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Rolf Berndt

Representations of Linear Groups Rolf Berndt

Representations of Linear Groups

An Introduction Based on Examples from Physics and Number Theory Bibliografische information published by Die Deutsche Nationalbibliothek Die Deutsche Nationalbibliothek lists this publication in the Deutschen Nationalbibliografie; detailed bibliographic data is available in the Internet at .

Prof. Dr. Rolf Berndt Department of Mathematics University of Hamburg Bundesstraße 55 D-20146 Hamburg Germany [email protected]

Mathematics Subject Classification 20G05, 22E45, 20C25, 20C35, 11F70

First edition, July 2007

All rights reserved © Friedr. Vieweg & Sohn Verlag | GWV Fachverlage GmbH, Wiesbaden 2007 Editorial Office: Ulrike Schmickler-Hirzebruch | Susanne Jahnel Vieweg is a company in the specialist publishing group Springer Science+Business Media. www.vieweg.de

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ISBN 978-3-8348-0319-1 Preface

There are already many good books on representation theory for all kinds of groups. Two of the best (in this author’s opinion) are the one by A.W. Knapp: “Representation Theory for Semisimple Groups. An Overview based on Examples” [Kn1] and by G.W. Mackey: “Induced Representations in Physics, Probability and Number Theory” [Ma1]. The title of this text is a mixture of both these titles, and our text is meant as a very elementary introduction to these and, moreover, to the whole topic of group representations, even infinite-dimensional ones. As is evident from the work of Bargmann [Ba], Weyl [Wey] and Wigner [Wi], group representations are fundamental for the theory of atomic spectra and elementary physics. But representation theory has proven to be an inavoidable ingredient in other fields as well, particularly in number theory, as in the theory of theta functions, automorphic forms, Galois representations and, finally, the Langlands program. Hence, we present an approach as elementary as possible, having in particular these applications in mind.

This book is written as a summary of several courses given in Hamburg for students of Mathematics and Physics from the fifth semester on. Thus, some knowledge of linear and multilinear algebra, calculus and analysis in several variables is taken for granted. Assuming these prerequisites, several groups of particular interest for the applications in physics and number theory are presented and discussed, including the symmetric group Sn as the leading example for a finite group, the groups SO(2), SO(3), SU(2), and SU(3) as examples of compact groups, the Heisenberg groups and SL(2, R), SL(2, C), resp. the Lorentz group SO(3, 1) as examples for noncompact groups, and the Euclidean groups E(n) = SO(n) Rn and the Poincarégroup P = SO(3, 1)+ R4 as examples for semidirect products.

This text would not have been possible without the assistance of my students and colleagues; it is a pleasure for me to thank them all. In particular, D. Bahns, S. B¨ocherer, O. v. Grudzinski, M. Hohmann, H. Knorr, J. Michaliˇcek,H. M¨uller,B. Richter, R. Schmidt, and Chr. Schweigert helped in many ways, from giving valuable hints to indicating several mistakes. Part of the material was treated in a joint seminar with Peter Slodowy. I hope that a little bit of his way of thinking is still felt in this text and that it is apt to participate in keeping alive his memory. Finally, I am grateful to U. Schmickler-Hirzebruch and S. Jahnel from the Vieweg Verlag for encouragement and good advice. Contents

Introduction ix

0 Prologue: Some Groups and their Actions 1 0.1 Several Matrix Groups ...... 1 0.2 Group Actions ...... 3 0.3 The Symmetric Group ...... 5

1 Basic Algebraic Concepts 7 1.1 Linear Representations ...... 7 1.2 Equivalent Representations ...... 9 1.3 First Examples ...... 10 1.4 Basic Construction Principles ...... 14 1.4.1 Sum of Representations ...... 14 1.4.2 Tensor Product of Representations ...... 14 1.4.3 The Contragredient Representation ...... 15 1.4.4 The Factor Representation ...... 16 1.5 Decompositions ...... 16 1.6 Characters ...... 21

2 Representations of Finite Groups 23 2.1 Characters as Orthonormal Systems ...... 23 2.2 The Regular Representation ...... 27 2.3 Characters as Orthonormal Bases ...... 28

3 Continuous Representations 31 3.1 Topological and Linear Groups ...... 31 3.2 The Continuity Condition ...... 33 3.3 Invariant Measures ...... 38 3.4 Examples ...... 40

4 Representations of Compact Groups 43 4.1 Basic Facts ...... 43 4.2 The Example G = SU(2) ...... 48 4.3 The Example G = SO(3) ...... 52

5 Representations of Abelian Groups 59 5.1 Characters and the Pontrjagin Dual ...... 59 5.2 Continuous Decompositions ...... 60 viii CONTENTS

6 The Inﬁnitesimal Method 63 6.1 Lie Algebras and their Representations ...... 63 6.2 The Lie Algebra of a Linear Group ...... 67 6.3 Derived Representations ...... 70 6.4 Unitarily Integrable Representations of sl(2, R)...... 73 6.5 The Examples su(2) and heis(R) ...... 82 6.6 Some Structure Theory ...... 84 6.6.1 Speciﬁcations of Groups and Lie Algebras ...... 85 6.6.2 Structure Theory for Complex Semisimple Lie Algebras ...... 89 6.6.3 Structure Theory for Compact Real Lie Algebras ...... 93 6.6.4 Structure Theory for Noncompact Real Lie Algebras ...... 95 6.6.5 Representations of Highest Weight ...... 97 6.7 The Example su(3)...... 104

7 Induced Representations 117 7.1 The Principle of Induction ...... 117 7.1.1 Preliminary Approach ...... 118 7.1.2 Mackey’s Approach ...... 120 7.1.3 Final Approach ...... 125 7.1.4 Some Questions and two Easy Examples ...... 126 7.2 Unitary Representations of SL(2, R) ...... 130 7.3 Unitary Representations of SL(2,C) and of the Lorentz Group ...... 143 7.4 Unitary Representations of Semidirect Products ...... 147 7.5 Unitary Representations of the Poincar´eGroup ...... 154 7.6 Induced Representations and Vector Bundles ...... 161

8 Geometric Quantization and the Orbit Method 173 8.1 The Hamiltonian Formalism and its Quantization ...... 173 8.2 Coadjoint Orbits and Representations ...... 178 8.2.1 Prequantization ...... 178 8.2.2 Example: Construction of Line Bundles over M = P1(C) .....181 8.2.3 Quantization ...... 184 8.2.4 Coadjoint Orbits and Hamiltonian G-spaces ...... 186 8.2.5 Construction of an Irreducible Unitary Representation by an Orbit 196 8.3 The Examples SU(2) and SL(2, R) ...... 197 8.4 The Example Heis(R) ...... 202 8.5 Some Hints Concerning the Jacobi Group ...... 209

9 Epilogue: Outlook to Number Theory 215 9.1 Theta Functions and the Heisenberg Group ...... 216 9.2 Modular Forms and SL(2,R) ...... 221 9.3 Theta Functions and the Jacobi Group ...... 236 9.4 Hecke’s Theory of L−Functions Associated to Modular Forms ...... 239 9.5 Elements of Algebraic Number Theory and Hecke L-Functions ...... 246 9.6 Arithmetic L-Functions ...... 250 9.7 Summary and Final Reﬂections ...... 256

Bibliography 261

Index 266 Introduction

In this book, the groups enumerated in the Preface are introduced and treated as matrix groups to avoid as much as possible the machinery of manifolds, Lie groups and bundles (though some of it soon creeps in through the backdoor as the theory is further developed). Parallel to information about the structure of our groups we shall introduce and develop elements of the representation theory necessary to classify the unitary representations and to construct concrete models for these representations. As the main tool for the classification we use the infinitesimal method linearizing the representations of a group by studying those of the Lie algebra of the group. And as the main tools for the construction of models for the representations we use – tensor products of the natural representation, – representations given by smooth functions (in particular polynomials) living on a space provided with an action of the group, and – the machinery of induced representations. Moreover, because of the growing importance in physics and the success in deriving branching relations, the procedure of geometric quantization and the orbit method, developed and propagated by Kirillov, Kostant, Duflo and many others shall be explained via its application to some of the examples above.

Besides the sources already mentioned, the author was largely influenced by the now classical book of Kirillov: “Elements of the Theory of Representations” [Ki] and the more recent “Introduction to the Orbit Method” [Ki1]. Other sources were the books by Barut and Raczka: “Theory of Group Representations and Applications” [BR], S. Lang: “SL(2, R)” [La], and, certainly, Serre: “Linear Representations of Finite Groups” [Se]. There is also the book by Hein: “Einführungin die Struktur- und Darstellungs- theorie der klassischen Gruppen” [Hei], which follows the same principle as our text, namely to do as much as possible for matrix groups, but does not go into the infinite- dimensional representations necessary for important applications. Whoever is further interested in the history of the introduction of representation theory into the theory of automorphic forms and its development is referred to the classical book by Gelfand, Graev and Pyatetskii-Shapiro: “Representation Theory and Automorphic Forms” [GGP], Gel- bart’s: “Automorphic Forms on AdèleGroups” [Ge], and Bump’s: “Automorphic Forms and Representations” [Bu]. More references will be given at the appropriate places in our text; as already said, we shall start using material only from linear algebra and analysis. But as we proceed more and more elements from topology, functional analysis, complex function theory, differential and symplectic geometry will be needed. We will try to introduce these as gently as possible but often will have to be very rudimentary and will have to cite the hard facts without the proofs, which the reader can find in the more refined sources. To sum up, this text is prima facie about real and complex matrices and the nice and sometimes advanced things one can do with them by elementary means starting from a certain point of view: Representation theory associates to each matrix from a given group G another matrix or, in the infinite-dimensional case, an operator acting on a Hilbert space. One may want to ask, why study these representations by generally more complicated matrices or operators if the group is already given by possibly rather simple matrices? An answer to this question is a bit like the one for the pudding: the proof is in the eating. And we hope our text will give an answer. To the more impatient reader who wants an answer right away in order to decide whether to read on or not, we offer the following rough explanation. Certain groups G appear in nature as symmetry groups leaving invariant a physical or dynamical system. For example, the orthogonal group O(3) is the symmetry group for the description of the motion of a particle in a central symmetric force field, and the Poincarégroup P is the symmetry group for the motion of a free particle in Minkowski space. Then the irreducible unitary representations of G classify indivisible intrinsic descriptions of the system and, boldly spoken, can be viewed as “elementary particles” for the given situation. Following Wigner and his contemporaries, the parameters classifying the representations are interpreted as quantum numbers of these elementary particles. . .

The importance of representations for number theory is even more difficult to put into a nutshell. In the Galois theory of algebraic number fields (of finite degree) Galois groups appear as symmetry groups G. Important invariants of the fields are introduced via certain zeta- or L-functions, which are constructed using finite-dimensional representations of these Galois groups. Another aspect comes from considering smooth (holomorphic resp. meromorphic) functions in several variables which are periodic or have a more general covariant transformation property under the action of a given discrete subgroup of a continuous group G, like for instance G = SL(2, R)orG a Heisenberg or a symplectic group. Then these functions with preassigned types, e.g., theta functions or modular forms, generate representation spaces for (infinite-dimensional) representations of the respective group G.

Finally, we will give an overview over the contents of our text: In a prologue we will fix some notation concerning the groups and their actions that we later use as our first examples, namely, the general and special linear groups over the real and complex numbers and the orthogonal and unitary groups. Moreover, we present the symmetric group Sn of permutations of n elements and some facts about its structure. We stay on the level of very elementary algebra and stick to the principle to introduce more general notions and details from group theory only when needed in our development of the representation theory. We follow this principle in the first chapter where we introduce the concept of linear representations using only tools from linear algebra. We define and discuss the fundamental notions of equivalence, irreducibility, unitarity, direct sums, tensor product, characters, and give some first examples.

The theory developed thus far is applied in the second chapter to the representations of finite groups, closely following Serre’s exposition [Se]. We find out that all irreducible representations may be unitarized and are contained in the regular representation. In the next step we move on to compact groups. To do this we have to leave the purely algebraic ground and take in topological considerations. Hence, in the third chapter, we define the notion of a topological and of a (real or complex) linear group, the central notion for our text. Following this, we refine the definition of a group representation by adding the usual continuity condition. Then we adequately modify the general concepts of the first chapter. We try to take over as much as possible from finite to compact groups. This requires the introduction of invariant measures on spaces with a (from now on) continuous group action, and a concept of integration with respect to these measures. In the forth chapter we concentrate on compact groups and prove that the irreducible representations are again unitarizable, finite-dimensional, fixed by their characters and contained in the regular representation. But their number is in general not finite, in contrast to the situation for finite groups. We state, but do not prove, the Peter-Weyl Theorem. But to get a (we hope) convincing picture, we illustrate it by reproducing Wigner’s discussion of the representaions of SU(2) and SO(3). We use and prove that SU(2) is a double cover of SO(3). Angular momentum, magnetic and spin quantum numbers make an appearance, but for further application to the theory of atomic spectra we refer to [Wi] and the physics literature.

In a very short ﬁfth chapter, we assemble some material about the representations of locally compact abelian groups. We easily get the result that every unitary irreducible representation is one-dimensional. But as can be seen from the example G = R, their number need not be denumerable. More functional analysis than we can oﬀer at this stage is needed to decompose a given reducible representation into a direct integral of irreducibles, a notion we not consider here.

Before starting the discussion of representations of other noncompact groups, we present in chapter 6 an important tool for the classification of representations, the infinitesimal method. Here, at first, we have to explain what a Lie algebra is and how to associate one to a given linear group. Our main ingredient is the matrix exponential function and its properties. We also reflect briefly on the notion of representations of Lie algebras. Here again we are on purely algebraic, at least in examples, easily accessible ground. We start giving examples by defining the derived representation dπ of a given group representation π. We do this for the Schrödingerrepresentation of the Heisenberg group and the standard representation π1 of SU(2). Then we concentrate on the classification of all unitary irreducible representations of SL(2, R) via a description of all (integrable) representations of its Lie algebra. Having done this, we consider again the examples su(2) and heis(R) (relating them to the theory of the harmonic oscillator) and give some hints concerning the general structure theory of semisimple Lie algebras. The way a general classification theory works is explained to some extent by considering Lie SU(3); we will see how quarks show up.

Chapters 7 and 8 are the core of our book. In the seventh chapter we introduce the concept of induced representations, which allows for the construction of (sometimes infinite-dimensional) representations of a given group G starting from a (possibly one- dimensional) representation of a subgroup H of G. To make this work we need again a bit more Hilbert space theory and have to introduce quasi-invariant measures on spaces with group action. We illustrate this by considering the examples of the Heisenberg group and G = SU(2), where we rediscover the representations which we already know. Then we use the induction process to construct models for the unitary representations of SL(2, R) and SL(2, C). In particular, we show how holomorphic induction arises in the discussion of the discrete series of SL(2, R) (here we touch complex function theory). We insert a brief discussion of the Lorentz group GL = SO(3, 1)0 and prove that SL(2, C) is a double cover of GL. To get a framework for the discussion of the representations of the Poincarégroup GP , which is a semidirect product of the Lorentz group with R4,we define semidirect products and treat Mackey’s theory in a rudimentary form. We outline a recipe to classify and construct irreducible representations of semidirect products if one factor is abelian. We do not prove the general validity of this procedure as Mackey’s Imprimitivity Theorem is beyond the scope of our book, but we apply it to determine the unitary irreducible representations of the Euclidean and the Poincarégroup, which are fundamental for the classification of elementary particles.

Under the heading of Geometric Quantization, in the eighth chapter we take an alternative approach to some material from chapter 7 by constructing representations via the orbit method. Here we have to recall (or introduce) more concepts from higher analysis: manifolds and bundles, vector ﬁelds, diﬀerential forms, and in particular the notion of a symplectic form. We can again use the information and knowledge we already have of our examples G = SL(2, R), SU(2) and the Heisenberg group to get a feeling what should be done here. We identify certain spheres and hyperboloids as coadjoint orbits of the respective groups, and we construct line bundles on these orbits and representation spaces consisting of polarized sections of the bundles.

Finally, in the nineth and last chapter, we give a brief outlook on some examples where representations show up in number theory. We present the notion of an automorphic representation (in a rudimentary form) and explain its relation with theta functions and automorphic forms. We have a glimpse upon Hecke’s and Artin’s L-functions and mention the Artin conjecture.

We hope that some of the exercises and/or omitted proofs may give a starting point for a bachelor thesis, and also that this text motivates further studies in a master program in theoretical physics, algebra or number theory.

¡ Libro, af´an de estar en todas partes, en soledad!

J. R. Jim´enez Chapter 0

Prologue: Some Groups and their Actions

This text is mainly on groups which some way or another come from physics and/or number theory and which can be described in form of a real or complex matrix group.

0.1 Several Matrix Groups

We will use the following notation: The letter K indicates a field. The reader is invited to think of the field R of real or C of complex numbers. Most of the things we do at the beginning of our text are valid also for more general fields at least if they are algebraically closed and of characteristic zero, but as this is only an introduction for lack of space we will not go into this to a greater depth. Mm,n(K) denotes the K-vector space of m × n matrices A =(aij ) with aij ∈ K (i = 1,...,m,j =1,...,n) and Mn(K) stands for Mn,n(K). Our groups will be (for some n) subgroups of of the general linear group of invertible n × n-matrices GL(n, K):={A ∈ Mn(K); det A =0 }. As usual, we will denote the special linear group by

SL(n, K):={A ∈ Mn(K); det A =1}, the orthogonal group by

t O(n):={A ∈ Mn(R); AA = En }, resp. for n = p + q

t O(p, q):={A ∈ Mn(R); ADp,qA = Dp,q }, where Dp,q is the diagonal matrix having p times 1 and q times −1 in its diagonal, and the unitary group t U(n):={A ∈ Mn(C); AA¯ = En }, 2 0. Prologue: Some Groups and their Actions resp. for n = p + q t U(p, q):={A ∈ Mn(R); ADp,qA¯ = Dp,q }. Again, addition of the letter S to the symbol of the group indicates that we take only matrices with determinant 1, e.g. SO(n):={A ∈ O(n); det A =1}. These groups together with some other families of groups, in particular the symplectic groups showing up later, are known as classical groups.

Later on, we will often use subgroups consisting of certain types of block matrices, e.g. the group of diagonal matrices ∗ An := {D(a1,...,an); a1,...,an ∈ K }, where D(a1,...,an) denotes the diagonal matrix with the elements a1,...,an in the diagonal, or the group of upper triangular matrices (or standard Borel group) Bn consisting of matrices with zeros below the diagonal and the standard unipotent group Nn, the subgroup of Bn where all diagonal elements are 1. In view of the importance for applications, moreover, we distinguish several types of Heisenberg groups: Thus the group N3 we just deﬁned, is mostly written as ⎛ ⎞ 1 xz Heis(K):={ g = ⎝ 1 y ⎠ ; x, y, z ∈ K }. 1 In the later application to theta functions it will become clear that, though it may seem more complicated, we shall better use the following description for the Heisenberg group. ⎛ ⎞ 100 µ ⎜ λ 1 µκ⎟ Heis(K):={ g =(λ, µ, κ):=⎜ ⎟ ; µ, λ, κ ∈ K }, ⎝ 001−λ ⎠ 000 1 and the “higher dimensional” groups, which for typographical reasons we here do not write as matrix groups Heis(Kn):={ g =(x, y, z); x, y ∈ Kn,z ∈ K } with the multiplication law given by gg =(x + x,y+ y,z+ z + txy − tyx). We suggest to write elements of Kn as columns.

Exercise 0.1: Write Heis(Kn) in matrix form. Show that Heis(Kn) for n = 1 and the other two Heisenberg groups above are isomorphic.

Exercise 0.2: Verify that the matrices 01 z 0 , ,z∈ C∗ := C \{0} −10 0¯z generate (by taking all possible ﬁnite products) a non-abelian subgroup of GL(2, C). This is called the Weil group of R and plays an important role in the epilogue at the end of our text. 0.2 Group Actions 3

0.2 Group Actions

Most groups “appear in nature” as transformation groups acting on some set or space. This motivates the introduction of the following concepts.

Let G be a group with neutral element e and let X be a set.

Deﬁnition 0.1: G acts on X from the left iﬀ a map

G ×X −→X, (g, x) −→ g · x is given, which satisﬁes the conditions

g · (g · x)=(gg) · x, e · x = x for all g, g ∈ G and x ∈X.

Remark 0.1: If Aut X denotes the group of all bijections of X onto itself, the deﬁnition says that we have a group homomorphism G −→ Aut X associating to every g ∈ G the transformation x −→ g · x.

In this case the set X is also called a left G-set.

The group action is called eﬀective iﬀ no element except the neutral element e acts as the identity, i.e. the homomorphism G −→ Aut X is faithful.

The group action is called transitive iﬀ for every pair x, x ∈X there is a g ∈ G with x = g · x.

For x0 ∈X we call the subset of X

G · x0 := {g · x0; g ∈ G } an orbit of G (through x0) and the subgroup

{ ∈ · } Gx0 := g G; g x0 = x0 the isotropy group or the stabilizing group of x0.

Example 0.1: G = GL(n, K) and its subgroups act on X = Kn from the left by matrix multiplication (A, x) −→ Ax for x ∈ Kn (viewed as a column).

Exercise 0.3: Assure yourself that GL(n, K) acts transitively on X = Kn \{0}. Describe the orbits of SO(n). 4 0. Prologue: Some Groups and their Actions

Example 0.2: A group acts on itself from the left in three ways: a) by left translation (g, g0) −→ g · g0 = gg0 =: λgg0, −1 b) by (the inverse of) right translation (g, g0) −→ g · g0 = g0g =: ρg−1 g0, −1 c) by conjugation (g, g0) −→ g · g0 = gg0g =: κgg0. Left- and right translations are obviously transitive actions. For a given group, the determination of its conjugacy classes, i.e. the classiﬁcation of the orbits under conjugation is a highly interesting question, as we shall see later.

Exercise 0.4: Determine a family of matrices containing exactly one representative for each conjugacy class of G1 = GL(2, C) and G2 = GL(2, R).

Deﬁnition 0.2: A set is called a homogeneous space iﬀ there is a group G acting transitively on X .

Remark 0.2: In this case one has X = G · x for each x ∈X and any two stabilizing groups Gx and Gx are conjugate. (Prove this as Exercise 0.5.)

Deﬁnition 0.3: For any G-set we shall denote by X /G or XG the set of G-orbits in X and by X G the set of ﬁxed points for G, i.e. those points x ∈X for which one has g · x = x for all g ∈ G.

If the set X has some structure, for instance, if X is a vector space, we will (tacidly) additionally require that a group action preserves this structure, i.e. in this case that x −→ g · x is a linear map for each g ∈ G (or later on, is continuous if X is a topological space). It is a fundamental question whether the orbit space X /G inherits the same properties as the space X may have. For instance, if X is a manifold, is this true also for X /G? We will come back to this question later several times. Here let us only look X ∈X at the following situation: Let be a homogeneous G-space, x0 , and H = Gx0 the isotropy group. Then we denote by G/H the set of left cosets gH := {gh; h ∈ H}. These are also to be seen as H-orbits H · g where H acts on G by right translation. G/H is a G-set, G acting by (g, g0H) −→ gg0H. Then we shall identify G/H and X via the map

G/H −→ X ,gH−→ g · x0. This map is an example for the following general notion. Definition 0.4: Let X and X be G-sets and f : X−→X be a map. The map f is called G-equivariant or a G-morphism iff one has for every g ∈ G and x ∈X g · f(x)=f(g · x). Now we come back to the question raised above. Exercise 0.6: Prove that G/H has the structure of a group iff H is not only a subgroup but a normal subgroup, i.e. one has ghg−1 ∈ H for all g ∈ G, h ∈ H.

We mention here another useful fact: If H is a subgroup of G, one deﬁnes the normalizer of H in G as −1 NG(H):={g ∈ G; gHg = H}.

It is clear that NG(H) is the maximal subgroup in G that has H as a normal subgroup. Then the group Aut (G/H)ofG-equivariant bijections of G/H is isomorphic to NG(H)/H. (Prove this as Exercise 0.7.) 0.3 The Symmetric Group 5

Parallel to the normalizer, one deﬁnes the centralizer

−1 CG(H):={g ∈ G; ghg = h for all h ∈ H}.

In particular for H = G, the centralizer specializes to the center

CG(G)={g ∈ G; gh = hg for all h ∈ G} =: C(G).

As well as left actions one deﬁnes and studies right actions of a group G on a set or a space X :

Deﬁnition 0.5: G acts from the right iﬀ a map

G ×X −→X, (g, x) −→ x · g, is given that satisﬁes the conditions

(x · g) · g = x · (gg),x· e = x for all g, g ∈ G and x ∈X.

Remark 0.3: Obviously it is easy to switch from right action to left action and vice versa using the antiautomorphism g −→ g−1 of G. Namely, on any right G-space X one can naturally deﬁne a left action by g · x := x · g−1.

Right actions often show up in number theory and they are written there as xg := x · g (for instance in the action of the Galois group on a number ﬁeld).

0.3 The Symmetric Group

Though this text is meant to treat mainly continuous groups like the matrix groups set up in Section 0.1, some discrete groups as, for instance, Z, SL(n, Z) (where the elements are matrices with determinant one and integers as entries), and in particular the symmetric group Sn, are unavoidable.

Deﬁnition 0.6: The symmetric group Sn is the group of permutations, i.e. bijections, of a set of n elements.

We already introduced Aut X as the group of bijections of a set X , so we can take Xn := {1,...,n} and have Sn = Aut Xn. We emphasize that (with the notions from Section 0.2) we treat permutations as left actions on the set Xn. There are several fundamental facts concerning this group, which the reader is supposed to know from Linear Algebra:

Remark 0.4: The number of elements of the symmetric group Sn is # Sn = n!

Remark 0.5: Each permutation σ ∈ Sn can be written as a product of transpositions, i.e. permutations, which interchange exactly two elements. 6 0. Prologue: Some Groups and their Actions

For every σ ∈ Sn one deﬁnes the signum of σ by σ(i) − σ(j) sgn(σ):=(σ):= . i − j 1≤i

Remark 0.6: The map σ −→ (σ) is a group homomorphism of Sn to the subgroup {1, −1} of the multiplicative group R∗ = R \{0}. Thus, the number of transpositions in a representation of σ as a product of transpositions is either even or odd. The kernel of is usually written as An. It is a normal subgroup and called the alternating group.

Permutations are often written as products of cycles, i.e. permutations (i1,...,ir) with ij −→ ij+1 for j1 and the identity for r = 1. More precisely, one has

Remark 0.7: Each permutation is (up to order) a unique product of disjoint cycles.

From here it is not far to the fundamental fact:

Theorem 0.1: The number of conjugacy classes of Sn is equal to the number p(n)of partitions of n.

A partition of n is a sequence (n1,...,nr) of natural numbers ni ∈ N with ni ≥ nj for i

Example 0.3: The partitions of n = 3 are (1,1,1), (2,1), and (3). The conjugacy classes of S3 are

– {id}

– the three two-element subgroups generated by transpositions, i.e., {(1, 2)}, {(2, 3)}, {(3, 1)},

– the two three-element subgroups generated by the 3-cycles (1, 2, 3) resp. (1, 3, 2).

Exercise 0.8: Determine the conjugacy classes of A4.

Sn can be realized as a matrix group: we associate to every σ ∈ Sn the matrix A = A(σ)=(aij) ∈ GL(n, Z) with aij = δi,σ−1j.

A matrix like this is called a permutation matrix.

Exercise 0.9: Write down the matrices A(σ) and verify that the map given by associating σ −→ A(σ) is in fact an isomorphism from S3 to the subgroup of GL(3, R) consisting of the permutation matrices. Chapter 1

Basic Algebraic Concepts for Group Representations

We first collect the basic definitions and concepts concerning representations of groups, using only algebraic methods. As is common, we introduce the concept of a linear representation, which is completely adequate and sufficient for the study of representations of finite groups. In a later chapter we will enrich the discussion with elements from topology to refine the notion of a representation to that of a continuous representation. We start by stating the essential definitions and only afterwards illustrate these by some examples and constructions.

1.1 Linear Representations

Let G be a group and V a K-vector space. Here, in principle, we mean K = C, but most things go through as well for an algebraically closed field of characteristic zero. There are interesting phenomenae in the other cases, which are beyond the scope of this introduction. For an interested reader wanting some impression of this, we recommend Chapter 12 ff from Serre’s book [Se], whose first Chapters together with §7-11 from Kir- illov’s book [Ki] are guidelines for this and our next section.

Definition 1.1: π is a linear representation of G in V iff π is a homomorphism of G into Aut V i.e. iff we have a map

π : G −→ Aut V, g −→ π(g) with π(gg)=π(g)π(g) for all g, g ∈ G.

Aut V is equivalently denoted by GL(V ) and stands for the group of all (linear) isomorphisms of V .

In the case of a ﬁnite-dimensional vector space V , say of dimension n, one says that π is of degree n or π is a n-dimensional representation. 8 1. Basic Algebraic Concepts

Let B =(v1,...,vn) be a basis of V . Then every F ∈ Aut V is represented with respect to B by an invertible n × n-matrix A := MB(F ), and we have an isomorphism of vector spaces V Kn and of groups Aut V GL(n, K). Essentially equivalent to the deﬁnition above, but perhaps more down-to-earth for some readers is the deﬁnition: An n-dimensional linear representation of a group G is a prescription π associating to each g ∈ G a matrix π(g)=A(g) ∈ GL(n, K) such that

A(gg)=A(g) A(g) holds for all g, g ∈ G.

As every group homorphism transforms the neutral elements of the groups into one another, it is clear that this prescription associates the unit matrix En to the neutral element e of the group G, and that we have π(e)=idV in the situation of the general deﬁnition above.

If G is a matrix group, G ⊂ GL(n, C), as in Section 0.1, obviously, we have the natural representation π0 given by π0(A)=A for each A ∈ G. Another even more ubiquous representation is the trivial representation π = id, associating the identity idV to each g ∈ G.

Using the notion of group action introduced in Section 0.2, we can also say that a linear representation of G in V is the same thing as a left action of G on the vector space V .

At a first sight, it appears to be the main problem of representation theory to determine all representations of a given group. To make this a more sensible and accessible problem, one has to find some building blocks or elementary particles to construct general representations from, and, moreover, decide when representations are really different. We begin by searching for the building blocks.

Definition 1.2: Let π be a linear representation of G in V . π is called irreducible,iff there is no genuine π-invariant subspace V0 in V . A subspace V0 ⊂ V is π-invariant iff we have π(g)v0 ∈ V0 for all g ∈ G and v0 ∈ V0. | If this is the case, π0 := π V0 is a representation of G in V0 and this is called a subrepresentation. So we can also say that π is irreducible, iff π has no genuine subrepresentation.

Next, we restrict our objects still more, in particular in view to the applications in physics.

We assume now that V is a unitary complex vector space, i.e. V is equipped with a scalar product <.,.>: V × V −→ C, (v, v) −→ which is – linear in the second variable and antilinear in the ﬁrst, – hermitian: for every v, v ∈ V one has = , and – positive deﬁnite: for every v ∈ V one has ≥ 0 and = 0 exactly for v =0. 1.2 Equivalent Representations 9

For V = Cn we use the standard scalar product

n n := x¯iyi for all x, y ∈ C . i=1

Definition 1.3: A representation π of G in V is unitary iff each π(g) is unitary, i.e. iff one has for every v, v ∈ V and every g ∈ G

<π(g)v, π(g)v > = .

1.2 Equivalent Representations

If two representations π and π of G in K-vector spaces V resp. V are given, it is obvious to look for G-equivariant maps F : V −→ V :

Deﬁnition 1.4: A K-linear map F : V −→ V is called an intertwining operator between π and π iﬀ one has for every g ∈ G

Fπ(g)=π(g)F, i.e. iﬀ the following diagram commutes

F // V V

π(g) π(g) F // V V .

π and π are called equivalent iﬀ there is an isomorphism

F : V −→ V intertwining π and π. In this case we write π ∼ π.

Remark 1.1: The space of intertwining operators between π and π is again a vector space. It is denoted by HomG(V,V )orC(V,V ). Moreover, we use the notation C(V ):=C(V,V ) and c(π, π)=c(V,V ) = dim C(V,V ). c(π, π) is also called the multiplicity of π in π and denoted by mult(π, π).

Representations π and π with c(π, π)=c(π,π) = 0 are called disjoint.

At this point it is possible to state as our principal task:

Determine the unitary (linear) dual Gˆ of a given group G, i.e. the set of equivalence classes of unitary irreducible representations of G. 10 1. Basic Algebraic Concepts

1.3 First Examples

We hope the reader gets the feeling that the following special cases prepare the ground for the later more general considerations.

Example 1.1: In Section 1.1 we already mentioned that every matrix group G has its natural representation π0, i.e. for each real or complex matrix group G ⊂ GL(n, K) one has the representation in V = Cn associating to every A ∈ G the matrix A itself. These natural representations are obviously unitary for G = SO(n)orSU(n) but in general neither unitary nor irreducible, as the following example shows:

Example 1.2: Let g be an element of the group G = SO(2). Then we usually write

αβ g = r(ϑ):=( ),α= cos ϑ, β = sin ϑ, ϑ ∈ R. −βα

For each k ∈ Z we put χk(r(ϑ)) := exp(ikϑ).

Hence π = χk is a one-dimensional unitary representation in V = C, as, by the addition theorems for the trigonometric functions, we have

χk(r(ϑ)r(ϑ )) = χk(r(ϑ + ϑ )) = exp(ik(ϑ + ϑ )) = χk(r(ϑ))χk(r(ϑ)). √ 1 i For U =(1/ 2)( ) we have the relation i 1

α − iβ 0 e−iϑ 0 Ur(ϑ)U −1 =( )=( ). 0 α + iβ 0 eiϑ

This can be interpreted as follows. The matrix U is the matrix for an intertwining operator F intertwining the natural representation π0 with the representation π1 on V = C2 given by e−iϑ 0 π (r(ϑ)) = ( ). 1 0 eiϑ This representation is reducible and we see that the natural representation is equivalent to a reducible one. Or we realize that the vectors √ 1 √ i e := Ue =(1/ 2) ,e := Ue =(1/ 2) 1 1 i 2 2 1

2 span nontrivial π0-invariant subspaces of V = C . But if we look at π0 as a real repre- 2 sentation, i.e. in the R-vector space V0 = R , π0 is in fact irreducible.

Example 1.3: In Section 0.3 we introduced the symmetric group Sn. G = S3 consists of the elements

id = (1), (1, 2) =: σ, (1, 3), (2, 3), (1, 2, 3) =: τ,(1, 3, 2).

As there are relations like σ2 = id, τ 3 = id and

στσ = τ 2 =(1, 3, 2),στ=(2, 3),τσ =(1, 3), 1.3 First Examples 11

we see that each element g ∈ S3 can be written as a product of (in general) several factors σ and τ. We indicate this here (and later on in a similar fashion in other cases) by the notation S3 =<σ,τ>. We easily ﬁnd one-dimensional representations in V = C, namely the trivial representation π1(g) = 1 for all g ∈ S3 and the signum representation

π2(g) = sgn g ∈{±1} for all g ∈ S3.

3 One also has a natural three-dimensional representation π0 given in V = C by the permutation matrices from Exercise 0.9. For instance, one has ⎛ ⎞ 001 ⎝ ⎠ π0(τ)=A(τ)= 100 . 010

This representation is also called the permutation representation. It can equivalently be described as follows. We write 3 3 V = C = eiC w = e1z1 + e2z2 + e3z3 i=1 t t with the canonical basis vectors e1 = (1, 0, 0),...,e3 = (0, 0, 1) and z1,z2,z3 ∈ C. Then π0 is given by π0(g)w = eg(i)zi = eizg−1(i). i Exercise 1.1: Verify that this is true.

π0 is unitary (verify this also) but not irreducible: V1 := (e1 + e2 + e3)C is an invariant subspace, and as we have

π0(g)(e1 + e2 + e3)=eg(1) + eg(2) + eg(3) = e1 + e2 + e3, | π0 V1 is the trivial representation π1. As one has for w = ziei = zi, { } −1 V3 := w, zi =0 is the orthogonal complement to V1 and as i zi = i zg (i) holds for all g ∈ S3, V3 is an invariant subspace.

2 2 Exercise 1.2: Show that a := e1ζ + e2 + e3ζ and b := e1 + e2ζ + e3ζ is a basis for 2πi/3 | V3,ifζ := e . Determine the matrices of π2 := π0 V3 with respect to this basis and show that π2 is irreducible.

Remark 1.2: By the general methods to be developed later, it will be clear that all irreducible representations of S3 are equivalent to π1,π2, or π3, i.e. the equivalence classes of πi (i=1,2,3) constitute the unitary dual of S3. 12 1. Basic Algebraic Concepts

Exercise 1.3: Prove this in a direct elementary way (see [FH] §1.3).

Example 1.4: A rather general procedure to construct representations is hidden in the following simple example.

⊂ We take G GL(2, C) and as vector space V the space Vm = C [u, v]m of homogeneous polynomials P of degree m in two variables u, v. By counting, we see dim V = m +1. There are two essentially equivalent ways to deﬁne a representation of G on Vm: ab a b For P ∈ V and g =( ) ∈ G resp. g−1 =( )weput m cd c d

(λm(g)P )(u, v):=P (a u + b v, c u + d v) or

(ρm(g)P )(u, v):=P (au + cv, bu + dv).

Obviously λ(g)P and ρ(g)P are both again homogeneous polynomials of degree m. The functional equations λ(gg)=λ(g)λ(g) and ρ(gg)=ρ(g)ρ(g) can be veriﬁed directly. It becomes also clear from the general discussion in the following Example 1.5. We remark that λm and ρm are irreducible unitary representations of G = SU(2), if Vm is provided with a suitable scalar product. This will come out later in section 4.2.

Example 1.5: Let X be a G-set with a left G-action x −→ g · x and V = F(X ) a vector space of complex functions f : X−→C such that for f ∈ V we also have fg ∈ V where fg is deﬁned by −1 fg(x)=f(g x).

Remark 1.3: The prescription

(λ(g)f)(x):=f(g−1x) deﬁnes a representation λ of G on V .

Proof: One has (gg) · x = g · g · x and hence

− − λ(gg)f(x)=f((gg)−1 · x)=f(g 1g−1 · x)=f(g 1 · g−1 · x) and −1 −1 −1 λ(g)λ(g )f(x)=λ(g)fg (x)=fg (g · x)=f(g · g · x).

An important special case is given by X = Cn and G ⊂ GL(n, C) acting by matrix multiplication g · x = gx (as usual x is here a column) on V , a space of continuous functions in n variables or, say, a subspace of homogeneous polynomials of ﬁxed degree (for n =2 we then have the last example).

Analogously, one treats a set X with a right G-action x −→ x · g and a vector space V = F(X ) of complex functions f on X closed under f −→ f g where f g is deﬁned by f g = f(x · g). 1.3 First Examples 13

Remark 1.4: The prescription

(ρ(g)f)(x):=f(x · g) deﬁnes a representation ρ of G on V .

Proof: One has x · (gg)=x · g · g and hence

ρ(gg)f(x)=f(x · (gg)) = f(x · g · g) and ρ(g)ρ(g)f(x)=ρ(g)f g (x)=f(x · g · g).

For X = Cn and V again a space of functions in n variables, this time viewed as a row, and G ⊂ GL(n, C), we have again the action x −→ x · g = xg by matrix multiplication. For n = 2 this leads naturally to the second version of Example 1.4.

Another (as, later to be seen, crucial) application of this procedure is given by making G act on itself, i.e. X = G, by left or right translation and V = L2(G), the space of square integrable functions with respect to a left resp. right invariant Haar measure (these notions will be explained later). We then get the left (right) regular representation of G.

Example 1.6: Let G = Heis(R) be the Heisenberg group from Section 0.1, in the form of the group of triples g =(λ, µ, κ),λ,µ,κ∈ R with the multiplication law

gg =(λ + λ,µ+ µ,κ+ κ + λµ − λµ), and V = L2(R) the space of quadratic Lebesgue integrable functions f on X = R (with 1/2 ∈ ∗ norm f 2 =( R f(x)f(x)dx) ). Then for m R ,π= πm given by

2πim(κ+(λ+2x)µ) πm(g)f(x):=e f(x + λ) is a unitary irreducible representation of Heis(R). We call it the Schr¨odingerrepresentation.

Exercise 1.4: Prove that πm is a unitary representation. (The proof of the irreducibility will become easier later when we have prepared some more tools.)

Example 1.7: Prove (as Exercise 1.5) that the Weil group of R, WR from Exercise 0.2 in Section 0.1, has the irreducible representations π = π0 and π = πm,m ∈ Z \{0} given by 01 01 z 0 z 0 π ( ) := det =1,π( ) := det = zz,¯ 0 −10 −10 0 0¯z 0¯z and for m =0 z 0 zm 0 01 01 π ( ):= ,π ( ):= . m 0¯z 0¯zm m −10 (−1)m 0 14 1. Basic Algebraic Concepts

1.4 Basic Construction Principles

Proceeding from a given representation (π, V ) of a group G one can construct several new representations using standard notions from (multi-)linear algebra. We suppose that the reader has some knowledge of direct sums, tensor products and dual spaces. If this is not the case, we hope that one can follow in spite of it as we stay on a very elementary level and do enough paraphrasing to get some understanding. But sooner or later the reader should consult any book where these notions appear, for instance take Fischer’s “Lineare Algebra” [Fi], Lang’s “Algebra” [La], or the appendices to [Ki1] or [Be].

1.4.1 Sum of Representations Let (π, V ) and (π,V) be (linear) representations of the same group G. Then we deﬁne the direct sum π ⊕ π of π and π by

(π ⊕ π)(g)(v ⊕ v):=π(g)v ⊕ π(g)v for all v ⊕ v ∈ V ⊕ V .

For V = Kn,V = Km and π(g)=A(g) ∈ GL(n, K),π(g)=A(g) ∈ GL(m, K)in matrix form, the sum is given by the block diagonal matrix

A(g)0 (π ⊕ π)(g)=( ) ∈ GL(n + m, K). 0 A(g)

1.4.2 Tensor Product of Representations Let (π, V ) and (π,V)beasaboveandV ⊗ V the tensor product of V and V . Then we deﬁne the tensor product π ⊗ π of π and π by

(π ⊗ π)(g)(v ⊗ v):=π(g)v ⊗ π(g)v for all v ⊗ v ∈ V ⊗ V .

