f) = G f(g0 1 gh 1 )dg = f(gh 1 )dg = Ĩh(f) G i.e. is a left Haar integral. By the uniqueness part of Haar s Theorem there exists a positive constant c such that Ĩh(f) = ci(f). We define the modular function : G R + by assigning this constant to the group element h i.e. f(gh 1 )dg = (h) f(g)dg. G It is not hard to see that this is indeed a homomorphism: on one hand we have f(g(hk) 1 )dg = (hk) f(g)dg, G and on the other hand we have that this equals f(gk 1 h 1 )dg = (h) f(gk 1 )dg = (h) (k) G G G G G.f(g)dg Since this holds for all integrable functions f we must have (hk) = (h) (k). One can show that this is in fact a continuous group homomorphism and thus in the case of G being a Lie group, a Lie group homomorphism. If is identically 1, that is if every right Haar integral satisfies f(hg)dg = f(g)dg (1.4) G for all h, then the group G is called unimodular. Eq. (1.4) says that an equivalent condition for a group to be unimodular is that all right Haar integrals are also left Haar integrals. As we have seen previously in this section abelian groups and compact groups are unimodular groups. G

18 18 Chapter 1 Peter-Weyl Theory 1.3 Matrix Coefficients Definition 1.22 (Matrix Coefficient). Let (π, V ) be a finite-dimensional representation of a compact group G. By a matrix coefficient for the representation π we understand a map G C of the form for fixed v V and ϕ V. m v,ϕ (g) = ϕ(π(g)v) If we pick a basis {e 1,..., e n } for V and let {ε 1,..., ε n } denote the corresponding dual basis, then we see that m ei,ε j = ε j (π(g)e i ) precisely are the entries of the matrix-representation of π(g), therefore the name matrix coefficient. If V comes with an inner product,, then by the Riesz Theorem all matrix coefficients are of the form m v,w = π(g)v, w for fixed v, w V. By Theorem 1.20 we can always assume that this is the case. Denote by C(G) π the space of linear combinations of matrix coefficient. Since a matrix coefficient is obviously a continuous map, C(G) π C(G) L 2 (G). Thus, we can take the inner product of two functions in C(G) π. Note, however that the elements of C(G) π need not all be matrix coefficients for π. The following technical lemma is an important ingredient in the proof of the Schur Orthogonality Relations which is the main result of this section. Lemma Let (π, H) be a finite-dimensional unitary representation of a compact group G. Define the map T π : End(H) C(G) by Then C(G) π = im T π. T π (A)(g) = Tr(π(g) A). (1.5) Proof. Given a matrix coefficient m v,w we should produce a linear map A : H H, such that m v,w = T π (A). Consider the map L v,w : H H defined by L v,w (u) = u, w v, the claim is that this is the desired map A. To see this we need to calculate Tr L v,w and we claim that the result is v, w. Since L v,w is sesquilinear in its indices (L av+bv,w = al v,w + bl v,w), it s enough to check it on elements of an orthonormal basis {e 1,..., e n } for H. while for i j Tr L ei,e i = Tr L ei,e j = L ei,e i e k, e k = k=1 L ei,e j e k, e k = k=1 Thus, Tr L v,w = v, w. Finally since we see that e k, e i e i, e k = 1 k=1 e k, e j e i, e k = 0. k=1 L v,w π(g)u = π(g)u, w v = u, π(g 1 )w v = L v,π(g 1 )wu T π (L v,w )(g) = Tr(π(g) L v,w ) = Tr(L v,w π(g)) = v, π(g 1 )w = π(g)v, w = m v,w (g). Conversely, we should show that any map T π (A) is a linear combination of matrix coefficients. Some linear algebraic manipulations should be enough to