For V = Kn,V = Km and π(g)=A(g) ∈ GL(n, K),π(g)=A(g) ∈ GL(m, K), the tensor product is given by the Kronecker product of the matrices A(g) and A(g) ⎛ ⎞ a1,1A (g) ... a1,nA (g) (π ⊗ π)(g)=⎝ ... ⎠ ∈ GL(nm, K) . an,1A (g) ... an,nA (g)

We will not go into the deﬁnition of the tensor product (as to be found in the afore mentioned sources) but simply recall the following. If V has a basis (ei)i∈I and V a basis (fj)j∈J , then V ⊗ V has a basis (ei ⊗ fj)(i,j)∈I×J . In V ⊗ V we have the rules

(v1 + v2) ⊗ v = v1 ⊗ v + v2 ⊗ v , ⊗ ⊗ ⊗ v (v1 + v2)=v v1 + v v2, ⊗ ⊗ ⊗ ∈ ∈ ∈ λ(v v )=(λv) v = v (λv ) for all v, v1,v2 V,v ,v1,v2 V ,λ K.

It is clear that one can also deﬁne tensor products of more than two factors, and in particular tensor powers like ⊗3V = V ⊗ V ⊗ V (It is not diﬃcult to see that these products are associative.) 1.4 Basic Construction Principles 15

Moreover, there are subspaces of tensor powers with a preassigned symmetry property. In particular we are interested in p-fold symmetric tensors SpV and p-fold antisymmetric tensors ∧pV . For instance, if V is three-dimensional with basis (e1,e2,e3),

•⊗2V has dimension 9 and a basis

(e1 ⊗ e1,e1 ⊗ e2,e1 ⊗ e3,e2 ⊗ e1,...,e3 ⊗ e3),

• S2V has dimension 6 and a basis

e1 ⊗ e1,e1 ⊗ e2 + e2 ⊗ e1,e1 ⊗ e3,e2 ⊗ e2,e2 ⊗ e3 + e3 ⊗ e2,e3 ⊗ e3.

•∧2V has dimension 3 and a basis

e1 ∧ e2 := e1 ⊗ e2 − e2 ⊗ e1,e1 ∧ e3 := e1 ⊗ e3 − e3 ⊗ e1,e2 ∧ e3 := e2 ⊗ e3 − e3 ⊗ e2.

More generally, one has symmetric and antisymmetric subspaces in ⊗pV with bases

····· ⊗···⊗ ≤···≤ ei1 eip := eig(1) eig(p) ,i1 ip ,

g∈Sp resp. ∧···∧ ⊗···⊗ ··· ei1 eip := sgngeig(1) eig(p) ,i1 <

g∈Sp p For instance, if V is three-dimensional, S V can be identified with the space K [u, v, w]p of homogeneous polynomials of degree p in three variables. And if π is a representation of G on V, the prescription ei −→ π(g)ei defines by linear continuation representations Spπ and ∧pπ on SpV resp. ∧pV . It is one of the most important construction principles for finite-dimensional representations to take the natural representation π0 and to get (other) irreducible representations by taking tensor products and reduce these to sums of irreducibles. This will be analysed later.

1.4.3 The Contragredient Representation We use the notation V ∗ to denote the dual space of a given K-vector space. This space consists of K-linear maps ϕ : V −→ K, is again a K-vector space (of the same dimension as V if the dimension is ﬁnite), and is written as

V ∗ = Hom(V,K)={ϕ : V −→ K, ϕ K − linear }.

We shall also use the notation

ϕ(v)=:<ϕ,v> for all ϕ ∈ V ∗,v ∈ V

∗ ∗ and, if dim V = n with basis (e1,...,en), we denote (e1,...,en) for the dual basis for ∗ ∗ V , i.e. with = δi,j(= 1 for i = j and = 0 for i = j). 16 1. Basic Algebraic Concepts

If π is a representation of G on V , we deﬁne the contragredient representation π∗ on V ∗ by (π∗(g)ϕ)(v):=ϕ(π(g−1)v) for all ϕ ∈ V ∗,v ∈ V, or equivalently by the condition

<π(g)∗ϕ, π(g)v>=<ϕ,v> for all ϕ ∈ V ∗,v ∈ V.

The proof that we really get a representation is already contained in the example 1.5 in section 1.3. It is clear from linear algebra that in matrix form we have A∗(g)=tA(g−1).

Exercise 1.6: Verify this.

1.4.4 The Factor Representation

If (π1,V1) is a subrepresentation of the representation (π, V )ofG, one has also an associated representation on the factor space V/V1, called the factor representation.

There is still another essential construction principle, namely the concept of induced representations. It goes back to Frobenius and is also helpful in our present algebraic context. But we leave it aside here and postpone it for a more general treatment in Chapter 7.

1.5 Decompositions

We now have at hand procedures to construct representations and we have the notion of irreducible representations which we want to look at as the building blocks for general representations. To be able to do this, we still have to clarify some notions. We deﬁned in Section 1.1 that (π, V ) is irreducible, iﬀ there is no nontrivial subrepresentation. Now we add:

Deﬁnition 1.5: (π, V )isdecomposable, iﬀ there exists an invariant subspace V1 ⊂ V which admits an invariant complementary subspace V2, i.e. with V = V1 ⊕ V2. We then write π = π1 + π2. Representations for which every nontrivial subrepresentation admits an invariant complement are called completely reducible.

Complete reducibility is desirable as it leads (at least in the case of ﬁnite-dimensional representations) immediately to the possibility to decompose a given representation into irreducible representations. Therefore facts like the following are very important.

Theorem 1.1: Let (π, V ) be the representation of a ﬁnite group G and (π1,V1) a subrepresentation. Then there exists an invariant complementary subspace V2 . 1.5 Decompositions 17

Proof: Let <, > be any scalar product for V (such a thing exists as is obvious at least for finite-dimensional V Kn) . We can find a G-invariant scalar product <, > , namely the one defined by := <π(g)v, π(g)v > for all v, v ∈ V. g∈G

Then V2 := {v ∈ V, < v,v1 > = 0 for all v1 ∈ V1} is a complementary subspace to V1 which is π-invariant: −1 For v ∈ V2,g ∈ G one has, as <, > is in particular π(g )-invariant, −1 −1 −1 <π(g)v, v1 > = <π(g )π(g)v, π(g )v1 > = =0

−1 if v1 ∈ V1 and hence also π(g )v ∈ V2.

Remark 1.5: If G is any group and (π, V ) a unitary representation of G, again every subrepresentation (π1,V1) has an invariant complement (π2,V2) given as in the proof ⊥ { ∈ ∈ } above by V2 := V1 = v V, < v,v1 > = 0 for all v1 V1 because V is already provided with a π-invariant scalar product.

Remark 1.6: If G is compact, the theorem holds as well, as the sum in the deﬁnition of the invariant scalar product can be replaced by an integral. This will be done later in Chapter 4.

Remark 1.7: As an immediate consequence we have that ﬁnite-dimensional unitary representations of an arbitrary group are completely reducible.

Remark 1.8: Every ﬁnite-dimensional representation of a ﬁnite (or compact) group is unitarizable as the representation space can be given a π-invariant scalar product by the Theorem 1.1 above.

Example 1.7: We have already seen an example for a decomposition of a representation in Example 1.3 in Section 1.3, namely the permutation representation of G = S3. The standard example that not every representation is completely reducible and not even decomposable is the following. Let G = R be the additive group of the real numbers and π a two-dimensional representation of G in V = C2 given by 1 b R b −→ ( )=:A(b). 01

Here V1 := e1C is an invariant subspace: we have A(b)e1 = e1 for all b ∈ R.If x V := v C = C 2 y is an invariant complementary subspace, we have x + by x A(b)v = = λ y y for a certain λ ∈ C, i.e. x + by + λx and y = λy.Asy must be nonzero, we have λ =1. This leads to by = 0 for all b and hence a contradiction. 18 1. Basic Algebraic Concepts

The following considerations are important in a general representation theory. But as we will not use them here so much they are only mentioned (for proofs see for instance [Ki] p.115 f). Every representation (π, V ) is either indecomposable or the sum of two representations π = π1 +π2. The same alternative applies to the representations π1 and π2 and so on. In the general case this process can continue infinitely. We say that π is finite, if every family of π-invariant subspaces Vi of V that is strictly monotone with respect to inclusion is finite. In this case there exists a strictly monotone finite collection of invariant subspaces

V = V0 ⊃ V1 ⊃···⊃Vn = {0} such that that the representations πi appearing on Vi/Vi+1 are irreducible. One has the Jordan - H¨olderTheorem: The length of such a chain of subspaces and the equivalence classes of the representations πi (up to order) are uniquely deﬁned by the equivalence class of π.

If we want to discuss the question whether the decomposition of a given representation (π, V ) is the direct sum of irreducibles, we need criteria for irreducibility. A central criterium is a consequence of the following famous statement.

Theorem 1.2: (Schur’s Lemma): If two linear representations (π, V ) and (π,V) are irreducible, then every intertwining operator F ∈C(π, π) is either zero or invertible. If V = V ,π = π and dim V = n, then F is a homothety, i.e. F = λ id, λ ∈ C.

Proof: i) As deﬁned in Section 1.2 an intertwining operator for π and π is a linear map F : V −→ V with

(1.1) π(g)F (v)=F (π(g)v) for all g ∈ G, v ∈ V.

Then Ker F = {v ∈ V,F(v)=0} is a π-invariant subspace of V because for v ∈ Ker F we have F (v) = 0 and by (1.1) F (π(g)v)=π(g)F (v) = 0, i.e. π(g)v ∈ Ker F .Bya similar reasoning it is clear that Im F = {F (v),v ∈ V } is an invariant subspace of V . The irreducibility of π implies Ker F = {0} or Ker F = V and by the irreducibility of π we have Im F = {0} or = V . Hence F is the zero map or an isomorphism. ii) If V = V , the endomorphism F : V −→ V has an eigenvalue λ ∈ C (as the charac- teristical polynomial PF (t) := det(F − tE) ∈ C[t] has a zero λ ∈ C by the fundamental theorem of algebra). Hence F := F − λE is a linear map with Ker F = {0}. From part i) we then deduce F = 0, i.e. F = λE.

Remark 1.9: The ﬁrst part of the Theorem is also expressed as follows: Two irreducible representations of the same group are either equivalent or disjoint. The second part can be written as C(π)=C.

There are several more reﬁned versions of Schur’s Lemma which come up if one sharpens the notion of a linear representation by adding a continuity requirement as we shall do later on. For the moment, though it is somewhat redundant, we add another version which we take over directly from Wigner ([Wi] p. 75) because it contains an irreducibility criterium used in many physics papers and its proof is a very instructive example for matrix calculus. 1.5 Decompositions 19

Theorem 1.3 (the ﬁnite-dimensional unitary Schur): Let (π, Cn) be a unitary matrix representation of a group G, i.e. with π(g)=A(g) ∈ U(n). Let M ∈ GL(n, Cn)bea matrix commuting with all A(g), i.e.

(1.2) MA(g)=A(g)M for all g ∈ G.

Then M is a scalar multiple of the unit matrix, M = λEn,λ∈ C.

Proof: i) First, we show that we may assume M is a Hermitian matrix: We take the adjoint (with A −→ A∗ := tA¯) of the commutation relation (1.2) and get

A(g)∗M ∗ = M ∗A(g)∗,

∗ multiply this from both sides by A(g) and, using the unitarity A(g)A(g) = En, get

M ∗A(g)=A(g)M ∗.

Hence, if M commutes with all A(g), not only M ∗ does but also the matrices

∗ ∗ H1 := M + M ,H2 := i(M − M ), which are Hermitian. It is therefore suﬃcient to show that every Hermitian matrix, which commutes with all A(g), is a scalar matrix, since if H1 and H2 are multiples of En,somustbe2M = H1 − iH2. ii) If M is Hermitian, one of the fundamental theorems from linear algebra (see for instance [Ko], p.256 or [Fi] Kapitel 4) tells that M can be diagonalized, i.e. conjugated into a diagonal matrix D by a matrix U ∈ U(n)

UMU−1 = D.

Then we put A˜(g):=UA(g)U −1. A˜(g) is unitary as can be easily veriﬁed. From the ˜ ˜ commutation relation (1.2), we have DA(g)=A(g)D. With D =(di,j),di,j = 0 for i = j, A˜(g)=(˜ai,j), this leads to

di,ia˜i,j =˜ai,jdj,j.

Hence for di,i = dj,j we havea ˜i,j =˜aj,i = 0, that is, all A˜(g) must have zeros at all intersections of rows and columns, where the diagonal elements are diﬀerent. This would mean that g −→ A˜(g) resp. g −→ A(g) are reducible representations. Hence, as this is not the case, we have that all diagonal elements are equal and D is a scalar matrix λEn,λ∈ C. The proof shows that we can sharpen the statement from the Theorem to the irreducibiliy criterium:

Corollary: If there exists a nonscalar matrix which commutes with all matrices of a ﬁnite-dimensional unitary representation π, then the representation is reducible. If there exists none, it is irreducible.

This Corollary will help to prove the irreducibility in many cases. Here we show the use- fulness of Schur’s Lemma by proving another fundamental fact (later we shall generalize this to statements valid also in the inﬁnite-dimensional cases): 20 1. Basic Algebraic Concepts

Theorem 1.4: Any ﬁnite-dimensional irreducible representation of an abelian group is one-dimensional.

Proof: Let (π, V ) be a representation of G. Then

F : V −→ V, v −→ π(g0)v

is an intertwining operator for each g0 ∈ G, as we have

π(g)F (v)=π(g)π(g0)v

= π(gg0)v, as π is a representation,

= π(g0g)v, as G is abelian,

= π(g0)π(g)v, = F (π(g)v).

The second part of Schur’s Lemma says that F is a homothety, i.e. we have a λ ∈ C with

F (v)=π(g0)v = λv for all v ∈ V.

As g0 ∈ G was chosen arbitrarily, we see that V0 = vC is an invariant subspace of V . For an irreducible representation π this implies V = V0, i.e. V is one-dimensional.

To finish our statements on decompositions in this section, we state here without proofs (for these see [Ki] p.121ff) some standard facts about the kind of uniqueness one can expect for a decomposition of a completely reducible representation. By the way, this theorem is a reflex of a central theorem from Algebra about the structure of modules over a ring (a representation (π, V ) will be seen also as module over the group ring C[G]).

Reminding the notions of disjoint representations from Section 1.2 and of ﬁnite representations from above, we add the following notion.

Deﬁnition 1.6: A representation π is said to be primary, iﬀ it cannot be represented as a sum of two disjoint representations.

As an example one may think here of a representation of the form πk, k ∈ N, the sum of k representations (π, V ) given on the k-th sum V k = V ⊕···⊕V . The central fact can be stated like this:

Theorem 1.5: Let π be a completely reducible ﬁnite representation in the space V . Then there exists a unique decomposition of V into a sum of invariant subspaces Wj, for | which the representations πj = π Wj are primary and pairwise disjoint. Every invariant subspace V ⊂ V has the property V = ⊕j(V ∩ Wj).

Among other things, the proof uses the notion of projection operators, known from linear algebra. As we have to use these here later on, we recall the following concept. 1.6 Characters 21

Deﬁnition 1.7: Suppose that the space V of a representation π is decomposed into a ⊕m direct sum of invariant subspaces Vi, i.e., V = i=1Vi. Then the projection operator Pi onto Vi parallel to the remaining Vj is deﬁned by Pi v = vi if v =(v1,...,vi,...,vm), vi ∈ Vi,i=1,...,m.

Pi has the following properties 2 -a)Pi = Pi, -b)PiPj = PjPi = 0 for i = j, m -c)i=1 Pi =1, -d)Pi ∈C(π). One can prove that every collection of operators (i.e. linear maps) in V that enjoys these properties produces a decomposition of V into a sum of invariant subpaces Vi = PiV .

Later on, we shall treat explicit examples of decompositions.

1.6 Characters of Finite-dimensional Representations

The following notions need the finite-dimensionality of a representation. For the infinite- dimensional cases there are generalizations using distributions and/or infinitesimal methods which we only touch briefly later on because they are beyond the scope of this book. So in this Section (π, V ) is always a finite-dimensional complex representation of the group G with dim V = n.

Deﬁnition 1.8: We call character of π the complex function χπ (or equivalently χV )on G given by χπ(g):=Trπ(g) for all g ∈ G.

As usual Tr denotes the trace.Ifπ is given by the matrices A(g)=(aij (g)) ∈ GL(n, C), one has n χπ(g)= aii. i=1 Conjugate matrices A ∼ A, i.e. with A = TAT−1,T ∈ GL(n, C) have the same trace. Therefore the trace of a representation is well deﬁned independently of the choice of the matrix representation. And for conjugate g, g ∈ G we have

χπ(g)=χπ(g ). This is also expressed by saying the character is a class function. Obviously one has for the neutral element e of G

χπ(e)=n = dim V. We assume moreover that π is unitary (we shall see soon that this is not really a restriction if the group is ﬁnite or compact). Then π(g) is in matrix form A(g) conjugate to a diagonal matrix D = D(λ1,...,λn), where λ1,...,λn are the eigenvalues of A(g). Hence: n Remark 1.10: χπ(g)= i=1 λi. 22 1. Basic Algebraic Concepts

If G is ﬁnite, each element g has ﬁnite order and so has π(g). Thus the eigenvalues λi | | −1 must have λi = 1 and we have λi = λi.

Using this, from Remark 1.10 we deduce an important fact.

−1 Remark 1.11: χπ(g )=χπ(g).

Finally we state two very useful formulae.

Theorem 1.6: Let (π, V ) and (π,V) be two ﬁnite-dimensional representations of a group G. Then we have

(1.3) χπ⊕π = χπ + χπ

(1.4) χπ⊗π = χπ × χπ .

Proof: The ﬁrst equation is simply the expression of the fact that the trace of a block A matrix ( ) is the sum of the traces of the two matrices A and A. The second A relation can be seen by inspection of the trace of the Kronecker product in 1.4.2.

Exercise 1.7: Let (π, V ) be a ﬁnite-dimensional representation of G. Show that for every g ∈ G one has the equations

2 2 χ∧2π(g)=(1/2)(χπ(g) − χπ(g )),

2 2 χS2π(g)=(1/2)(χπ(g) + χπ(g )). As we shall find out in the next chapters, the irreducible representations of finite and compact groups (where in the last case we have to add a continuity condition) are all finite-dimensional. Then the characters of these representations already contain enough information to fix these representations in a way which we will discuss more precisely now. Chapter 2

Representations of Finite Groups

As we are mainly interested in continuous groups like SO(3) etc., this topic plays no central rôlein our text. But here we can learn a lot about how representation theory works. Thus we present the basic facts. We will not do it by using the abstract notions offered by modern algebra (as for instance in [FH]), where eventually the whole theory can be put into the nutshell of one page. Instead we follow very closely the classic presentation from the first pages of Serre’s book [Se].

In this chapter G is always a ﬁnite group of order #G = m.

2.1 Characters as Orthonormal Systems

We introduce the group ring (or group algebra): Let K be a ﬁeld and G a group with #G = m. Then we denote the space of maps from G to K by

KG := {u : G −→ K}. KG is a K-vector space as we have an addition u + u deﬁned by (u + u)(g):=u(g)+u(g) for all g ∈ G and scalar multiplication λu by (λu)(g):=λu(g) for u ∈ KG and λ ∈ K. But we also have a multiplication uu deﬁned by (uu)(g):= u(a)u(b). a,b∈G,ab=g Remark 2.1: One can show that KG is an associative K-algebra (for this notion see Section 6.1). There is an alternative way to describe KG, namely as a set of formal sums g∈G u(g)g built from the values u(g)ofu in g ∈ G, i.e. KG := { u(g)g, u(g) ∈ K}. g∈G 24 2. Representations of Finite Groups

Here we have addition u(g)g + u(g)g := (u(g)+u(g))g and multiplication given by u(g)g · u(g)g := v(g)g, with v(g):= u(a)u(b). ab=g

As we restrict our interest in this text to complex representations, in the sequel we write H = CG and H0 = CclG for the algebra of class functions, i.e. with u(g)=u(g ) for g ∼ g.If(π, V ) is a representation of G, by Remark 1.8 in 1.5 we may assume that it is unitary. Conjugate elements of G have the same character. Hence the character u = χπ is not only an element in H but also in H0. Our ﬁrst aim is to prove that, for a complete system π1,...,πh of irreducible represen- H tations, the characters χπ1 ,...,χπh are an orthonormal system (=: ON-system) in . Here the scalar product <, > is given by := (1/m) u(t)v(t) for all u, v ∈H. t∈G

(It is clear that this is a scalar product.) We will also use the following bilinear form (u, v):=(1/m) u(t−1)v(t). t∈G

−1 If u = χπ is a character, by the Remark 1.11 in 1.6 we have u(t)=u(t ) and hence (χπ,v)=<χπ,v > for v ∈H.

Theorem 2.1: Let π and π be irreducible representations of G and χ = χπ as well as χ = χπ their characters. Then we have

<χ,χ > =0, if π ∼ π, <χ,χ> =1.

This is also written as <χπ,χπ > = δπ,π .

The proof will be a consequence of two propositions.

Proposition 2.1: Let π, π be irreducible, F : V −→ V a linear map, and F 0 := (1/m) π(g−1)Fπ(g). g∈G

Then one has

1) F 0 =0, if π ∼ π (“case 1”), 2) F 0 = λ id, if V = V ,π = π, where λ =(1/ dim V )Tr F (“case 2”). 2.1 Characters as Orthonormal Systems 25

Proof: F 0 is an intertwining operator for π and π:Wehave π(g)−1F 0π(g)=(1/m) π(g)−1π(t)−1Fπ(t)π(g) ∈ tG =(1/m) π(tg)−1Fπ(tg)=F 0. t∈G

By application of Schur’s Lemma, we have F 0 =0orF 0 = λid in the respective cases, and because of Tr F 0 =(1/m) Tr(π(t−1)Fπ(t)) = Tr F t∈G and Tr id = n = dim V, we get λ =(1/n)Tr F .

Proposition 2.2: For π and π as in Proposition 2.1 given in matrix form by π(g)=A(g) and π(g)=B(g), we have −1 (1/m) Bij (t )Akl(t) = 0 for all i, j, k, l in case 1 , ∈ tG −1 (1/m) Aij (t )Akl(t)=(1/n)δilδjk for all i, j, k, l in case 2 . t∈G

Proof: With F =(Fij ) Proposition 2.1 expressed in matrix form −1 0 (1/m) Bij (t )FjkAkl(t)=Fil t∈G jk states −1 (1/m) Bij(t )FjkAkl(t) = 0 for all i, l in case 1, t∈G jk and this requires that all the coeﬃcients of all Fjk are zero. Hence, we have the ﬁrst relation. In the second case we have −1 (1/m) Aij (t )FjkAkl(t)=λδil =(1/n) Fjkδjkδil. t∈G jk jk

Comparing the coeﬃcients of Fjk on both sides leads to the second relation. Proof of the Theorem: We have χ (t)= Bii(t) and χ(t)= Aii(t) and hence −1 −1 <χ,χ> =(1/m) χ(t )χ(t)=(1/m) Aii(t )Ajj(t) ∈ ∈ t G t G ij =(1/n) δijδij =1 ij by the second relation in Proposition 2.2, and −1 −1 <χ,χ >=(1/m) χ (t )χ(t)=(1/m) Bii(t )Ajj(t) t∈G t∈G ij by the ﬁrst relation in Proposition 2.2. 26 2. Representations of Finite Groups

Corollary 1: Let (π, V ) be a ﬁnite-dimensional representation of G with character

χ := χπ and (πi,Vi) irreducible representations with characters χi := χπi and

V = V1 ⊕···⊕Vk.

Let moreover be (π ,V ) an irreducible representation with character χ = χπ . Then we have

(2.1) # {Vi,Vi V } = <χ,χ>.

From 1.2 we know that this number is also recognized as mult(π,π). Proof: By Theorem 1.6 in 1.6 we have χ = i χi. Theorem 2.1 in 2.1 says that the summands in <χ,χ>= <χ,χi > i

are either 0 or 1 if χ ∼ χi or χ ∼ χi. Hence <χ,χ> counts the number of irreducible components πi contained in π which are equivalent to π .

Corollary 2: mult(π,π) is independent of the decomposition as in Corollary 1.

Proof: <χ,χ> does not depend on the decomposition.

Corollary 3: If two representations π and π have the same character χπ = χπ , they are equivalent.

Proof: By Corollary 1 the multiplicities are equal for all irreducible components of π and π and so both representations are equivalent.

Corollary 4: If π decomposes into the irreducible representations π1,...,πh with respective multiplicities mi, i.e. one has

V = m1V1 ⊕···⊕mhVh, or, as we also write m1 ⊕···⊕ mh V = V1 Vh , one has h 2 <χ,χ>= mi . i=1 Proof: This follows immediately from Theorem 2.1 in 2.1, as χ = miχi and the χi are an ON-system.

This Corollary specializes to the following important irreducibility criterium.

Corollary 5: π is irreducible exactly if <χ,χ>=1. 2.2 The Regular Representation 27

2.2 Regular Representation and its Decomposition

The regular representation λ of the finite group G with #G = m is defined on the vector space V spanned by a basis et indexed by the elements t of G, i.e. V =t∈G by the prescription λ(g)et := egt, −1 ∈ i.e. we have λ(g)v = t∈G z(g t)et for v = t∈G ztet,zt C. V can be identified with H concerning the vector space structure. Then we have λ(g)u = u(g−1t)t for all u = u(t)t ∈H. t∈G t∈G If g = e,wehavegt = t for all t ∈ G, which shows that the diagonal elements of λ(g) are zero. In particular we have Tr λ(g) = 0 and Tr λ(e)=TrEm = m. So we have proven: Proposition 2.3: The character χλ of the regular representation λ of G is given by

χλ(e)=#G = m,

χλ(g) = 0 for all g = e.

Proposition 2.4: Every irreducible representation πi is contained in the regular representation with multiplicity equal to its dimension ni, i.e. mult(πi,λ)=ni = dim Vi.

Proof: According to Corollary 1 in 2.1 we have

mult(πi,λ)=<χπi ,χλ > and by the deﬁnition of the scalar product also −1 mult(πi,λ)=(1/m) χλ(t )χπi (t). t∈G

By Proposition 2.3 and the general fact that the character of the neutral element equals the dimension, we ﬁnally get

· · mult(πi,λ)=(1/m) m χπi (e)=ni.

Proposition 2.5: Let π1,...,πh be a complete system representing the equivalence classes of irreducible representations of G. Then the dimensions ni = dim Vi satisfy the relations h 2 a) i=1 ni = m, and, if t ∈ G, t = e, h b) i=1 niχπi (t)=0.

Proof: By Proposition 2.4 we have χλ(t)=Σniχi(t) for all t ∈ G. Taking t = e we obtain a) and for t = e we obtain b).

This result is very useful for the determination of the irreducible representations of a ﬁnite group: suppose one has constructed some mutually nonequivalent irreducible representations of dimensions n1,...,nh. In order that they be all irreducible representations (up 2 ··· 2 to equivalence) it is necessary and suﬃcient that one has n1 + + nh = m. 28 2. Representations of Finite Groups

In 1.3 we already discussed the example G = S3. Here we have m = 6. And we singled out the two one-dimensional representations π1 = 1 and π2 = sgn and the two- 2 2 2 dimensional representation π3. As we have here 1 +1 +2 =6=m, we see that we got all irreducible representations. Moreover, we have a simple way to check the irreducibility of π3 by using our Corollary 5. in 2.1 As an easy Exercise 2.1, we recommend to do this and to determine the characters of all the representations π1,π2,π3, and π0 introduced there. One can see that that one has π0 = π1 + π3 consistent with our Corollary 4 in 2.1.

One can prove more facts in this context, for instance the dimensions ni all divide the order m of G (see [Se] p.53]).

2.3 Characters as Orthonormal Bases and Number of Irreducible Representations

We sharpen Theorem 2.1 from 2.1 to the central fact that the characters χ1,...,χh belonging to a complete set of equivalence classes of irreducible representations of G form an orthonormal basis of the space of class functions H0. We recall that a function f deﬁned on G is a class function if one has f(g)=f(tgt−1) for all g, t ∈ G.

Proposition 2.6: Let f be a class function on G and (π, V ) a representation of G. Let πf be the endomorphism of V deﬁned by πf := f(t)π(t). t∈G

If π is irreducible with dim V = n and character χ, then πf is a scalar multiple of the identity, i.e. πf = λidV , with λ =(1/n) χ(t)f(t)=(m/n) < χ,¯ f > . t∈G

Proof: πf is an intertwining operator for π: We have −1 −1 −1 π(s) πf π(s)= f(t)π(s) π(t)π(s)= f(t)π(s ts). t∈G t∈G

As f is a class function, we get by t −→ s−1ts = t −1 π(s) πf π(s)= f(t )π(t )=πf . t∈G

By the second part of Schur’s Lemma, πf = λid is a scalar multiple of the identity. We have Tr λid= nλ and Tr πf =Σf(t)Tr π(t)=Σf(t)χ(t). Hence λ =(1/n)Σf(t)χ(t)=(m/n) < χ,¯ f > .

As the characters are class functions, they are elements of H0. 2.3 Characters as Orthonormal Bases 29

Theorem 2.2: The characters χ1,...,χh form an ON-basis of H0.

Proof: Theorem 2.1 from Section 2.1 says that the (χi) form an ON-system in H0.We have to prove that they generate H0. That is, we have to show that every f ∈H0, orthogonal to allχ ¯i, is zero. Let f be such an element. For each representation π of G, we put πf =Σf(t)π(t) as above. Proposition 2.6 shows that πf is zero if π is irreducible. But as each π may be decomposed into irreducible representations, πf is always zero. We apply this to the regular representation , i.e. to π = λ, and compute for the ﬁrst basis vector of the representation space for λ πf e1 = f(t)λ(t)e1 = f(t)et. t t

Since πf is zero, we have πf e1 = 0 and this requires f(t) = 0 for all t ∈ G. Hence f is zero.

As each element of H0 is ﬁxed by associating a complex number to each conjugacy class g∼ of G, the dimension of H0 as a C-vector space equals the number of conjugacy classes. Hence we have:

Corollary 1: The number of irreducible representations of G (up to equivalence) is equal to the number of conjugacy classes.

Corollary 2: Let g be an element of G and c(g):=#{g; g ∼ g} the number of elements in the conjugacy class of g. Then we have h a) i=1 χi(g)χi(g )=m/c(g)ifg = g, b)=0ifg ∼ g.

Proof: Let fg be the characteristic function of the class of g, i.e. fg(g )=1ifg ∼ g and = 0 else. Since we have fg ∈H0, by the Theorem, we can write

h fg = λiχi with λi = <χi,fg > =(c(g)/m)χi(g). i=1

For each t ∈ G we then have

h fg(t)=(c(g)/m) χi(g)χi(t). i=1

And because of fg(t) = 1 for t ∼ g and = 0 for t ∼ g, we get a) and b).

Remark 2.2: We recall the statement from 0.3 that the number of conjugacy classes of the symmetric group Sn equals the number p(n) of partitions of n. It is a nice excercise to redo the example of G = S3 in the light of the general facts now at hand: We have p(3) = 3 and thus know of three irreducible representations π1,π2,π3 with respective characters χ1,χ2,χ3. Moreover we know that G = S3 can be generated by the transposition σ =(1, 2) and the cycle τ =(1, 2, 3). The three conjugacy classes are represented by e = id, σ, and τ.Wehave

σ2 = e, τ 3 = e, στ = τ 2σ. 30 2. Representations of Finite Groups

Hence, for each representation π the equation π(σ)2 = id holds. So there are two one- dimensional representations, namely π1 = 1 and π2 = sgn with

χ1(e)=1,χ1(σ)=1,χ1(τ)=1, χ2(e)=1,χ2(σ)=−1,χ2(τ)=1.

2 2 2 The third representation π3 must have dim π3 = n with 1 +1 + n = 6, i.e. n =2,as we already concluded at the end of 2.2. The value of χ3 can be deduced (Proposition 2.4 in 2.2) from the relation with the character χλ of the regular representation λ

χ1 + χ2 +2χ3 = χλ.

As we have χλ(e) = 6 and χλ(g) = 0 for g = e, we conclude

χ3(e)=2,χ3(σ)=0,χ3(τ)=−1.

These formulae should also be read oﬀ from the results of Exercise 2.1.

There is a lot more to be said about the representations of finite groups in general and of the symmetric group in particular. Here, we only mention that the irreducible representations of Sn are classified by the Young tableaus realizing a permutation of n. And the following theorem about the canonical decomposition of a representation of a finite group, which is a special case of our Theorem 1.5 in 1.5 and can be proved rather easily (see [Se] p. 21): Let π1,...,πh be as before irreducible representations representing each exactly one equivalence class and let (π, V ) be any representation. Let V = U1 ⊕···⊕Uk be a direct sum decomposition of V into spaces belonging to irreducible representations. For i =1,...,hdenote by Wi the direct sum of those of the U1,...,Uk which are isomorphic to Vi (the Wi belong to the primary representations introduced at the end of 1.5). Then we have V = W1 ⊕···⊕Wh, which is called the canonical decomposition. Its properties are as follows:

Theorem 2.3: i) The decomposition V = W1 ⊕···⊕Wh does not depend on the initially chosen decomposition of V into irreducible subspaces. ii) The projection Pi of V onto Wi associated to the decomposition is given by Pi =(ni/m) χi(t)π(t),ni = dim Vi. t∈G

We stop here our treatment of ﬁnite groups and propose to the reader to treat the following examples.

Exercise 2.2: Determine the irreducible representations of the cyclic group Cn and of the dihedral group Dn (This is the group of rotations and reflections of the plane which preserve a regular polygon with n vertices, it is generated by a rotation r and a reflection s fulfilling the relations rn = e, s2 = e, srs = r−1).

Exercise 2.3: Determine the characters of S4. Chapter 3

Continuous Representations

Among the topological groups, compact groups are the most easy ones to handle. Since we would like to treat their representations next, it is quite natural that the appearance of topology leads to an additional requirement for an appropriate definition, namely a continuity requirement. We will describe this first and indicate the necessary changes for the general concepts from Sections 1.1 to 1.5 (to be used in the whole text later). In the next chapter we specialize to compact groups. We will find that their representation theory has a lot in common with that of the finite groups, in particular the fact that all irreducible representations are finite-dimensional and contained in the regular representation (the famous Theorem of Peter and Weyl). But there is an important difference: the number of equivalence classes of irreducible representations may be infinite.

We do not prove all the modiﬁcations in the general theorems when linear representations are specialized to continuos representations, but concentrate in the next chapters on an explicit description of the representation theory for SU(2) resp. SO(3) and the other examples mentioned in the introduction.

3.1 Topological and Linear Groups

We follow [Ki] p.22: Deﬁnition 3.1: A topological group is a set that is simultaneously a group and a topological space in which the group and topological structures are connected by the following condition:

The map G × G −→ G, (a, b) −→ ab−1 is continuous.

Remark 3.1: This requirement is equivalent to the following three conditions which are more convenient for checking

1.) (a, b) −→ ab is continuous in a and in b, 2.) a −→ a−1 is continuous at the point a = e, 3.) (a, b) −→ ab is continuous in both variables together at the point (e, e). 32 3. Continuous Representations

Every abstract group can be regarded as a topological group if one gives it the discrete topology where every subset is open.

All the matrix groups presented in Section 0.1 and later on are topological groups: As n2 n2 they are subsets of Mn(R) R , they inherit the (standard) topology from R .Open balls in Mn(R) (and analogously in Mn(C)) are

Kρ(A):={B ∈ Mn(R), B − A <ρ},ρ∈ R>0,A∈ Mn(R), where (3.1) A := ,:= Re Tr tAB.¯

This topology is consistent with the group structure: multiplication of two matrices gives a matrix where the coefficients of the product matrix are polynomials in the coefficients of the factors and thus the two continuity requirements 1.) and 3.) in the remark above are fulfilled. Similarly with condition 2.), as the inverse of a matrix has coefficients which are polynomials in the coefficients of the matrix divided by the determinant, another polynomial here not equal to zero, i.e. we have again a continuous function.

For A ∈ U(n)wehavetAA¯ = E, i.e. A 2 = n, so we have an example of a compact group. Heis(R), SL(2, R) and SO(3, 1) are not compact but these groups together with all others in this text are locally compact, i.e. each element admits a neighbourhood with compact closure (as Rn does). Moreover, they are closed subgroups of some GL(N,R). This, in principle, has to be veriﬁed in each case. A practical tool is the following standard criterium. Remark 3.2: G is closed if for every convergent sequence (Aj), Aj ∈ G with lim Aj ∈ GL(n, R), we have lim Aj ∈ G.

An example for a diﬀerent procedure is the following: SL(n, R) is a closed subgroup of GL(n, R) as det: GL(n, R) −→ R∗ is a continuous map and SL(n, R) is the inverse image of the closed subset {1}⊂R∗.

Now we have the possibility to deﬁne the class of groups we will restrict to in this text:

Deﬁnition 3.2: A group is called a linear group if there is an n ∈ N such that G is isomorphic (as abstract group) to a closed subgroup of GL(n, R)orGL(n, C).

Remark 3.3: More generally, one calls also linear those groups, which are isomorphic to a closed subgroup of GL(n, H), H the quaternion skewﬁeld, which in this text will only appear in a later application.

Remark 3.4: In another kind of generality one treats Lie groups, which are topological groups and at the same time diﬀerentiable manifolds such that

G × G −→ G, (a, b) −→ ab and G −→ G, a −→ a−1, are diﬀerentiable maps. As we try to work without the (important) notion of a manifold as long as possible, it is quite adequate to stay with the concept of linear groups as deﬁned above (which comprises all the groups we are interested in here). 3.2 The Continuity Condition 33

While treating topological notions, let us remind that a group G is called (path-)connected if for any two a, b ∈ G there is a continuous map of a real interval mapping one endpoint of the interval to a and the other to b.

Exercise 3.1: Show that SU(2) and SL(2, R) are connected and GL(2, R) is not.

Before we come to the deﬁnition of continuous representations, let us adapt the notion of a G-set from Section 0.2 to the case that G is a topological group.

Deﬁnition 3.3: A set X is called a (left-)topological G-space iﬀ it is a left G-set as in 0.2 and X is also provided with a topology such that the map

G ×X −→X, (g, x) −→ gx, is continuous.

In the sequel, G-space will abbreviate the content of this deﬁnition.

If the space X is Hausdorff (as all our examples will be), then the stabilizing subgroup Gx ⊂ G of each point x ∈X is a closed subgroup of G. Conversely, if H is a closed subgroup of a Hausdorff group G, then the space G/H of left cosets gH, g ∈ G,isa homogeneous Hausdorff G-space when given the usual quotient space topology. If X is any other topological G-space with the same stabilizing group Gx = H for x ∈X, then the natural map Φ:G/H −→ X ,gH−→ gx is 1-1 and continuos. If G and X are locally compact, then Φ is a homeomorphism and allows an identification of both spaces.

3.2 The Continuity Condition

From now on a group will mean a locally compact topological group. Then an appropriate representation space V should be a linear topological space. To make things easier for us, we will suppose that V = H is a separable complex Hilbert space (i.e. having de- 2 1 numerable Hilbert space basis (ej)j∈I ,I N), as for instance L (S ) with basis given by the functions ej(t) := exp(2πijt),j ∈ Z. For the finite-dimensional case this is nothing new: we have as before V = Cn. If we write again GL(V ), in the finite-dimensional case, we have the same set as before provided with the topology given by an isomorphism GL(V ) GL(n, C). In the infinite-dimensional case, V = H, GL(V ) = GL(H) is meant as the group L(H) of linear bounded operators in H. We recall here the following fundamental notions relevant for the infinite-dimensional cases (from e.g. [BR] p.641f) :

• An operator T with domain D(T ) ⊂His said to be continuous at a point v0 ∈ D(T ) iﬀ for every >0 there exists a δ = δ() > 0 such that from v − v0 <δwith v ∈ D(T ) we can conclude Tv− Tv0 <. 34 3. Continuous Representations

• An operator T is said to be bounded iﬀ there exists a constant C such that one has Tv ≤ C v for all v ∈ D(T ). • The norm T of a bounded operator T is deﬁned as the smallest C such that

Tv ≤C v for all v ≤1

. Remark 3.5: A bounded linear operator is uniformly continuous. Conversely, if a linear operator T is continuous at a point v0 (e.g. v0 = 0), then T is bounded.

Exercise 3.2: Prove this.

• Let A be a linear operator in H with dense domain D(A) ⊂H. Then one can prove that there are v, v ∈Hsuch that < Au, v > = holds for all u ∈H. One sets v =: A∗v, verifies that A∗ is a linear operator, and calls A∗ the adjoint of A. • For operators A and B in H one writes B ⊃ A iff one has D(B) ⊃ D(A) and Bu = Au for all u ∈ D(A). Then B is called an extension of A. • A linear operator A with dense domain D(A) is called symmetric iff one has A∗ ⊃ A and < Au, v > = for all u, v ∈ D(A).

• A linear operator A with dense domain is called self-adjoint iﬀ one has A∗ = A. One can prove that a linear symmetric operator A with D(A)=H is bounded and self-adjoint.

If not mentioned otherwise, L(H) is provided with the strong operator topology ([BR]), i.e. we have as basis of open neighbourhoods

Bρ,u(T0):={T ∈ L(H); Tu− T0u <ρ},ρ∈ R>0,u∈H,T0 ∈ L(H). In the sequel a representation of G shall mean the following.

Deﬁnition 3.4: A linear representation (π, H) is said to be continuous iﬀ π is strongly continuous, i.e. the map G −→ H ,g−→ π(g)v is continuous for each v ∈ V .

Remark 3.6: It is quite obvious that one could also ask for the following continuity requirements: a) The map G ×H−→H, (g, v) −→ π(g)v is continuous. b) The map G −→ GL(H),g−→ π(g) is continuous. and c) For each v ∈ V, the map G −→ H ,g−→ π(g)v is continuous in g = e and there is a uniform bound for π(g) in some neighbourhood of e ([Kn] p.10).

Deﬁnition 3.5: A representation (π, V ) is bounded iﬀ we have

sup π(g) < ∞. g∈G 3.2 The Continuity Condition 35

In all examples in this section and in most cases of the following sections all these conditions are fulﬁlled. We will not discuss the hypotheses necessary so that all these conditions are equivalent and refer to this to [Ki] p.111, [Kn] p.10 and [BR] p.134. But we will show for some examples how the continuity of a representation comes out. Later we will not examine continuity and leave this to the reader’s diligence. (To check this and the question whether a representation from the examples to be treated is bounded or not may even be (part of) the subject of a bachelor thesis.)

Example 3.1: The linear representation π = χk,k∈ Z of G = SO(2) in V = C given by cos ϑ sin ϑ χ (r(θ)) := eikϑ for all g = r(ϑ)=( ),ϑ∈ R, k − sin ϑ cos ϑ is obviously continuous: The map (g, z) −→ eikϑz is continuous in g = r(ϑ) for each z ∈ C or even contiuous in both variables g and z, because the exponential function is (in particular) continuous and an open neighborhood of e ∈ SO(2) is homeomorphic to an open interval in R containing 0.

Later we will modify the notions of reducibility and irreducibility for continuous representations. But here we can already prepare the way how to do this and prove the following: Remark 3.7: Each irreducible continuous representation π of G = SO(2) is equivalent to some χk,k ∈ Z.

Proof: Welookat(π, V ) only as a linear representation and apply Schur’s Lemma from 1.5. As G is abelian, we can conclude by Theorem 1.4 following from Schur’s Lemma in 1.5 that V = C. Hence we can assume π : SO(2) −→ C∗. As SO(2) is homeomorphic to R/Z, we have a continuous map r : R −→ R/Z. Now we use the assumption that π is continuous and we have a sequence of continuous maps R −→ SO(2) −→ C∗,t−→ r(t) −→ π(r(t)), and in consequence a continuous map R t −→ π(r(t)) =: χ(t) ∈ C∗. As π is a linear representation, χ has the property

χ(t1 + t2)=χ(t1)χ(t2),χ(t) = 1 for all t ∈ 2πZ. Here we have to recall a fact from standard calculus (see for instance [Fo] p.75): Every ∗ continuous function χ : R −→ C obeying the functional equation χ(t1 +t2)=χ(t1)χ(t2) is an exponential function, i.e. χ(t)=c exp(tζ) for c ∈ C∗,ζ ∈ C. The condition χ(2πn) = 1 for all n ∈ Z ﬁxes c = 1 and ζ = ki for k ∈ Z.

By the way, already here, we can can anticipate a generalization of Proposition 2.4 in 2.2 from ﬁnite to compact groups: L2(SO(2)) can be identiﬁed with a space of periodic functions. And from the theory of Fourier series one knows that this space has as a Hilbert space basis the trigonometric functions ej,j ∈ Z, with 2πijt ej(t):=e ,t∈ R. 36 3. Continuous Representations

This can be translated into the statement that each irreducible continuous representation of SO(2) is contained (with multiplicity one) in the left- (or right-)regular representation λ resp. ρ of SO(2) and λ and ρ are direct sums of the χj,j ∈ Z.

Example 3.2: Let G be SO(3) and, for ﬁxed m ∈ N, let πm be the representation of G given on the space Vm of all homogeneous polynomials of degree m in 3 variables, i.e.

Vm := {P ∈ C[x1,x2,x3]; P homogeneous, deg P = m}, by −1 t π(g)P (x):=P (g x),x= (x1,x2,x3) (a column).

πm is a linear representation as is already clear from the general consideration in 1.3. The continuity condition is fulﬁlled since SO(3) inherits the topology from GL(n, R) and matrix multiplication x −→ g−1x is a continuous map and polynomials P are continuous functions in their arguments.

This example is similarly constructed as the example π = πj for G = SU(2) introduced in 1.3, which is continuous by the same reasoning as are also the other examples in 1.3 and later on. So we will now take the continuity of our representations for granted.

Moreover, let us announce here that we shall have to discuss the relationship of the two examples G = SO(3) and G = SU(2) very thoroughly later. The representations πj of SU(2) will come out as irreducible, but here we see already that for instance for m =2 2 2 2 ∈ (πm,Vm) has an invariant subspace V0 := (x1 + x2+ x3)C since each element g SO(3) 3 2 2 acts on R leaving invariant the spheres given by xi = ρ .

The addition of the continuity condition to the definition of the representation has natural consequences for the notions related to the definition of the linear representation in Sections 1.1 and 1.2: Here only closed subspaces are relevant. Hence in the future, we use definitions modified as follows.

• Suppose that there is a closed subspace V1 of the space V of a representation (π, V ) of G, invariant under all π(g),g ∈ G. Then π1 := π|V is called a (topological) subrepresentation of π.

• The representation π2 on the factor (or quotient) space V/V1 is called a (topological) factor representation.

The representation (π, V ) is called

• (topologically) irreducible iﬀ it admits no nontrivial subrepresentation,

• (topologically) decomposable iﬀ there exist closed subspaces V1 and V2 in V such ⊕ | that V = V1 V2. In this case we write π = π1 + π2, where πi := π Vi , 3.2 The Continuity Condition 37

• completely reducible (or discretely decomposable) iﬀ it can be expressed as a direct sum of irreducible subrepresentations,

• unitary iﬀ V is a Hilbert space H and every π(g) respects the scalar product in H.

• If (π, V ) and (π,V) are continuous representations of G, then a bounded linear map (or operator) F : V −→ V is called an intertwining operator for π and π iﬀ one has Fπ(g)=π(g)F for all g ∈ G.

Then Schur’s Lemma here appears in the following form.

Theorem 3.1: If π and π are unitary irreducible representations in the Hilbert spaces H resp. H and F : H−→H is an intertwining operator, then F is either an isomorphism of Hilbert spaces or F =0.

This theorem has as a consequence the following irreducibility criterium, which is a generalization to the inﬁnite-dimensional case of the similar one we proved for the ﬁnite- dimensional linear representations without the continuity condition in 1.5. We will call it the unitary Schur:

Theorem 3.2: A unitary representation (π, H) is irreducible iﬀ the only operators commuting with all the π(g) are multiples of the identity.

The proofs of these theorems (see e.g. [BR] p. 143/4 or [Kn] p.12) use some tools from functional analysis (the spectral theorem) which we do not touch at this stage.

We indicate another very useful notion (in particular concerning unitary representations).

Deﬁnition 3.6: A representation π of G in V is said to be cyclic iﬀ there is a v ∈ V (called a cyclic vector for V ) such that the closure of the linear span of all π(g)v is V itself.

Theorem 3.3: Every unitary representation (π, H)ofG is the direct sum of cyclic representations.

As the proof is rather elementary and gives some indications how to work in the inﬁnite- dimensional case and how the continuity condition works, we give it here (following [BR] p.146): H ∈ ∈H Let v1 be the closure of the linear span of all π(g)v1,g G for any 0 = v1 . H H Then v1 is π-invariant: Indeed, let v1 be the linear span of all π(g)v1, then for each ∈H ∈H u v1 we have a sequence (un),un v1 which converges to u. Obviously, one has ∈H −→ π(g)un v1 . The continuity of each π(g) implies π(g)un π(g)u and hence we have ∈H H | π(g)u v1 and v1 is invariant. Thus the subrepresentation π1 = π Hv1 is cyclic with cyclic vector v1. H H ∈H⊥ H\H H If v1 = , choose 0 = v2 v1 = v1 , consider the closed linear span v2 which H H H ⊕H is π-invariant and orthogonal to v1 , and continue like this if = v1 v2 . {H } Let ξ denote the family of all collections vi , each composed of a sequence of mutually orthogonal, invariant and cyclic subspaces. We order the family by means of the 38 3. Continuous Representations inclusion relation. Then ξ is an ordered set to which Zorn’s Lemma applies: It assures {H } H the existence of a maximal collection vi max. By the separability of , there can {H } be at most a countable number of subspaces in vi max and their direct sum, by the {H } H maximality of vi max must coincide with .

As a consequence of this theorem, we have a very convenient irreducibility criterium for unitary representations : Corollary: A unitary representation (π, H)ofG is irreducible iﬀ every nonzero u ∈H is cyclic for π.

Proof:i)Ifπ is irreducible, every 0 = u ∈His cyclic as to be seen from the ﬁrst part of the proof above. ii) Suppose that H1 ⊂His a nontrivial invariant subspace of H and 0 = u ∈H1. Due to the invariance of H1 we have π(g)u ∈H1. Moreover, the closure of the linear span of all π(g)u is contained in H1 but by assumption is also equal to H. Hence, we have a contradiction and π is irreducible.

To close these general considerations let us recall the following. If H and H are Hilbert spaces with scalar products <, > resp. <, >, then we can define a scalar product in H⊗H by the formula :=< v, w >< v,w >. If either H or H is finite-dimensional, then the space H⊗H equipped with this scalar product is complete. If both H and H are infinite-dimensional, we complete H⊗H by the norm defined by the scalar product above and denote this by H⊗Hˆ .

3.3 Invariant Measures

We want to extend the results from finite groups to compact groups. The tool to do this is given by replacing the finite sum g∈G by an integration G. To be a bit more precise, one has to recall some notions from measure and integration theory, which moreover are inevitable in all the following sections. For the whole topic we recommend the classical books Measure Theory by Halmos ([Ha]) and Abstract Harmonic Analysis by Hewitt-Ross ([HR]) or, for summaries, the appropriate chapters in [Ki] p.129ff and [BR] p.67ff.

If X is a topological space with the family V = {Oi}i∈I of open sets, it is also a Borel space with a family B of Borel sets B, where these sets are obtained from open and closed sets by the operation of countable unions, countable intersections and complementation. A little more generally, a Borel space is a set with a family B of subsets B such that B is a σ-algebra, i.e. one has

X\B ∈ B for B ∈ B, ∈ ∈ ∈ i∈I Bi B for Bi B,i I N, ∈ ∈ ∈ i∈I Bi B for Bi B,i I N. Let (X , B) and (X , B) be two Borel spaces. A map F : X−→X is called a Borel map iﬀ one has F −1(B) ∈ B for all B ∈ B. 3.3 Invariant Measures 39

Deﬁnition 3.7: We call µ a measure on the Borel space (X , B)ifµ is a map

µ : B −→ [0, ∞]:=R≥0 ∪ {∞}, where µ is σ-additive, i.e. i) µ(∪Bi)= µ(Bi) if the Bi are pairwise disjoint (i ∈ I N), and ii) there is a covering of X by sets Bi of ﬁnite measure, i.e. we have Bi ∈ B,i∈ I N with µ(Bi) < ∞ and ∪Bi = X .

The most obvious example is X = R provided with the Lebesgue-measure µ =: λ, which is deﬁned by λ([a, b)) := b − a for a, b ∈ R,a≤ b.

As we know from elementary integration theory (or see immediately), λ is translation invariant. This is a special incidence of the following general machinery. Let X be a space with a family V of open sets and a family B of Borel sets B and let µ be a measure on X . We then say (X , B,µ)isameasure space. If we have a right G-action (G ×X) −→ X , (g, x) −→ xg, we can deﬁne a measure µg by

µg(B)=µ(Bg) for all B ∈ B.

Then the measure µ is called right-invariant iﬀ one has µg = µ for all g ∈ G. Similarly µ is called left-invariant iﬀ we have a left action and gµ = µ for all g ∈ G where gµ(B)=µ(gB) for B ∈ B.

Now we take X = G, a locally compact group. Then a right- or left-invariant measure on G is called a right- or left-Haar measure. And we have as central result ([Ki] p.130):

Theorem 3.4 (Haar): On every locally compact group with countable basis for the topology, there exists a nonzero left-invariant (and similarly a right-invariant) measure. It is deﬁned uniquely up to a numerical factor.

We will not go into the proof of the Theorem (see for instance [HR] Ch.IV, 15) but give later on several concrete examples for its validity. The most agreable case is the following:

Deﬁnition 3.8: G is called unimodular iﬀ it has a measure which is simultaneously left- and right-invariant. Such a measure is also called biinvariant or simply invariant.

To make these up to now only abstract concepts work in our context, we have to recall a more practical aspect due to the general integration theory: Let (X , B,µ) be a measure space and f : X−→C a measurable function, i.e. f is the sum f = f1 − f2 + i(f3 − f4) of four real nonnegative functions fj (j =1, .., 4) having the property

{x ∈X; fj(x) >a}∈B for all a>0. 40 3. Continuous Representations

For measurable nonnegative functions f general integration theory provides a linear map f −→ µ(f)= f(x)dµ(x) ∈ C ∪ {∞}. X We call a measurable function f integrable iﬀ |f(x)| dµ(x) < ∞. If X = G as above, one usually writes

dµ(g)=:drg if µ is right − invariant, =: dlg if µ is left − invariant, =: dg if µ is bi − invariant.

The following fact is central for applications to representation theory.

Proposition 3.1: Let G be a locally compact group. Then there is a continuous homomorphism ∆G, called the modular function of the group G, into the multiplicative group of positive real numbers for which the following equalities hold

−1 dr(gg )=∆G(g)drg ,dl(gg )=∆G(g ) dlg,

−1 drg = const.∆G(g)dlg, dlg = const.∆G(g) drx,

−1 −1 −1 dr(g )=∆G(g) drg = const.dlg, dl(g )=∆G(g)dlg. As Kirillov does in [Ki] p.130, we propose the proof as an exercise based on the application of Haar’s Theorem. There is a proof in [BR] p.68/9, but the modular function there is the inverse to our function, which here is deﬁned following Kirillov.

Proposition 3.2: If G is compact, then the Haar measure is ﬁnite and biinvariant.

Proof: The image of a compact group under the continuous homomorphism ∆G is again a compact group. The multiplicative group R>0 has only {1} as a compact subgroup. So by Proposition 3.1, we have a biinvariant measure which can be normalized to G dg =1.

3.4 Examples

Most of the noncompact groups to be treated later on are also unimodular, e.g. the group SL(2, R) or the Heisenberg group Heis(R). We will show here how this can be proved.

Example 3.3: G = Heis(R)={g =(λ, µ, κ); λ, µ, κ ∈ R} with

gg =(λ + λ,µ+ µ,κ+ κ + λµ − λµ).

As Heis(R) is isomorphic to R3 as a vector space, we try to use the usual Lebesgue measure for R3. So we take dg := dλdµdκ. −→ The left translation λg0 : g g0g =: g acts as

(λ, µ, κ) −→ (λ := λ0 + λ, µ := µ0 + µ, κ := κ0 + κ + λ0µ − λµ0). 3.4 Examples 41

The Jacobian of this map is ⎛ ⎞ 1 −µ0 ⎝ ⎠ Jg0 (g)= 1 λ0 1

with det Jg0 (g) = 1. Hence we see that dg is left-invariant. Similarly we prove the right- invariance.

Remark 3.8: Without going into the general theory of integration on manifolds, we indicate that, for calculations like this, it is very convenient to associate the inﬁnitesimal measure element with an appropriate exterior or alternating diﬀerential form and then check if this form stays invariant. In the example above we take

ω := dλ ∧ dµ ∧ dκ and calculate using the the general rules for alternating diﬀerential forms (we shall say a bit more about this later in section 8.1), in particular

dµ ∧ dλ = − dλ ∧ dµ, dµ ∧ dµ =0, as follows

◦ ∧ ∧ − ∧ ∧ ω λg0 = dλ dµ (dκ + λ0dµ µ0dλ)=dλ dµ dκ = ω.

Example 3.4: G = SO(2)

We introduced as standard notation

cos ϑ sin ϑ SO(2) r(ϑ)=( ). − sin ϑ cos ϑ

Here we have dg = dϑ as biinvariant measure with 2π dg = dϑ =2π. G 0

Example 3.5: G = SU(2)

This group will be our main example for the representation theory of compact groups. As standard parametrization we use the same as Wigner does in [Wi] p.158: The general form of a two–dimensional unitary matrix

ab g =( ). cd of determinant one is

ab g =( ),a,b∈ C with | a2| + | b2| =1. −¯b a¯ 42 3. Continuous Representations

With eiα cos ϑ sin ϑ s(α):=( ) and r(ϑ)=( ) e−iα − sin ϑ cos ϑ one has a decomposition

ab SU(2) g =( )=s(−α/2)r(−β/2)s(−γ/2),α,γ∈ [0, 2π],β ∈ [0,π]. −¯b a¯

In particular, one can see that SU(2) is generated by the matrices s(α),r(β).

Exercise 3.3: Prove this.

Later on we shall see later that the angles α, β, γ correspond to the Euler angles for the three-dimensional rotation introduced by Wigner in [Wi] p.152. In Hein [He] one ﬁnds all these formulas, but unfortunately in another normalization, namely one has zHein = a, uHein = −b, αHein = β,βHein = −α, γHein = −γ. Our parametrization is one-to-one for α, γ ∈ [0, 2π],β ∈ [0,π] up to a set of measure zero. An invariant normalized measure is given by

dg := (1/16π2) sin βdαdβdγ.

Exercise 3.4: Verify this.

Example 3.6: An example for a group G which is not unimodular is the following

ab G = {g =( ); a ∈ R∗,b∈ R} 1

aa ab + b We have gg =( )=:g, hence 1

da ∧ db = ada ∧ (bda + db)=ada ∧ db

aaab + b so that d g = a−1dadb and, from gg =( )=:g∗ r 1

da∗ ∧ db∗ = a2da ∧ db,

−2 −1 we see dlg = a dadb. Hence, we have here ∆G(g)=a .

Exercise 3.5: Determine ∆G for

1 x y1/2 G = {g =( )( ); y>0,x∈ R}. 1 y−1/2 Chapter 4

Representations of Compact Groups

As already mentioned, the representation theory of compact groups generalizes the theory of finite groups. We shall closely follow the presentation in [BR], and cite from [BR] p.166: “The representation theory of compact groups forms a bridge between the rela- tively simple representation theory of finite groups and that of noncompact groups. Most of the theorems for the representations of finite groups have direct analogues for compact groups and these results in turn serve as the starting point for the representation theory of noncompact groups.”

4.1 Basic Facts

Let G be a compact group provided with an invariant measure µ normalized such that H G dg = 1 and let π be a representation of G in a Hilbert space . We recall that all our representations are meant to be linear und continuous. The following statements are easy to prove for the finite-dimensional case dim H < ∞. Most times this is quite sufficient because it will come out that all irreducible representations of compact groups are finite- dimensional. To prepare a feeling for the noncompact cases where infinite-dimensionality is essential, we state the results here for general H. But we will report on the proofs only when not too much general Hilbert space theory is needed.

Proposition 4.1: Let (π, H) be a representation of G in the Hilbert space H with scalar product <, >. Then there exists a new scalar product <, > deﬁning a norm equivalent to the initial one, relative to which the map g −→ π(g) deﬁnes a unitary representation of G.

Proof: i) As to be expected from the case of ﬁnite groups, we deﬁne <, > by := <π(g)u, π(g)v> dg. G Sesquilinearity and hermiticity of <, > are obvious. Let us check that we can conclude u = 0 from = <π(g)u, π(g)u> dg =0: G 44 4. Representations of Compact Groups

From G <π(g)u, π(g)u> dg = 0 one deduces that <π(g)u, π(g)u>= 0 almost everywhere. If g ∈ G is such that π(g)u = 0, then π(g)−1π(g)u = u =0. ii) <,> is π-invariant: <π(g)u, π(g)v> = <π(g)π(g)u, π(g)π(g)v> dg G = <π(gg)u, π(gg)v> dg = , G because dg is left-invariant. iii) The norms . and . induced by <, > resp. <, > are equivalent: We have u 2 = <π(g)u, π(g)u> dg G ≤ sup π(g) 2 dg = N 2 u 2 g∈G G

with N := supg∈G π(g) . And from

u 2 = <π(g−1)π(g)u, π(g−1)π(g)u> ≤ sup π(g−1) 2<π(g)u, π(g)u>= N 2 π(g)u 2 g∈G it follows that u 2 = dg G ≤ N 2 <π(g)u, π(g)u> dg = N 2 = N 2 u 2 . G

Hence we have N −1 u ≤ u ≤ N u and . and . are equivalent. iv) Equivalent norms deﬁne the same families of open balls and thus equivalent topolo- gies, i.e. if g −→ π(g)v, v ∈His continuous for the topology belonging to . , it is also continuous for the topology deduced from . .

We now turn to the already announced result that every unitary irreducible representation is ﬁnite-dimensional. In its proof we use an interesting tool, namely the Weyl operator Ku deﬁned for elements u, v from a Hilbert space H with π-invariant scalar product by

Kuv := <π(g)u, v > π(g)udg. G Remark 4.1: The Weyl operator has the following properties i) Ku is bounded, ii) Ku commutes with every π(g), i.e. Kuπ(g)=π(g)Ku. 4.1 Basic Facts 45

Proof: We perform the standard calculations using Cauchy–Schwarz inequality. ≤ | | i) Kuv G <π(g)u, v > π(g)u dg ≤ π(g)u 2 v dg = u 2 v , G ii) π(g )Kuv = G <π(g)u, v > π(g g)udg = G <π(g g)u, π(g )v>π(g g)udg = G <π(g)u, π(g )v>π(g)udg = Kuπ(g )v for all g ∈ G and v ∈H.

Theorem 4.1: Every irreducible unitary representation π of G in a Hilbert space H is ﬁnite-dimensional.

Proof: By the last remark, Ku is an intertwining operator for π. Hence by Schur’s Lemma, Ku is a homothety, i.e. Ku = λ(u)idH and, in consequence,

(4.1) = <π(g)u, v > dg G = |<π(g)u, v >|2 dg = λ(u) v 2 . G By interchanging the rˆolesof u and v and using the equality (which is equivalent to the unimodularity of G) f(g−1)dg = f(g)dg, we get λ(v) u 2 = |<π(g)v, u >|2 dg = ||2 dg = |<π(g−1)v, u >|2 dg = |<π(g)u, v >|2 dg

= λ(u) v 2 .

Hence we have λ(u)=c u 2 for all u ∈Hwith a constant c ∈ C. In (4.1) we put u = v, v = 1 and get |<π(g)u, u >|2 dg = λ(u)=c.

As the nonnegative continuous function g −→ | <π(g)u, u >| assumes the value u =1 at g = e, we must have c>0. Let {ei}i=1,...,n be a set of ON-vectors in H. In (4.1) we now put u = ek and v = e1 to obtain 2 2 |<π(g)ek,e1 >| dg = λ(ek) e1 = c. and n n 2 2 nc = |<π(g)ek,e1 >| dg = |<π(g)ek,e1 >| dg G G k=1 k=1 2 ≤ e1 dg =1. G 46 4. Representations of Compact Groups

The last inequality is a special case of Parseval’s inequality saying that for a unitary ma- −→ | |2≤ ≤ trix representation g A(g) one has k Ak1(g) 1. So ﬁnally, one has n 1/c, i.e. the dimension of H must stay ﬁnite.

We already proved that a ﬁnite-dimensional unitary representation is completely reducible. This result can be sharpened for compact groups G: Theorem 4.2: Every unitary representation π of G is a direct sum of irreducible ﬁnite- dimensional unitary subrepresentations.

The proof uses some more tools from functional analysis and so we skip it here (see [BR] p.169/170). But we mention the appearance of the important notion of a Hilbert–Schmidt operator, which is an operator A in H, s.t. for an arbitrary basis {ei}i∈I of H we have 2 Aei < ∞. i∈I

Again, as in the theory of ﬁnite groups, we have ON-relations for the matrix elements of an irreducible representation.

Theorem 4.3: Let π and π be two irreducible unitary representations of G and A(g) { } H { } H resp. A (g) their matrices with respect to a basis ei of and ek of . Then one has relations ∼ Aij(g)Akl(g)dg =0 ifπ π =(1/n)δikδjl if π ∼ π and n := dim H.

Proof: As to be expected, one applies Schur’s Lemma (Theorem 3.1): We introduce a matrix fij with entries (fij )kl = δikδjl and an operator −1 Fij := π(g)fijπ (g )dg. G

This is an operator intertwining the representations π and π since we have −1 π(g )Fij = π(g g)fijπ (g )dg G −1 = π(g)fijπ (g g )dg = Fijπ (g ). G

Hence, if π ∼ π ,wehaveFij = 0 or in matrix form for all (r, t) −1 (Fij )rt = Ari(g)Ajt(g )dg = Ari(g)A tj(g)dg =0.

For π = π , Schur’s Lemma requires Fij = λijid. Hence for (r, t) =( i, j), the orthogo- nality relations just obtained are still satisﬁed. For (r, t)=(i, j), we have (Fij )ij = Aii(g)Ajj(g)dg = λij =0 if i = j, 2 = | Aii(g) | dg = λii if i = j. 4.1 Basic Facts 47

We have Tr Fiiid = nλii and from the deﬁnition of Fij , −1 Tr Fii = Tr (A(g)fiiA(g )dg =Trfii =1 G and hence λii =1/n.

Similar to the case of ﬁnite groups we have a nice fact:

Theorem 4.4: Every irreducible unitary representation π of G is equivalent to a subrepresentation of the right regular representation.

H Proof: Let A(g)=(Ajk(g))j,k=1,...,n be a matrix form√ of π(g) and let π be the subspace 2 of L (G) spanned by the vectors ek with ek(g):= nA1k(g),k =1,...,n (these are an ON-system by the preceding theorem). We have √ ρ(g )e (g)=e (gg )= nA(gg ) 0 k √k 0 0 √ = n Σj A1j(g)Ajk(g0)=Σj nA1j(g)Ajk(g0)

=Σj ej(g)Ajk(g0).

Hence ρ restricted to Hπ is a subrepresentation equivalent to π.

Exercise 4.1: Show the same fact for the left regular representation.

Remark 4.2: With the same reasoning as in the case of finite groups, the characters χπ of finite-dimensional irreducible unitary representations π, defined by χπ(g):=Trπ(g) resp. χπ(g):= i Aii(g)= i for a matrix form with respect to a basis e1,...,en, have the following properties.

−1 ∈ 1) The characters are class functions, i.e. χ(g0gg0 )=χ(g) for all g, g0 G. 2) We have χ(g−1)=χ(g). ∼ 3) If π π , then χπ = χπ. ∼ ∼ 4) χπ(g)χπ(g)dg =0if π π and = 1 if π π .

If π is any ﬁnite-dimensional representation of G, then we can decompose it into irreducible representations πi appearing with multiplicities mi and we have

χπ = miχπi . i=1 Due to the ON-relation 4) in Remark 4.2 above, we obtain

mi = χπ(g)χπi (g)dg G and h 2 mi = χπ(g)χπ(g)dg. i=1 G This formula specializes to the very useful irreducibility criterium: 48 4. Representations of Compact Groups

Corollary: The ﬁnite-dimensional representation π is irreducible iﬀ

χπ(g)χπ(g)dg =1. G The big difference to the theory of finite groups is that for compact groups the number of equivalence classes of irreducible representations no longer necessarily is a finite number. We already know this from the example G = SO(2) in section 3.2. The central fact is now the famous Peter–Weyl Theorem:

Theorem 4.5: Let Gˆ = {πi}i∈I be the unitary dual of G, i.e. a complete set of representatives of the equivalence classes of irreducible unitary representations of G. For every ∈ i i i I let A (g)=(Ajk(g))j,k=1,...,ni be a matrix form of πi and √ i i Yjk(g):= ni Ajk(g). Then we have i ∈ 2 1) The functions Yjk,i I,j,k =1,...,ni form a complete ON-system in L (G). 2) Every irreducible unitary representation π of G occurs in the decomposition of the right regular representation with a multiplicity equal to the dimension of π, i.e.

mult (πi,ρ)=ni. 3) Every C-valued continuous function f on G can be uniformly approximated by a i linear combination of the (Yjk).

4) The characters (χπi )i∈I generate a dense subspace in the space of continuous class functions on G.

The proof of these facts use standard techniques from higher analysis. We refer to [BR] p. 173 - 176 or [BtD] p.134.

4.2 The Example G = SU(2)

Representation theory owes a lot to the work of H. Weyl and E. Wigner. As we have the feeling that we can not do better, we will discuss the example G = SU(2) following Wigner’s presentation in [Wi] p.163 - 166:

We put ab G = SU(2) g =( ), | a | 2 + | b | 2 =1 −¯b a¯

(j) and for j a half integral nonnegative number, we let V be the space C[x, y]2j of homogeneous polynomials of degree 2j in two variables. Thus we have dim V (j) =2j +1. (j) We choose as a basis of V monomials fp, which are conveniently normalized by xj+p yj−p fp(x, y):= ,p= −j, −j +1,...,j. (j + p)! (j − p)!

We deﬁne the representation πj by the action x x (g, ) −→ g−1 , y y 4.2 The Example G = SU(2) 49 i.e. we put ¯ πj(g)fp(x, y):=fp(¯ax − by, bx + ay), and get by a straightforward computation

j πj(g)fp(x, y)=Σp fp (x, y)App(g) with j+p − − j k (j + p)!(j p)!(j + p )!(j p )! j−p −k j+p−k k k+p −p A (g)= (−1) a a¯ b ¯b p p (j − p − k)!(j + p − k)!k!(k + p − p)! k=0 in particular (2j)! (4.2) Aj (g)= . jp (j + p)!(j − p)!

Exercise 4.2: Verify these formulae (at least) for j =1.

Theorem 4.6: These representations πj,j ∈ (1/2) N0, are unitary, irreducible, and - up to equivalence - there are no other such representations of SU(2).

Proof:i)Unitarity

Wigner’s proof relies on the fact that the polynomials fp are normalized so that we have

j j 1 (| x2 | + | y2 |)2j f¯ f = | x2 |j+p| y2 |j−p= . p p (j + p)!(j − p)! (2j)! p=−j p=−j

By a similar computation, we conclude from the deﬁnition of πj

j j | ax¯ − by |2(j+p)| ¯bx − ay |2(j−p) | π (g)f (x, y) |2 = j p (j + p)!(j − p)! p=−j p=−j (| ax¯ − by |2 + | ¯bx − ay |2)2j (| x |2 + | y |2)2j = = , (2j)! (2j)! where the last equality follows from the unitarity of g. So we have the invariance

2 2 Σp | π(g)fp | =Σp | fp | .

j Substituting here π(g)fp =Σp fp App(g), we get

j j ¯ ¯ ¯ Σp(Σp fp App(g)Σp fp App(g)) = Σpfpfp. 2 ¯ If we know that the (2j +1) functions fp fp are linearly independent, we see that we have j j j j ¯ ¯ t ΣpApp(g)App(g)=ΣpApp(g) App (g)=δp p , i.e. the unitarity of Aj(g). Hence we still have to show the linear independence of the ¯ functions fp fp : We look at a relation j+p j−p j+p j−p cpp x¯ y¯ x y =0,cpp ∈ C. p,p 50 4. Representations of Compact Groups

Then in particular for x ∈ R with q =2j + p + p, the coeﬃcient of xq has to vanish. After division byy ¯jy3j−q this requires for p cp,q−2j−p (y/y¯) =0. p

But this condition implies that cp,q−2j−p = 0 (and thus the linear independence of the ¯ it fp fp ), because we can write y/y¯ = e ,t∈ R, and a relation itp Σcp,q−2j−p e = 0 for all t ∈ R requires that all the coeﬃcients c must vanish. ii) Irreducibility

Wigner’s proof of the irreducibility of πj is by direct application of the criterium which here appears as Corollary to the ﬁnite-dimensional unitary Schur (Theorem 1.3) in Sec- tion 1.5, namely by showing that any matrix M, which commutes with all Aj(g), must necessarily be a constant matrix: We take e−iα/2 g = s(−α/2) = ( ). eiα/2 Then the matrix Aj(g) specializes to the diagonal matrix j ipα − (4.3) App(s( α/2)) = δp pe . It is a standard fact that each matrix M commuting with all such matrices must be diagonal. Just after the introduction of the matrices Aj(g) we remarked (in (9.9)) that no element in the last row of Aj(g) vanishes identically. Then by equating the elements of the j-th row of Aj(g)M and MAj(g) we conclude that j j Ajk(g)Mkk = MjjAjk(g), i.e.Mjj = Mkk, and M is a scalar multiple of the unit matrix.

One can also ﬁnd a proof by application of the criterium given in the Corollary at the end of Section 4.1 using the character of πj. We will see this as a byproduct of the following third part. iii) Completeness

Wigner’s proof that there are - up to equivalence - no other irreducible representations of SU(2) than the πj can be seen as a direct proof of (part of) the Peter–Weyl Theorem from 4.1 in this special case: Characters are class functions. Since each unitary matrix can be diagonalized by conjugation with a unitary matrix, in each conjugacy class of SU(2) we ﬁnd a diagonal matrix of the type eiα s(α)=( ),α∈ [0, 2π). e−iα 1 Conjugation by w =( ) transforms −1 1 ζ−1 −1 ζ ( )( )( )=( ). −1 ζ 1 ζ−1 4.2 The Example G = SU(2) 51

Hence every conjugacy class is exactly represented by a matrix

eiα s(α):=( ),α∈ [0,π), resp.s(α/2),α∈ [0, 2π). e−iα

As we have (see (4.3) in ii))

j ipα − App(s( α/2)) = δp pe its trace is j − ipα χπj (s( α/2)) = e =: ξj(α). p=−j

So we have

ξ0(α)=1, −iα/2 iα/2 ξ1/2(α)=e + e = 2 cos(α/2), −iα iα ξ1(α)=e +1+e = 2 cos α +1,

ξ3/2(α) − ξ1/2(α) = 2 cos(3α/2), etc.

It is now evident that SU(2) can have no other representations than the πj,j ∈ (1/2)N0: For the character of such a representation must, after multiplication by a weighting function, be orthogonal to all ξj, and therefore to ξ0,ξ1/2,ξ1 − ξ0,ξ3/2 − ξ1/2,.... But a function, which is orthogonal to 1, 2 cos(α/2), 2 cos α, 2 cos(3α/2),... in [0, 2π) must vanish according to Fourier’s Theorem.

Exercise 4.3: Verify the irreducibilty of πj using the character criterium (Corollary to Remark 4.2 in Section 4.1).

The explicit determination of the characters χπj = ξj of the irreducible representations πj is very useful if one wants to decompose a given representation π.

Example 4.1: For π = π1 ⊗ π1, by Theorem 1.6 in Section 1.6, we have

2 iα −iα 2 χπ(s(α/2)) = χπ1 (s(α/2)) =(e +1+e )

= ξ2(α)+ξ1(α)+ξ0(α), i.e.

π1 ⊗ π1 = π0 + π1 + π2.

Exercise 4.4: Verify the decomposition ([Wi] p.185)

j+j πj ⊗ πj = π. =|j−j| 52 4. Representations of Compact Groups

4.3 The Example G = SO(3)

One can construct representations of SO(3) by a procedure similar to the one used in the case of SU(2), namely by using homogeneous polynomials in three variables. But as we already remarked in the Example 3.2 in Section 3.2, we do not get irreducible representations without some further considerations. So let us discuss this a bit later and ﬁrst follow a method suggested by H. Weyl deducing the representation theory of SO(3) from the result just obtained in 4.2 and the fact that SU(2) is a double cover of SO(3):

The elements g of SO(3) are real 3×3-matrices consisting of three ON-rows (or columns) (a1,a2,a3), i.e. we have six conditions = δjk for the nine entries in g. Hence we expect three free parameters. It is a classical fact that the elements of SO(3) are parametrized by three Euler angles α, β, γ. Again we follow Wigner’s notation ([Wi] p.152):

Proposition 4.2: For every g ∈ SO(3) there exist angles α, β, γ ∈ R, such that

g = S3(α)S2(β)S3(γ).

3 Here S3(α) and S3(γ) are rotations about the z-axis (or e3 ∈ R ) through α resp. γ and 3 S2(β) is a rotation about the y-axis (or e2 ∈ R ), i.e. we put

⎛ ⎞ cos α − sin α ⎝ ⎠ S3(α):= sin α cos α , 1 ⎛ ⎞ cos α − sin α ⎝ ⎠ S2(α):= 1 , sin α cos α ⎛ ⎞ 1 ⎝ ⎠ S1(α):= cos α − sin α . sin α cos α Exercise 4.5: Prove this.

Remark 4.3: i) There are several notational conventions for the Euler angles in common use. In Hein [He] p.56 we ﬁnd a statement (with proof) analogous to our Proposi- tion 4.2 with S2(β) replaced by S1(β). For a parametrization of SO(3) one needs only α, γ ∈ [0, 2π) and β ∈ [0,π]. This parametrization is not everywhere injective: for β =0 α and γ are only ﬁxed up to their sum. Goldstein [Go] p.145-148 gives an overview of other descriptions. ii) SO(3) is generated by rotations S3(t) and S2(t) or by rotations of type S3(t) and S1(t).

The key fact for our discussion here is given by the following theorem (we shall later see that this is a special case of an analogous statement about SL(2, C) covering the (restricted) Lorentz group SO(3, 1)+): 4.3 The Example G = SO(3) 53

Theorem 4.7: There is a surjective homomorphism

ϕ : SU(2) −→ SO(3) with ker ϕ = {±E2}. This homomorphism can be chosen such that

ϕ(s(α/2)) = S3(α) and ϕ(r(−β/2)) = S2(β).

Proof: As in [Wi] p.158 we consider the Pauli matrices

1 i −1 s := ( ),s := ( ),s := ( ). x 1 y −i z 1

The three-dimensional R-vector space V of hermitian 2 × 2-marices H with trace zero is the linear span of the Pauli matrices

t V := {H ∈ M2(C); H = H,¯ Tr H =0} −zx+ iy = {H =( )=xs + ys + zs ; x, y, z ∈ R}. x − iy z x y z

G = SU(2) acts on V by conjugation, i.e. we have a map

G × V −→ V, (g, H) −→ gHg−1 = gHtg¯ =: H,

ab which for H as above and g =( ) gives by a straightforward computation −¯b a¯

−z x + iy H =( ) x − iy z with

x =(1/2)(a2 +¯a2 − b2 − ¯b2)x +(i/2)(a2 − a¯2 + b2 − ¯b2)y +(¯a¯b + ab)z (4.4) y =(i/2)(¯a2 − a2 + b2 − ¯b2)x +(1/2)(a2 +¯a2 + b2 + ¯b2)y + i(¯a¯b − ab)z z = −(¯ab + a¯b)x + i(¯ab − a¯b)y +(aa¯ − b¯b)z.

E.g. from the fact that the map det is multiplicative, we deduce

det H = x2 + y2 + z2 = det H = x2 + y2 + z2 and if we write (4.4) as a matrix relation with column t(x, y, z)=x

x = A(g)x, we see that multiplication with A(g) respects the euclidean quadratic form, i.e. we have A(g) ∈ O(3). Moreover, by a continuity argument, we have A(g) ∈ SO(3): as every g ∈ SU(2) can be deformed contiuously to g = E2 and det is a continuous function, we cannot have det A(g)=−1. 54 4. Representations of Compact Groups

From the fact that conjugation is a group action, we see that ϕ : g −→ A(g) ∈ SO(3) is a group homomorphism. As ker ϕ = {g ∈ SU(2); gHtg¯ = H for all H ∈ V }, we can deduce easily that ker ϕ = {±E2} (by choosing appropriate matrices H). The fact that ϕ is surjective is more delicate: If g = s(α/2), i.e. a = eiα/2,b= 0, we see from (4.4) that we have A(s(α/2)) = S3(α). If g = r(β/2), we see from (4.4) that we have

A(r(β/2)) = S2(−β).

We already know that SO(3) is generated by matrices of type S2 and S3 and so we are done.

Now we use this result to describe representations of SO(3) :

If a representation π : SU(2) −→ GL(V ) factorizes through SO(3), i.e. one has a homomorphism π : SO(3) −→ GL(V ) with π = π · ϕ resp. a commutative diagram π SU(2) - GL(V ) @

ϕ@ @ π @R SO(3) then, this way, π induces the representation π of SO(3). Obviously this is the case for a representation π with π(−E2)=id, where we get a unique prescription by putting π(A):=π(g)ifA = ϕ(g).

For π(−E2)=−id one gets a double valued representation by putting π(A):=±π(g)ifA = ϕ(g)=ϕ(−g). (This is a special case of a projective representation, which will be discussed below.)

In consequence, we know the unitary dual of SO(3): The representations πj of SU(2) with j ∈ N0 are exactly those with π(−E2)=π(E2)=id and thus induce (up to equivalence) all irreducible unitary representations π of SO(3): If there were any π not in this family, it would composed with ϕ produce a representation of SU(2), which by our completeness result for SU(2) would be equivalent to a πj. j For g ∈ SU(2) and ϕ(g)=g , Wigner ([Wi] p.167) conjugates the π (g ):=πj(g)=A (g) −2k j from 4.2 by the diagonal matrix M =((i) δkl) and calls the result D ({α, β, γ}) with {α, β, γ} representing the Euler angles of g ∈ SO(3) in the parametrization chosen above. Hence we have from the explicit form of the matrices Aj(g) obtained in 4.2 the explicit form − − j k (j + p)!(j p)!(j + p )!(j p )! D ({α, β, γ}) = (−1) p p (j + p − k)!(j + p − k)!k!(k + p − p)! k × eip α(cos(β/2))2j+p+p −2k(sin(β/2))2k+p −p. 4.3 The Example G = SO(3) 55

The characters of these (projective) representations are

j ipα ··· χπ j (S3(α)) = e = 1 + 2 cos α + + 2 cos(jα),jintegral p=−j = 2 cos(α/2) + 2 cos(3α/2) + ···+ 2 cos(jα),jhalf − integral.

The double valued representations π j,j half-integral, led to H. Weyl’s explanation of the spin of an electron. As said above, they are special examples of projective representations, which are very natural for reasons from physics: In quantum theory the state of a system is not so much described by an element v of a Hilbert space H but by an element v∼ of the associated projective space P(H) consisting of the one-dimensional subspaces of H. In general, one writes for a K-vector space P(V ):={v∼; v ∈ V,v =0 } where v∼ is the class of v for the equivalence v ∼ v if there is a λ ∈ K∗ such that v = λv. Having this in mind, it is appropriate to introduce the following.

Deﬁnition 4.1: A projective representation π˜ of G in the K-vector space V is a map

π˜ : G −→ GL(V ), which enjoys the property π˜(gg)=c(g, g)˜π(g)˜π(g), where c : G × G −→ K∗ with c(g, g)c(gg,g)=c(g, gg)c(g,g) for all g, g,g ∈ G. c is also called a system of multipliers.

Remark 4.4: The functional equation for c is equivalent to the associativity

π˜(g(gg)) =π ˜((gg)g).

Exercise 4.6: Prove this.

Every linear representation is also a projective representation. And analogously to the case of linear representations, one also adds a continuity condition and discusses in general these representations.