19 1.3 Matrix Coefficients 19 convince the reader that we for any A End(H) have A = n i,j=1 Ae j, e i L ei,e j w.r.t some orthonormal basis {e 1,..., e n }. But then we readily see ( n ) T π (A)(g) = T π Ae j, e i L ei,e j (g) = = i,j=1 Ae j, e i m ei,e j (g). i,j=1 n i,j=1 Ae j, e i T π (L ei,e j )(g) Theorem 1.24 (Schur Orthogonality I). Let (π 1, H 1 ) and (π 2, H 2 ) be two unitary, irreducible finite-dimensional representations of a compact group G. If π 1 and π 2 are equivalent, then we have C(G) π1 = C(G) π2. If they are not, then C(G) π1 C(G) π2 inside L 2 (G). Before the proof, a few remarks on the integral of a vector valued function would be in order. Suppose that f : G H is a continuous function into a finite-dimensional Hilbert space. Choosing a basis {e 1,..., e n } for H we can write f in it s components f = n i=1 f i e i, which are also continuous, and define f(g)dg := f i (g)dg e i. G i=1 It s a simple change-of-basis calculation to verify that this is independent of the basis in question. Furthermore, one readily verifies that it is left-invariant and satisfies G f(g)dg, v = f(g), v dg and A f(g)dg = G G when A End(H). G Af(g)dg Proof of Theorem If π 1 and π 2 are equivalent, there exists an isomorphism T : H 1 H 2 such that T π 1 (g) = π 2 (g)t. For A End(H 1 ) we see that T π2 (T AT 1 )(g) = Tr(π 2 (g)t AT 1 ) = Tr(T 1 π 2 (g)t A) = Tr(π 1 (g)a) = T π1 (A)(g). Hence the map sending T π1 (A) to T π2 (T AT 1 ) is the identity id : C(G) π1 C(G) π2 proving that the two spaces are equal. Now we show the second claim. Define for fixed w 1 H 1 and w 2 H 2 the map S w1,w 2 : H 1 H 2 by S w1,w 2 (v) = π 1 (g)v, w 1 π 2 (g 1 )w 2 dg. S w1,w 2 G is in Hom G (H 1, H 2 ) since by left-invariance S w1,w 2 π 1 (h)(v) = π 1 (gh)v, w 1 π 2 (g 1 )w 2 dg = π 1 (g)v, w 1 π 2 (hg 1 )w 2 dg G G = π 2 (h) π 1 (g)v, w 1 π 2 (g 1 )w 2 dg G = π 2 (h)s w1,w 2 (v). Assume that we can find two matrix coefficients m v1,w 1 and m v2,w 2 for π 1 and π 2 that are not orthogonal, i.e. we assume that 0 m v1,w 2 (g)m v2,w 2 (g)dg = π 1 (g)v 1, w 1 π 2 (g)v 2, w 2 dg G G = π 1 (g)v 1, w 1 π 2 (g 1 )w 2, v 2 dg. G

20 20 Chapter 1 Peter-Weyl Theory From this we read S w1,w 2 v 1, v 2 0, so that S w1,w 2 0. Since it s an intertwiner, Schur s Lemma tells us that S w1,w 2 is an isomorphism. By contraposition, the second claim is proved. In the case of two matrix coefficients for the same representation, we have the following result Theorem 1.25 (Schur Orthogonality II). Let (π, H) be a unitary, finitedimensional irreducible representation of a compact group G. For two matrix coefficients m v1,w 1 and m v2,w 2 we have m v1,w 1, m v2,w 2 = 1 dim H v 1, v 2 w 2, w 1. (1.6) Proof. As in the proof of Theorem 1.24 define S w1,w 2 : H H by S w1,w 2 (v) = π 1 (g)v, w 1 π 2 (g 1 )w 2 dg = π(g 1 )L w2,w 1 π(g)v dg. G G We see that m v1,w 1, m v2,w 2 = π(g)v 1, w 1 π(g)v 2, w 2 dg G = π(g)v 1, w 1 π(g 1 )w 2, v 2 dg G = π(g)v 1, w 1 π(g 1 )w 2, v 2 G = S w1,w 2 v 1, v 2. Furthermore, since S w1,w 2 commutes with π(g), Schur s Lemma yields a complex number λ(w 1, w 2 ), such that S w1,w 2 = λ(w 1, w 2 ) id H. The operator S w1,w 2 is linear in w 2 and anti-linear in w 1, hence λ(w 1, w 2 ) is a sesquilinear form on H. We now take the trace on both sides of the equation S w1,w 2 = λ(w 1, w 2 ) id H : the right hand side is easy, it s just λ(w 1, w 2 ) dim H. For the left hand side we calculate That is, we get λ(w 1, w 2 ) = (dim H) 1 w 1, w 2, and hence S w1,w 2 = (dim H) 1 w 1, w 2 id H. By substituting this into the equation m v1,w 1, m v2,w 2 = S w1,w 2 v 1, v 2 the desired result follows. 1.4 Characters Definition 1.26 (Class Function). For a group G, a class function is a function on G which is constant on conjugacy classes. The set of square-integrable resp. continuous class functions on G are denoted L 2 (G, class) and C(G, class). It is not hard to see that the closure of C(G, class) inside L 2 (G) is L 2 (G, class). Thus, L 2 (G, class) is a Hilbert space. Given an irreducible finite-dimensional representation the set of continuous class functions inside C(G) π is very small: Lemma Let (π, H) be a finite-dimensional irreducible unitary representation of a compact group G, then the only class functions inside C(G) π are complex scalar multiples of T π (id H ).