In the second part of his book [Wi], Wigner describes the application of these results to the theory of spin and atomic spectra. Though we are tempted to do more, from this we only take over that, roughly, an irreducible unitary (projective) representation πj of SO(3) corresponds to a particle having three-dimensional rotational symmetry. The number j specifying the representation then is its angular momentum quantum number or its spin and the indices numbering a basis of the representation space are interpreted as magnetic quantum numbers. We strongly recommend to the reader interested in this topic to look into Wigner’s book or any text from physics. 56 4. Representations of Compact Groups

To finish this chapter, we briefly indicate another approach to the determination of the representations of G = SO(3), which is a first example for the large theory of eigenspace representations promoted in particular by Helgason.

From analysis it is well known that the three-dimensional euclidean Laplace operator

2 2 2 ∆=∂x + ∂y + ∂z is SO(3) invariant, i.e. for f ∈C∞(R3)wehave

(∆f)(gx)=∆f(x), for all g ∈ G, x = t(x, y, z) ∈ R3.

Hence the space V of homogeneous polynomials of degree in three variables, which satisfy the Laplace equation ∆f =0, is invariant under the action f −→ f g with f g(x)=f(g−1x)ofG = SO(3). We denote g by D (g) the matrix form of the representation π of G on V given by π(g)f = f . To solve the Laplace equation, one usually introduces spherical coordinates (r, ϕ, ϑ) with

x = r sin ϑ cos ϕ, y = r sin ϑ sin ϕ, z = r cos ϑ.

Homogeneous polynomials f of degree in x, y, z have the form f = rY (ϑ, ϕ). If this f is introduced into the Laplace equation written in the spherical coordinates

2 2 2 2 2 (∂r +(2/r)∂r +(1/r )(∂ϑ + cot ϑ∂ϑ +(1/(sin ϑ) )∂ϕ))f =0 r drops out and the diﬀerential equation

2 2 2 (( +1)+(∂ϑ + cot ϑ∂ϑ +(1/(sin ϑ) )∂ϕ))Y =0 in the variables (ϑ, ϕ) results. The (2+1) linearly independent solutions Y = Ym (with m = −, − +1,...,,) are known as spherical harmonics of the −th degree. They have the form

Ym(ϑ, ϕ)=Φm(ϕ)Θm(ϑ) where

1 imϕ Φm(ϕ):=√ e , 2π 2 +1( − m)! dm Θ (ϑ):=(−1)m( )1/2(sin ϑ)m P (cos ϑ), m 2 ( + m)! d(cos ϑ)m l − m 2 +1( m)! 1/2 m d Θ − (ϑ):=( ) (sin ϑ) P (cos ϑ), , m 2 ( + m)! d(cos ϑ)m l with m ≥ 0 in the last two equations. The P are the Legendre polynomials deﬁned by

1 d P (x):= (x2 − 1). 2! dx 4.3 The Example G = SO(3) 57

This leads to the matrix form D for π with

π(g)r Ym(ϑ, ϕ)= Ym (ϑ, ϕ)Dmm(g). m=−

As one easily deduces, we have in particular

imα Dmm(S3(α)) = e δmm.

By applying Schur’s commutation criterium, one can prove here, like we did already above, that D is irreducible for all =0, 1, 2,.... The conjugacy classes of SO(3) are parametrized by the angle α, α ∈ [0,π). For the characters we obtain the formula

() imα χ (S3(α)) = e = 1 + 2 cos α + ···+ 2 cos(α). m=−

{ } Again we can see that we got all irreducible representations as cos nϕ n∈N0 is a complete system in L2([0,π)) by Fourier’s Theorem. Chapter 5

Representations of Abelian Groups

We have seen that – finite groups have (up to equivalence) finitely many irreducible representations and these are all finite-dimensional, and – compact groups have (up to equivalence) denumerably many irreducible representations and these are also all finite-dimensional. Abelian topological groups G (with prototype G = R or = C) are something of an extreme in the opposite direction. Here we know that unitary irreducible representations are all one-dimensional (we proved this as a consequence of the “Unitary Schur” in 1.5). But in general the unitary dual is not denumerable. Since in this text we are mainly interested in groups like SL(2, R) and Heis(R), where unitary irreducible representations will turn out to be infinite-dimensional, we discuss here only briefly some notions of general interest, following [BR] p.159 - 165 and [Ki] p.167 - 173.

5.1 Characters and the Pontrjagin Dual

Deﬁnition 5.1: A character of an abelian locally compact group G is a continuous function χ : G −→ C, which satisﬁes | χ(g) | = 1 and χ(gg)=χ(g)χ(g) for all g, g ∈ G, i.e. a character is a (one-dimensional) continuous irreducible unitary representation of G.

Remark 5.1: We already deﬁned the unitary dual Gˆ of any group G as the set of equivalence classes of unitary irreducible representations of G.ForG abelian, Gˆ consists just of all characters of G. It is also called the Pontrjagin dual of G.

Remark 5.2: For G abelian, Gˆ is also a group: We deﬁne the composition of two characters χ and χ by (χχ)(g):=χ(g)χ(g) for all g ∈ G. If G is a topological group, Gˆ is also. More precisely, if G is locally compact (resp. com- ˆ pact, resp. discrete), Gˆ is locally compact (resp. discrete, resp. compact). One has G G.ˆ 60 5. Representations of Abelian Groups

Example 5.1: For G = Rn every character χ has the form

n n χ(x) = exp(i(ˆx1x1 + ···+ˆxnxn)) = exp(i), withx ˆ ∈ R , for x ∈ R , i.e. we have Gˆ = G. Example 5.2: For G = S1 = {ζ ∈ C; | ζ | =1} every character has the form

χ(ζ)=ζk,k∈ Z, as we discussed already in 3.2. I.e., here we have Gˆ = Z. Example 5.3: For G = Z, every character has the form

χ(m)=eimϕ,ϕ∈ [0, 2π), i.e. we have Gˆ = S1 (and, hence, the self-duality of Z).

5.2 Continuous Decompositions

While studying representation theory for compact groups, we generalized the Fourier series 2π f(t)= c(k)eikt,c(k):=(1/(2π)) e−iktf(t)dt k∈Z 0 for G = SO(2) S1 to other compact groups. Now, the analogon of Fourier series for G = R is the Fourier transform associating to a suﬃciently nice function f its Fourier transform √ ∞ F ˆ ikt (5.1) f(k)=f(k):=(1/ 2π) χk(t)f(t)dt, χk(t):=e . −∞ This may be interpreted in the following way: In the case of compact groups we had a (denumerable) direct sum decomposition of the regular representation ρ into irreducible subrepresentations πj ρ = πj. j∈J For G = R we analogously have the regular representation ρ given on H = L2(R,dx)by

ρ(t)f(x)=f(x + t) for all x, t ∈ R.

Suppose H1 would be the space of a one-dimensional subrepresentation. Then we have for every f1 ∈H1

ρ(t)f1(x)=f1(x + t)=λ(t)f1(x) for all x, t ∈ R.

2 Hence f1 has to be an exponential function and we get a contradiction, because L (R) contains no exponential function = 0. But (5.1) indicates that ρ can be decomposed into a direct integral of the irreducible one-dimensional representations χk,k ∈ R. For a proper generalization one here needs more tools from functional analysis, in particular replacement of the usual measure (as recalled in 3.3) by the notion of a spectral measure E(.), i.e. a measure which takes operators as values and allows to decompose (linear) 5.2 Continuous Decompositions 61 operators into a direct integral. Following [BR] p.649 resp. [Ki] p.57, we present them brieﬂy as they are also essential for subsequent chapters.

Spectral Measure

Let [a, b] ⊂ R be a finite or infinite interval and E be a function on [a, b] with values E(λ) for λ ∈ [a, b], which are operators in a Hilbert space H. E is called a spectral function if it satisfies the following conditions: i) E(λ) is self-adjoint for each λ, ii) E(λ)E(µ)=E(min(λ, µ)), iii) The operator function E is strongly right continuous, i.e. one has lim E(λ + )u = E(λ)u for all u ∈H, →0+ iv) E(−∞)=0,E(∞)=Id, i.e. one has lim E(λ)u =0, lim E(λ)u = u for all u ∈H. λ →−∞ λ →∞ Conditions i) and ii) mean that E(λ),λ ∈ [a, b], are (refining Definition 1.7) orthogonal projection operators. For an interval ∆ = [λ1,λ2] ⊂ [a, b] one denotes the difference E(λ2) − E(λ1)byE(∆). For ∆1, ∆2 ⊂ [a, b] one has

E(∆1)E(∆2)=E(∆1 ∩ ∆2), in particular E(∆1)E(∆2)=0if∆1 ∩ ∆2 = ∅, i.e. the subspaces H1 = E(∆1)H and H2 = E(∆2)H are orthogonal.

Example 5.4: We take G = R, H = L2(R) and π(a)u(x):=u(x + a) for u ∈H. Then λ ∞ R λ −→ E(λ):=(2π)−1 dλ e−iλ aπ(a)da −∞ −∞ is a spectral function.

The properties of the spectral function imply that, for any u ∈H, the function

σu(λ):= is a right continuous, non-decreasing function of bounded variation with

2 σu(−∞)=0,σu(∞)= u .

Moreover it is denumerable additive: For a pairwise disjoint decomposition {∆n} with ∪∞ ∆= n=1∆n, one has ∞ ∞ E(∆) = E(∆n) and σu(∆) = σu(∆n). n=1 n=1

The function σu is called the spectral measure. We now can state the fundamental theorem of spectral decomposition theory as follows: 62 5. Representations of Abelian Groups

Theorem 5.1: Every self-adjoint operator A in H has the representation ∞ A = λdE(λ). −∞ Continuous Sums of Hilbert Spaces

The operation of direct sum of Hilbert spaces admits a generalization: Let there be given a set X with measure µ and a family {Hx}x∈X of Hilbert spaces. It is natural to try to deﬁne a new space

H = Hxdµ(x), X the elements of which are to be functions f on X, assuming values in Hx for all x ∈ X, and the scalar product being deﬁned by the formula

:= Hx dµ(x). X The difficulty consists in the fact that the integrand in this expression may be non- measurable. In the case where all of the spaces Hx are separable, one can introduce an appropriate definition of measurable vector-functions f, which will guarantee this measurability of the numerical function x −→ : In the special example that all Hx coincide with a fixed separable H0 with basis {ej}, the numerical functions x −→ are measurable (for more general cases see [Ki] p.57). Once an appropriate concept of measurable vector-functions has been adopted one defines the H H continuous sum (or direct integral ) = X xdµ(x) as the set of equivalence classes of measurable vector-functions f with summable square of norms. H H { } An operator A in = X xdµ(x) is called decomposable iff there exists a family A(x) of operators in the spaces Hx such that

(Af)(x)=A(x)f(x) almost everywhere on X.

By a suitable generalization of the notion of spectral measure to the dual Gˆ (which for an abelian locally compact group G can be given the structure of a topological space), one has the general statement:

Theorem 5.2: Let π be a unitary continuous representation of an abelian locally compact group G in a Hilbert space H. Then there exists a spectral measure E(.)onthe character group Gˆ such that π(g)= χ(g)dE(χ). Gˆ This theorem is called SNAG-theorem, as it goes back to Stone, Naimark, Ambrose, and Godement. One can find a proof in [BR] p.160ff. We also recommend the Chapter on locally compact commutative groups in [Ma] p.37ff, which is centered on the notion of projection valued measure. This is a function E −→ PE associating a projection operator PE in H to each Borel set E from a Borel space S with the properties i) P∅ =0,PS = Id, ii) PE∩F =PEPF for all E and F, iii) PE = j∈N PEj for every disjoint decomposition E = j∈N Ej. Chapter 6

The Inﬁnitesimal Method

Up to now, with exception of the Schrödingerrepresentation of the Heisenberg group, all irreducible representations treated here were finite-dimensional. Before we go further into the construction of infinite-dimensional representations, we discuss a method which allows a linearization of our objects and hence simplifies the task to determine the structure of the representations and classify them. As we shall see, this is helpful for compact and noncompact groups as well. The idea is to associate to the linear topological group G a linear object, its Lie algebra g = Lie G, and study representationsπ ˆ of these algebras, which are open to purely algebraic and combinatorial studies. One can associate to each representation π of G an infinitesimal representation dπ of g. Conversely, one can ask which representationsπ ˆ may be integrated to a unitary representation π of G, i.e., which are of the formπ ˆ = dπ, π ∈ Gˆ. As we will see in several examples, this method allows us to classify the (unitary) irreducible representations and, hence, gives a parametrization of Gˆ. It will further prove to be helpful for the construction of explicit models for representations (π, H).

6.1 Lie Algebras and their Representations

We introduce some algebraic notions, which will become important in the sequel. The interested reader can learn more about these in any text book on Algebra (e.g. Lang’s Algebra [La]).

Deﬁnition 6.1: Let K be a ﬁeld. A K-algebra is a K-vector space A provided with a K-bilinear map (multiplication)

A×A−→A, (x, y) −→ xy.

A is called

– associative iﬀ one has

(xy)z = x(yz) for all x, y, z ∈A,

– commutative iﬀ one has

xy = yx for all x, y ∈A, 64 6. The Inﬁnitesimal Method

–aLie algebra iﬀ the multiplication xy, usually written as xy =: [x, y] (the Lie bracket), is anti-symmetric, i.e.

[x, y]=−[y, x] for all x, y ∈A, and fulﬁlls the Jacobi identity, i.e. one has

[x, [y, z]]+[y, [z,x]]+[z,[x, y]] = 0 for all x, y, z ∈A.

–aJordan algebra iﬀ one has

xy = yx and x2(xy)=x(x2y) for all x, y ∈A,

–aPoisson algebra iﬀ A is a commutative ring with multiplication

A×A−→A, (x, y) −→ xy and one has a bilinear map

A×A−→A, (x, y) −→ { x, y}

(the Poisson bracket) fulﬁlling the identities

{x, y} = −{y, x}, {x, yz} = {x, y}z + y{x, z}, {x, {y, z}} = {{x, y},z} + {y, {x, z}} for all x, y, z ∈A.

The dimension of A as a K-vector space is called the dimension of the algebra.

A map ϕ : A−→A is an algebra homomorphism iﬀ it is a K-linear map and respects the respective composition, i.e. if one has

ϕ(xy)=ϕ(x)ϕ(y) for all x, y ∈A.

Exercise 6.1: Show that C∞(R2n) is a Poisson algebra with

n ∂f ∂g ∂f ∂g {f,g} := ( − ) for all f,g ∈C∞(R2n), ∂q ∂p ∂p ∂q i=1 i i i i

2n where the coordinates are denoted by (q1,...,qn,p1,...,pn) ∈ R .

Exercise 6.2: Verify that each Poisson algebra is also a Lie algebra.

In the following, we are mainly interested in Lie algebras over K = R or K = C, which we usually denote by the letter g . We recommend as background any text book on this topic, for instance Humphreys’ Introduction to Lie Algebras and Representation Theory [Hu].

Example 6.1: Every Mn(K) is an associative noncommutative algebra with composition XY of two elements X, Y ∈ Mn(K) given by matrix multiplication. 6.1 Lie Algebras and their Representations 65

Example 6.2: Every associative algebra A is a Lie algebra with

[X, Y ]=XY − YX for all X, Y ∈A, the commutator of X and Y . In particular, one denotes

gl(n, K):=Mn(K).

Example 6.3: If V is any K-vector space, the space End V of linear maps F from V to V is a Lie algebra with

[F, G]:=FG− GF for all F, G ∈ End V.

Example 6.4: If g is a Lie algebra and g0 is a subspace with [X, Y ] ∈ g0 for all X, Y ∈ g0, g0 is again a Lie algebra. One easily veriﬁes this and that

sl(n, K):={X ∈ gl(n, K); Tr X =0}, so(n):={X ∈ gl(n, K); X = −tX}, su(n):={X ∈ gl(n, K); X = −tX,¯ Tr X =0} are examples of Lie algebras.

Example 6.5: If A is a K-algebra,

DerA := {D : A−→A K − linear; D(ab)=aDb+ Da b for all a, b ∈A} is a Lie algebra, the algebra of derivations.

Exercise 6.3: Verify that the matrices

1 1 F := ( ),G:= ( ),H:= ( ) 1 −1 are a basis of sl2(K) with the relations

[H, F]=2F, [H, G]=−2G, [F, G]=H.

Exercise 6.4: Verify that the matrices

−i −1 i X := (1/2)( ),X := (1/2)( ),X := (1/2)( ) 1 −i 2 1 3 −i are a basis of su(2) with the relations

[Xi,Xj]=−Xk, (i, j, k)=(1, 2, 3), (2, 3, 1), (3, 1, 2).

Exercise 6.5: Verify that the matrices ⎛ ⎞ ⎛ ⎞ ⎛ ⎞ 0 −1 −1 ⎝ ⎠ ⎝ ⎠ ⎝ ⎠ X1 := −1 ,X2 := 0 ,X3 := 1 1 1 0 66 6. The Inﬁnitesimal Method are a basis of so(3) with the relations

[Xi,Xj]=−Xk, (i, j, k)=(1, 2, 3), (2, 3, 1), (3, 1, 2).

Exercise 6.6: Show that g = R3 provided with the usual vector product as composition is a Lie algebra isomorphic to so(3).

Exercise 6.7: Show that g = R3 provided with the composition

((p, q, r), (p,q,r)) −→ (0, 0, 2(pq − pq)) is a Lie algebra, the Heisenberg algebra heis(R), with basis

P =(1, 0, 0),Q=(0, 1, 0),R=(0, 0, 1) and relations [R, P ]=[R, Q]=0, [P, Q]=2R. Remark 6.1: It will soon be clear that gl(n, R), sl(n, R), so(n), su(n), and heis(R) are associated to the groups GL(n, R), SL(n, R), SO(n), SU(n), resp. Heis(R).

In analogy with the concept of the linear representation of a group G in a vector space V , one introduces the following notion.

Deﬁnition 6.2: A representation of the Lie algebra g in V is a Lie algebra homorphism πˆ : g −→ End V , i.e.π ˆ is K-linear and one has

πˆ([X, Y ]) = [ˆπ(X), πˆ(Y )] for all X, Y ∈ g.

Here we have no topology and, hence, no continuity condition.

Each Lie algebra has a trivial representation πˆ =ˆπ0 withπ ˆ0(X) = 0 for all X ∈ g and an adjoint representation πˆ = ad given by

ad X(Y ):=[X, Y ] for all X, Y ∈ g.

Exercise 6.8: Verify this.

In analogy with the respective notions for group representations, we have the following concepts. –(ˆπ0,V0)isasubrepresentation of (ˆπ, V )iﬀV0 ⊂ V is aπ ˆ-invariant subspace, i.e.π ˆ(X)v0 ∈ V0 for all v0 ∈ V0, andπ ˆ0 =ˆπ |V . –ˆπ is irreducible iﬀπ ˆ has no nontrivial subrepresentation. Ifπ ˆ1 andπ ˆ2 are representations of g we have – the direct sum πˆ1 ⊕ πˆ2 as the representation on V1 ⊕ V2 given by

(ˆπ1 ⊕ πˆ2)(X)(v1,v2):=(ˆπ1(X)v1, πˆ2(X)v2) for all v1 ∈ V1,v2 ∈ V2, and – the tensor product πˆ1 ⊗ πˆ2 as the representation on V1 ⊗ V2 given by

(ˆπ1 ⊗ πˆ2)(X)(v1 ⊗ v2):=ˆπ1(X)v1 ⊗ v2 + v1 ⊗ πˆ2(X)v2 for all v1 ∈ V1,v2 ∈ V2. 6.2 The Lie Algebra of a Linear Group 67

Exercise 6.9: Verify that these are in fact representations.

Exercise 6.10: Find the prescription to deﬁne a contragredient representation (ˆπ∗,V∗) to a given representation (ˆπ, V )ofg.

There is a lot of material about the structure and representations of Lie algebras, much of it going back to Elie and Henri Cartan. As already indicated, a good reference for this is the book by Humphreys [Hu]. We will later need some of this and develop parts of it. Here we only mention that one comes up with facts like this: If g is semisimple (i.e. has no nontrivial ideals), then each representationπ ˆ of g is completely reducible. Before we discuss more examples, we establish the relation between linear groups and their Lie algebras.

6.2 The Lie Algebra of a Linear Group

In Section 3.1 we introduced the concept of a linear group G as a closed subgroup of a matrix group GL(n, R)orGL(n, C) for a suitable n ∈ N. To such a group we associate a Lie algebra g = Lie G using as essential tool the exponential function for matrices.We rely again on some experience from calculus in several complex variables and we deﬁne the exponential function exp by ∞ k exp X := (1/k!)X for all X ∈ Mn(C). k=0 Remark 6.2: One has the following facts:

1. exp X is absolutely convergent for each X ∈ Mn(C). 2. exp X : Mn(C) −→ Mn(C) is continuously diﬀerentiable. 3. There is an open neighbourhood U of 0 ∈ Mn(R), which is diﬀeomorphic to an open neighbourhood V of E ∈ GL(n, R). 4. log X := Σ((−1)k/k)(X − E)k converges for X − E < 1 and we have

exp log X = X for X − E < 1, log exp X = X for exp X − E < 1. 5. One has

i) exp(X + Y ) = exp X exp Y for commuting X, Y ∈ Mn(C), −1 ii) (exp X) = exp(−X) for X ∈ Mn(C), −1 −1 iii) A exp XA = exp(AXA ) for X ∈ Mn(C),A∈ GL(n, C), iv) det(exp X) = exp Tr X for X ∈ Mn(C). The concientious reader should prove all this as Exercise 6.11. She/he should remember that we introduced in (3.1) in Section 3.1

t ¯ 1/2 =ReTr XY, X = for all X, Y ∈ Mn(C), and that one has the fundamental inequalities

X + Y ≤ X + Y , the triangle inequality, ||≤ X Y , the Cauchy Schwarz inequality. 68 6. The Inﬁnitesimal Method

Everyone interested in concrete examples could do the following Exercise 6.12 preparing the ground for the next deﬁnition: Verify 1 t 1 exp(tX)= , for X = , 1 cos t sin t 1 exp(tX)= − , for X = − , sin t cos t 1 et 1 exp(tX)= −t , for X = − , ⎛ e ⎞ ⎛ 1 ⎞ cos t sin t 1 exp(tX)=⎝ − sin t cos t ⎠ , for X = ⎝ −1 ⎠ . 1 0 Exercise 6.13: Verify that one has the following relations {exp(tX),X ∈ so(3); t ∈ R} = SO(3), {exp(tX),X ∈ su(2); t ∈ R} = SU(2), {exp(tX),X ∈ sl(2, R); t ∈ R} = SL(2, R). This exercise shows that there are groups, which are examples for groups of exponential type, i.e. images under the expontial map of their Lie algebras. For G = SL(2, R) one has {exp(tX),X ∈ sl(2, R),t∈ R} = SL(2, R), but the matrices exp(tX),X ∈ sl(2, R),t ∈ R, generate the identity component of G = SL(2, R).

We take this as motivation to associate a Lie algebra to each linear group by a kind of inversion of the exponential map:

Deﬁnition 6.3: Let G be a linear group contained in GL(n, K), K = R or K = C. Then we denote

g = Lie G := {X ∈ Mn(K); exp(tX) ∈ G for all t ∈ R}. Theorem 6.1: g is a real Lie algebra.

A hurried or less curious reader may simply believe this and waive the proof. (See for instance [He] p.114: it is, in principle, not too diﬃcult but lengthy. It shows why (in our approach) we had to restrict ourselves to closed matrix groups.) We emphasize that it is essentially the heart of the whole theory that not only a linear group but also every Lie group has an associated Lie algebra. This is based on another important notion, which we now introduce because it gives us a tool to determine explicitly the Lie algebras for those groups we are particularly interested in (even if we we have skipped the proof of the central theorem above).

Deﬁnition 6.4: A one-parameter subgroup γ of a topological group G is a continuous homomorphism γ : R −→ G, t −→ γ(t).

This notation is a bit misleading: to be precise, G0 = γ(R) is indeed a subgroup of G. γ is often abbreviated as a 1-PUG (from the German term “Untergruppe”). 6.2 The Lie Algebra of a Linear Group 69

Remark 6.3: Each X ∈ g produces a 1-PUG γ, namely the subgroup given by γ(t) := exp(tX). This is obvious since exp(tX) is continuous (and even diﬀerentiable) in t and fulﬁlls the functional equation i) in part 5 of the Remark 6.2 above.

If the 1-PUG γ = γ(t) is differentiable in t, we assign to it an infinitesimal generator X = Xγ by d X := γ(t) | . dt t=0 If γ is given as γ(t) = exp(tX), we have X as infinitesimal generator.

This suggests the following procedure to determine the Lie algebra g of a given matrix group G (which is also the background of our Exercise 6.13 above): Look for “sufficiently many independent” one-parameter subgroups γ1,...,γd and determine their generators X1,...,Xd. Then g will come out as the R-vector space generated by these matrices Xi. Obviously, one needs some explanation: – Two 1-PUGs γ and γ are independent iff their infinitesimal generators are linearly independent over R as matrices in Mn(C). – The number d of the sufficiently many independent (γi) is just the dimension of our group G as a topological manifold: Up to now we could work without this notion, but later on we will even have to consider differentiable manifolds. Therefore, let us recall (or introduce) that the notion of a real manifold M of dimension d includes the condition that each point m ∈ M has a neighbourhood which is homeomorphic to an open set in Rd. This amounts to the practical recipe that the dimension of a group G is the number of real parameters needed to describe the elements of G. For instance for G = SU(2) and SO(3) we have as parameters the three Euler angles α, β, γ, which we introduced in 4.3 and 4.2. Hence, here we have d = 3, as also in the next example.

Example 6.6: For G = SL(2, R) we take as a standard parametrization (a special instance of the important Iwasawa decomposition) ab (6.1) SL(2, R) g =( )=n(x)t(y)r(ϑ) cd with 1 x y1/2 cos ϑ sin ϑ n(x):=( ),t(y):=( ),r(ϑ):=( ), 1 y−1/2 − sin ϑ cos ϑ where x ∈ R,y >0,ϑ∈ [0, 2π). Therefore, we take 1 t et γ (t):=( ),γ(t):=( ),γ(t):=r(t), 1 1 2 e−t 3 and get by diﬀerentiating and then putting t =0 1 1 1 X =( ),X =( ),X =( ). 1 2 −1 3 −1 Obviously, we have with the notation introduced in Exercise 6.3 in 6.1

Lie SL(2, R)= = = sl2(R). 70 6. The Inﬁnitesimal Method

Example 6.7: For ⎛ ⎞ 1 xz G = Heis(R):={⎝ 1 y ⎠ ; x, y, z ∈ R} 1 we take ⎛ ⎞ ⎛ ⎞ ⎛ ⎞ 1 t 1 1 t ⎝ ⎠ ⎝ ⎠ ⎝ ⎠ γ1(t):= 1 ,γ2(t):= 1 t ,γ3(t):= 1 , 1 1 1 and get ⎛ ⎞ ⎛ ⎞ ⎛ ⎞ 1 1 ⎝ ⎠ ⎝ ⎠ ⎝ ⎠ X1 = ,X2 = 1 ,X3 = .

By an easy computation of the commutators we get the Heisenberg commutation relations

[X1,X2]=X3, [X1,X3]=[X2,X3]=0. Exercise 6.14: Repeat this for G = Heis(R) as a subgroup of GL(4, R).

Exercise 6.15: Determine Lie G for G = {g = n(x)t(y); x ∈ R,y > 0}.

6.3 Derived Representations

Now that we have associated a Lie algebra g = Lie G to a given linear group (using a diﬀerentiation procedure), we want to associate an algebra representationπ ˆ of g to a given representation (π, H)ofG. Here we will have to use diﬀerentiation again. To make this possible, we have to modify the notion of a group representation:

Deﬁnition 6.5: Let (π, H) be a continuous representation of a linear group G. Then its associated smooth representation (π, H∞) is the restriction of π to the space H∞ of smooth vectors, i.e. those v ∈Hfor which the map

G g −→ π(g)v ∈H is diﬀerentiable.

This definition makes sense if we accept that a linear group is a differentiable manifold and extend the notion of differentiability to vector valued functions. In our examples things are fairly easy as we have functions depending on the parameters of the group where differentiability is not difficult to check.

Deﬁnition 6.6: Let (π, H) be a continuous representation of a linear group G. Then its associated derived representation is the representation dπ of g given on the space H∞ of smooth vectors by the prescription

d dπ(X)v := π(exp(tX))v | for all v ∈H∞. dt t=0 6.3 Derived Representations 71

If it is clear, which representation π is meant, one abbreviates Xvˆ := dπ(X)v. It is not diﬃcult to verify that dπ(X) stays in H∞ and rather obvious that dπ(X)is linear. But there is some more trouble to prove (see [Kn] p.53, [BR] p.320, or [FH] p.16) that one has [dπ(X),dπ(Y )] = dπ([X, Y ]) for all X, Y ∈ g.

Example 6.8: We take G = SU(2) and π = π1 given by

π(g)f(x, y):=f(¯ax − by, ¯bx + ay) for (1) f ∈ V = with √ √ 2 2 f−1(x, y)=(1/ 2)y ,f0(x, y)=xy, f1(x, y)=(1/ 2)x . As we did in Exercise 6.4 in 6.1, we take as basis of su(2)

−i −1 i X := (1/2)( ),X := (1/2)( ),X := (1/2)( ). 1 −i 2 1 3 −i

Then we have the one-parameter subgroups

cos (t/2) −i sin (t/2) exp(tX )=( ), 1 −i sin (t/2) cos (t/2) cos (t/2) − sin (t/2) exp(tX )=( ), 2 sin (t/2) cos (t/2) eit/2 exp(tX )=( ). 3 e−it/2

It is clear that V (1) consists of smooth vectors and hence we can compute

d Xˆ f− = π(exp tX )f− | 1 1 dt √ 1 1 t=0 = d (1/ 2)(i sin(t/2)x + cos(t/2)y)2 | dt √ √ t=0 =(i/ 2)xy =(i/ 2)f0, and ˆ d | X1f0 = dt π(exp tX1)f0 t=0 = d (cos(t/2)x + i sin(t/2)y)(i sin(t/2)x + cos(t/2)y) | dt √ t=0 2 2 =(i/2)(x + y )=(i/ 2)(f−1 + f1), Xˆ f = d π(exp tX )f | 1 1 dt √ 1 1 t=0 = d (1/ 2)(cos(t/2)x + i sin(t/2)y)2 | dt √ √ t=0 =(i/ 2)xy =(i/ 2)f0. Similarly we get √ √ √ Xˆ2f−1 = −(1/ 2)f0, Xˆ2f0 =(1/ 2)(f−1 − f1), Xˆ2f1 =(1/ 2)f0, Xˆ3f−1 = if−1, Xˆ3f0 =0, Xˆ3f1 = −if−1. 72 6. The Inﬁnitesimal Method

Exercise 6.16: i) Verify this and show that for √ X0 := iX3,X± := (1/ 2)(−iX1 ∓ X2) one has Xˆ0fp = pfp, Xˆ±fp = fp±1,p= −1, 0, 1(f2 = f−2 =0). ii) Do the same for π = πj,j >1.

Example 6.9: We take the Heisenberg group G = Heis(R) ⊂ GL(4, R) and the Schr¨odingerrepresentation π = πm,m∈ R \{0}, given by

π(g)f(x)=em(κ +(2x + λ)µ)f(x + λ) with g =(λ, µ, κ) ∈ G, x ∈ R,f ∈H= L2(R),em(u):=e2πimu. Here H∞ is the (dense) subspace of diﬀerentiable functions in L2(R). We realize the Lie algebra g = heis(R) by the space spanned by the four-by-four matrices ⎛ ⎞ ⎛ ⎞ ⎛ ⎞ 1 ⎜ 1 ⎟ ⎜ 1 ⎟ ⎜ 1 ⎟ P = ⎜ ⎟ ,Q= ⎜ ⎟ ,R= ⎜ ⎟ . ⎝ −1 ⎠ ⎝ ⎠ ⎝ ⎠

Then we have the Heisenberg commutation relations

[P, Q]=2R, [R, P ]=[R, Q]=0 and (using again the notation introduced in 0.1)

exp tP =(t, 0, 0), exp tQ =(0,t,0), exp tR =(0, 0,t).

For diﬀerentiable f we compute

ˆ d | d | Pf = dt π(t, 0, 0)f t=0 = dt f(x + t) t=0 = ∂xf, ˆ d | d m | Qf = dt π(0,t,0)f t=0 = dt e (2xt)f(x) t=0 =4πimxf(x), ˆ d | d m | Rf = dt π(0, 0,t)f t=0 = dt e (t)f(x) t=0 =2πimf(x), i.e. we have Pˆ = ∂x, Qˆ =4πimx, Rˆ =2πim. Exercise 6.17: i) Verify that these operators really fulﬁll the same commutation relations as P, Q, R. −πmx2 ii) Take f0(x):=e and

Yˆ0 := −iR,ˆ Yˆ± := (1/2)(Pˆ ± iQˆ) and determine ˆ p ˆ ˆ fp := Y+f0, Y−fp and Y0fp for p =0, 1, 2,.... These Examples show how to associate a Lie algebra representation to a given group representation. Now the following question arises naturally: Is there always a nontrivial subspace H∞ of smooth vectors in a given representation space H? We would even want 6.4 Unitarily Integrable Representations of sl(2, R)73 that it is a dense subspace. The answer to this question is positive. This is easy in our examples but rather delicate in general and one has to rely on work by Garding, Nelson and Harish Chandra. It is not very surprising that equivalent group representations are inﬁnitesimally equivalent, i.e. their derived representations are equivalent algebra representations. The converse is not true in general. But as stated in [Kn] p.209, at least for semisimple groups one has that inﬁnitesimally equivalent irreducible representations are unitarily equivalent. Taking these facts for granted, a discussion of the unitary irreducible representations of a given linear group G can proceed as follows: We try to classify all irreducible representations of the Lie algebra g = Lie G of G and determine those, which can be realized by derivation from a unitary representation of the group. In general, this demands for some more information about the structure of the Lie algebra at hand. Before going a bit into this structure theory, we discuss in extenso three fundamental examples whose understanding prepares the ground for the more general cases.

6.4 Unitarily Integrable Representations of sl(2, R)

A basic reference for this example is the book “SL(2, R)” [La1] by Serge Lang but in some way or other our discussion can be found in nearly any text treating the representation theory of semisimple algebras or groups. As we already know from Exercise 6.3 in 6.1 and Example 6.5 in 6.2, g = sl(2, R) is the Lie algebra of G = SL(2, R) and we have

g = sl(2, R)={X ∈ M2(R); Tr X =0} = with 1 1 F =( ),G=( ),H=( ) 1 −1 and the relations [H, F]=2F, [H, G]=−2G, [F, G]=H. The exercises at the end of the examples in the previous section should give a feeling that it is convenient to complexify the Lie algebra, i.e. here we go over to

gc := g ⊗R C = C = C where 1 ±i −i (6.2) X± := (1/2)(H ± i(F + G)) = (1/2)( ),Z:= −i(F − G)=( ) ±i −1 i with [Z, X±]=±2X±, [X+,X−]=Z.

This normalization will soon prove to be adequate and the operators X± will get a meaning as ladder operators: We consider a representationπ ˆ of gc = on a C-vector space V =i∈I where we abbreviate

Xv :=π ˆ(X)v for all v ∈ V,X ∈ gc.

We are looking for representations (ˆπ, V )ofgc, which may be integrated, i.e. which are complexiﬁcations of derived representations dπ for a representation π of SL(2, R). 74 6. The Inﬁnitesimal Method

This has the following consequence: π|K is a representation of K := SO(2). From Example 3.1 in 3.2 we know that all irreducible unitary representations of SO(2) are given by ikϑ χk(r(ϑ)) = e ,k∈ Z, hence one has the derived representation dχk with 1 dχ (Y )=ik for Y =( )=iZ. k −1 This motivates the idea to look for each k ∈ Z at the subspace of V consisting of the eigenvectors with eigenvalue k with respect to the action of Z, namely

Vk := {v ∈ V ; Zv = kv}.

This space Vk is called a K-isotypic subspace of V . Each v ∈ Vk is said to have weight k. Now we can explain the meaning of the term ladder operator: The isotypic spaces Vk may be seen as rungs of a ladder numerated by the weight k and the operators X± change the weights in a controlled way:

Remark 6.4: We have X±Vk ⊂ Vk±2. Proof: For v ∈ Vk we put v := X±v and check that v is again a Z-eigenvector of weight k ± 2: Zv = ZX±v =(X±Z ± 2X±)v = X±(k ± 2)v =(k ± 2)v .

Here we used the Lie algebra relation [Z, X±]=±2X±, which for the operators acting on V is transformed into the relation

ZX± − X±Z = ±2X±. This remark has immediate consequences: i) If V =ΣVk is the space of an irreducible gc-representation, there are only nontrivial Vk where k is throughout odd or even. { } ≥ ≤ ii) If there is a Vk0 = 0 , then all Vk with k k0 or all Vk with k k0 have to be zero. Hence, there can be only four different configurations for our irreducible gc-representations. The nontrivial weights k, i.e. those for which Vk = {0}, compose – case 1: a finite interval n { ≤ ≤ ≡ ≡ } Im := m k n; m k n mod 2 of even or odd integers, – case 2: a half line with an upper bound In := {k ≤ n; k ≡ n mod 2} consisting of even or odd integers, – case 3: a half line with lower bound

Im := {m ≤ k; m ≡ k mod 2} consisting of even or odd integers, – case 4: a full chain of even or odd integers

I+ := {k ∈ 2Z} or I− := {k ∈ 2Z +1}. 6.4 Unitarily Integrable Representations of sl(2, R)75

ss s s s s s s s s s s s s s s s s s 6 case 3 X X− + s ? s s s s s s s s s s case 1 case 2 case 4 s s s s s s

It is quite natural to call an element 0 = v ∈ Vk with k = n in conﬁguration 1 or 2 a vector of highest weight and similarly for k = m in conﬁguration 1 or3avector of lowest weight.

The following statement oﬀers the decisive point for the reasoning that we really get all representations by our procedure. It is a special case of a more general theorem by Godement on spherical functions. It also is a ﬁrst example for the famous and important multiplicity-one statements.

Lemma: Ifπ ˆ is irreducible, we have dim Vk ≤ 1.

For a proof we refer to Lang [La1] p.24 or the article by van Dijk [vD].

We look at the conﬁgurations 1 or 2: Let 0 = v ∈ Vn be a vector of highest weight, i.e. with

Zvn = nvn,X+vn =0, and put

k vn−2k := X−vn. 76 6. The Inﬁnitesimal Method

As we want an irreducible representation and we have accepted to take the muliplicity- one statement from the Lemma for granted, we can take this vn−2k as a generator for Vn−2k with vn−2k = 0 for all k ∈ N0 in configuration 1 and all k with n − 2k ≥ m in configuration 2. We check what X+ does to vn−2k: Remark 6.5: For k as fixed above, we have

X+vn−2k = akvn−2k+2 with ak := kn − k(k − 1).

Proof: Since X+vn = 0, we know that a0 = 0. As in the proof of Remark 6.4, for the operators realizing the commutation relation [X+,X−]=Z we calculate

X+vn−2 = X+X−vn =(X−X+ + Z)vn = nvn.

Hence we get a1 = n. Then we verify by induction

X+vn−2(k+1) = X+X−vn−2k =(X−X+ + Z)vn−2k = X−((kn − k(k − 1))vn−2k+2 +(n − 2k)vn−2k =((k +1)n − k(k + 1))vn−2k = ak+1vn−2k.

We see that the irreducibility ofπ ˆ forces ak to be nonzero for all k ∈ N in configuration 2 and for 2k ≤ n − m in configuration 1. Since ak =0iffn − (k − 1) = 0, we conclude that n has to be negative in case 2 and m = −n with n ∈ N in case 1. Just as well, we can start to treat the configurations 1 and 3 with a vector 0 = vm ∈ Vm k of lowest weight. Here we put vm+2k := X+vm. In parallel to Remark 6.5 we get:

Remark 6.6: For k ∈ N0 in conﬁguration 3 and m +2k ≤ n in conﬁguration 1, we have

X−vm+2k = bkvm+2k−2 with bk = −(km + k(k − 1)). Exercise 6.18: Prove this.

As above, we see that we have m = −n with negative n in configuration 1 and that n has to be positive in configuration 3. For instance, the following are possible configurations.

s s 6 s 4 ss n =2 m =2 s 6 X X− + s ? case 2 case 3 s

m = −2 s n = −2 s −4 s s case 1 −6 s 6.4 Unitarily Integrable Representations of sl(2, R)77

In conﬁguration 4 we have no distinguished vector of highest or lowest weight. In this case we choose for each even (or odd) integer n a nonzero vn ∈ Vn, i.e. with Zvn = nvn, such that they are related as follows:

Remark 6.7: For all even (resp. odd) n ∈ Z we have

± ± ± X±vn = an vn±2 with an =(1/2)(s +1 n), ∈ ± ∈ ∈ ∈ where s C, such that an = 0 for all n 2Z (resp. 2Z + 1), i.e. s 2Z + 1 for n 2Z and s ∈ 2Z for n ∈ 2Z +1.

Proof: From irreducibility and multiplicity-one we conclude that each X±vn has to be ± a nonzero multiple of vn±2. We verify that the choice of an is consistent with the commutation relations:

− ± − ± [Z, X±]vn =(ZX± X±Z)vn = Zan vn±2 an nvn±2 ± ± = 2an vn±2 = ±2X±vn, and [X+,X−]vn = Zvn = nvn. Verify this as Exercise 6.19.

± ∓ The conditions for s are obvious since we have an =0iﬀs +1= n.

We chose a symmetric procedure for our conﬁguration 4. But as well we could have ∈ j j ∈ chosen any nonzero vn Vn, and then vn+2j := X+vn,vn−2j := X−vn for j N would constitute an equivalent representation. Exercise 6.20: Check this.

We denote the representations obtained above as follows

–ˆπs,+ orπ ˆs,− for the even resp. odd case in configuration 4 with s =2 Z + 1 forπ ˆs,+ and s =2 Z forπ ˆs,−,s∈ C, + ∈ –ˆπn for configuration 3 with lowest weight n, n N0, − − ∈ –ˆπn for configuration 2 with highest weight n, n N0, –ˆσn for configuration 1 with highest weight n and lowest weight −n, n ∈ N0.

Remark 6.8: If inπ ˆs,+ andπ ˆs,− we do not exclude the values of s as above, we get all representations as sub - resp. quotient representations of theseπ ˆs,± .

Proposition 6.1: We have the following equivalences

πˆs,+ ∼ πˆ−s,+ andπ ˆs,− ∼ πˆ−s,−.

All other representations listed above are inequivalent.

Proof: i) Let V = and V = be representation spaces forπ ˆs,+ resp.π ˆs,+ and F : V −→ V an intertwining operator. Because of Zvn = nvn, we have for each n

ZFvn = FZvn = nF vn. 78 6. The Inﬁnitesimal Method

∈ ∈ Hence one has Fvn Vn, i.e. Fvn = bnvn with bn C. Moreover, from

X±Fvn = FX±vn we deduce ± ± (s +1 n)bnvn±2 = bn±2(s +1 n)vn±2, i.e. bn/bn−2 =(s +1− n)/(s +1− n) and bn+2/bn =(s +1+n)/(s +1+n) resp. bn/bn−2 =(s + n − 1)/(s + n − 1). Now we see that we have

(s +1− n)(s + n − 1) = (s + n − 1)(s +1− n), i. e.s = ±s.

The same conclusions can be made forπ ˆs,−. ii) The inequivalence of theσ ˆn’s among themselves and with respect to all the other representations is clear because the dimensions are diﬀerent. The other inequivalences have to be checked individually. As an example let V = and V = be + −→ spaces of representationsπ ˆk andπ ˆs,+ and F : V V an intertwining operator. As ≥ ∈ above, for k n we have the relation Fvn = bnvn with bn C. Hence F is not surjective and both representations are not equivalent. The same reasoning goes through in all the other cases. Exercise 6.21: Realize this.

Classiﬁcation of the Unitarily Integrable Representations of sl(2, R)

At ﬁrst, we derive a necessary condition for a general representationπ ˆ of a Lie algebra g = Lie G to be integrable to a unitary representation π of G.

Proposition 6.2: If (π, H) is a unitary representation of G and X ∈ g = Lie G, the operator Xˆ := dπ(X)onH∞ is skew-Hermitian, i.e. one has Xˆ = −Xˆ ∗.

Proof: For v, w ∈H∞, unitarity of π implies

= <π(g−1)v, w > for all g ∈ G and hence, for t ∈ R,t=0 , π(exp(tX))w − w π(exp(−tX))v − v = . t −t Recalling the deﬁnition d π(exp tX)v − v dπ(X)v := π(exp(tX))v |t=0= lim , dt t→0 t we get from the last equation in the limit t −→ 0

= − . 6.4 Unitarily Integrable Representations of sl(2, R)79

Therefore we have = − , i.e. Xˆ = dπ(X) is skew-Hermitian.

Now, we apply this to g = sl(2, R). The Proposition leads to the condition:

∗ Remark 6.9: We have dπ(X±) = −dπ(X∓).

Proof: Using the notation from the beginning of this section we have 1 ±i X± =(1/2)(H ± i(F + G)) = (1/2)( )=:(X ± iX ) ±i −1 1 2 with X1 = H, X2 = F + G ∈ sl(2, R). Hence, the skew-Hermiticity of Xî leads to ˆ ˆ ∗ ˆ ˆ (X1 ± iX2) = −(X1 ∓ iX2). Now, if we ask for unitarity, we have to check whether we find a scalar product <, > defined on the representation space V such that we get

< Xˆ±v, w > = − for all v, w ∈ V.

For V = j∈J this condition implies

(6.3) < Xˆ+vj,vj+2 > = − for all j ∈ J. i) In conﬁguration 1 and 2 we have

Xˆ−vj+2 = vj and from Remark 6.5 for j = n − 2k

Xˆ+vj = akvj+2 with ak = k(n − k +1), i.e. 2 2 ak vj+2 = − vj .

An equation like this is only possible for ak < 0, which is true for the ak in conﬁguration 2 but not for conﬁguration 1, where we have seen that all ak are positive.

Similarly Remark 6.6 shows that in conﬁguration 3 the condition (6.3) is no obstacle.

Exercise 6.22: Verify this. ii) Conﬁguration 4 is more delicate: From Remark 6.7 we have ˆ ± ± ± X±vj = aj vj±2 with aj =(1/2)(s +1 j), i.e. the condition (6.3) has the form

2 2 (6.4) (¯s +1+j) vj+2 = −(s − 1 − j) vj .

At ﬁrst we treat the even case V =j∈2Z. Condition (6.4) requires for j =0

2 2 (¯s +1) v2 = −(s − 1) v0 . 80 6. The Inﬁnitesimal Method

As the norm has to be positive, a relation like this is only possible if one has

(1 − s)/(¯s +1)> 0, i. e. (1 − s)(1 + s) > 0.

We put s = σ + iτ, σ, τ ∈ R, and get the condition

s2 = σ2 − τ 2 +2στi < 1, i.e. unitarity demands for

σ = 0 and τ ∈ R arbitrary or

τ = 0 and σ2 < 1.

In the odd case V = j∈2Z+1 we get for j = −1 from (j )

2 2 s¯ v1 = −s v−1 .

This is possible only for

−s/s¯ > 0, i. e. s¯2 = σ2 − τ 2 − 2στi < 0.

Therefore, here we have only one possibility, namely

σ = 0 and τ ∈ R.

Putting all this together, we have proved the following statement.

Theorem 6.2: Unitary irreducible representations π of G = SL(2, R) can exist only if their derived representationπ ˆ = dπ is among the following list: –ˆπ the trivial representation, ± ∈ –ˆπ =ˆπk ,k N, –ˆπ =ˆπis,±,s∈ R (s = 0 forπ ˆis,−) 2 –ˆπ =ˆπs,s∈ R, 0

In the next chapter we will prove that, in fact, all these representationsπ ˆ can be integrated to unitary representations π of G. They get the following names ∓ ∓ – π = πk discrete series representation of highest (lowest) weight k, – π = πis,± even (odd ) principal series representation, – π = πs complementary (or supplementary) series representation.

~~Because of the equivalences stated in Proposition 6.1, we have for the unitary dual of G = SL(2, R) ˆ { ± ∈ ∈ } G = π0; πk ,k N; πis,±,s R>0; πs, 0~~

~~πis,+~~

~~π− π+ k π k s sssssss s sssssss~~

~~−3 −2 −1 123~~

~~πis,−~~

~~We can observe a very nice fact, which will become important later and is the starting point for a general procedure in the construction of representations:~~

~~Exercise 6.23: Show that the operator~~

~~2 Ω:=X+X− + X−X+ +(1/2)Z acts as a scalar for the three types of representations in Theorem 6.2 and determine these scalars.~~

Our inﬁnitesimal considerations showed that there is no nontrivial ﬁnite-dimensional unitary representation of SL(2, R). There are other proofs for this fact. Because it is so elegant and instructive, we repeat the one from [KT] p.16.

~~Theorem 6.3: Every ﬁnite-dimensional representation π of G = SL(2, R) is trivial.~~

~~Proof: If G has an n-dimensional representation space, we have a homomorphism~~

~~π : SL(2, R) −→ U(n).~~

Since a 1 b a 1 a2b ( )( )( )−1 =( ) a−1 1 a−1 1 1 b all n(b)=( ) are conjugate. So all π(n(b)) with b>0 are conjugate. It is a not too 1 diﬃcult topological statement that in a compact group all conjugacy classes are closed. As U(n) is compact, we can deduce

~~lim π(n(b)) = π(E2)=En, b→0+ 82 6. The Inﬁnitesimal Method~~

~~and therefore π(n(b)) = En for all b>0. But we can repeat this conclusion for b<0 1 and as well forn ¯(b)=( ) and get b 1~~

~~π(n(b)) = π(¯n(b)) = En for all b ∈ R.~~

~~Now the proof is ﬁnished if we recall the following~~

~~Lemma: SL(2, R) is generated by the elements n(b) andn ¯(b),b∈ R.~~

~~Exercise 6.24: Prove this.~~

~~6.5 The Examples su(2) and heis(R)~~

In section 6.3 we computed derived representations for G = SU(2) and Heis(R). We look at this again on the ground of what we did in the last section. We refer to Exercise 6.4 in 6.1 and Example 6.8 in 6.3 and (having in mind later applications) change the basis (Xj)ofsu(2) we used there to

−i −i −1 (6.5) H =( ),H =( ),H =( ), 1 i 2 −i 3 1 with [Hj,Hk]=2Hl, (j, k, l)=(1, 2, 3), (2, 3, 1), (3, 1, 2). su(2) and sl(2, R) have the same complexiﬁcations. Hence we can express the matrices Z, X± from (6.2) in the previous section as

Z = iH3,X± =(1/2)(iH1 ∓ H2) and repeat the discussion of the representation space V = j∈J to get the conﬁgu- rations 1 to 4. Let us look at conﬁguration 1: We take a vector vn ∈ V of highest weight n ∈ N, put k vn−2k = Xˆ−vn, Xˆ+vn−2k =: akvn−2k+2, get ak = kn − k(k − 1) and m = −n for the lowest weight. But in this case the skew- hermiticity of the Hj changes things and Remark 6.9 in 6.4 is to be replaced by:

~~Remark 6.10: For a unitary representation π of SU(2) we must have~~

~~∗ dπ(X±) = dπ(X∓).~~

Hence, the discussion in 6.4 shows that for unitarity in this case the ak have to be positive as, in fact, they are for n ≥ k ≥ 0. We see that our discussion rediscovers the representation space V = V (j) from section 4.2 with n =2j. By the way, we get here for free another proof that the representation πj from 4.2 is irreducible. Conﬁgurations 2 to 4 are not unitarizable.

~~For G = Heis(R), as in Example 6.8 in 6.3, we have~~

~~g = heis(R)= 6.5 The Examples su(2) and heis(R)83 with the Heisenberg commutation relations~~

~~[P, Q]=2R, [R, P ]=[R, Q]=0.~~

~~We complexify g and take as a C-basis~~

~~Y0 := iR, Y± := (1/2)(P ± iQ) to get the relations [Y+,Y−]=Y0, [Y0,Y±]=0.~~

We want to construct a representationπ ˆ of g on a space V = j∈J , (which we continue linearly to a representation of gc,) such thatπ ˆ can be unitarily integrated, i.e. we can ﬁnd a unitary representation π of Heis(R) with dπ =ˆπ : In every group one has as a distinguished subgroup, the center C(G), deﬁned by

~~C(G):={g ∈ G; gg0 = g0g for all g0 ∈ G}.~~

~~Obviously for G = Heis(R)wehave~~

~~C := C(Heis(R)) = {(0, 0,κ); κ ∈ R}R.~~

~~Hence, a representation π of our G restricted to the center C is built up (see Chapter 5) from the characters of R. We write here~~

m ψ(κ):=ψm(κ) := exp(2πimκ)=e (κ),m∈ R \{0} with derived representation dψ(κ)=2πim. Having this in mind (and the discussion of Example 6.9 in 6.3), we propose the following construction of a representation (ˆπ, V ), where we abbreviate againπ ˆ(Y )v =: Yv.

~~We suppose that we have a vacuum vector, i.e. an element v0 ∈ V with~~

~~Y0v0 = µv0,Y−v0 =0,µ=2πm.~~

~~The first relation is inspired by differentiation of the prescription of the Schrödinger representation π((0, 0,κ))f = ψ(κ)f where we have (0, 0,κ) = exp κR and Y0 = iR.~~

~~j We put vj := Y+v0.AsY0 commutes with Y±,wehaveY0vj = µvj. Then we try to realize a relation Y−vj = ajvj−1 :~~

~~From the commutation relation [Y+,Y−]=Y0 we deduce~~

~~Y−v1 = Y−Y+v0 =(Y+Y− − Y0)v0 = −µ, Y−vj = Y−Y+vj−1 =(Y+Y− − Y0)vj−1 =(aj−1 − µ)vj−1 = ajvj−1, and, hence, by induction aj = −jµ.~~

~~Thus we have V spanned by v0,v1,v2,... with~~

~~Y0vj = µvj,Y+vj = vj+1,Y−vj = −jµvj−1. 84 6. The Inﬁnitesimal Method~~

~~We check whether this representation can be unitarized: We look at the necessary condition in Proposition 6.2 in 6.4. The skew-Hermiticity of P, Q, R leads to~~

~~∗ ∗ ∗ ∗ Yˆ± = dπ(Y±) =(1/2)(P ∓ iQ )=−(1/2)(P ∓ iQ)=−Yˆ∓.~~

~~Hence, as in the sl-discussion, we get the necessary condition for a scalar product <, > on V < Yˆ±v, w > = − , in particular, for v = vj−1,w = vj, we have~~

~~2 2 vj = jµ vj−1 .~~

We see that unitarity is possible iﬀ µ =2πm > 0. A model, i.e. a realization of this representationπ ˆ by integration, is given by the Schr¨odingerrepresentation πm, which we discussed already in several occasions (see Example 6.9 and Exercise 6.17 in 6.3).

~~Remark 6.11: Our inﬁnitesimal considerations show that πm is irreducible.~~

Remark 6.12: This discussion above is already a good part of the way to a proof of the famous Stone-von Neumann Theorem stating that up to equivalence there is no other irreducible unitary representation π of Heis(R) with π |C = ψm for m = 0 (for a complete proof see e.g. [LV] p.19ﬀ).

Remark 6.13: What we did here is essentially the same as that appears in the physics −2πmx2 literature under the heading of harmonic oscillator. Our v0 = f0 with f0(x)=e describes the vacuum, and Y+ and Y− are creation resp. annihilation operators producing higher resp. lower excitations.

~~Exercise 6.25: Use Exercise 6.17 in 6.3 to verify the equation~~

~~2 (Y+Y− + Y−Y+)f =(1/2)(f − (4πmx) f)=−2πmf and show that f0 is a solution. Recover the Hermite polynomials.~~

~~6.6 Roots, Weights and some Structure Theory~~

In the examples discussed above in 6.4 and 6.5 we realized the representations of the complexification gc of the given Lie algebra g = Lie G by ladder operators X± resp. Y± changing generators vj of our representation space V by going to vj±2 resp. vj±1 in a controlled way. For G = SL(2, R) and SU(2) the indices of the generators were related to the weights, i.e. the eigenvalues of the operator Z.ForG = Heis(R) we observe a different behaviour: We have an operator Y0 producing the same eigenvalue when applied to all generators of our representation space. This is expression of the fact that sl(2, R) and su(2) resp. the groups SL(2, R) and SU(2) on the one hand and heis(R) resp. Heis(R)on the other are examples of two different types for which we have to expect different ways of generalizations. Let us already guess that we expect to generalize the machinery of the weights in the first case (the semisimple one) by assuming to have a greater number of commuting operators whose eigenvalues constitute the weights and to have two different kinds of operators raising or lowering the weights (ordered, say, lexicographically). 6.6 Some Structure Theory 85

We have to introduce some more notions to get the tools for an appropriate structure theory. The reader interested mainly in special examples and eager to see the representation theory of the other groups mentioned in the introduction may want to skip this section. Otherwise she or he is strongly recommended to a parallel study of a more detailed source like [Hu], [HN] or [Ja]. Here we follow [Ki] p.92f, [KT], and [Kn1], which don’t give proofs neither (for these see [Kn] p.113ﬀ, and/or [Kn2] Ch.I - VI).

6.6.1 Specifications of Groups and Lie Algebras In this section we mainly treat Lie algebras. But we start with some definitions for groups, which often correspond to related definitions for algebras.

The reports [Kn1] and [KT] take as their central objects real or complex linear connected reductive groups. The deﬁnition of a reductive group is not the same with all authors but the following deﬁnition ([Kn] p.3, [KT] p.25) is very comprehensive and practical for our purposes.

Definition 6.7: A linear connected reductive group is a closed connected group of real or complex matrices that is stable under conjugate transpose, i.e. under the Cartan involution t −1 Θ:Mn(C) −→ Mn(C),X−→ ( X¯) . A linear connected semisimple group is a linear connected reductive group with finite center. More standard is the following definition: A group is called simple iff it is non-trivial, and has no normal subgroups other than {e} and G itself.

Example 6.10: The following groups are reductive a) G = GL(n, C), b) G = SL(n, C), c) G = SO(n, C), E d) G = Sp(n, C):={g ∈ SL(2n, C); tgJg = J := ( n )}. −En The center of GL(n, C) is isomorphic to C∗, so GL(n, C) is not semisimple. The other groups are semisimple (with exception n = 2 for c)). More examples come up by the groups of real matrices in the above complex groups. GL(n, R) is disconnected and therefore not reductive in the sense of the above deﬁnition, however its identity component is. We will come back to a classiﬁcation scheme behind these examples in the discussion of the corresponding notions for Lie algebras.

For the moment we follow another approach (as in [Ki] p.17). Non-commutative groups can be classiﬁed according to the degree of their non-commutativity: Given two subsets A and B of the group G, let [A, B] denote the set of all elements of the form aba−1b−1 as a runs through A and b runs through B. We deﬁne two sequences of subgroups:

~~Let G0 := G and for n ∈ N let Gn be the subgroup generated by the set [Gn−1,Gn−1]. Gn is called the n-th derived group of G. 86 6. The Inﬁnitesimal Method~~

~~Let G0 := G and for n ∈ N assume Gn to be the subgroup generated by the set [G, Gn−1]. We obviously obtain the following inclusions~~

~~G = G0 ⊃ G1 ⊃···⊃Gn ⊃ ..., G = G0 ⊃ G1 ⊃···⊃Gn ⊃ .... n For a commutative group these sequences are trivial, we have Gn = G = {e} for all n ∈ N.~~

~~Definition 6.8: G is called solvable of class k iff Gn = {e} beginning with n = k. G is called nilpotent of class k iff Gn = {e} beginning with n = k.~~

~~Exercise 6.26: Show that Heis(R) is nilpotent and solvable of class 1.~~

~~Now we pass to the corresponding concepts for Lie algebras (following [Ki] p.89).~~

~~Deﬁnition 6.9: A linear subspace g0 in a Lie algebra is called a subalgebra iﬀ~~

[X, Y ] ⊂ g0 for all X, Y ∈ g0. A linear subspace a in a Lie algebra is called an ideal iﬀ [X, Y ] ⊂ a for all X ∈ g and Y ∈ a. If a is an ideal in g, then the factor space g/a is provided in a natural way with the structure of a Lie algebra, which is called the factor algebra of g by a.

~~In every Lie algebra g we can deﬁne two sequences of subspaces, where here the expression [a, b] denotes the linear hull of all [X, Y ],X∈ a,Y ∈ b:~~

~~g1 := [g, g], g2 := [g, g1],...,gn+1 := [g, gn], g1 := [g, g], g2 := [g1, g1], ..., gn+1 := [gn, gn]. It is clear that one has the following inclusions~~

~~n n+1 n gn ⊃ gn+1, g ⊃ g , gn ⊃ g ,n∈ N.~~

~~n n n+1 Exercise 6.27: Prove that all g and gn are ideals in g and that gn/gn+1 and g /g are commutative.~~

~~n For dim g < ∞ the sequences (gn) and (g ) must stabilize, beginning with a certain n n n+1 ∞ we have gn = gn+1 = ···= g∞ and g = g = ···= g .~~

~~∞ Deﬁnition 6.10: g is called solvable resp. nilpotent iﬀ g = {0} resp. g∞ = {0}. Obviously every nilpotent algebra is solvable.~~

~~Example 6.11: heis(R) is nilpotent, and hence solvable as we have more generally~~

bn(K):={X =(xij) ∈ Mn(K); xij = 0 for i>j} is solvable, nn(K):={X =(xij) ∈ Mn(K); ,xij = 0 for i ≥ j} is nilpotent. n It is clear that every X ∈ nn is nilpotent, i.e. has the property X = 0. This is generalized by the following statement justifying the name nilpotent algebra: 6.6 Some Structure Theory 87

Theorem 6.4 (Engel): g is nilpotent iﬀ the operator ad X is nilpotent for all X ∈ g, i.e. for every X there is an n ∈ N with (ad X)n =0. We recall that ad X is deﬁned by ad X(Y ):=[X, Y ] for all Y ∈ g.

~~Now we pass to the other side of Lie algebra structure theory. As in [Kn1] p.1 ﬀ we restrict ourselves to the treatment of ﬁnite dimensional real or complex algebras g.~~

~~Deﬁnition 6.11: g is said to be simple iﬀ g is non-abelian and g has no proper non-zero ideals.~~

~~In this case one has [g, g]=g, which shows that we are as far from solvability as possible.~~

~~Deﬁnition 6.12: g is said to be semisimple iﬀ g has no non-zero abelian ideal.~~

~~There are other (equivalent) definitions, for instance g is said to be semisimple iff one has rad g = 0. In this definition the radical rad g is the sum of all solvable ideals of g.~~

~~Semisimple and simple algebras are related a follows.~~

Theorem 6.5: g is semisimple iﬀ g is the sum of simple ideals. In this case there are no other simple ideals, the direct sum decomposition is unique up to order of summands and every ideal is the sum of the simple ideals. Also in this case [g, g]=g.

~~Finally, from [KT] p.29 we take over~~

~~Deﬁnition 6.13: A reductive Lie algebra is a Lie algebra that is the direct sum of two ideals, one equal to a semisimple algebra and the other to an abelian Lie algebra.~~

~~We have a practical criterium:~~

Theorem 6.6: If g is a real Lie algebra of real or complex (even quaternion) matrices closed under conjugate transposition, then g is reductive. If moreover the center of g is trivial, i.e. Zg := {X ∈ g;[X, Y ] = 0 for all Y ∈ g} = 0, then g is semisimple.

~~Reductive Lie algebras have a very convenient property:~~

~~Proposition 6.3: A Lie algebra g is reductive iﬀ each ideal a in g has a complementary ideal, i.e. an ideal b with g = a ⊕ b.~~

~~As to be expected, to some extent the notions just deﬁned for groups and algebras ﬁt together and one has statements like the following:~~

Proposition 6.4 ([KT] p.29): If G is a linear connected semisimple group, then g = Lie G is semisimple. More generally, if G is linear connected reductive, then g is reductive with g = Zg ⊕ [g, g] as a direct sum of ideals. Here Zg denotes the center of g, and the commutator ideal is semisimple.

~~Example 6.12: gl(n, R)={scalars}⊕sl(n, R). 88 6. The Inﬁnitesimal Method~~

~~The main tool for the structure theory of semisimple Lie algebras is the following bilinear form that was ﬁrst deﬁned by Killing and then extensively used by E. Cartan.~~

~~Deﬁnition 6.14: The Killing form B is the symmetric bilinear form on g deﬁned by~~

~~B(X, Y ) := Tr (adXadY ) for all X, Y ∈ g.~~

~~Remark 6.14: B is invariant in the sense that~~

~~B([X, Y ],Z]) = B(X, [Y,Z]) for all X, Y, Z ∈ g.~~

~~Exercise 6.28: Determine the matrix of B with respect to the basis a) F, G, H (from Exercise 6.3 in 6.1) for g = sl(2, R), b) P, Q, R (from Exercise 6.7 in 6.1) for g = heis(R).~~

~~The starting point for structure theory of semisimple Lie algebras is Cartan’s Criterium for Semisimplicity.~~

~~Theorem 6.7: g is semisimple if and only if the Killing form is nondegenerate, that is B(X, Y ) = 0 for all Y ∈ g implies X =0.~~

~~The proof uses the remarkable fact that ker B is an ideal in g. There is another important and perhaps a bit more accessible bilinear form on g: The trace form B0 is given by~~

~~B0(X, Y ):=Tr(XY ) for all X, Y ∈ g. The trace form is invariant in the same sense as the Killing form. Both forms are related (see e.g. [Fog] IV.4):~~

~~Remark 6.15: We have B(X, Y )=2nTr (XY ) for g = sl(n, K) and n ≥ 2, =(n − 2)Tr (XY ) for g = so(n, K) and n ≥ 3, =2(n + 1)Tr (XY ) for g = sp(2n, K) and n ≥ 1.~~

~~A variant of the trace form already appeared in our deﬁnition of topology in matrix spaces. We introduced for X, Y ∈ Mn(C) := Re Tr tXY,¯ and X 2 := Re Tr tXX.¯~~

~~As inﬁnitesimal object corresponding to the Cartan involution for our matrix groups G~~

~~Θ : GL(n, C) −→ GL(n, C),g−→ (tg¯)−1, we have the map for the Lie algebra g = Lie G~~

~~t θ : Mn(C) −→ Mn(C),X−→ − X.¯ Hence we have := −Re B0(X, θY ) as a scalar product on g as a real vector space. This will become important later on. 6.6 Some Structure Theory 89~~

6.6.2 Structure Theory for Complex Semisimple Lie Algebras A complete classification exists for semisimple Lie algebras, as one knows their building blocks, the simple algebras: Over C there exist four infinite series of classical simple Lie algebras and five exceptional simple Lie algebras. Over R we have 12 infinite series and 23 exceptional simple algebras. From all these, here we reproduce only the following standard list of classical complex simple algebras

1) An := {X ∈ Mn+1(C); Tr X =0},n=1, 2, 3,... t 2) Bn := {X ∈ M2n+1(C); X = − X},n=2, 3,... En 3) Cn := {X ∈ M2n(C); XJ2n + J2nX =0},n=3, 4,...,J2n := ( ), −En t 4) Dn := {X ∈ M2n(C); X = − X},n=4, 5,....

~~We have the isomorphisms~~

~~B1 A1 C1,C2 B2,D2 A1 ⊕ A1,D3 A3 and D1 is commutative. In the sequel we shall in most cases return to our previous notation and write~~

An = sl(n +1, C),Bn = so(2n +1, C),Cn = sp(n, C),Dn = so(2n, C). All these algebras are also simple algebras over R. The remaining real classical algebras remain simple upon complexiﬁcation. For a complete list see for instance [Ki] p.92/3. By the way, we introduce here the following also otherwise important concept:

~~Deﬁnition 6.15: Given a complex Lie algebra g, a real algebra g0 is called a real form of g iﬀ (g0)c := g0 ⊗ C = g.~~

~~Example 6.13: g = sl(2, C) has just two real forms g0 = sl(2, R) and su(2). More about this is to be found in [Kn1] p.15.~~

The main ingredients for a classification scheme are the root and Dynkin diagrams.We give a sketch starting by the case of complex semisimple algebras where the theory is most accessible, and then go to compact real and finally briefly to noncompact real algebras. The central tool to define roots and their diagrams is a certain distinguished abelian subalgebra.

~~Deﬁnition 6.16: Let g be a complex semisimple Lie algebra. A Cartan subalgebra h is a maximal abelian subspace of g in which every ad H, H ∈ h, is diagonizable.~~

~~There are other equivalent ways to characterize a Cartan subalgebra h, for instance (see [Kn1] p.2): h is a nilpotent subalgebra whose normalizer Ng(h) satisﬁes~~

Ng(h):={X ∈ g, [X, H] ∈ h for all H ∈ h} = h. Each semisimple complex algebra has a Cartan subalgebra. Any two are conjugate via Int g ([Kn1] p.24), where Int g is a certain analytic subgroup of GL(g) with Lie algebra ad g. We refer to [Kn2] p.69/70 for the notion of an analytic subgroup, which comes up quite naturally when one analyzes the relation between Lie algebras and Lie groups. 90 6. The Inﬁnitesimal Method

The theory of Cartan subalgebras for the complex semisimple case extends to a complex reductive Lie algebra g by just saying that the center of g is to be adjoined to a Cartan subalgebra of the semisimple part of g.

~~Example 6.14: For g = sl(n, C) we have as Cartan subalgebra~~

~~h := {diagonal matrices in g}.~~

~~Using our Cartan subalgebra h, we generalize the decomposition we introduced at the beginning of Section 6.4 for g = sl(2, C)~~

~~g = + + ,~~

[Z0,X±]=±2X±, [X+,X−]=Z0, to the root space decomposition g = h ⊕ gα α∈∆ as follows: For all H ∈ h the maps ad H are diagonizable, and, as the elements of h all commute, they are simultaneously diagonizable (by the finite-dimensional spectral theorem from Linear Algebra). Let V1,...,V be the eigenspaces in g for the different systems of eigentupels. If h := and ad Hi acts as λij · id on Vj, define a linear functional λj on h by λj(Hi)=λij . If H := ΣciHi, then ad H acts on Vj by multiplication with ciλij = ciλj(Hi)=:λj(H). i i In other words, ad h acts in simultaneously diagonal fashion on g and the simultaneous ∗ eigenvalues are members of the dual vector space h := HomC(h, C). There are finitely many such simultaneous eigenvalues and we write

gλ := {X ∈ g;[H, X]=λ(H)X for all H ∈ h} for the eigenspace corresponding to λ ∈ h∗. The nonzero such λ are called roots and the corresponding gλ a root space. The (ﬁnite) set of all roots is denoted by ∆.

~~The following are some elementary properties of root space decompositions (for proofs see for instance [Kn2] II.4):~~

Proposition 6.5: We have a)[gα, gβ] ⊂ gα+β. b) If α, β ∈ ∆ \{0} and α + β =0 , then B(gα, gβ) = 0, i.e. root spaces are orthogonal with respect to the Killing form. c) B is nonsingular on gα × g−α if α ∈ ∆. d) −α ∈ ∆ifα ∈ ∆. e) B |h×h is nondegenerate. We deﬁne Hα to be the element of h paired with α. f) ∆ spans h∗.

~~Some deeper properties of root space decompositions are assembled in the following statement (see again [Kn2] II.4). 6.6 Some Structure Theory 91~~

Theorem 6.8: Root space decompositions have the following properties: a) dim gα = 1 for α ∈ ∆. b) nα ∈ ∆ for α ∈ ∆ and any integer n ≥ 2. c) [gα, gβ]=gα+β for α + β =0. | d) The real subspace h0 of h on which all roots are real is a real form of h and B h0×h0 is an inner product.

| ∗ We transfer B h0×h0 to the real span h0 of the roots obtaining a scalar product <, > and a norm . . It is not too diﬃcult to see that for α ∈ ∆ there is an orthogonal ∗ transformation sα given on h0 by

~~2 <ϕ,α> s (ϕ):=ϕ − α for ϕ ∈ h∗. α α 2 0~~

~~⊥ sα is called a root reﬂection in α and the hyperplane α a mirror.~~

~~The analysis of the root space shows that it has very nice geometrical and combinatorical properties. The abstraction of these is the following:~~

Definition 6.17: An abstract root system is a finite set ∆ of nonzero elements in a real inner product space V such that a) ∆ spans V , b) all reflections sα for α ∈ ∆ carry ∆ to itself, c) 2 <β,α>/ α 2∈ Z for all α, β ∈ ∆. The abstract root system is called reduced iff α ∈ ∆ implies 2α ∈ ∆. And it is called reducible iff∆=∆ ∪ ∆ with ∆ ⊥ ∆, otherwise it is irreducible.

The root system of a complex semisimple Lie algebra g with respect to a Cartan subal- ∗ gebra h forms a reduced abstract root system in h0. And a semisimple Lie algebra g is simple if the corresponding root system is irreducible.

Deﬁnition 6.18: The dimension of the underlying space V of an abstract root system ∆ is called its rank, and if ∆ is the root system of a semisimple Lie algebra g, we also refer to r = dim h as the rank of g.

~~To give an illustration, we sketch the reduced root systems of rank 2.~~

~~case 1 case 2 case 3 case 4~~

~~6 J] @I 6 Qk 3 6 J - @ - QJ]J - - Q J @ + J Qs J^ © ?@R J^ ? ? 92 6. The Inﬁnitesimal Method~~

~~Ordering of the Roots, Cartan Matrices, and Dynkin Diagrams~~

∗ We fix an ordered basis α1,...,αr of h0 and define λ =Σciαi to be positive if the first nonzero ci is positive. The ordering comes from saying λ>µif λ − µ is positive. Let ∆+ be the set of positive members in ∆.

Arootα is called simple if α>0 and α does not decompose as α = β1 + β2 with β1 and β2 positive roots. And a root α is called reduced if (1/2)α is not a root. Relative to a given simple system α1,...,αr, the Cartan matrix C is the r × r-matrix with entries 2 cij =2<αi,αj >/ αi . It has the following properties a) cij ∈ Z for all i, j, b) cii = 2 for all i, c) cij ≤ 0 for all i = j, d) cij =0iﬀcji =0, e) there exists a diagonal matrix D with positive diagonal entries such that DCD−1 is symmetric positive deﬁnite.

An abstract Cartan matrix is a square matrix satisfying properties a) through e) as above. To such a matrix we associate a diagram usually called a Dynkin diagram (historically more correct is the term CDW-diagram, C indicating Coxeter and W pointing to Witt): To the elements α1,...,αr of a basis of a root system correspond bijectively r points of a plane, which are also called α1,...,αr.Fori = j one joins the point αi to αj by cijcji lines, which do not touch any αk,k = i, j. For cij = cji the lines are drawn as arrows pointing in the direction to αj if cij

The main facts are that a Cartan matrix and equivalently the CDW-diagram determine the Lie algebra uniquely (up to isomorphism). In particular this correspondence does not depend on the choice of a basis of the root system.

~~Example 6.15: For g = sl(r +1, C),r ≥ 1 and its standard Cartan subalgebra~~

h := {D(d1,...,dr); d1,...,dr ∈ C, Σdi =0} we have the Cartan matrix ⎛ ⎞ 2 −1 ⎜ ⎟ ⎜ −12 ⎟ ⎜ ⎟ ⎜ • ⎟ C := ⎜ ⎟ ⎜ • ⎟ ⎝ 2 −1 ⎠ −12 and the corresponding CDW-diagram

~~rrrrrr Ar ... r ≥ 1. α1 α2 αr 6.6 Some Structure Theory 93~~

~~The diagrams for the other classical algebras are the following~~

~~rrrrrr- Br ... r ≥ 2, α1 α2 αr−1 αr~~

~~rrrrrr Cr ... r ≥ 3, α1 α2 αr−1 αr r αr−1~~

~~rrrrrr Dr ... r ≥ 4. α1 α2 αr−2 αr~~

~~The remaining exceptional graphs of indecomposable reduced root systems are the following~~

~~rrrrr E6~~

~~rrrrrr E7~~

~~rrrrrrr E8~~

~~rrrrrr- F4 G2 -~~

~~6.6.3 Structure Theory for Compact Real Lie Algebras~~

Up to now, we treated complex Lie algebras. We go over to a real Lie algebra g0 and (following [Kn1] p.16) we call a subalgebra h0 of g0 a Cartan subalgebra if its complexification is a Cartan subalgebra of the complex algebra g = g0 ⊗ C. At first we look at the compact case: If g0 is the Lie algebra of a compact Lie group G and if t0 is a maximal abelian subspace of g0, then t0 is a Cartan subalgebra. We have already discussed the example g0 = su(2) in 4.2 and we will discuss g0 = su(3) in the next section and see that we have dim t0 = 2 in this case. To illustrate the general setting, we collect some more remarks about the background notions (from [Kn1] p.15): 94 6. The Infinitesimal Method

Theorem 6.9: If g0 is semisimple, then the following conditions are equivalent: a) g0 is the Lie algebra of some compact Lie group. b) Int g0 is compact. c) The Killing form of g0 is negative deﬁnite.

If g is semisimple complex, a real form g0 of g is said to be compact if the equivalent conditions of the theorem hold. Our main example su(n) is a compact form of sl(n, C). The fundamental result is here:

~~Theorem 6.10: Each complex semisimple Lie algebra has a compact real form.~~

~~Another important topic is maximal tori.~~

~~Deﬁnition 6.19: Let G be a compact connected linear group. A maximal torus in G is a subgroup T maximal with respect to the property of being compact connected abelian.~~

~~From [Kn2] Proposition 4.30, Theorem 4.34 and 4.36 we take over:~~

Theorem 6.11: If G is a compact connected linear group, then a) the maximal tori in G are exactly the analytic subgoups corresponding to the maximal abelian subalgebras of g0 = Lie G; b) any two maximal abelian subalgebras of g0 are conjugate via Ad G and hence any two maximal tori are conjugate via G.

~~Theorem 6.12: If G is compact connected and T a maximal torus, then each element of G is conjugate to a member of T .~~

~~Example 6.16: 1. For G = SU(n) one has as a maximal torus, its Lie algebra and its complexiﬁed Lie algebra~~

~~iϑ1 iϑ T = {D(e ,...,e n ); Σϑj =0}, t0 = {D(iϑ1,...,iϑn); Σϑj =0},~~

~~t = {D(z1,...,zn); Σzj =0}.~~

~~2. For G = SO(2n) and SO(2n + 1) one has as a maximal torus T the diagonal block matrices with 2 × 2-blocks r(ϑj),j =1,...,n, in the diagonal, resp. additionally 1 in the second case.~~

~~We go back to the general case and use the notation we have introduced. In this setting, we can form a root-space decomposition g = t ⊕ gα. α∈∆~~

~~Each root is the complexiﬁed diﬀerential of a multiplicative character χα of the maximal torus T that corresponds to t0, with~~

~~Ad(t)X = χα(t)X for all X ∈ gα. 6.6 Some Structure Theory 95~~

Another central concept is the one of the Weyl group. We keep the notation used above: G is compact connected, g0 = Lie G, T a maximal torus, t0 = Lie T , t = t0 ⊗ C,∆(g, t)is the set of roots, and B is the negative of a G invariant scalar product on g0. We define ∗ tR = it0. As roots are real on tR, they are in tR. The form B, when extended to be ∗ complex bilinear, is positive definite on tR, yielding a scalar product <, > on tR. Now, the Weyl group W = W (∆(g, t)) is in this context defined as the group generated by the root ∈ ∗ − 2<λ,α> reflections sα,α ∆(g, t), given (as already fixed above) on tR by sα(λ)=λ |α|2 α. This is a finite group, which also can be characterized like this: One defines W (G, T ) as the quotient of the normalizer by the centralizer

~~W (G, T ):=NG(T )/ZG(T ).~~

~~By Corollary 4.52 in [Kn2] p.260 one has ZG(T )=T and hence also the formula W (G, T )=NG(T )/T.~~

~~∗ Theorem 6.13: The group W (G, T ), when considered as acting on tR, coincides with W (∆(g, t)).~~

~~6.6.4 Structure Theory for Noncompact Real Lie Algebras~~

Now, in the ﬁnal step, we brieﬂy treat the general case of a real Lie algebra g0 of a noncompact semisimple group G. We already introduced the Killing form B, the t Cartan involution θ with θX = − X¯ for a matrix X, and the fact that if g0 consists of real, complex or quaternion matrices and is closed under conjugate transpose, then it is reductive. More generally we call here a Cartan involution θ any involution of g0 such that the symmetric bilinear form

~~Bθ(X, Y ):=−B(X, θY ) is positive deﬁnite. Then we have the Cartan decomposition~~

~~g0 = k0 ⊕ p0, k0 := {X ∈ g0; θX = X}, p0 := {X ∈ g0; θX = −X} with the bracket relations~~

~~[k0, k0] ⊆ k0, [k0, p0] ⊆ p0, [p0, p0] ⊆ k0.~~

One has the following useful facts ([Kn 2] VI.2): a) g0 has a Cartan involution. b) Any two Cartan involutions of g0 are conjugate via Int g0. c) If g is a complex semisimple Lie algebra, then any two compact real forms of g are conjugate via Int g. d) If g is a complex semisimple Lie algebra and is considered as a real Lie algebra, then the only Cartan involutions of g are the conjugations with respect to the compact real forms of g.

~~The Cartan decomposition of Lie algebras has a global counterpart (see for instance [Kn1] Theorem 4.3). Its most rudimentary form is the following: 96 6. The Inﬁnitesimal Method~~

Proposition 6.6: For every A ∈ GL(n, C) resp. GL(n, R), there are S1,S2 ∈ O(n) resp. U(n) and a diagonal matrix D with real positive elements in the diagonal such that A = S1DS2. This decomposition is not unique.

~~Proof as Exercise 6.29 (use the fact that to each positive deﬁnite matrix B there is a unique positive deﬁnite matrix P with B = P 2 and/or see [He] p.73).~~

~~Restricted Roots~~

We want to ﬁnd a way to describe Cartan subalgebras in our general situation: Let G be semisimple linear group with Lie G = g0, g = g0 ⊗ C, θ a Cartan involution of g0, and g0 = k0 ⊕ p0 the corresponding Cartan decomposition. Let B be the Killing form (or more generally any nondegenerate symmetric invariant bilinear form on g0 with B(θX, θY )=B(X, Y ) such that Bθ(X, Y )=−B(X, θY ) is positive deﬁnite).

~~Deﬁnition 6.20: Let a0 be a maximal abelian subspace of p0. Restricted roots are the ∈ ∗ nonzero λ a0 such that~~

~~(g0)λ := {X ∈ g0; (ad H)X = λ(H)X for all H ∈ a0} = {0}.~~

Let Σ be the set of restricted roots and m0 := Zk0 . We ﬁx a basis of a0 and an associated ∗ + lexicographic ordering in a0 and deﬁne Σ as the set of positive resticted roots. Then n0 = (g0)λ λ∈Σ+ is a nilpotent Lie subalgebra, and we have the following Iwasawa decomposition:

~~Theorem 6.14: The semisimple Lie algebra g0 is a vector space direct sum~~

~~g0 = k0 ⊕ a0 ⊕ n0.~~

~~Here a0 is abelian, n0 is nilpotent, a0 ⊕ n0 is a solvable subalgebra of g0, and a0 ⊕ n0 has [a0 ⊕ n0, a0 ⊕ n0]=n0.~~

For a proof see for instance [Kn2] Proposition 6.43. There is also a global version ([Kn2] Theorem 6.46) stating (roughly) that one has a diﬀeomorphism of the corresponding groups K × A × N −→ G given by (k, a, n) −→ kan. We already know all this in our standard example G = SL(2, R): If we take θX = −tX, we have

~~g0 = sl(2, R)=k0 ⊕ a0 ⊕ n0, with 1 1 1 k =< ( ) >, a =< ( ) >, n =< ( ) > 0 −1 0 −1 0 and as in Example 6.6 in 6.2~~

~~G = KAN with K = SO(2), A = {t(y); y > 0 }, N = {n(x); x ∈ R}. 6.6 Some Structure Theory 97~~

As to be expected, roots and restricted roots are related to each other. If t0 is a maximal ⊕ abelian subspace of m0 := Zk0 (a0), then h0 := a0 t0 is a Cartan subalgebra of g0 in the sense we deﬁned at the beginning ([Kn2] Proposition 6.47). Roots are real valued on a0 and imaginary valued on t0. The nonzero restrictions to a0 of the roots turn out to be restricted roots. Roots and restricted roots can be ordered compatibly by taking a0 before it0. Cartan subalgebras in this setting are not always unique up to conjugacy.

~~1 1 Exercise 6.30: Show that h :=< ( ) > and h :=< ( ) > are Cartan 0 1 0 −1 subagebras of g0 = sl(2, R) and determine the corresponding Iwasawa decomposition.~~

~~Every Cartan subalgebra of g0 is conjugate (via Intg0) to this h0 or the h0 above.~~

6.6.5 Representations of Highest Weight After this excursion into structure theory of complex and real Lie algebras, we ﬁnally come back to show a bit more of the general representation theory, which is behind the examples discussed in our sections 6.4 and 6.5 and which we shall apply in our next section to the example g0 = su(3). We start by following the presentation in [Kn1] p.8. Let at ﬁrst g be a complex Lie algebra, h a Cartan subalgebra, ∆ = ∆(g, h) the set of roots, h0 the real form of h where roots are real valued, B the Killing form (or a more ∈ ∈ ∗ general form as explained above), and Hλ h0 corresponding to λ h0. Letπ ˆ : g −→ End V be a representation. For λ ∈ h∗ we put

~~n Vλ := {v ∈ V ;(ˆπ(H) − λ(H)1) v = 0 for all H ∈ h and some n = n(H, V ) ∈ N}.~~

If Vλ = {0}, Vλ is called a generalized weight space and λ a weight.IfV is ﬁnite dimensional, V is the direct sum of its generalized weight spaces. This is a generalization of the fact from linear algebra about eigenspace decompositions of a linear transformation on a ﬁnite-dimensional vector space. If λ is a weight, then the subspace

0 { ∈ ∈ } Vλ := v V ;ˆπ(H)v = λ(H)v for all H h is nonzero and called the weight space corresponding to λ. One introduces a lexicographic ordering among the weights and hence has the notion of highest and lowest weights. The set of weights belonging to a representationπ ˆ is denoted by Γ(ˆπ). For dim h =1we have h h∗ C and so the weights are simply the complex numbers we met in our examples in 6.4 and 6.5, for instance for g = sl(2, C) we found representationsσ ˆN with dim VN = N + 1 and weights −N,−N +2,...,N − 2,N. In continuation of this, we treat a more general example.

Example 6.17: Let G = SU(n) and, hence, g = su(n)c = sl(n, C) and the Cartan subalgebra the diagonal subalgebra h :=. We choose as generators of h the matrices Hj := Ejj − Ej+1,j+1 and (slightly misusing) we write also Hj = ej − ej+1 n−1 where ej denote the canonical basis vectors of C h. We have three natural types of representations : At ﬁrst, let V be the space of homogeneous polynomials P of degree N in z1,...,zn and their conjugates and take the action of g ∈ SL(n, C) given by

~~−1 −1 t (π(g)P )(z,z¯):=P (g z,g¯ z¯),z= (z1,...,zn). 98 6. The Inﬁnitesimal Method~~

~~Hence for H = D(iϑ1,....iϑn) with ϑj ∈ R, Σϑj =0, we come up with~~

~~n n − − (dπ(H)P )(z,z¯)= ( iϑjzj)∂zj P (z,z¯)+ ( iϑjz¯j)∂z¯j P (z,z¯). j=1 j=1~~

If P is a monomial n k1 ····· kn · l1 ····· ln P (z,z¯)=z1 zn z¯1 z¯n with (kj + lj)=N, j=1 then we get n dπ(H)P = (lj − kj)(iϑj)P. j=1 Exercise 6.31: Describe the weights for this representation with respect to the lexicographic ordering for the basis elements of h0 =R. Do this also for the two other “natural” representations, namely on the subspaces V1 of holomorphic and V2 of antiholomorphic polynomials in V (i.e. polyomials only in z1,...,zn resp.z ¯1,...,z¯n). We will come back to this for the case n = 3 in the next section.

In our examples g0 = sl(2, R) and g0 = su(2) in 6.4 resp. 6.5 we saw that the ﬁnite- dimensionality of the representation space V was equivalent to an integrality condition for the weight. Now we want to look at the general case where the weights are l-tupels ∗ ∗ if one ﬁxes a basis of a Cartan subalgebra h resp. its dual h and the real form h0.

~~Deﬁnition 6.21: λ ∈ h∗ is said to be algebraically integral if~~

~~2 <λ,α>/|α|2 ∈ Z for all α ∈ ∆.~~

~~Then as a generalization of what we saw in 6.4 and 6.5, we have the elementary properties of the weights for a ﬁnite-dimensional representationπ ˆ on a vector space V :~~

Proposition 6.7: a)π ˆ(h) acts diagonally on V , so that every generalized weight vector is a weight vector and V is the direct sum of all weight spaces. b) Every weight is real valued on h0 and algebraically integral. c) Roots and weights are related byπ ˆ(gα)Vλ ⊂ Vλ+α.

We ﬁx a lexicographical ordering and take ∆+ to be the set of positive roots with Π={α1,...,αl} as the corresponding simple system of roots. We say that λ ∈ V is dominant if <λ,αj > ≥ 0 for all αj ∈ Π. The central fact is here the beautiful Theorem of the Highest Weight ([Kn2] Th. 5.5):

Theorem 6.15: The irreducible ﬁnite-dimensional representationsπ ˆ of g stand (up to equivalence) in one-one correspondence with the algebraically integral dominant linear functionals λ on h, the correspondence being that λ is the highest weight ofπ ˆλ. The highest weight λ ofπ ˆλ has these additional properties: a) λ depends only on the simple system Π and not on the ordering used to deﬁne Π. b) The weight space Vλ for λ is one-dimensional. + c) Each root vector Eα for arbitrary α ∈ ∆ annihilates the members of Vλ, and the members of Vλ are the only vectors with this property. 6.6 Some Structure Theory 99

− l ∈ ∈ d) Every weight ofπ ˆλ is of the form λ i=1 niαi with ni N0 and αi Π. e) Each weight space Vµ forπ ˆλ has dimVwµ = dim Vµ for all w in the Weyl group W (∆), and each weight µ has | λ |≥| µ | with equality only if µ is in the orbit W (∆)λ.

~~We already introduced in 3.2 the concept of complete reducibility. Here we can state the following fact ([Kn2] Th. 5.29).~~

Theorem 6.16: Letπ ˆ be a complex linear representation of g on a ﬁnite-dimensional complex vector space V . Then V is completely reducible in the sense that there exist invariant subspaces U1,...,Ur of V such that V = U1 ⊕···⊕Ur and such that the restriction of the representation to each Ui is irreducible.

The proofs of these two theorems use three tools, which are useful also in other contexts: – Universal enveloping algebras, – Casimir elements, – Verma modules. Again following [Kn1] p.10f, we brieﬂy present these as they will help us to a better understanding in our later chapters.

~~The Universal Enveloping Algebra~~

~~This is a general and far reaching concept applicable for any complex Lie algebra g:~~

~~We take the tensor algebra~~

~~T (g):=C ⊕ g ⊕ (g ⊗ g) ⊕ .... and the two-sided ideal a in T (g) generated by all~~

~~X ⊗ Y − Y ⊗ X − [X, Y ],X,Y∈ T 1(g).~~

~~Here T 1(g) denotes the space of ﬁrst order tensors.~~

~~Then the universal enveloping algebra is the associative algebra with identity given by~~

~~U(g):=T (g)/a.~~

This formal deﬁnition has the practical consequence that U(g) consists of sums of mono- j1 jn ∈ ∈ mials usually written (slightly misusing) as a(j)X1 ...Xn ,a(j) C,X1,...,Xn g with the usual addition and a multiplication coming up from the Lie algebra relations by XiXj = XjXi +[Xi,Xj]. More carefully and formally correct, one has the following: Let ι : g −→ U(g) be the composition of natural maps

~~ι : g T 1(g) → T (g) −→ U(g), so that ι([X, Y ]) = ι(X)ι(Y ) − ι(Y )ι(X). 100 6. The Inﬁnitesimal Method~~

~~ι is in fact injective as a consequence of the fundamental Poincar´e-Birkhoﬀ-WittTheorem:~~

~~Theorem 6.17: Let {Xi}i∈I be a basis of g, and suppose that a simple ordering has been imposed on the index set I. Then the set of all monomials~~

~~j1 ····· jn (ιXi1 ) (ιXin ) with i1 < ···~~

~~The name universal enveloping algebra stems from the following universal mapping property.~~

~~Theorem 6.18: Whenever A is a complex associative algebra with identity and we have a linear mapping ϕ : g −→ A such that~~

~~ϕ(X)ϕ(Y ) − ϕ(Y )ϕ(X)=ϕ([X, Y ]) for all X, Y ∈ g, then there exists a unique algebra homomorphismϕ ˜ : U(g) −→ A such thatϕ ˜(1)=1 and ϕ =˜ϕ ◦ ι.~~

~~The mapϕ ˜ from the theorem may be thought of as an extension of ϕ from g to all of U(g). This leads to the following useful statement.~~

Theorem 6.19: Representations of g on complex vector spaces stand in one-one correspondence with left U(g) modules in which 1 acts as the identity. This fact is essential for the construction of representations and implicit in our examples in 6.4 and 6.5.

We now come back to the case that g is semisimple and to the notation introduced above. We enumerate the positive roots as α1,...,αm and we let H1,...,Hl be a basis of h. Then for the construction of highest weight representations it is appropriate to use the ordered basis

~~E−α1 ,...,E−αm ,H1,...,Hl,Eα1 ,...,Eαm in the Poincar´e-Birkhoﬀ-WittTheorem. The theorem says that~~

Ep1 ...Epm Hk1 ...Hkl Eq1 ...Eqm −α1 −αm 1 l α1 αm is a basis of U(g). If one applies members of this basis to a nonzero highest weight vector v of V, one gets control of a general member of U( )v : Eq1 ...Eqm will act as 0 if 0 g 0 α1 αm ··· k1 kl q1 + + qm > 0, and H1 ...Hl will act as a scalar. Thus one has only to sort out the effect of Ep1 ...Epm and most of the conclusions of the Theorem of the Highest −α1 −αm Weight follow readily. If one looks at our examples in 6.4 and 6.5, one can get an idea how this works even in the case of representations, which are not finite-dimensional, and how the integrality of the weight leads to finite-dimensionality. 6.6 Some Structure Theory 101

~~The Casimir Element~~

For a complex semisimple Lie algebra g with Killing form B, the Casimir element Ωis the element ˜ ˜ Ω0 := B(Xi,Xj)XiXj i,j of U(g), where (Xi) is a basis of g and (X˜i) is the dual basis relative to B. One can show that Ω0 is deﬁned independently of the basis (Xi) and is an element of the center Z(g) of U(g).

~~Exercise 6.32: Check this for the case g = sl(2, C) and determine the Casimir element.~~

~~In the general case one has the following statement.~~

Theorem 6.20: Let Ω0 be the Casimir element, (Hi)i=1,.,l an orthogonal basis of h0 relative toB, and choose root vectors Eα so that B(Eα,E−α) = 1 for all roots α. Then l 2 a) Ω0 = i=1 Hi + α∈∆ EαE−α. 2 2 2 b) Ω0 operates by the scalar |λ| +2<λ,δ>= |λ + δ| −|δ| in an irreducible ﬁnite- dimensional representation of g of highest weight λ, where δ is half the sum of the positive roots. c) The scalar by which Ω0 operates in an irreducible ﬁnite-dimensional representation of g is nonzero if the representation is not trivial.

~~The main point is that ker Ω0 is an invariant subspace of V if V is not irreducible.~~

Remark 6.16: The center of U(g) is important also in the context of the determination of infinite-dimensional representations. We observed in Exercise 6.23 in 6.4 that Ω, a multiple of the Casimir element Ω0, acts as a scalar for the unitary irreducible representations of sl(2, R). For more general information we recommend [Kn] p.214 where as a special case of Corollary 8.14 one finds the statement: If π is unitary, then each member of the center Z(gc)ofU(gc) acts as a scalar operator on the K−finite vectors of π. We will come back to this in 7.2 while explicitly constructing representations of SL(2, R).

~~The Verma Module~~

We fix a lexicographic ordering and introduce b := h ⊕ gα. α>0 ∈ ∗ For ν h , make C into a one-dimensional U(b) module Cν by defining the action of ∈ ∈ ∈ ∗ H h by Hz = ν(H)z for z C and the action of α>0 gα by zero. For µ h ,we define the Verma module V (µ)by

~~V (µ):=U(g) ⊗U(h) Cµ−δ. where δ is again half the sum of the positive roots and this term is introduced to simplify calculations with the Weyl group. 102 6. The Inﬁnitesimal Method~~

Verma modules are essential for the construction of representations. They have the following elementary properties: a) V (µ) =0 . b) V (µ) is a universal highest weight module for highest weight modules of U(g) with highest weight µ − δ. c) Each weight space of V (µ) is ﬁnite-dimensional. d) V (µ) has a unique irreducible quotient L(µ).

~~If λ is dominant and algebraically integral, then L(λ+δ) is the irreducible representation of highest weight λ looked for in the theorem of the Highest Weight (Theorem 6.15).~~

In our treatment of finite and compact groups we already saw the effectiveness of the theory of characters of representations. As for the moment we look at finite-dimensional representations we can use characters here too. To allow for more generalization, we treat them for now as formal exponential sums (again following [Kn1] p.12/3): Let again g be a semisimple Lie algebra, h a Cartan subalgebra, ∆ a set of roots provided with a lexicographic ordering, α1,...,αl the simple roots, and W (∆) the Weyl group. ∗ We regard the set Zh of functions f from h∗ to Z as an abelian group under pointwise addition. We write f = f(λ)eλ. λ∈h∗

∗ The support of f is deﬁned to be the set of λ ∈ h∗ for which f(λ) = 0. Within Zh , let Z[h∗] be the subgroup of all f of ﬁnite support. The subgroup Z[h∗] has a natural commutative ring structure, which is determined by eλeµ = eλ+µ. Moreover, we introduce a larger ring Z < h∗ > with

~~∗ Z[h∗] ⊆ Z < h∗ > ⊆ Zh~~

∗ consisting of all f ∈ Zh whose support is contained in the union of ﬁnitely many sets + ∗ νi − Q , νi ∈ h and l + Q := { niαi; ni ∈ N0}. i=1 Multiplication in Z < h∗ > is given by λ µ ν ( cλe )( cµe ):= ( cλcµ)e . λ∈h∗ µ∈h∗ λ∈h∗ λ+µ=ν

If V is a representation of g (not necessarily ﬁnite-dimensional), one says that V has a character if V is the direct sum of its weight spaces under h, i.e., V = ⊕µ∈h∗ Vµ , and if ∗ dim Vµ < ∞ for µ ∈ h . In this case the character is µ char(V ):= (dim Vµ)e µ∈h∗

~~∗ as an element of Zh . This deﬁnition is meaningful if V is ﬁnite-dimensional or if V is a Verma module. 6.6 Some Structure Theory 103~~

~~We have two more important notions: The Weyl denominator is the element d ∈ Z[h∗] given by~~

~~δ −α d := e Πα∈∆+ (1 − e ).~~

Here δ is again half the sum of the positive roots. The Kostant partition function P is the function from Q+ to N that tells the number of ways, apart from order, that a member of Q+ can be written as the sum of positve roots. −γ ∗ We put P(0) = 1 and deﬁne K := Σγ∈Q+ P(γ)e ∈ Z < h >. Then one can prove that one has Ke−δd = 1 in the ring Z < h∗ > , hence d−1 ∈ Z < h∗ >. Then we have as the last main theorem in this context the famous Weyl Character Formula:

~~Theorem 6.21: Let (π, V ) be an irreducible ﬁnite-dimensional representation of the complex semisimple Lie algebra g with highest weight λ. Then char(V )=d−1 (det w)ew(λ+δ). w∈W (∆)~~

Now we leave the treatment of the complex semisimple case and go over to the following situation: G is compact connected, g0 := Lie G, g the complexification of g0, T a maximal torus, t0 := Lie T ,∆(g, t) the set of roots, and B the negative of a G invariant ∗ inner product on g0, and tR := it0. As we know, roots are real on tR, hence are in tR. The form B, when extended to be complex bilinear, is positive definite on tR, yielding an ∗ inner product <, > on tR. W (∆(g, t)) is the Weyl group generated by the root reflections sα for α ∈ ∆(g, t). Besides the notion of algebraic integrality already exploited in the complex semisimple case above, we have here still another notion of integrality: We say that λ ∈ t∗ is analytically integral if the following equivalent conditions hold: 1) Whenever H ∈ t0 satisfies exp H = 1, then λ(H)isin2πiZ. λ(H) 2) There is a multiplicative character ψλ of T with ψλ(exp H)=e for all H ∈ t0. In [Kn1] p.18 one finds a list of properties of these notions. We cite part of it: a) Weights of finite-dimensional representations of G are analytically integral. In particular every root is analytically integral. b) Analytically integral implies algebraically integral. c) If G is simply connected and semisimple, then algebraically integral implies analytically integral.

~~For instance, the half sum δ of positve roots is algebraically integral but not analytically if G = SO(3).~~

~~In our situation the Theorem 6.15 (Theorem of the Highest Weight) comes in the following form:~~

Theorem 6.22: Let G be a compact connected Lie group with complexified Lie algebra g, let T be a maximal torus with complexified Lie algebra t, and let ∆+(g, t)bea positive system for the roots. Apart from equivalence the irreducible finite-dimensional representations π of G stand in one-one correspondence with the dominant analytically integral linear functionals λ on t, the correspondence being that λ is the highest weight of π. 104 6. The Infinitesimal Method

~~And we restate Theorem 6.21 (Weyl’s Character Formula) in the form:~~

Theorem 6.23: The character χλ of the irreducible ﬁnite-dimensional representation of G with highest weight λ is given by w∈W (det w)ψw(λ+δ)−δ(t) χλ = Πα∈∆+ (1 − ψ−α(t)) at every t ∈ T where no ψα takes the value 1 on t.

~~In the next section we shall illustrate this by treating an example, which was a milestone in elementary particle physics.~~

~~6.7 The Example su(3)~~

In 1962 Gell-Mann proposed in a seminal paper [Gel] a symmetry scheme for the description of hadrons, i.e. certain elementary particles (defined by interacting by strong interaction), which had a great influence in elementary physics and beyond, as it led to the notion of quarks and the eightfold way. We can not go too much into the physical content but discuss the mathematical background to give another example of the general theory. Respecting the historical context, we adopt the notation introduced by Gell-Mann and used in Cornwell’s presentation in [Co] vol II, p.502ff: As su(3) consists of tracelass skew hermitian three-by-three matrices, one can use as a basis ⎛ ⎞ ⎛ ⎞ ⎛ ⎞ i 1 i ⎝ ⎠ ⎝ ⎠ ⎝ ⎠ A1 := i ,A2 := −1 ,A3 := −i , ⎛ ⎞ ⎛ ⎞ ⎛ ⎞ i 1 ⎝ ⎠ ⎝ ⎠ ⎝ ⎠ A4 := ,A5 := ,A6 := i , − ⎛ i ⎞ ⎛1 ⎞ i i A := ⎝ 1 ⎠ ,A := √1 ⎝ i ⎠ . 7 8 3 −1 −2i

~~This basis is also a basis for the complexiﬁcation su(3)c = sl(3, C). One may be tempted to use the elementary matrices Eij with entries (Eij )kl = δikδjl and take as a basis~~

~~Eij,i= j, (i, j =1, 2, 3) H1 := E11 − E22,H2 := E22 − E33.~~

~~One has the commutation relations [Eij ,Ekl]=δjkEil − δilEkj. For a diagonal matrix H := D(h1,h2,h3) we get~~

[H, Eij]=(ei(H) − ej(H))Eij,ei(H)=hi, and hence [H1,E12]=2E12, [H2,E12]=−E12, [H1,E13]=E13, [H2,E13]=E13, [H1,E23]=−E23, [H2,E23]=2E23, and [E12,E21]=H1, [E23,E32]=H2, [E13,E31]=E11 − E33 =: H3. 6.7 The Example su(3) 105

~~We see that h := is a Cartan subalgbra,~~

~~gα1 := , gα2 := , gα3 :=~~

~~∗ are root spaces, i.e. nontrivial eigenspaces for the roots α1,α2,α3 ∈ h given by~~

~~α˜1 := (α1(H1),α1(H2)) = (2, −1), α˜2 := (α2(H1),α2(H2)) = (1, 1), α˜3 := (α3(H1),α3(H2)) = (−1, 2).~~

~~(In the physics literature these tuplesα ˜j themselves sometimes are called roots.) We choose these roots αj as positive roots and then get in this coordinization a slightly unsymmetric picture:~~

~~r α˜ 2 3 AK A A r α˜ A 2 A rr 12rr AH -1-2 H HH r HHj -1 α˜1~~

Hence we better use Gell-Mann’s matrices and change to Xj := −iAj. Then we get ± ± H1 = X3,H2 := X8. Moreover, for X± := (1/2)(X1 iX2),Y± := (1/2)(X6 iX7), and Z± := (1/2)(X4±iX5), we have X+ = E12,X− = E21 etc. and the commutation relations

~~[H1,X±]=2X±, [H2,X±]=0,√ ∓ ± [H1,Y±]= Y±, [H2,Y±]= √3Y±, ± ± [H1,Z±]= Z±, [H2,Z±]= 3Z±. Now the Cartan subalgebra is h = and the root spaces are~~

~~g±α1 =, g±α2 =, g±α3 = with positive roots given by~~

~~α¯1 := (α1(H1),α1(H2)) = (2, 0)√, − α¯2 := (α2(H1),α2(H2)) = ( 1√, 3), α¯3 := (α3(H1),α3(H2)) = (1, 3).~~

~~α1 and α2 are simple roots, we have α1 + α2 = α3 and we get a picture with hexagonal symmetry: r 2 α¯ α¯ 2 AK ¡ 3 A ¡ A r ¡ A ¡ A ¡ α¯1 rr A¡ rr- -1-2 12~~

~~r -1 106 6. The Inﬁnitesimal Method~~

~~In the literature (see for instance [Co] p.518) we ﬁnd still another normalization, which is motivated like this: The Killing form B for su(3) is given by diagonal matrices~~

~~B(Ap,Aq)=−12δpq, resp.B(Xp,Xq)=12δpq.~~

~~Exercise 6.33: Verify this.~~

~~One uses B to introduce a nondegenerate symmetric bilinear form <, > on h∗ as follows: We deﬁne a map~~

~~∗ h ⊃ ∆ α −→ Hα ∈ h by B(Hα,H)=α(H) for all H ∈ h, and then <α,β>:= B(Hα,Hβ) for all α, β ∈ ∆.~~

~~This leads to inner products on hR = R (α1,...,αl simple positive roots) and its dual space. For su(3) we have l = 2 and with α1,α2 from above~~

~~<α1,α1 >=<α2,α2 >=1/3,<α1,α2 >=<α2,α1 >= −1/6.~~

~~It is convenient to introduce an orthonormal basis Hˇ1,...,Hˇl of h, i.e. with B(Hˇp, Hˇq)= δpq. Then we have the simple rule ˇ ˇ ˇ ˇ <α,β>= αˇjβj, αˇj = α(Hj), βj = β(Hj). j~~

~~In our case this leads to √ √ ˇ ˇ H1 =(1/(2 3))H1, H2 =(1/2)H2 =(1/(2 3))X8 and √ √ √ √ αˇ1 =( 3/6)¯α1 =(1/ 3, 0), αˇ2 =( 3/6)¯α2 =(−1/(2 3), 1/2).~~

We see that, for the Cartan matrix C =(Cpq)=(2<αp,αq >/<αp,αp >) from 6.6.2, we have in this case 2 −1 C =( ). −12 The Weyl group W (see the end of 6.6.3) is generated by the reﬂections − sαp (ˇαq)=ˇαq Cpqαˇp

~~∗ l at the planes in hR R orthogonal to theα ˇp with l = 2 and p, q =1, 2 in our case. For g0 = su(3) we have 6 elements in W .~~

~~Exercise 6.34: Determine these explicitely.~~

Now we proceed to the discussion of the representations (ˆπ, V )ofsu(3). The general theory tells us that we can assume thatπ ˆ(H) is a diagonal matrix for each H ∈ h. The diagonal elementsπ ˆ(H)jj =: λj(H) fix the weights λj, i.e. linear functionals on h.We write λ =(λ(Hˇ1),...,λ(Hˇl)) for an ON-basis of h. By the Theorem of the Highest Weight (Theorem 6.15 and Theorem 6.22), an irreducible representation is determined by its highest weight Λ and this highest 6.7 The Example su(3) 107 weight is simple, i.e. has multiplicity one. Moreover the general theory says ([Co] p.568) that these highest weights are linear combinations with non negative integer coefficients of fundamental weights Λ1,...,Λl defined by

l −1 Λj(H)= (C )kjαk(H) k=1 where α1,...,αl are the positive simple roots, ﬁxed at the beginning. And with δ =(1/2)Σαj the half sum of these positive simple roots Weyl’s dimensionality formula says ([Co] p.570) that the irreducible ﬁnite-dimensional representation (ˆπ, V ) with highest weight Λ has dimension

~~l < Λ+δ, αj > d =Πj=1 . <δ,αj >~~

~~In our case we have 2/31/3 C−1 =( ). 1/32/3 and the fundamental weights √ Λ 1 =(2/3)ˇα1 +(1/3)ˇα2 =(1/6)( 3, 1), Λ 2 =(1/3)ˇα1 +(2/3)ˇα2 =(1/6)(0, 2).~~

~~We writeπ ˆ(n1,n2) for the representation with highest weight Λ = n1Λ1 + n2Λ2 and get by Weyl’s formula for its dimension~~

~~(6.6) d =(n1 + 1)(n2 + 1)((1/2)(n1 + n2)+1).~~

In the physics literature one often denotes the representation by its dimension. Then one has a)π ˇ(0, 0) = {1}, the trivial representation, which has only one weight, namely the highest weight Λ = 0, b)π ˇ(1, 0) = {3} with highest weight Λ = Λ1 and dimension 3, ∗ c)π ˇ(0, 1) = {3 } with highest weight Λ = Λ2 and dimension 3, d)π ˇ(2, 0) = {6} with highest weight Λ = 2Λ1 and dimension 6, ∗ e)π ˇ(0, 2) = {6 } with highest weight Λ = 2Λ2 and dimension 6, f)π ˇ(1, 1) = {8} with highest weight Λ = Λ1 +Λ2 and dimension 8, g)π ˇ(3, 0) = {10} with highest weight Λ = 3Λ1 and dimension 10, and so on. The weights appearing in the representations can be determined by the fact that they are of the form

λ =Λ− m1α1 − m2α2,m1,m2 ∈ N0, and that the Weyl group transforms weights into weights (preserving the multiplicity). We sketch some of the weight diagrams and later discuss an elementary method to get these diagrams, which generalizes our discussion in 6.4 and 6.5. 108 6. The Inﬁnitesimal Method

~~πˇ(1, 0) = {3} λ(Hˇ ) 6 2 √ Λ=(1/6)( 3, 1)=(2/3)α1 +(1/3)α2 t t √ − − sα1 Λ=(1/6)( 3, 1) = (1/3α1 +(1/3)α2 -~~

~~λ(Hˇ1)~~

~~t − − sα2 sα1 Λ=(1/6)(0, 2) = (1/3)α1 (2/3)α2 πˇ(0, 1) = {3∗} λ(Hˇ ) 6 2 Λ=(1/6)(0, 2) = (1/3)α1 +(2/3)α2 t~~

~~ˇ λ(H1) - √ − − − sα2 Λ=(1/6)( 3, 1) = (1/3α1 (1/3)α2 t t √ − − − − sα1 sα2 Λ=(1/6)( 3, 1) = (2/3)α1 (1/3)α2~~

~~πˇ(2, 0) = {6} λ(Hˇ2) 6 Λ − α1 =(1/6)(0, 2) = (1/3)α1 +(2/3)α2 t tt √ − sα1 Λ= (2/3)α1 +(2/3)α2 Λ=(1/6)(2 3, 2) = (4/3)α1 +(2/3)α2~~

~~- − − − ˇ sα1 sα2 (Λ α1)= (2/3)α1 (1/3)α2 λ(H1) t t √ − − − sα2 (Λ α1)=(1/6)( 3, 1) = (1/3)α1 (1/3)α2~~

~~− − − sα2 sα1 Λ=(1/6)(0, 4) = (2/3)α1 (4/3)α2 t 6.7 The Example su(3) 109~~

~~λ(Hˇ ) 6 2 πˇ(1, 1) = {8} u u √ sα1 Λ=α2 Λ=(1/6)( 3, 3) = α1 + α2~~

~~λ(Hˇ1) uuuj - − sα1 sα2 Λ= α1 0=(0, 0) sα2 Λ=α1~~

~~√ − − − − sα1 sα2 sα1 Λ= α1 α2 sα2 sα1 Λ=(1/6)( 3, 3) = α2 uu~~

~~πˇ(3, 0) = {10} 6 λ(Hˇ2)~~

~~uuuu √ − α1 + α2 α2 α1 + α2 Λ=(1/6)(3 3, 3) = 2α1 + α2~~

~~λ(Hˇ1) uuu -~~

~~−α1 0=(0, 0) α1~~

~~−α1 − α2 −α2~~

~~u −α1 − 2α2~~

~~In {8} we find a first example of a weight, which has not multiplicity one, namely the weight (0, 0). 110 6. The Infinitesimal Method~~

Similar to our discussion for G = SU(2) and g = su(2) in 4.2, one has here the problem of the explicit decomposition of representations into irreducible components. This goes under the heading of Clebsch Gordon series and can be found for instance in [Co] p.611ﬀ. As examples we cite

~~{3}⊗{3∗}{8}⊕{1} and {3}⊗{3}⊗{3}{10}⊕2{8}⊕{1}.~~

The proofs of these formulae need some skill or at least patience. But it is very tempt- ing to imagine the bigger weight diagrams in our examples as composition of the tri- angles {3} and {3∗}. This leads to an important interpretation in the theory of elementary particles: We already remarked that the discussion of the irreducible representations of SU(2) resp. SO(3) is useful in the description of particles subjected to SO(3)-symmetry (in particular an electron in an hydrogen atom), namely a representation πj,j =0, 1/2, 1, 3/2,... describes a multiplet of states of the particle with angular momentum resp. spin 2j + 1 and the members of the multiplet are distinguished by a magnetic quantum number m ∈ Z with m = −(2j +1), −(2j − 1),...,(2j + 1). As one tried to describe and classify the heavier elementary particles, one observed patterns, which could be related to the representation theory of certain compact groups, in particular G = SU(3): In Gell-Mann’s eightfold way we take SU(3) as an internal symmetry group to get the ingredients of an atom, the proton p and the neutron n as members of a multiplet of eight states, which correspond to our representation {8}. To be little bit more precise, one looks at a set of hadrons, i.e. particles characterized by an experimentally ﬁxed behaviour (having strong interaction), and describes these by certain quantum numbers, namely electric charge Q, hypercharge Y, baryon number B, strangeness S, isospin I with third component I3. The baryon number is B = 1 for baryons like proton or the neutron, -1 for antibaryons, and zero for other particles like the π-mesons. In this context, one has the following relations

~~Y = B + S, Q = I3 +(1/2)Y~~

and one relates particle multipets to representations of su(3) by assigning an I3 - axis to the axis measuring the weight λ(H1) and an Y -axis to λ(H2). This way, the lowest dimensional nontrivial (three-dimensional) representations correspond to two triples (u, d, s) and (¯u, d,¯ s¯), named quarks resp. antiquarks (with u for up, d for down, and s for nonzero strangeness). We give a list of the quantum numbers characterizing the quarks

quark BI I3 YSQ u 1/31/2 −1/21/302/3 d 1/31/21/21/30−1/3 s 1/30 0 −2/3 −1 −1/3 and for more information refer to p. 50 f of [FS]. In this picture the weight diagrams look like this: 6.7 The Example su(3) 111

~~{3} =: quark {3∗} =: antiquark 6Y 6Y 2/3 u s¯~~

~~u d uu1/3 r 1/3 r~~

~~rrrr- I3~~

~~−1/21/2 I3 −1/21/2~~

~~−1/3 r uu−1/3 r u¯ d¯ −2/3 u s~~

Up to now the experimental evidence for the existence of the quarks (as far as we know) is still indirect: they are thought of as building blocks of the more observable particles like the proton and the neutron. The following diagram shows the quantum numbers of the baryon octet {8}. By a simple addition process (comparing the (I3,Y)-coordinates) one can guess the quark content of the neutron as n =(udd) and of the proton as p =(uud). One can do as well for the other particles showing up in the diagram.

~~πˇ(1, 1) = {8}~~

~~6 Y u r u 1 n p~~

~~0 −1/21∆ /2 I3 uurruj -~~

~~Σ− Σ0 Σ+~~

~~Ξ− Ξ+ uur −1 112 6. The Inﬁnitesimal Method~~

We shall come back to these interpretations of representations later on when we discuss the Euclidean groups and the Poincar´egroup. A broader discussion of the physical content of these schemes is beyond the scope of this book. But we still reproduce the baryon decuplet {10}. It became famous in history because it predicted the existence of an Ω−-particle (the bottom of the diagram) with ﬁxed properties, which after this prediction really was found experimentally.

~~πˇ(3, 0) = {10} 6Y uur uu 1 ∆− ∆0 ∆+ ∆++~~

~~−3/2 −1/21/23/2 rrrruuu - I3 Σ− Σ0 Σ+~~

~~uur −1 Ξ− Ξ0~~

~~− Ω u −2 6.7 The Example su(3) 113~~

To ﬁnish this section, let us look what happens when we apply to our example the procedure, which was successful in our constructions of the representations of su(2) and sl(2, R) in 6.4 and 6.5. We stay in the unnormalized picture with

~~g = with the commutation relations √ √ − [X+,X−]=0, [Y+,Y−]=(1/2)( 3H2 H1), [Z+,Z−]=(1/2)( 3H2 + H1),~~

[X+,Y+]=Z+, [X+,Z−]=−Y−, [X−,Z+]=Y+, [ X−,Y−]=−Z−, [ Y+ ,Z+]=X−, [ Y− ,Z+]=−X+, [ H1,X±]=2X±, [H2,X±]= 0√, ∓ ± [ H1,Y±]= Y±, [ H2,Y±]= √3Y±, ± ± [ H1,Z±]= Z±, [ H2,Z±]= 3Z±, (the others are zero) and the roots with coordinates √ √ α¯1 =(2, 0), α¯2 =(−1, 3), α¯2 =(1, 3). √ (We then get the weights from our diagrams above by multiplication with ( 3/6).) And as we prefer to think positively, we start our construction of a representation space V by taking a vector v ∈ V,v =0 , of lowest weight Λ = (k, l), i.e. with

~~H1v = kv, H2 = lv, X−v = Y−v = Z−v =0.~~

We try to construct V by applying the plus or creation operators X+,Y+,Z+ to v, i.e. we try r s t V = vrstC,vrst := X+Y+Z+v. From the PBW-Theorem we recall that we have to be careful with the order in which we apply the creation operators. For instance, one has − v110 := Y+X+v =(X+Y+ +[Y+,X+])v = v110 v001 since [X+,Y+]=Z+, but Z+X+v = X+Z+v = v101 as X+ and Z+ commute. From the commutation relations it is clear that vrst has weight Λ + rα¯1 + sα¯2 + tα¯3, so, in particular, the elements v100,v010,v001√ of the ﬁrst shell√ have the respective weights coordinatized by (k +2,l), (k − 1,l + 3), (k +1,l + 3). If we apply the minus or annihilation operators X−,Y−,Z−, using the commutation relations, we get

X−v100 = X−X+v =([X−,X+]+X+X−)v = −H1v +0=−kv, and the same way Y−v100 =0, Z−v100 =0, X−v010 =0, √ Y−v010 = −(1/2)(l 3 − k)v, Z−v010 =0, X−v001 = v010, − Y−v001 = v100, √ Z−v001 = −(1/2)(l 3+k)v, 114 6. The Inﬁnitesimal Method

~~Before we interpret this and get into danger to lose track in the next shells, let us see how the natural representation of su(3) looks in this picture: Here we have~~

3 t t t V = ejC,e1 = (1, 0, 0),e2 = (0, 1, 0),e3 = (0, 0, 1), j=1 and hence ⎛ ⎞ ⎛ ⎞ 1 1 ⎝ ⎠ ⎝ ⎠ H1e1 = −1 0 = e1, 0 0 ⎛ ⎞ ⎛ ⎞ √ 1 1 √ ⎝ ⎠ ⎝ ⎠ H2e1 =(1/ 3) 1 0 =(1/ 3)e1. −2 0

~~In the same way we get √ − H1e2 = e2,H2e2 =(1/ √3) e2 − H1e3 =0,H2e3 = (2/ 3) e3.~~

~~Recalling X+ = E12,X− = E21 and the analogous relations for the other matrices, one has~~

~~X+e1 =0,Y+e1 =0,Z+e1 =0, X−e1 = e2,Y−e1 =0,Z−e1 = e3, X+e2 = e1,Y+e2 =0,Z+e2 =0, X−e2 =0,Y−e2 = e3,Z−e2 =0, X+e3 =0,Y+e3 = e2,Z+e3 = e1, X−e3 =0,Y−e3 =0,Z−e3 =0.~~

These relations√ show that, in the coordinates√ fixed above, e1 is a highest weight vector of weight (1√, (1/ 3)), e2 has weight (−1, 3) and e3 is a lowest weight vector with weight (0, −(2/ 3). Thus, as to be expected, we find the weight diagram of {3}. This fits into the construction we started above as follows: we put v1 := v = e3 and get in the first shell translating the equations just obtained

~~X+v = v100 =0,Y+v = v010 = e2 := v2,Z+v = v001 = e1 := v3.~~

Moreover, the equations translate to the fact that the creation operators applied to the highest weight vector v3 and to v2 give zero with the exception X+v2 = v3. The only nontrivial actions of the annihilation operators are Y−v2 = v1, X−v3 = v2 and Z−v3 = v1. We compare this to the general form for the action obtained at the beginning (now with v100 =0,v010 = v2,v001 = v3) and ﬁnd as neccessary condition√ for a three-dimensional representation in our general scheme the condition k =0,l = −2/ 3, just as it should be. 6.7 The Example su(3) 115

~~To show how to come to higher dimensional representations, one has to go to higher shells:~~

2 We have v200 = X+v, v110 = X+Y+v, v101 = X+Z+v and so on. Here we remember to pay attention to the order if the creation operators do not commute: As already − remarked, we have v110 := Y+X+v = v110 v001 since one has [X+,Y+]=Z+. Though it is a bit lengthy we give the list of the action of the annihilation operators (to be obtained parallel to the case X−v100 above and using the actions of the minus operators on the ﬁrst shell) − − X−v110 = (k 1)√v010, − − Y−v110 = (1/2)(l√3 k)v100, Z−v110 = −(1/2)(l 3 − k)v, X−v101 = −(k +1)v001 + v110, − Y−v101 = v200, √ Z−v101 = −(1/2)(l 3+k +2)v100, X−v011 = v200, √ − − − Y−v011 = (1/2)(l√3 k)v001 v110, Z−v011 = −(1/2)(l 3+k +2)v010, X−v200 = −2(k +1)v100, Y−v200 =0, Z−v200 =0, X−v020 =0, √ Y−v020 = −(l 3 − k +2)v010, Z−v020 =0, X−v002 =2v001, − Y−v002 = 2v√101, Z−v002 = −(l 3 − k +2)v001. Just as well one can determine the action of the minus operators in the next shell and so on. As an example, we show how the eight-dimensional representation {8} (with highest weight Λ = α1 + α2) comes up in this picture: As above we start with a lowest weight vector v1 := v, which has in our coordinates the weight Λ0 =(k, l). In the ﬁrst shell we have v2 := X+v = v100,v3 := Y+v = v010,v4 = Z+v = v001 of respective weights Λ0 +α1, Λ0 +α2, Λ0 +α1 +α2. For the second shell we put (secretely looking at the weight diagram we already constructed using the results from the general theory, otherwise the reasoning would take a bit more time)

~~2 X+v = v200 =0,Y+X+v = v110 =: v5,Z+X+v =: v6, − and using the relation v110 = v110 v001 of weight Λ0 +α1 +α2 obtained at the beginning~~

~~2 X+Y+v = v110 = v5 + v4,Y+v =0,Z+Y+v = v011, and, since X+ and Z+ commute~~

2 X+Z+v = v101 = v6,Y+Z+v =: v7,Z+v = v002 =: v8. If we look at the relations coming from the application of the minus operators to the elements of the second shell listed above, we ﬁnd in particular the equations √ X−v200 = −2(k +1)v100,Y−v020 =(l 3 − k +2)v010. 116 6. The Inﬁnitesimal Method

√ Since we have fixed v200 = v020 = 0, these equations lead to k = −1 and l =2/ 3. One verifies easily that all the other relations are fulfilled and the weight diagram for the eightfold way from above in this context leads to the following picture where only part of the relations between the operators and generators are made explicit:

~~πˇ(1, 1) = {8} 6 Y u r √ u 2/ 3~~

~~v7 = Y+v4 v8 = Z+v4~~

~~v5 = Y+v2 1 uurruj - I3 v3 = Y+v1 −1 v4 = Z+v1 v6 = Z+v2~~

~~v1 v2 = X+v1 uur √ −2/ 3~~

~~Exercise 6.35: Do the same to construct the representations {6} and {10}.~~

We hope that these examples help to get a feeling how the discreteness of the weights is related to the ﬁnite-dimensionality of the representations. And, as this procedure gets more and more complicated, one gets perhaps some motivation to look into the proofs of the general theorems in the previous section about the integrality of the weights, the existence of fundamental weights, the action of the Weyl group etc. Chapter 7

~~Induced Representations~~

Induction is in this context a method to construct representations of a group starting by a representation of a subgroup. As we shall see, in its ultimate specialization one rediscovers Example 1.5 in our section 1.3, where we constructed representations in function spaces on G-homogeneous spaces. Already in 1898, Frobenius worked with this method for finite groups. Later on, Wigner, Bargman, and Gelfand-Neumark used it to construct representations of special groups, in particular the Poincarégroup. But it was Mackey, who, from 1950 on, developed a systematic treatment using essentially elements of functional analysis. Because it is beyond the scope of this book, we do not describe these here as carefully as we should. In most places we simply give recipes so that we can construct the representations for the groups we are striving for. We refer to Mackey [Ma], Kirillov [Ki] p.157ff, Barut-Raczka [BR] p.473ff, or Warner [Wa] p.365ff for the necessary background.

~~7.1 The Principle of Induction~~

Let G be a group, H a subgroup, and π0 a representation of H in a space H0, all this with certain properties to be speciﬁed soon. From these data one can construct representations of G in several ways. The guiding principle is the following. We look at a space of functions H consisting of functions φ : G −→ H 0, which allow for some additional conditions also to be speciﬁed below and for the fundamental functional equation

~~1/2 (7.1) φ(hg)=δ(h) π0(h)φ(g), for all h ∈ H, g ∈ G.~~

Here δ : H −→ R>0 is a normalizing function, which has several diﬀerent descriptions. It is identically one in many important cases, so we do not bother with it for the moment (if δ is left out, one has unnormalized induction, which later on will appear in connection with an interpretation via line bundles). If H is closed under right translation, this space is the representation space for the representation

G π = indH π0 given by right translation π(g0)φ(g)=φ(gg0). Before we make all this more precise, in the hope to add some motivation, we treat as an example a representation, which we already treated as one of our ﬁrst examples in 1.3. 118 7. Induced Representations

Example 7.1: Let G be the Heisenberg group Heis(R)={g := (λ, µ, κ); λ, µ, κ ∈ R}, H the subgroup consisting of the elements h := (0,µ,κ) and π0 the one-dimensional representation of H, which for a ﬁxed m ∈ R∗ is given by

~~m 2πimκ π0(h):=e (κ)=e .~~

~~As we know from 3.3 that the Heisenberg group and the subgroup H are unimodular, one has δ ≡ 1 (we shall see this below in 7.1.2) and we have to deal with the functional equation (7.1)~~

~~m φ(hg)=π0(h)φ(g)=e (κ)φ(g), for all h ∈ H, g ∈ G.~~

~~By this functional equation the complex function φ(g)=φ(λ, µ, κ) is forced to have the form~~

~~(7.2) φ(λ, µ, κ)=em(κ + λµ)f(λ) where f is a one-variable function since every g ∈ Heis(R) has a decomposition~~

g =(λ, µ, κ)=(0,µ,κ+ λµ)(λ, 0, 0) =: h0s(λ) with h0 ∈ H and s(λ)=(λ, 0, 0) ∈ G. Application of the right translation with g0 ∈ G (via the muliplication law in the Heisenberg group) leads to

~~(π(g0)φ)(g)=φ(gg0)=φ(λ + λ0,µ+ µ0,κ+ κ0 + λµ0 − λ0µ) m = e (κ + κ0 + λµ + λ0µ0 +2λµ0)f(λ + λ0) m m = e (κ + λµ)e (κ0 +(λ0 +2λ)µ0)f(λ + λ0).~~

~~Comparing with (7.2) above, we rediscover the Schr¨odingerrepresentation from Section 1.3: If we choose H such that in the decomposition of the elements φ above we have f ∈ L2(R), then~~

m f −→ π(g0)f with π(g0)f(x)=e (κ0 +(λ0 +2x)µ0)f(x + λ0) is the prescription for the unitary representation from Example 1.6 in 1.3. The representation given by right translation on the space of functions φ on G of the form (7.2) is called Heisenberg representation.

7.1.1 Preliminary Approach Now we want to see what is behind this example and try a more systematic treatment. At ﬁrst we do not consider the most general case but (following [La] p.43f) one suﬃcient for several applications. Let • G be an unimodular connected linear group, • H be a closed subgroup, • K be an unimodular closed subgroup such that the map

~~H × K −→ G, (h, k) −→ hk is a topological isomorphism, • δ =∆H be the modular function of H, and • π0 a representation of H in H0 . 7.1 The Principle of Induction 119~~

~~Then the homogeneous space of right H-cosets X = H\G can be identiﬁed with K (provided with an action of K from the right). Let Hπ be the space of functions~~

φ : G −→ H 0 satisfying the functional condition (7.1) 1/2 φ(hg)=δ(h) π0(h)φ(g) for all h ∈ H, g ∈ G 2 and the ﬁniteness condition that the restriction φ|K =: f is in L (K). We deﬁne a norm for Hπ by 2 2 2 2 | | (7.3) φ Hπ := f L (K)= f(k) H0 dk, K ˜ ˜ and with φ |K =: f the scalar product by <φ,φ>˜ :=.˜ G We denote π = indH π0 for π given by right translation (π(g0)φ)(g)=φ(gg0) and call this the representation of G induced by π0. We have to show that this really makes sense and shall see at the same time why the strange normalizing function δ has to be introduced.

~~π Theorem 7.1: If π0 is bounded, π deﬁnes a bounded representation of G on H . π is unitary if π0 is.~~

~~g0 g0 π Proof: We have to show that φ given by φ (g):=φ(gg0)isinH . One has~~

g0 1/2 1/2 g0 φ (hg)=φ(hgg0)=δ(h) π0(h)φ(gg0)=δ(h) π0(h)φ (g), i.e. φg0 satiﬁes the functional equation. Now to the ﬁniteness condition: We write g = hk ∈ ∈ and kg0 =: hkk with h, hk H, k, k K. Then we have (with δ =∆H )

~~g0 1/2 φ (k)=φ(kg0)=φ(hkk )=δ(hk) π0(hk)φ(k ),~~

g0 2 and, since π is bounded, φ π is bounded by a constant times 0 H | |2 ∆H (hk) φ(k ) dk, K and equality holds if π0 is unitary. The proof is done if we can show that for f ∈Cc(K) one has f(k)dk = ∆H (hk)f(k )dk. K K We take φ ∈Cc(G) and use Fubini’s Theorem. Since G is unimodular, we get φ(hk)dkdh = φ(g)dg = φ(gg )dg H K G G 0 = φ(hkg )dkdh H K 0 = φ(hh k)dhdk K H k = φ(hk)∆ (h )dhdk K H H k = H K φ(hk )∆H (hk)dkdh. Here we take φ such that

~~φ(hk)=ϕ(h)f(k) with f ∈C(K),ϕ∈Cc(H), ϕ(h)dh =1, H and get the desired result. 120 7. Induced Representations~~

~~Exercise 7.1: Verify that the case of the Heisenberg group and its Schr¨odingerrepresentation is covered by this theorem.~~

Remark 7.1: Using the notions from our section 6.6 where we introduced a bit more structure theory, one can also state a useful variant of Theorem 7.1 (as in Flicker [Fl] p.39): If G has an Iwasawa decomposition G = NAK,K a maximal compact subgroup, A the maximal torus in H = NA, N the unipotent radical, and χ a character of H, the representation space consists of smooth functions φ : G −→ C with

~~φ(nak)=(δ1/2χ)(a)φ(k), for all a ∈ A, n ∈ N,k ∈ K, where the normalizing function can be described by~~

~~δ(a)=| det(Ad a |LieN ) | .~~

~~The representation is given again by right translation.~~

7.1.2 Mackey’s Approach We shall afterwards come back to more easily accessible special cases but now go (following Mackey) to a more general situation (see [Ki] p.187ﬀ, [BR] p.473) where we have to intensify the measure theoretic material from section 3.3. Let • G be a connected linear group, • H be a closed subgroup, • π0 a unitary representation of H in H0 (later on, in some cases, we will also treat non-unitary representations), • δ := ∆H /∆G where ∆H and ∆G are the modular functions of H resp. G introduced in 3.3, •Xthe space of right H-cosets X = H\G, • p : G −→ X the natural projection g −→ Hg =: x ∈X.

~~Theorem 7.2: The normalized induced representation~~

G π := indH π0 is given by right translation on the space Hπ, which is deﬁned as the completion of its dense subspace spanned by the continuous functions φ : G −→ H 0 satisfying the functional equation (7.1)

~~1/2 φ(hg)=(δ(h)) π0(h)φ(g) for all h ∈ H, g ∈ G and the ﬁnite norm condition 2 2 ∞ (7.4) φ Hπ := φ(s(x)) H0 dµs(x) < . X~~

~~Before we elaborate on the fact that we really get a unitary representation, this condition introducing the norm . Hπ needs some explanation: 7.1 The Principle of Induction 121~~

~~Quasi-invariant Measures, Master Equation and the Mackey Decomposition~~

As above, we denote by X the space of left H-cosets, X = H\G, and p : G −→ X the natural projection given by g −→ Hg =: x ∈X. Then it is a general fact that the homogeneous space X admits quasi-invariant measures µ. These are deﬁned by the condition that, for every g ∈ G, µ and µg (with µg(B):=µ(Bg) for all Borel sets B) have the same null sets, i.e. µ(B) = 0 is equivalent to µg(B) = 0. Again we do not prove this general statement but later on give enough examples to illuminate the situation (we hope). For f ∈Cc(X )wehave −1 −1 µg(f)= f(xg)dµ(x)= f(x)dµ(xg ) with dµ(xg )=:dµg(x) X X and for g ∈ G we have a density function ρg, which may be understood as the quotient of dµg and dµ (in the general theory this goes under the name of a Radon-Nikodym derivative). The construction of quasi-invariant measures can be done as follows. We take over (from [BR] p.70:) [Ma1] part I, Lemma 1.1:

Theorem 7.3 (Mackey’s Decomposition Theorem): Let G be a separable locally compact group and let H be a closed subgroup of G. Then there exists a Borel set B in G such that every element g ∈ G can be uniquely represented in the form

~~g = hs, h ∈ H, s ∈ B.~~

~~In our situation where we have~~

p : G −→ X ,g−→ Hg =: x, this leads to the existence of a Borel section s : X−→G, i.e. a map preserving Borel sets with p ◦ s = idX . Hence for g ∈ G, we have a unique Mackey decomposition

~~(7.5) G g = h(x)s(x),h(x) ∈ H, x ∈X.~~

~~Then the invariant measures drh and drg on H resp. G are related to a quasi-invariant measure dµs associated to the section ϕ by~~

∆G(h) drg =: drhdµs(x) ∆H (h) (by a reasoning, which is a reﬁnement of the last part of the proof of Theorem 7.1, or by solving the problems in [Ki] p.132) and we have the quasi-measure relation

dµs(xg) ∆H (h(g, x)) (7.6) = ρg(x)= = δ(h(g, x)). dµs(x) ∆G(h(g, x)) To someone eventually thinking this is complicated, we recommend to look at the case of the Heisenberg group we treated above: We have the section s : X−→G given by X x −→ s(x)=(x, 0, 0) and the Mackey decomposition

~~Heis(R) g =(λ, µ, κ)=h(x)s(x) with h(x)=(0,µ,κ+ λµ),s(x)=(λ, 0, 0),x= λ. 122 7. Induced Representations~~

~~And, since the groups G and H are both unimodular, we simply have~~

~~drg = dλdµdκ, drh = dµdκ, dµs(x)=dx.~~

In this special case, one easily verifies directly that dx is not only quasi-invariant but invariant. And the general finite norm condition above reduces to the condition that f is an L2-function on XR, i.e. the condition we found above while rediscovering the Schrödingerrepresentation.

~~The First Realization~~

In the general case we will need the explicit form of the action of G on X in the framework of the Mackey decomposition: We apply the Mackey decomposition (7.5) to s(x)g0 and ∗ get a consistent picture by chosing xg0 such that, for h := h(xg0) ∈ H (which, for reasons to be seen later, we also write as h(g0,x)), we have the Master Equation (the name is from [Ki1] p.372)

~~∗ (7.7) s(x)g0 = h s(xg0).~~

Now, let us verify that the space Hπ, as deﬁned above, is in fact the space for our induced g0 representation with π(g0)φ = φ : If φ satisﬁes the functional equation (7.1), by the same reasoning as in the proof of g0 Theorem 7.1, so does φ for every g0 ∈ G. The main point is the norm condition. The more experienced reader will see that, in principle, we do the same as in the proof of Theorem 7.1: Using (7.7) above, the functional equation (7.1) and the unitarity of π0, one has from (7.4) g0 2 g0 2 φ π = φ (s(x)) dµ (x) H X H0 s = φ(s(x)g ) 2 dµ (x) X 0 H0 s = φ(h∗s(xg )) 2 dµ (x) X 0 H0 s 2 = X φ(s(xg0)) H0 δ(h(g0,x))dµs(x).

−→ −1 Replacing x xg0 , we see that translation does not change the norm − − g0 2 2 1 1 φ Hπ = X φ(s(x)) H0 δ(h(g0,xg0 ))dµs(xg0 ) 2 2 = X φ(s(x)) H0 dµs(x)= φ Hπ , since from the Master Equation (7.7) (as expression of the associativity of the group operation) one deduces the cocycle condition

~~(7.8) h(g1,x)h(g2,xg1)=h(g1g2,x).~~

~~Via the chain rule the quasi-measure relation (7.6) dµs(xg0)/dµs(x)=ρg0 (x)=δ(h(x, g0)) provides for −1 −1 δ(xg0 ,g0)dµs(xg0 )=dµs(x). It is now evident that π is unitary if we put~~

~~<φ,ψ>:= <φ(s(x)),ψ(s(x)) >H0 dµs(x). X 7.1 The Principle of Induction 123~~

Thus we have the essential tools to extend the validity of Theorem 7.1 to the more general hypothesis stated above (we leave out to show that changing the section ϕ leads to equivalent representations). This realization of the induced representation by functions living on the group is usually called the ﬁrst realization or induced picture.

Before we go over to another realization, let us point out that the representation by right translation has an equivalent version where G acts by left translation, i.e. we have −1 Hπ π (g0)φ(g)=φ(g0 g) where here we have the space consisting of functions φ fulﬁlling the fundamental functional equation

~~−1 1/2 −1 (7.9) φ(gh)=δ(h ) π0(h )φ(g) for all h ∈ H, g ∈ G, and the same ﬁniteness condition.~~

~~Exercise 7.2: Fill in missing details.~~

~~The Second Realization~~

~~We stay with the same hypotheses as above and turn to functions living on the homogeneous space X = H\G. We denote by~~

2 Hπ := L (X ,dµs, H0) the space of H0-valued functions f on X , which are quadratic integrable with respect to the quasi-invariant measure dµs. In this context we have as fundamental fact the following

~~Lemma: The map π ψ : H −→ H π,φ−→ f given by f(x):=φ(s(x)) for x ∈X is bijective and an isometry.~~

~~Proof: ψ is linear and has as its inverse the map~~

~~π ϕ : Hπ −→ H ,f −→ φf~~

1/2 for g = h(x)s(x),x∈X given by φf (g):=δ(h(x)) π0(h(x))f(x). One has 2 2 φ π = φ (s(x)) dµ (x) f H X f H0 s = f(x) 2 dµ (x)= f 2 . X H0 s Hπ Hence the maps preserve the norm. Both spaces have compatible Hilbert space structure since in a complex inner product space V for u1,u2 ∈ V one has the relation

~~2 2 2 4 = u1 + u2 − u1 − u2 −i u1 + iu2 +i u1 − iu2 .~~

~~Exercise 7.3: Prove this.~~

~~π Using this, we can carry over the induced representation from H to Hπ and from our Theorem 7.2 come to an equivalent picture as follows. 124 7. Induced Representations~~

~~G H Theorem 7.4 (Second Realization): π = indH π0 is realized on π byπ ˜ with~~

~~π˜(g0)f(x):=A(g0,x)f(xg0) for all f ∈Hπ with 1/2 A(g0,x):=δ(h(g0,x)) π0(h(g0,x)) for s(x)g0 = h(g0,x)s(xg0),x∈X,g0 ∈ G.~~

~~Proof: We evaluate the commutative diagram:~~

~~π(g0) Hπ - Hπ 6 ϕ ψ~~

~~π˜(g ) ? 0 - Hπ Hπ~~

~~π • f ∈Hπ is mapped by ϕ onto φ ∈H with~~

~~1/2 φ(g)=φ(h(x)s(x)) = δ(h(x)) π0(h(x))f(x) for g = h(x)s(x) ∈ G, x ∈X.~~

~~g0 g0 • π(g0) maps φ to φ with φ (g)=φ(gg0). • ψ maps φg0 to f˜ with~~

~~˜ g0 f(x)=φ (s(x)) = φ(s(x)g0).~~

~~By the Master Equation s(x)g0 = h(g0,x)s(xg0) and the deﬁnition of φ this can be written as~~

~~˜ 1/2 f(x)=φ(h(x, g0)s(xg0)) = δ(h(g0,x)) π0(h(g0,x))f(xg0). ˜ ˜ Hence,π ˜(g0) transports f to f =˜π(g0)f with f as given in the Theorem.~~

Exercise 7.4: Repeat this discussion to get a representation space consisting of functions f living on the space of left cosets X = G/H by starting from the induced representation π given by left translation on the space Hπ. Deduce the representation prescription

~~−1/2 −1 −1 (7.10) π˜(g0)f(y):=(δ(h(g0,y)) π0((h(g0,y) )f(g0 y) based on the Master equation~~

~~−1 −1 g0 s(y)=s(g0 y)h(g0,y).~~

Remark 7.2: In the special case that π0 is the trivial one-dimensional representation with π0(h) = 1 for all h ∈ H, we fall back to the situation of our Example 1.5 in Section 1.3 where we constructed a representation on a space of functions on a homogeneous space X = H\G. This may be seen as starting point of the construction of induced representations. 7.1 The Principle of Induction 125

~~Remark 7.3: From the cocycle condition (7.8) for h = h(g, x), one can deduce that the multiplier system A in Theorem 7.4 as well fulﬁlls this relation, i.e. we have~~

A(g1g2,x)=A(g1,x)A(g2,xg1) for all g1,g2 ∈ G, x ∈X. And, vice versa, one can take such a system as tool to construct a representation on a homogeneous space X with a right G-action (see for instance [Ki1] p.388).

For instance in [BR] p.478 a proof can be found that essentially all this does not depend on the choice of the section s. In most practical cases one has a more or less natural decomposition of the given group into subgroups as at the beginning of this section, so one can avoid to talk about diﬀerent sections, since one has a natural decomposition. We already discussed the Heisenberg group as an example for this. Here the Schr¨odinger 2 representation came out as the Second Realization withπ ˜ = πm acting on Hπ = L (R). The First Realization by functions living on the group is called Heisenberg representation .

7.1.3 Final Approach Before we give more concrete examples, we reproduce some general considerations (which may be skipped at first reading) linked to the report on structure theory in section 6.6 and take over the following situation over from [Kn] p.168: Let G be (not uniquely) decomposed as G = KH and H in Langlands decomposition H = MAN. We do not give all the details for a proper definition as to be found in [Kn] p.132. But it may suffice to say that one has a corresponding direct sum decomposition with corresponding Lie algebras g = k ⊕ m ⊕ a ⊕ n where a is abelian, n nilpotent, and m and a normalize n. (Our standard example is G = SL(2, R). Here we have as in section 6.6.4 the (unique) Iwasawa decomposition G = KAN with K = SO(2),Athe positive diagonal elements, + N = {n(x); x ∈ R}, and H = MAN with M = {±E2}.) Let ∆ denote the roots of (g, a) positive for N and ρ := (1/2)Σα∈∆+ (dim gα)α. Moreover, let σ be an irreducible unitary representation of M on V σ, and ν ∈ (a∗)C, i.e. a complex linear functional on a. Then in this context, the induced representation is written as G ⊗ ⊗ π = indMAN(σ exp ν 1) −1 and in the induced picture given by left translation π(g0)φ(g)=φ(g0 g) on the representation space, which is given by the dense subspace of continuous functions φ : G −→ V σ with functional equation φ(gman)=e−(ν+ρ) log aσ(m)−1φ(g) and the norm φ 2 := |φ(k)|2dk. K By restriction to the compact group K one gets the compact picture. A dense subspace is given by continuous functions f : K −→ V σ with f(km)=σ(m)−1f(k) and the same norm as above. Here the representationπ ˜ is given by

~~−1 −(ν+ρ)H(g0 k) −1 −1 −1 π˜(g0)f(k):=e σ(µ(g0 k)) f(κ(g0 k)) where g decomposes under G = KMAN as g = κ(g)µ(g)eH(g)n. 126 7. Induced Representations~~

~~In [Kn] p.169 there is a third picture, the noncompact picture, given by restriction to N¯ =ΘN. Here the representationπ ˜ is given by~~

−1 −(ν+ρ) log a(g0 n¯) −1 −1 −1 π˜(g0)f(¯n):=e σ(m(g0 n¯)) f(¯n(g0 n¯)) if one takes account of the fact that NMAN¯ exhausts G except for a lower dimensional set and that in general one has a decomposition g =¯n(g)m(g)a(g)n. The representation space is here L2(N,e¯ 2Re νH(¯n)dn¯).

~~7.1.4 Some Questions and two Easy Examples Several questions arise immediately:~~

π i) How to construct non-trivial functions in H resp. Hπ? ii) When is an induced representation π irreducible? iii) Which representations can be constructed by induction? iv) If one restricts an induced representation to the subgroup, does one get back the representation of the subgroup? v) What happens by iteration of the induction procedure? vi) What about the induced representation if the inducing representation is the sum or tensor product of two representations ?

~~Here we can not give satisfying answers to these questions (but hope that our examples later on will be suﬃciently illuminating). We only mention some remarks.~~

~~Ad i) One can prove ([BR] p.477) the following statement. Proposition 7.1: Let ϕ be a continuous H0-valued function on G with compact support. Deﬁne φϕ by −1 φϕ(g):= π0(h )ϕ(hg)drh. H~~

~~Then φϕ is a continuous function on G whose support goes to a compact set by the projection p : G −→ X and is an element of Hπ. The set~~

~~π C := {φϕ; ϕ(g)=ξ(g)v, ξ ∈Cc(G),v ∈H0} is a dense set in Hπ.~~

Ad ii) We have seen that the Schrödingerrepresentation of the Heisenberg group is an irreducible induced representation. In general, this is not the case and in our examples below we will have to discuss this. We shall find conditions to be added to the definition of our space if one wants to come to an irreducible representation (e.g. holomorphic induction). In [BR] p.499, there are general criteria to get irreducibility (using the concept of a canonical system of imprimitivity).

Ad iii) This concept of canonical system of imprimitivity is also used to give a criterium of inducibility stating under which conditions a unitary representation is equivalent to an induced representation. One ﬁnds versions of this in [BR] p.495, [Ki] p.191, and in [Ki1] p.389 (Mackey Inducibility Criterion). We do not reproduce these because the statements need more new notions. 7.1 The Principle of Induction 127

Ad iv) Induction and restriction are adjoint functors. We do not try to make this more precise. At least for ﬁnite groups, one ﬁnds statements of the Frobenius Reciprocity as in [Se] p.57 or [Ki] p.185, which, using our notation (see section 1.2), for a representation π of G can be written as G | c(π, indH π0)=c(π H ,π0). Ad v) One has the concept of induction in stages, which symbolically can be written as

~~G H G indH indK indK if we have a subgroup chain G ⊃ H ⊃ K ([Ki] p.184, [BR] p.489).~~

~~Ad vi) If π0 and π0 are representations of the subgroup H of G the operations of taking direct sum and induction are interchangeable~~

~~G ⊕ G ⊕ G indH (π0 π0) indH π0 indH π0~~

~~([BR] p.488). And if π1 and π2 are unitary representations of the closed subgroups H1 and H2 of the separable, locally compact groups G1 resp. G2, then one has ([BR] p.491)~~

G1×G2 ⊗ G1 ⊗ G2 indH1×H2 (π1 π2) indH1 π1 indH2 π2. Up to now, we treated only the example of the Heisenberg group. Before we do more serious work with G = SL(2, R), we discuss another example we already looked at, namely G = SU(2), and the two-dimensional form of the Poincar´egroup, which later on will be discussed more thoroughly in the four-dimensional form in the context of Mackey’s method for semidirect products.

~~Example 7.2: Let~~

~~An a G = {g =( ) ∈ SL(3, R); A ∈ SO(1, 1),n=( ) ∈ R2} 01 t be the two-dimensional Poincar´egroup. We abbreviate g =: (A, n), so that matrix multiplication leads to the composition law~~

~~gg =(AA,An + n).~~

~~In analogy to the case of SO(2), we write the elements of SO(1, 1) in the form~~

~~cosh ϑ sinh ϑ A =( )=:rh(ϑ),ϑ∈ R. sinh ϑ cosh ϑ~~

~~From the addition theorem we have AA = rh(ϑ)rh(ϑ)=rh(ϑ + ϑ), and dϑ is a biinvariant measure on SO(1, 1). We take the subgroup~~

~~E n H := {h =( 2 ); n ∈ R2} 01 and the one-dimensional representation π0 deﬁned by~~

~~t α π (h):=eπi(aα+tβ) = eπi nnˆ =: χ (n), nˆ := ( ) ∈ R2. 0 nˆ β 128 7. Induced Representations~~

Moreover, we denote by K the image of the injection of SO(1, 1) into G A 0 K := {k =( ); A ∈ SO(1, 1)} 01 Then every g ∈ G has a Mackey decomposition g =(A, n)=(E,n)(A, 0) and we have G −→ X = H\G K R,g=(A, n) −→ A = rh(ϑ) −→ ϑ.

~~The Master Equation s(x)g0 = h(g0,x)s(xg0) in this case simply is~~

(A, 0)(A0,n0)=(AA0,An0)=(E,An0)(AA0, 0). We see that dϑ is not only quasi-invariant but invariant and we have a trivial normalizing ∈ G function δ(h) = 1 for all h H. Hence, the induced representation π = indH χnˆ is given by right translation on the space Hπ, which is the completion of the space of continuous functions 2 φ : G −→ C,φ(hg)=χnˆ (h)φ(g), | φ((rh(ϑ), 0)) | dϑ < ∞. R 2 The induced representation has its Second Realization on the space Hπ = L (X ,dµ(x)) = L2(R,dϑ). We have the isomorphism given by π H φ −→ f ∈Hπ with f(ϑ):=φ((rh(ϑ), 0)), and the prescription for the representation from Theorem 7.4, namely

π˜(g0)f(x)=A(g0,x)f(xg0), in this case leads to t πi n0rh(ϑ)ˆn π˜((rh(ϑ0),n0))f(ϑ)=χnˆ (rh(ϑ)n0)f(ϑ + ϑ0)=e f(ϑ + ϑ0). The representation is irreducible. This is a byproduct of Mackey’s general theory for representations of semidirect products, upon which we will report later in section 7.3. The reader is invited to ﬁnd a direct proof using the inﬁnitesimal method analogous to the example in 6.5 (Exercise 7.5). In the next example the induced representation will come out to be reducible.

Example 7.3: For the group ab G = SU(2) = {g =( ); a, b ∈ C, | a |2 + | b |2=1} −¯b a¯ we obtained in 4.2 the result that every irreducible representation of G is equivalent to (j) a representation πj,j ∈ (1/2)N0, given on the space V = C[x, y]2j of homogeneous polynomials P of degree 2j by the prescription ¯ πj(g0)P (x, y):=P (¯a0x − by0, b0x + a0y). This formula is based on the action on functions produced by − x −→ −1 x t x a¯0x b0y ( ) g0 ( )= g¯0( )=( ¯ ). y y y b0x + a0y We want to see how this ﬁts into our scheme of construction of induced representations: It is very natural to take as subgroup H the group of diagonal matrices a H := {h =( ); a = eiϕ,ϕ∈ R}U(1) SO(2) a¯ 7.1 The Principle of Induction 129

ikϕ and as its representation π0 given by π0(h)=e =: χk(h),k ∈ N0. Since in πj we have a left action, we deal here also with an action of G by left translation, i.e. −1 π(g0)φ(g)=φ(g0 g) for functions φ from a representation space Hπ, which fulﬁll the functional equation −1 k −iϕ φ(gh)=π0(h )φ(g)=λ φ(g),λ= e . (We know that G and H are compact and, hence, unimodular so that the normalizing factor δ is trivial.) The homogeneous space X = G/H showing up here is one of the most fundamental examples of homogeneous spaces: One has SU(2)/H P1(C), the one-dimensional complex projective space. We already met with projective spaces in 4.3 while introducing the concept of projective representations. Here we use the following notation: G = SU(2) acts on V = C2 by x (g, v) −→ gv, v := ( ). y

1 ∗ We have P (C)=(V \0)∼ where for v, v ∈ V one has v ∼ v iﬀ there exists a λ ∈ C 1 with v = λv. The action of G on V induces an action on P (C) given by g(v∼):=(gv)∼. t t t We write (x : y)= (x, y)∼.Forv0 = (0, 1) we have gv0 = (b, a¯), i.e. g(0 : 1) = (b :¯a), and the stabilizing group of (v0)∼ = (0 : 1) is just H U(1). Hence we have ab SU(2)/H −→ P1(C),g=( ) −→ (b :¯a). −¯b a¯ A homogeneous function in two variables F (x, y) induces via f(x : y):=F (x, y)a 1 function f on P (C) , and this in turn induces a function φF living on G given by t t t φF (g):=F ( (g (0, 1))) = F ((0, 1) g). Hence we associate the function φP to a homogeneous polynomial P of degree k =2j from our representation πj, namely ab φ (g):=P (b, a¯) for g =( ). P −¯b a¯ The fact that P is homogeneous of degree k translates to the functional equation t −iϕ t −iϕk t k φP (gh)=P ((0, 1) (gh)) = P ((0,e ) g)=e P ((0, 1) g)=λ φP (g) eiϕ where h =( ) and λ = e−iϕ as above. One has e−iϕ ¯ −1 a¯0 −b0 ab a¯0a + b0b a¯0b − b0a¯ g0 g =( ¯ )( ¯ )=( ¯ ¯ ¯ ) b0 a0 −b a¯ b0a − a0b b0b + a0a¯ and hence −1 − ¯ φP (g0 g)=P (¯a0b b0a,¯ b0b + a0a¯)=φπ(g0)P (g).

We see that πj is a candidate for a subrepresentation of the induced representation G π = indH χk. We propose to show as Exercise 7.6 that the ﬁnite norm condition is fulﬁlled and to search for further conditions to be imposed such that we get a subspace π (j) of H , which is isomorphic to the space V of πj. We shall return to this question and the example in Section 7.6 in the framework of the more sophisticated and elegant concept of an interpretation of the induction process via line bundles. 130 7. Induced Representations

~~7.2 Unitary Representations of SL(2, R)~~

Serge Lang dedicated a whole book, namely “SL(2, R)” ([La1]), to this group and its representations. Also, there is a lot of refined material concerning the representations of SL(2, R) throughout Knapp’s book [Kn]. Summaries presenting models for the different representations can be found in many places, for instance in [KT] p.1–24 or in [Do]. The most classical reference is Bargmann’s article [Ba] from 1947. We shall need the representations of G = SL(2, R) later in the discussion of the representations of the Poincaré group and most essentially in the Outlook to Number Theory.

Using the Infinitesimal Method, in Theorem 6.2 in section 6.4, we obtained a list fixing the possible equivalence classes of irreducible unitary representations of G = SL(2, R). Here we will show that these representations really exist by presenting concrete realizations or, as one also says, models. In most occasions it is important to have different models, depending on the occasion where the representation is showing up. The method we follow here is just the application of the induction procedure that we developed in the preceding section. Thus, we have to start by choosing a subgroup H of our G. Already in several occasions in this text, we specified subgroups of G = SL(2, R). We recall (for instance from 6.6.4) the notation

cos ϑ sin ϑ K := SO(2) = {r(ϑ)=( ); ϑ ∈ R}, − sin ϑ cos ϑ y1/2 A := {t(y)=( ); y ∈ R,y >0}, y−1/2 1 x N := {n(x)=( ); x ∈ R}, 1 1 N¯ := {n¯(x)=( ); x ∈ R}, x 1 M := {±E2}, ab B := MAN = {h =( ); a, b ∈ R,a=0 }, a−1 B¯ := MAN.¯

The subgroup B of upper triangular matrices is a special example of a parabolic subgroup. It is also called the standard Borel group. One has the disjoint Iwasawa decomposition G = KAN = ANK and G = KB = BK with B ∩ K = {±E2}. It makes sense to start the induction procedure from all these subgroups but the best thing to do is parabolic induction:

~~1. The Principal Series~~

~~We take H := B and π0 the one-dimensional representation, which for ﬁxed s ∈ R and ∈{0, 1}, is given by is π0(h):=| a | (sgn a) . We put X := H\G and take the Mackey decomposition (7.5)~~

~~ab d−1 b 1 g =( )=( )( )=h(x)s(x) for d =0 , cd d c/d 1 b −a 1 =( )( )=h(x)s(x) for d =0. b−1 −1 7.2 Unitary Representations of SL(2, R) 131~~

1 1 Hence X is in bijection to the set S := {( ); x ∈ R}∪{( )}. As in our x 1 −1 1 context a point has measure zero, we can disregard the element ( ) and work −1 with G −→ X = H\G ⊃X R,g−→ x := c/d, d =0

~~1 a0 b0 with section s : X −→ G, x −→ s(x)=( ). For g0 =( ) our Master x 1 c0 d0 Equation (7.7) s(x)g0 = h(g0,x)s(xg0) is realized by~~

~~−1 (b0x + d0) b0 1 a0x + c0 h(x, g0)=( ),s(xg0)=( ),xg0 = . 0 b0x + d0 xg0 1 b0x + d0~~

We see that dµ(x)=dx is a quasi-invariant measure on X R with the quasi-measure relation (7.6) dµ(xg0) 1 = 2 =: ρg0 (x)=δ(h(g0,x)). dµ(x) (b0x + d0) 2 G Hence, we have δ(h)=a and the induced representation π = indH π0 is given in the First Realization by right translation on the space Hπ, which is the completion of the space of smooth functions φ : G −→ C with

1/2 (7.11) φ(hg)=(δ(h)) π0(h)φ(g) for all h ∈ H, g ∈ G and φ 2 := | φ(s(x)) |2 dx < ∞. R As we saw in our discussion of the representations of the Heisenberg group, it is more illuminating to look at the Second Realization. This is given by the representation space 2 2 Hπ = L (X ,dµ(x)) = L (R,dx). We see from Theorem 7.4 (or by a direct calculation) that the representation acts by

−is−1 a0x + c0 2 π˜(g0)f(x)=| b0x + d0 | (sgn (b0x + d0)) f( ) for f ∈ L (R,dx). b0x + d0 Exercise 7.7: Determine the derived representation and verify that this induced representation is a model for the up to now only hypothetical representation πis,± from Theorem 6.2 in 6.4.

~~Remark 7.4: This exercise provides as byproduct a proof that this induced representation is irreducible. There are more direct ways to see this (see for instance [Kn] p.36).~~

Exercise 7.8: We propose to verify in a special situation that the construction essentially does not depend on the choice of the quasi-invariant measure: As in [BR] p.483, repeat the former discussion for the quasi-invariant measure dµ(x)=ϕ(x) dx 2 −2 ˜ with ϕ(x)=(1+x ) to get a representation space Hπ linked to our Hπ by the intertwining operator f˜ −→ f, f(x)=f˜(x)/(1 + x2)1/2. 132 7. Induced Representations

To get more familiarity with these constructions and because of the general importance of the result, we also discuss the realization of our induced representation by left translation, −1 s.t., for functions φ living on G, we have a representation given by π (g0)φ(g)=φ(g0 g). ab We keep H and π as above and now put X = G/H.Forg =( ) with a =0,we 0 cd have a Mackey decomposition 1 ab g = s(y)h(y),s(y)=( ),y= c/a, h(y)=( ), y 1 a−1 i.e. a map p : G −→ G/H = X ⊃ X˙ R,g−→ y = c/a, 1 with section y −→ s(y)=( ). The Master Equation is in this case y 1 −1 −1 g0 s(y)=s(g0 y)h(y, g0) with − −1 1 −1 a0y c0 s(g0 y)=(−1 ),g0 y = , g0 y 1 −b0y + d0

~~d0 − b0y −b0 ∗ h(y, g0)=( −1 )=:h . (d0 − b0y)~~

We take dµ(y)=dy as quasi-invariant measure on X˙ R and have the quasi-measure relation −1 dµ(g0 y) 1 = 2 = ρg0 (y). dµ(y) (−b0y + d0) To be on the safe side and to provide for some more exercise in the handling of these notions, we verify that left translation preserves the norm: We take a function φ with

~~−1/2 −1 φ(gh)=(δ(h)) π0(h )φ(g) for all h ∈ H, g ∈ G,~~

−1 dµ(˜y) dµ(g0 y) 1 ∗ −1 = = 2 = δ(h ) . dµ(y) dµ(y) (−b0y + d0) So we have as the First Realization the induced representation deﬁned by functions φ on G with the functional equation

~~−1/2 −1 φ(gh)=(δ(h)) π0(h )φ(g) for all h ∈ H, g ∈ G, and the norm condition φ 2 = | φ(s(y)) |2 dµ(y) < ∞. R 7.2 Unitary Representations of SL(2, R) 133~~

~~1 From here, by the map φ −→ f, f(y):=φ(( )), we come to the Second Realization y 1 2 given on the space Hπ = L (R,dy) by the prescription~~

−1/2 −1 −1 π˜(g0)f(y)=(δ(h(y, g0))) π(h(y, g0) )f(g y) 0 − −is−1 a0y c0 = | d0 − b0y | (sgn(d0 − b0y)) f( ). −b0y + d0 This is the same formula as in [KT] p.17 and [Kn] p.36 and p.167 where this is an example for a noncompact picture. We remark that Knapp comes to this formula from his induced picture, which is prima facie different from our First Realization as he does not use the definition of the norm we used above but the same definition of the norm as in (7.3) φ 2 = | φ(k) |2 dk. K In [Kn] p.168, one finds a proof that this norm and the one we used above are equal. We propose to search here for a direct proof (Exercise 7.9).

~~Moreover, in [Kn] p.37, there is a proof of the irreducibility of these induced representations πis,± except for π0,−, which is based on euclidean Fourier Transform.~~

~~2. The Discrete Series~~

~~We will discuss several models, which realize the discrete series representations.~~

~~The Upper Half Plane Model~~

ikϑ We take H = K = SO(2) and π0(h)=χk(r(ϑ)) = e ,k ∈ Z, for h = r(ϑ) ∈ SO(2). ± It is a custom to construct the representations πk from our list in Theorem 6.2 in 6.4 by functions living on the space of left cosets G/K = SL(2, R)/SO(2). This space is again an eminent example of a homogeneous space. It is called the Poincar´eupper half plane and we denote it by H := {τ = x + iy ∈ C; y>0}. Since this is important for our constructions here but also as a standard example in complex function theory, we assemble some related standard notions and facts.

~~ab For g =( ) ∈ SL(2, R) and τ ∈ H, we put cd aτ + b g(τ):= cτ + d and call this a linear fractional transformation.~~

Exercise 7.10: Prove that this is well deﬁned and provides a transitive left action of G = SL(2, R) (even of GL(2, R)) on H. Verify that H = SO(2) is the stabilizing group of τ = i. Show that the automorphic factor

~~j(g, τ):=(cτ + d)−1 fulﬁlls the cocycle relation (7.8) (in the version for the action from the left)~~

~~j(gg,τ)=j(g, g(τ))j(g,τ). 134 7. Induced Representations~~

One has a map p : G = SL(2, R) −→ SL(2, R)/SO(2) H,g−→ g(i)=:τ = x + iy for the Iwasawa decomposition g = n(x)t(y)r(ϑ), which here can be interpreted as a Mackey decomposition g = s(τ)h, s(τ)=n(x)t(y),h= r(ϑ). Moreover, one easily veriﬁes the following formulae 1 ac + bd d − ic ab (7.12) y = ,x= ,eiϑ = √ for g =( ) c2 + d2 c2 + d2 c2 + d2 cd and, for g = r(ϑ)t(y)n(x), a −c 1 ab + cd (7.13) cos ϑ = √ , sin ϑ = √ ,y = ,x = . a2 + c2 a2 + c2 a2 + c2 a2 + c2 Exercise 7.11: Verify d(g(τ)) 1 = dτ (cτ + d)2 and show that dµ(τ)=y−2dxdy is a G−invariant measure on H.

This is also an occasion to practise again the calculation of infinitesimal operations initiated in 6.2 and 6.3: For a smooth complex function φ on G and U ∈ g, we get a right invariant differential operator RU defined by d R φ(g):= φ((exp tU)−1g) | U dt t=0 and a left invariant operator LU by d L φ(g):= φ(g (exp tU)) | . U dt t=0 From 6.4 we recall that we have

g = sl(2, R)={X ∈ M2(R); Tr X =0} = with 1 1 F =( ),G=( ),H=( ) 1 −1 and gc := g ⊗R C = C = C where 1 ±i −i X± =(1/2)(H ± i(F + G)) = (1/2)( ),Z = −i(F − G)=( ) ±i −1 i with [Z, X±]=±2X±, [X+,X−]=Z. Using the coordinates x, y, ϑ of the Iwasawa decomposition as parameters for g and the commutation rule t(y)n(x)=n(yx)t(y), one obtains

~~R d −1 | H φ(g)= dt φ((exp tH) g) t=0 e−t = d φ(( )n(x)t(y)r(ϑ)) | dt et t=0 d −2t −2t | = dt φ(n(e x)t(e y)r(ϑ)) t=0~~

~~= −(2x∂x +2y∂y)φ(g). 7.2 Unitary Representations of SL(2, R) 135~~

~~The same way one has~~

~~R d −1 | F φ(g)= dt φ((exp tF ) g) t=0 1 −t = d φ(( )n(x)t(y)r(ϑ)) | dt 1 t=0 d − | = dt φ(n(x t)t(y)r(ϑ)) t=0~~

= −∂xφ(g), and R d −1 | Gφ(g)= dt φ((exp tG) g) t=0 1 = φ(( )n(x)t(y)r(ϑ)) | −t 1 t=0 = φ(n(x )t(y )r(ϑ )) |t=0 2 2 =((x − y )∂x +2xy∂y + y∂ϑ)φ(g), where in the last formula we used (7.12) to get

~~x − t(x2 + y2) x = , t2y2 +(1− tx)2 y y = , t2y2 +(1− tx)2 1 − t(x − iy) eiϑ = eiϑ . t2y2 +(1− tx)2~~

~~Hence we have the diﬀerential operators~~

~~RF = −∂x, 2 2 (7.14) RG =(x − y )∂x +2xy∂y + y∂ϑ,~~

~~RH = −2x∂x − 2y∂y, which combine to~~

~~2 2 RZ = i((1 + x − y )∂x +2xy∂y + y∂ϑ), (7.15) R ± ± 2 − 2 ± X± = (i/2)(((x i) y )∂x +2(x i)y∂y + y∂ϑ).~~

~~A similar calculation (Exercise 7.12) leads to~~

~~LZ = −i∂ϑ, (7.16) L ± ±2iϑ ∓ − X± = (i/2)e (2y(∂x i∂y) ∂ϑ).~~

~~In 6.4 we already met with the (multiple of the Casimir) element~~

2 Ω:=X+X− + X−X+ +(1/2)Z , which, by the way, is also (Exercise 7.13)Ω=FG+ GF +(1/2)H2. By the natural extension of the deﬁnition of the diﬀerential operators from g to U(gc) (via RUU := RU RU ) and a small calculation we get the SL(2, R)-Laplacian R L 2 2 2 − (7.17) ∆ := Ω = Ω =2y (∂x + ∂y ) 2y∂x∂ϑ. 136 7. Induced Representations

After these general preparations, we finally come to the discrete series representations. There are a lot of different models in the literature. We present the most customary one and then discuss how it fits into the scheme of the induction procedure. For instance in [La1] p.181, we find the representation πm with the representation space H 2 ≥ m := Lhol(H,µm),m 2. This space consists of holomorphic (i.e. complex differentiable) functions f on the upper half plane H, which have a finite norm f m < ∞, i.e. with dxdy 2 | |2 m ∞ f m := f(x, y) y 2 < H y and one has an scalar product defined by m dxdy := f1(x, y)f2(x, y)y 2 . H y

As will be apparent soon (even for those readers, which are not so familiar with complex function theory), the main point is here reduced to the fact that holomorphic functions f are essentially smooth complex valued functions characterized by the Cauchy Riemann diﬀerential equation

(∂x + i∂y)f =0. ab We write f(τ):=f(x, y) and put g−1 =: ( ) (this is some trick to get nicer formulae cd but, if not remembered carefully, sometimes leads to errors). Then the representation πm is in the half plane model given by

~~−1 −1 −1 −m (7.18) πm(g)f(τ):=f(g (τ))jm(g ,τ)=f(g (τ))(cτ + d) .~~

~~Here we have used the important notation of the automorphic factor~~

~~m jm(g, τ):=j(g, τ) extending the one introduced in Exercise 7.10.~~

~~Remark 7.5: In [Kn] p.35, one ﬁnds a representation given on the same space by the prescription ab aτ − c πˇ ( ))f(τ):=(−bτ + d)−mf( ). m cd −bτ + d~~

~~And in [GGP] p.53, the authors deﬁne a representation Tm given by~~

~~ab aτ + c T ( ))f(τ):=(bτ + d)−mf( ). m cd bτ + d~~

The reader is invited to compare this to our deﬁnition. We stick to it because we want to take over directly some proofs from Lang’s presentation in [La1] p.181ﬀ. Namely, one has to verify several things:

~~– Hm is complete and non-trivial. – π fulﬁlls the algebraic relation π(gg)=π(g)π(g). – π is continuous and unitary. 7.2 Unitary Representations of SL(2, R) 137~~

The algebraic relation is an immediate consequence of the facts stated as Exercise 7.10. We verify the unitarity using Exercise 7.11: ∞ ∞ π(g)f 2 = | f(g−1τ) |2| cτ + d |−2m ym−2 dxdy 2 0 −∞ ∞ ∞ | |2 m−2 = 0 −∞ f(τ) y dxdy 2 = f 2. For the other statements we refer to Lang. The idea is to transfer the situation from the upper half plane to the unit disc where the function theory is more lucid. Since this is an important technique, we report on some facts:

~~The Unit Disc Model~~

By the Cayley transform c the Poincar´ehalf plane H is (biholomorphically) equivalent to the unit disc D := {ζ ∈ C; | ζ | < 1} : In all cases, this bijective map sends the point i to 0 but otherwise is not uniquely ﬁxed in the literature. We take the map τ − i 1 −i c : H −→ D,τ−→ ζ := = C(τ) for C =( ), τ + i 1 i which has as its inverse the map given by ζ +1 ii ζ −→ τ = −i = C−1(ζ) with C−1 =(1/(2i))( ). ζ − 1 −11 It is a standard fact that G = SL(2, R) is isomorphic to αβ G := SU(1, 1) = {g =( ); | α |2 −|β |2=1}. β¯ α¯ We take ab αβ c˜ : SL(2, R) −→ SU(1, 1),g=( ) −→ g := CgC−1 =( ) cd β¯ α¯ with (7.19) α = α(g)=(1/2)(a + d + i(b − c)),β= β(g)=(1/2)(a − d − i(b + c)). Then we have cos ϑ sin ϑ eiϑ r(ϑ):=( ) −→ ( )=:s(ϑ), − sin ϑ cos ϑ e−iϑ i.e.c ˜(SO(2)) = {s(ϑ); ϑ ∈ R} = K, which we identify with U(1). By the linear fractional transformation G × D −→ D, (g,ζ) −→ g(ζ),G acts (transitively) on D. The stabilizing group of ζ = 0 is U(1) = K. Hence in analogy to the map p : G −→ G/SO(2),g −→ g(i)=τ = x + iy with the Iwasawa decomposition g = n(x)t(y)r(ϑ), we have the map p : G −→ G/K = D,g −→ g(0) = ζ = u + iv and the commutative diagram 138 7. Induced Representations

~~c˜ G = SL(2, R) - G = SU(1, 1)~~

~~p p~~

~~? c ? H = SL(2, R)/SO(2) - D = SU(1, 1)/U(1)~~

~~with 1+iτ¯ −1 − iτ c˜(n(x)t(y)) = (1/2)y−1/2 −1+iτ¯ 1 − iτ~~

~~τ−i and c(τ)=ζ = τ+i .~~

~~Now, it is a rather easily acceptable fact that the space of holomorphic functions on D n is spanned by the monomials ζ ,n∈ N0. And some calculation shows that~~

dudv dudv dν(ζ):= ,dν (ζ):=41−m(1−|ζ |2)m , 1−|ζ |2 m 1−|ζ |2 is a G−invariant measure resp. a quasi-invariant measure on D and one has an isomorphism between the corresponding spaces of holomorphic functions on H and D

H 2 −→ 2 −→ ˜ Cm : m = Lhol(H,dµm) Lhol(D,dνm),f Cmf := f given by ζ +1 −2i f˜(ζ)=f(−i )( )m. ζ − 1 ζ − 1 Using a convenient extension of the group action given by (7.18), this can also be written as ˜ −1 −1 f(ζ)=f(C (ζ))jm(C ,ζ).

The map Cm intertwines the representation πm from the half plane model given by (7.18) ˜ ∈ 2 with the equivalent representationπ ˜m on the space of functions f Lhol(D,dνm) given by the prescription ˜ ˜ −1 −1 π˜m(g)f(ζ):=f(˜c(g) (ζ))jm(˜c(g) ,ζ).

~~We call this the unit disc model. By the way, if one replacesc ˜(g)byg ∈ G, one also has a representation of G = SU(1, 1).~~

2 Already above, we stated the (plausible) fact that Lhol(D,dνm) is spanned by functions ˜ ˜ n fn with fn(ζ)=ζ ,n∈ N0. As to be seen easily, via Cm, these functions correspond to functions fn living on H with

~~τ − i f (τ):=( )n(τ + i)−m. n τ + i 7.2 Unitary Representations of SL(2, R) 139~~

~~The Group Model~~

Intertwined with these two equivalent models, there is a third one, which we will call the group model because its representation space consists of functions living on the group G = SL(2, R). We go back to the mapc ˜ : G −→ G˜. In the formulae (7.19) one has the elements of G˜ as functions of g ∈ G. We introduce the parameters x, y, ϑ describing g in the Iwasawa decomposition as in (7.12) and get by a small calculation

~~α(g)=(1/2)(a + d + i(b − c)) = (1/2)y−1/2eiϑi(¯τ − i), β(g)=(1/2)(a − d − i(b + c)) = (1/2)y−1/2e−iϑ(−i)(τ − i), and by another one~~

~~n −(n+m) m m/2 imϑ φn(g):=β(g) α¯(g) (1/(2i)) = y e fn(τ).~~

~~Inspired by this formula, we deﬁne a lift ϕm, which associates functions φ living on G to functions f on H via~~

~~m/2 imϑ f(τ) −→ φf (g)=(ϕmf)(g):=f(g(i))jm(g, i)=f(τ)y e .~~

−m m/2 imϑ Here we use the fact that one has jm(g, i)=(cτ + d) = y e if g is ﬁxed by our Iwasawa parametrization (7.12). With the help of the cocycle condition for the automorphic factor (or multiplier) jm, we deduce for these lifted functions

~~−1 −1 −1 φ(g0 g)=f(g0 g(i))jm(g0 g, i) −1 −1 = f(g0 τ)jm(g0 ,g(i))jm(g, i) =(πm(g0)f)(g(i))jm(g, i).~~

This shows that the lift ϕm intertwines (algebraically) the representation πm on the space 2 Lhol(H,dµm) with the left regular representation λ on the space of functions φ living on G and coming out by the lift ϕm. In [La1] p.180 one ﬁnds a proof that the functions φn are square integrable with respect to the (biinvariant) measure dg on G given (again in our Iwasawa parameters) by dg = y−2dxdydϑ. We call this representation, given by left translation on the space spanned by the functions φn living on the group, the group model.

Already earlier, while introducing the other two models, we could have identified the − − class of our representation with the one of the representation πm of highest weight m from the list of (possible) irreducible unitary representations obtained by infinitesimal considerations in 6.4. We will do this now. The reader is invited to watch very closely because Lang in [La1] p.184 (erroneously if our reasoning is correct) identifies it as a lowest weight representation. There are several ways to proceed: For the isomorphismc ˜ : G = SL(2, R) −→ SU(1, 1) one hasc ˜(r(ϑ)) = s(ϑ), c˜(r(ϑ)−1g)) = s(−ϑ)˜c(g). Hence we deduce

α(r(ϑ)−1g)=e−iϑα(g),β(r(ϑ)−1g)=e−iϑβ(g), and −1 −(2n+m)iϑ φn(r(ϑ) g)=e φn, i.e. our representation has the weights −m, −(m +2), −(m +4),.... We compare this with our results from 6.4 and conclude that we have a representation equivalent to the 140 7. Induced Representations

− highest weight representation πm. But we can do more: In (7.15) we determined the op- R R R R R erators F , G, H and Z , X± . These operators are just the operators producing the derived representation of the left regular representation λ. Thus, by a small calculation (Exercise 7.14), one can verify the formulae

~~R − R − R Z φn = (2n + m)φn, X+ φn = nφn+1, X− φn =(n + m)φn+1.~~

~~We can get corresponding formulae by determination of the derived representations of the other models. We propose the following~~

~~Exercise 7.15: Verify the formulae~~

Rm 2 Z =ˆπm(Z)=i((1 + τ )∂τ + mτ), Rm ± ± 2 ± X± =ˆπm(X±)= (i/2)((i τ) ∂τ + m(τ i)), and Rm − Z fn = (m +2n)fn, Rm − X+ fn = nfn−1, Rm X− fn =(n + m)fn+1. + Similarly the representations πm of lowest weight can be realized by holomorphic functions on the lower half plane H¯ = {τ = x+iy ∈ C,y <0} or by antiholomorphic functions f, i.e. with (∂x − i∂y)f =0.

~~Exercise 7.16: Determine the class of the representationπ ˇm taken over from Knapp in the Remark 7.5 above.~~

~~The Holomorphically Induced Model~~

Finally, we want to show how these realizations of the discrete series representations ﬁt into the discussion of the induction procedure. Namely one can obtain them as subrepresentations of suitable induced representations:

~~We start with the subgroup H = K = SO(2) and its representation π0 given by~~

~~ikϑ π0(r(ϑ)) = χk(r(ϑ)) = e ,k∈ Z.~~

~~The Iwasawa decomposition can be understood as a Mackey decomposition~~

g = s(τ)r(ϑ),s(τ):=n(x)t(y), and we have our standard projection p : G −→ G/SO(2) = H,g−→ g(i)=x + iy = τ −→ −→ ind G with section s : H G, τ s(τ). Then the induced representation πk := indSO(2)χk is given by (inverse) left translation λ on the completion Hχk of the space of smooth functions φ on G with functional equation

~~φ(gr(ϑ)) = e−ikϑφ(g), for all g ∈ G, r(ϑ) ∈ SO(2) and the norm condition 2 | |2 −2 ∞ φ 2 = φ(s(τ)) y dxdy < . H 7.2 Unitary Representations of SL(2, R) 141~~

From the functional equation we learn that the functions φ in the representation space are all of the type φ(g)=e−ikϑF (x, y) with a function F depending only on x, y if g is given in our Iwasawa parametrization. Comparing this to the functions establishing the group model above, we see that we have to ﬁnd a condition, which further restricts these functions F . Here we remember from the inﬁnitesimal considerations in 6.4 that functions spanning the space of a discrete ± series representation πk have to be eigenfunctions of the Casimir element

~~2 Ω=X+X− + X−X+ +(1/2)Z .~~

~~In (7.17) we already determined that Ω acts on smooth functions φ living on G as the diﬀerential operator~~

~~R L 2 2 2 − ∆= Ω = Ω =4y (∂x + ∂y ) 4y∂x∂ϑ. The preceding discussion in mind, we try the ansatz~~

~~φ(g)=e−ikϑy−k/2f(x, y).~~

~~We get~~

2 2 2 −k/2 −ikϑ ∆φ = [(4y (∂x + ∂y )+4yik(∂x + i∂y)+(k/2)((k/2) + 1))f]y e , i.e. we have ∆φ =(k/2)((k/2)+1)φ if 2 2 2 ∆kf := 4y (∂x + ∂y )+4yik(∂x + i∂y)f =0, and this is obviously the case if f fulﬁlls the equation

∂τ¯f := (1/2)(∂x + i∂y)f =0, i.e. if f is holomorphic. Hence for k = −m, we have recovered the lift of the representation ind πm as a subrepresentation of our induced representation πk . χ There is a shorter way to single out in H k a subspace carrying πm: To the two conditions characterizing the space of the induced representation, namely the functional equation and the norm condition, we add simply the condition L X− φ =0.

~~This comes out as follows: From (7.16) we have~~

~~L − −2iϑ − X− = (i/2)e (2y(∂x + i∂y) ∂ϑ).~~

~~−ikϑ −k/2 For φ(g)=e y f(x, y) this again leads to the holomorphicity condition ∂τ¯f =0.~~

~~Remark 7.6: This is a special example of a general procedure propagated in particular by W. Schmid and called holomorphic induction.~~

Remark 7.7: Here we began by stating the prescription for the representation πm in a space of functions living on the homogeneous space H = G/SO(2). It is a nice exercise to 142 7. Induced Representations rediscover this formula by our general procedure to change from the First to the Second Realization of an induced representation: ab We start with Mackey’s decomposition as above, g = s(τ)r(ϑ), and for g−1 =( ) 0 cd the Master Equation, using s(τ)=n(x)t(y), states

~~−1 −1 ∗ g0 s(τ)=s(g0 (τ))r(ϑ ) with ∗ cτ + d eiϑ = . cτ¯ + d Moreover one has the standard formulae~~

~~dµ(g−1(τ) 1 y 0 = , Im (g−1(τ)) = , dµ(τ) (cτ + d)2 0 | cτ + d |2~~

~~−1 such that the left regular representation π(g0)φ(g)=φ(g0 g) in the Second Realization with F (τ)=φ(s(τ)) translates to the prescription~~

~~dµ(g−1(τ)) π(g )F (τ)=( 0 )1/2π−1(r(ϑ∗))F (g−1(τ)). 0 dµ(τ) 0 0~~

−k/2 With π0 = χk and F (τ)=y f(τ) for k = −m by a small calculation this leads to the prescription (7.18) −1 −m πm(g0)f(τ)=f(g0 (τ))(cτ + d) . Remark 7.8: The functions φ in the representation space of the Fﬁrst Realization and the matrix coeﬃcients <π(g)φ, φ > are square-integrable for m ≥ 2. This is a general criterium to distinguish discrete series representations from the other representations.

± The representations π1 are called mock discrete series representations or limits of discrete ± ≥ series. They are realized by the same prescriptions as the πm with m 0 but here the − space of holomorphic functions f on H for π1 gets the new norm f 2 = sup | f(x + iy) |2 dx, y>0 R and analoguously for the other case.

~~The courageous reader may be tempted to apply this principle of holomorphic induction to reconstruct as Exercise 7.17 the irreducible unitary representations of G = SU(2).~~

~~3. The Complementary Series~~

For the sake of completeness, we indicate here also a model, but only brieﬂy, as done in most sources. (The best idea is to go back to Bargmann’s original paper [Ba].) We take up again the prescriptions as in the construction of the principal series but to get a model for πs, 0

~~ab For g−1 =( ) the prescription is here cd~~

~~−1−s −1 πs(g)f(x)=| cx + d | f(g (x)).~~

~~Some more models can be found in [vD] p.465 and [GGP] p.36.~~

~~A Final General Remark~~

We close this section by pointing out to the Subrepresentation Theorem, which goes back to Harish–Chandra and roughly states (for a precise formulation and proof see [Kn] p.238) that a certain class of representations, the admissible representations,maybe realized as subrepresentations of parabolically induced representations starting eventually by non-unitary representations of the subgroup. Admissible representations are very important, in particular in number theory. An admissible representation (π, V )ofG is (in ﬁrst approximation) characterized as follows ([Ge] p.55): Let K be a maximal compact subgroup of G. For an irreducible representation χ of K, denote by V (χ) the subspace of vectors in V, which transform according some multiple of χ. Then one has a decomposition V = ⊕χ V (χ) where each V (χ) is ﬁnite dimensional.

Irreducible unitary representations of reductive groups are admissible. So all representations in this text have this property (with exception of the representations of the (non-reductive) Jacobi group appearing later).

The interested reader can exemplify the statement of the subrepresentation theorem in the case of G = SL(2, R) by constructing as Exercise 7.18 another model of the discrete series representation by starting the induction procedure from

~~s H = {h = n(x)t(y); x, y ∈ R,y >0},π0(h)=y ,s∈ C.~~

~~7.3 Unitary Representations of SL(2,C) and of the Lorentz Group~~

The representations of the group G = SL(2, C) are of particular importance for physicists because this group is the double cover of the identity component of the complete homogeneous Lorentz group in four dimensions (to which we will restrict ourselves in this section). At ﬁrst, we will discuss the representations of SL(2, C) and then the homomorphism into the Lorentz group.

As central fact we have: Theorem 7.5: For G = SL(2, C) there are (up to equivalence) only two series of irreducible unitary representations: – the unitary principal series πk,iv depending on two parameters k ∈ Z,v ∈ R, and – the complementary series π0,u depending on a real parameter u with 0

Thus, in contrast to the smaller group SL(2, R), we have no discrete series representations. We reproduce models for these representations from [Kn] p. 33/4 and discuss their construction via the induction procedure afterwards.

As representation space we take the space H = L2(C) of square-integrable complex functions f on C with measure dzdz.¯ For z ∈ C,f∈H, the prescription for the unitary principal series representation is given by

~~ab −bz + d az − c (7.20) π (( ))f(z):=|−bz + d |−2−iv ( )−kf( ). k,iv cd |−bz + d | −bz + d~~

~~Here, the notion of the fractional linear transformation used in the last section as a map SL(2, R) × H −→ H, (g, τ) −→ g(τ), is naturally extended to a map~~

~~SL(2, C) ×{C ∪∞}−→{C ∪∞} with ab ( )(z):= az+b for z ∈ C,cz+ d =0 , cd cz+d := ∞ for z ∈ C,cz+ d =0,~~

~~a ∞ := c for z = .~~

~~In a similar way, for k ∈ Z,w = u + iv ∈ C, there is a non–unitary principal series πk,w given by the prescription~~

~~ab −bz + d az − c π (( )f(z):=|−bz + d |−2−w ( )−kf( ) k,w cd |−bz + d | −bz + d~~

2 2 u for functions f in the space L (C,dµw) with dµw(z) = (1+ | z | ) dzdz.¯ However, in the special case k =0,w = u ∈ R, 0 := 2−u dzdζ. C C | z − ζ | This is our model for the complementary series.

As to the construction, these representations are induced representations where one takes the subgroup αβ H := {h =( ); α, β ∈ C,α=0 } α−1 | |w α k and the representation π0 with π0(h):= α ( |α| ) . We have the Mackey decomposition

~~ab g =( )=s(τ)h cd where (as in our discussion of the principal series for SL(2, R) in the previous section) it suﬃces to treat the case a = 0, such that one has~~

~~1 ab s(τ)=( ),z:= b/a and h =( ), z 1 a−1 7.3 Unitary Representations of SL(2,C) and of the Lorentz Group 145 and the Master Equation g−1s(z)=s(z∗)h∗ with~~

~~az − c −bz + d −b z∗ = ,h∗ =( ). −bz + d (−bz + d)−1~~

Moreover, we have dz∗dz¯∗ = |−bz + d |−4 dzdz.¯ Hence, since the induced representation in left regular version in its Second Realization on the space H = L2(C) has the general prescription dµ(z∗) π(g)f(z)= π (h∗)f(z∗), for all f ∈H, dµ(z) 0 with w = iv this leads to the formula (7.20).

In [Kn] p.33/4 Knapp proves the irreducibility of πk,iv using Schur’s Lemma in the unitary version cited in 3.2. One could also do this (and verify that really we only have the above mentioned irreducible unitary representations) by inﬁnitesimal considerations as in our discussion of SL(2, R). But since sl(2, C) has real dimension 6, this is leads to considerable non-trivial computations (to be found for instance in Naimark [Na] or Cornwell [Co] p.668ﬀ in the framework of the discusssion of the Lorentz group).

The interest in the representations of G = SL(2, C) is enlarged by the fact that they immediately lead to projective representations of the proper orthochronous Lorentz group, resp. those, which are trivial on −E2, to genuine representations, the same way as in our discussion of SU(2) and SO(3) in 4.3. Before we describe the relation between SL(2, C) and the Lorentz group we have to ﬁx some notation.

We write ⎛ ⎞ 1 ⎜ 1 ⎟ GL := {A ∈ GL(4, R); tAD A = D = ⎜ ⎟} 3,1 3,1 ⎝ 1 ⎠ −1 for the complete homogeneous Lorentz group, i.e. the group of linear transformations, which leave invariant the Minkowski metric from the theory of special relativity

~~2 2 2 2 − 2 ds = dx1 + dx2 + dx3 dx4.~~

One may be tempted to replace (x1,x2,x3,x4)by(x, y, z, ct), (c the velocity of light) and put ds2 = c2dt2 − dx2 − dy2 − dz2, as often done in the physics literature, but we will not do this here. In our former notation L L from 0.1 we also have G = O(3, 1) and G+ = SO(3, 1) for the proper homogeneous Lorentz group, i.e. the invariant subgroup consisting of matrices A with det A =1.We write the matrices in block form Bx A =( ),B∈ GL(3, R),x,y ∈ R3,a∈ R. tya 146 7. Induced Representations

~~L G+ has two connected components distinguished by a>0 and a<0 since one has tBy E Bx tBB − yty tBx − ya ( )( 3 )( )=( ), txa −1 ty a txB − aty txx − a2~~

~~E and the condition that this should be equal to ( 3 ) asks for −1~~

~~t t t t 2 (7.21) BB − y y = E3, Bx − ya =0, xx − a = −1, and, hence, in particular a2 ≥ 1.~~

L + The subgroup G+ with a>0 is called the proper orthochronous homogeneous Lorentz group. It is the connected component of the identity in SO(3, 1). We will abbreviate this 0 here to Lorentz group and write G0 := SO(3, 1) for it. As easily to be seen, one can imbed SO(3) into G0 by putting B SO(3) B −→ B˜ := ( ). 1

~~These elements are called Lorentz rotations. Moreover, we introduce the Lorentz boosts, which are of the type ⎛ ⎞ cosh t sinh t ⎜ 1 ⎟ B(t)=⎜ ⎟ ,t∈ R. ⎝ 1 ⎠ sinh t cosh t~~

~~The structure of G0 is clariﬁed by the following remark.~~

~~Proposition 7.2: For every A ∈ G0 there are R, S ∈ SO(3) and t ∈ R such that A = RB˜ (t)S,˜ i.e. G0 is generated by Lorentz rotations and Lorentz boosts.~~

~~A proof can be found in any source treating the Lorentz group, for instance in [He] p.75. Our main fact in this context is the next theorem.~~

~~Theorem 7.6: There is a surjective homomorphism~~

~~0 ρ : G = SL(2, C) −→ G0 = SO(3, 1) with ker ρ = {±E2}.~~

~~Proof (sketch): We put~~

~~αz V := {X ∈ M (C); X = tX¯ =: X∗} = {( ); α, β ∈ R,z ∈ C} 2 zβ¯ and look at the map~~

~~−a + db+ ic ϕ : R4 −→ V, v = t(a, b, c, d) −→ ( ). b − ic a + d 7.4 Unitary Representations of Semidirect Products 147~~

~~For X = ϕ(v),Y = ϕ(w) this map transforms the Minkowski (quasi-)inner product~~

~~= aa + bb + cc − dd into the corresponding one deﬁned on V by~~

σ : V × V −→ R, (X, Y ) −→ − (1/2)(det(X + Y ) − det X − det Y ) as easily to be veriﬁed by a small calculation. Hence our ϕ induces a bijection between O(3, 1) and O(V,σ), which can be recognized as an isomorphism and we take this to identify both sets. We deﬁne a map

~~ρ : SL(2, C) −→ GL(V ),A−→ ρ(A) with ρ(A)X := AXA∗ for X ∈ V, which obviously is a homomorphism. We have~~

~~ker ρ = {A; AXA∗ = X for all X ∈ V }~~

1 and deduce ker ρ = {±E } by evaluating the deﬁning condition for X = E , ( ), 2 2 −1 1 and ( ). The main task is to prove the surjectivity of ρ. We go back to the proof 1 of Theorem 4.7 in 4.3 and use the fact that G0 is generated in GL(V ) by elements S˜2(α), S˜3(α) and B(t). By some computation, one veriﬁes that these are images under ρ of respectively r(α/2),s(α/2) and s(−it/2).

This theorem justifies the statement made above that the representations π of SL(2, C) 0 with π(−E2)=id create the representations of SO(3, 1) . E.g., in [Co] p.671ff one finds a discussion of the finite-dimensional (non-unitary) representations of the Lorentz group.

~~7.4 Unitary Representations of Semidirect Products~~

The semidirect products form a class of groups whose importance is also enhanced by its applications in physics as the Euclidean and Poincar´egroups are prominent examples, which ﬁt into the following general concept:

~~We take two groups G0 and N and a (left) action of G0 on N given by~~

ρ : G0 × N −→ N, (a, n) −→ ρ(a)n with the additional condition that every ρ(a) is an automorphim of N, i.e. ρ(a) ∈ Aut N for all a ∈ G0. Then we deﬁne the associated semidirect product G := G0 N as the set of pairs g =(a, n),a∈ G0,n∈ N provided with the composition law

~~(7.22) gg := (aa,nρ(a)n).~~

G comes out as a group with neutral element e consisting of the pair of neutral elements −1 −1 −1 −1 in G0 and in N. The inverse to g =(a, n)isg := (a ,ρ(a )n ). (Veriﬁcation of the associative law as Exercise 7.19.) 148 7. Induced Representations

~~Examples:~~

E Example 7.4: The Euclidean group G (n)inn dimensions: We take G0 = SO(n) and N = Rn. For A ∈ SO(n),x ∈ Rn, the action ρ is given by matrix multiplication (A, x) −→ Ax. We denote the semidirect product SO(n) Rn by GE(n) and write this again as a matrix group

~~Aa GE(n)={g =(A, a)=( ); A ∈ SO(n),a∈ Rn}. 1~~

Here the matrix multiplication in Mn+1(R) Aa A a AA a + Aa gg =( )( )=( ) 1 1 1 reﬂects exactly the composition law (7.22) leading in this case to gg =(AA,a+ Aa). This law comes out very naturally if one asks for the composition of the aﬃne transfor- −→ 2 2 ··· 2 mations x Ax + a, which preserve the euclidean norm x = x1 + + xn.

Example 7.5: The Poincar´egroup GP (n)inn dimensions: Paralell to the case above n we take G0 = SO(n − 1, 1) and N = R and get Aa GP (n)={g =(A, a)=( ),A∈ SO(n − 1, 1),a∈ Rn} 1 with the same composition law. This is the invariance group of the aﬃne transformations 2 2 ··· 2 − 2 preserving the Minkowski (pseudo-)norm x = x1 + + xn−1 xn. ab Example 7.6: Interprete the group B := {g =( ); a, b ∈ R,a>0} as a semidi- a−1 rect product (Exercise 7.20).

~~Example 7.7: Interprete the Heisenberg group ⎛ ⎞ 1 xz Heis(R):={g = ⎝ 1 y⎠ ; x, y, z ∈ R} 1~~

~~2 as semidirect product of G0 = R acting on N = R . Can you do the same with Heis(R) as realized in 0.3? (Exercise 7.21)~~

~~J Example 7.8: The Jacobi group G is the semidirect product of G0 = SL(2, R) acting on the Heisenberg group N = Heis(R) via~~

~~ab (( ), (λ, µ, κ)) −→ (dλ − cµ, −bλ + aµ, κ). cd~~

~~This group can be presented in several ways as a matrix group (see our Section 8.5 or for instance [EZ] or [BeS] p.1).~~

With exception of the last one, in all these examples the group N is abelian. In the sequel we will restrict our treatment to this case as things become easier under this hypothesis and use additive notation in the second item of the pair g =(a, n) (as we already 7.4 Unitary Representations of Semidirect Products 149 did in the examples). Whoever is interested in more general cases is refered to Mackey’s original papers or the treatment of the representation theory of the Jacobi group in [BeS].

~~Remark 7.9: We use the following embeddings~~

G0 −→ G = G0 N, a −→ (a, 0) =: a and N −→ G = G0 N, n −→ (e, n)=:n. Here one has to be a bit careful because we have n · a =(e, n)(a, 0)=(a, n)=g but a · n =(a, 0)(e, n)=(a, ρ(a)n) =( a, n)=g in general. There is the useful relation

~~(a, 0)(e, n)(a−1, 0) = (e, ρ(a)n), which, using our embeddings, can be stated as ana−1 = ρ(a)n.~~

~~Example 7.9: We take~~

~~G := Heis(R)={g =(x, y, z); x, y, z ∈ R},~~

~~G0 := {a =(x, 0, 0); x ∈ R}, and N := {n =(0,y,z); y, z ∈ R}. |= Using the multiplication law gg =(x + x,y+ y,z+ z + xy), one calculates~~

~~ana−1 =(0,y,z+ xy)=ρ(a)n and this is consistent with the usual semidirect product composition law (7.22) given by gg =(a, n)(a,n)=(aa,nρ(a)n).~~

~~Example 7.10: For a presentation as semidirect product of the group~~

ab B := {g =( )=:(a, b); a, b ∈ R,a>0} a−1 one may be tempted to try G0 := {a =(a, 0); a>0} and N := {n =(1,b); b ∈ R}. − From matrix multiplication we have gg =(aa,ab + ba 1) and (using the embedding introduced above) ana−1 =(1,a2b)=ρ(a)n. But then the composition law would give gg =(a, b)(a,b)=(aa,nρ(a)n)=(aa,b+a2b) which is not the multiplication law in this group. To make things consistent (and thus solve the problem from Exercise 7.20), one constructs the semidirect product G = R>0 R = {[a, x],a,x∈ R,x > 0} with ρ(a)x = a2x and deﬁnes an isomorphism

~~ϕ : B −→ G, (a, b) −→ [a, ab].~~

The agreable thing about these semidirect products G = G0 N, N abelian, is the fact that Mackey’s Theory provides an explicit way to construct their irreducible unitary representations as induced representations. We describe this method without going everywhere into the intricacies of the proofs, which rely on some serious functional analysis: 150 7. Induced Representations

~~We start by an easy but important fact relating the representations of G = G0 N with G0 acting on N by ρ to those of its constituents G0 and N.~~

Proposition 7.3: Let π be a representation of G and π,π the restrictions of π to N resp. G0, i.e π (n)=π((e, n)) for n ∈ N and π (a)=π((a, 0)) for a ∈ G0. Then π and π completely determine π and are related by the condition

~~−1 (7.23) π (ρ(a)n)=π (a)π (n)π (a ) for all a ∈ G0,n∈ N.~~

~~Proof: π is determined by π and π as one has~~

~~π((a, n)) = π((e, n)(a, 0)) = π((e, n))π((a, e)) = π(n)π(a).~~

~~And from the composition law in G~~

~~(a1,n1)(a2,n2)=(a1a2,n1 + ρ(a1)n2) we deduce~~

~~π (n1)π (a1)π (n2)π (a2)=π (n1)π (ρ(a1)n2)π (a1)π (a2) and with a1 = a, n2 = n this leads immediately to (7.23).~~

Hence to find representations of G, one has to find representations of G0 and N fulfilling (7.23). This condition is important in the context of Mackey’s system of imprimitivity, which is the main tool in the proof that the construction we describe in the sequel leads to a way to construct all irreducible unitary representations of semidirect products. We essentially follow [BR] p.503ff and [Ki] p.195ff (where the more general case of representations of group extensions is treated). A decisive element in the construction comes from the fact that the action ρ of G0 on N leads also to an action on the unitary dual Nˆ of N, which in our abelian case consists of (classes of) charactersn ˆ of N. Namely, we have G0 × Nˆ −→ N,ˆ (a, nˆ) −→ anˆ given by anˆ(n):=ˆn(ρ(a−1)n). Since the (abelian) group N acts (trivially) on Nˆ by −1 conjugation, i.e. via n0nˆ(n):=ˆn(n0 nn0), one has an action of the semidirect product G on N.ˆ The G−orbit ofn ˆ ∈ Nˆ is written as

~~Oˆnˆ := Gn.ˆ~~

Mackey’s construction works under the assumption that the semidirect product is regular. All the groups we treat as examples are of this type. Hence, since we will not give complete proofs anyway, a not so ambitious reader at ﬁrst sight may skip the following rather technical condition (from [BR] p.506).

Deﬁnition 7.1: We say that G is a regular semidirect product of N and G0 if Nˆ contains a countable family Z1,Z2,... of Borel subsets, each a union of G−orbits, such that every ˆ orbit in N is the intersection of the members of a subfamily Zn1 ,Zn2 ,... containing that orbit. Without loss of generality, we can suppose that the intersection of a ﬁnite number of Zi is an element of the family (Zi). This is equivalent to the assumption that any orbit is the limit of a deceasing subsequence of (Zi). 7.4 Unitary Representations of Semidirect Products 151

~~One also has a slightly diﬀerent looking condition for regularity used by Mackey ([Ma] p.77): There is a subset J ⊂ Nˆ, which meets each G−orbit Oˆ exactly once and is an analytic subset.~~

~~Now as our central prescription for the construction of representations in this section, we take over from [BR] Theorem 4 on p.508:~~

Theorem 7.7: Let G = G0 N be a regular semidirect product of separable, locally compact groups G0 and N, N abelian. Then every irreducible unitary representation G of G is unitarily equivalent to an induced representation π = indH π0, which can be constructed as follows. – Step 1: Determine the set Nˆ of charactersn ˆ of N, i.e. the dual group of N. – Step 2: Classify and determine the G0−orbits Oˆ in Nˆ. – Step 3: Choose an elementn ˆ0 in each orbit Oˆ and determine the stabilizing group

G0nˆ0 =: H0. (This group is called the little group in the physics literature.) – Step 4: Determine the unitary dual Hˆ0 of H0. ˆ – Step 5: From every element of H0 take a representation π1 of H0 and extend it to a representation π0 of H := H0 N by

~~π0((a, n)) :=n ˆ0(n)π1(a) for all a ∈ H0,n∈ N.~~

~~– Step 6: Take H and π0 from the last step and proceed to (normalized) induction G π = indH π0.~~

It is quite evident that this recipe produces unitary representations. But it is far from evident that these are irreducible and that we really get the unitary dual of the semidirect product and, hence, have a classification scheme at hand. As already said, we can not go into the proofs here (which essentially use Mackey’s imprimitivity theorem) and refer to the Chapters 16 and 17 of [BR] or §13 of [Ki] and Section 10 of [Ma] where slightly different approaches are adopted. Now, we discuss some examples to show how the method works. In these examples one can verify directly the validity of the procedure encapsulated in Theorem 7.7 for instance with the help of the infinitesimal method from our Chapter 6.

~~Examples:~~

Example 7.11: G := Heis(R)={g =(x, y, z); x, y, z ∈ R} with gg =(x + x,y+ y,z+ z + xy) is isomorophic to the semidirect product of G0 := {a =(x, 0, 0); x ∈ R} acting on N := {n =(0,y,z); y, z ∈ R} via ρ(a)n := (0,y,z + xy) as seen in the Example 7.9 above. – Step 1: The dual Nˆ of N R2 is again to be identiﬁed with R2. With (l, m) ∈ R2 we write i(ly+mz) nˆ(n)=e =: χ(l,m)(y, z). ˆ 2 – Step 2: a ∈ G0 R acts onn ˆ ∈ N R by

~~−1 i(ly+m(z−xy)) anˆ(n):=ˆn(ρ(a )n)=e = χ(l−xm,m)(y, z). 152 7. Induced Representations~~

~~Hence Nˆ is disjointly decomposed into exactly two types of G0−orbits:~~

~~• ˆ { − ∈ } Type 1: For m = 0 andn ˆ0 =(0,m) one has the line On0 = ( xm, m),x R , • ˆ { } Type 2: For m = 0 andn ˆ0 =(l, 0) there is the point On0 = (l, 0) .~~

~~We get the following picture m 6 ˆ O(0,m) rrrrrrrrrr - r l ˆ O(l,0)~~

One can see that G is a regular semidirect product: The set {(l, m) ∈ R2; lm =0} is an analytic set, which meets each orbit exactly once. – Step 3: The stabilizing group ofn ˆ0 =(0,m),m =0is H0 = G0(0,m) = {0} and that ofn ˆ0 =(l, 0),l ∈ R is H0 = G0(l,0) = G0 R. – Step 4: In the ﬁrst case the unitary dual of H0 simply consists of the trivial identity representation and in the second case we have Hˆ0 R with representatives given by ijx π1(a)=e =: χj(x),j ∈ R. – Step 5 and 6: In the second case, the type 2 orbits, the product of the stabilizing group H0 = G0 with N already exhausts the whole group G. Hence there is no induction 2 necessary and one has for (j, l) ∈ R a one-dimensional representation π of G = G0 N given by π((x, y, z)) = eijxeily. 2 In the ﬁrst case, belonging to the type 1 orbits, one has H = H0 N N R with a imz representation π0 given by π0(h)=e for all h =(0,y,z). Here the induction makes G sense and we have a representation π = indH π0 (for instance) given by right translation on the completion H of the space of smooth functions φ : G −→ C with functional equation φ(hg)=eimzφ(g) for all h ∈ H, g ∈ G, 2 | |2 ∞ and φ = R φ((x, 0, 0) dx < . As the composition law says

~~gg0 =(x + x0,y+ y0,z+ z0 + xy0) and (0,y,z)(x, 0, 0) = (x, y, z), an element φ in H is of the form φ(g)=eimzf(x) with f ∈ L2(R) and one has~~

~~imz im(z0+xy0) φ(gg0)=e e f(x + x0).~~

~~So we have a second realization of this representation given by~~

~~im(z0+xy0) 2 π˜(g0)f(x)=e f(x + x0) for f ∈ L (R).~~

~~This is the Schr¨odingerrepresentation πm from our Example 1.6 in 1.3 translated to Heis(R) by the isomorphism~~

~~ϕ : Heis(R) −→ Heis(R),g=(x, y, z) −→ (λ, µ, κ)=(x, y, 2z − xy), from Exercise 0.1. Hence the unitary dual of the Heisenberg group G is~~

~~Gˆ = R2 (R \{0}). 7.4 Unitary Representations of Semidirect Products 153~~

E 2 2 Example 7.12: G := G (2) = SO(2) R has G0 = SO(2) and N = R . We write t g =(a, n) with a = r(ϑ) and n = x := (x1,x2). One has ρ(a)n := r(ϑ)x 2 2 – Step 1: As in example 7.11 we have Nˆ = R . This time, for y =(y1,y2) ∈ R we write i(y1x1+y2x2) nˆ(n)=e = χ(y1,y2)(x1,x2).

~~– Step 2: a = r(ϑ) ∈ G0 = SO(2) acts onn ˆ ∈ Nˆ by~~

~~−1 iyr(−ϑ)x anˆ(n):=ˆn(ρ(a )n)=e = χyr(−ϑ)(x).~~

~~Hence Nˆ is again decomposed into two types of orbits: • Type 1: Forn ˆ0 =(r, 0),r >0, the circle ˆ { ∈ 2 2 2} On0 = y R , y = r . • ˆ { } Type 2: Forn ˆ =(0, 0), the point On0 = (0, 0) .~~

We can verify the regularity of the semidirect product: One has a countable family of ˆ ˆ { } Borel subsets of N consisting of the following sets: Z0,0 := the point On0 = (0, 0) and for any two positive rational numbers r1 0 the genuine subgroup H N and its one-dimensional representation π0 given by

~~irx1 π0((E2,x)) =n ˆ0(n)=e .~~

~~G Hence, for every r>0, we have an induced representation π = indH π0. In consequence, the unitary dual of G = GE(2) is~~

~~Gˆ = R>0 Z.~~

Example 7.13: As Exercise 7.22 apply our construction recipe to construct representations of ab G = B := {g =( )=:(a, b); a, b ∈ R,a>0} a−1 and ab G = B := {g =( )=:(a, b); a, b ∈ R,a=0 } a−1 and determine the unitary dual.

~~Exercise 7.23: Do the same for the Euclidean group in three dimensions GE(3). 154 7. Induced Representations~~

~~7.5 Unitary Representations of the Poincar´eGroup~~

~~The Poincar´eor inhomogeneous Lorentz group is generally known as the symmetry group of space-time. In continuation to our discussion of the Lorentz group in 7.3, we put~~

~~P 0 4 G = G (4) := G0 N with G0 := SO(3, 1) ,N:= R .~~

~~Sometimes it is necessary to take more generally G0 = SO(3, 1) or even G0 = O(3, 1), which also preserve the Minkowski (pseudo)norm |·|given by~~

~~E | x |2:= x2 + x2 + x2 − x2 = tx( 3 )x, for all x = t(x ,x ,x ,x ) ∈ R4. 1 2 3 4 −1 1 2 3 4~~

We restrict our treatment to a discussion of the simplest case (it should not be too difficult to extend our results to the more general cases). It should also be not difficult to translate our procedure to the situation apparently prefered by the physics literature ([Co] p.677ff, [BR] p.513ff]), namely that the coordinates are x0,x1,x2,x3 and the norm | |2 2 − 2 − 2 − 2 is given by x = x0 x1 x2 x3. Moreover, there are reasons from physics (see the hints in 4.3 while discussing the relation of the representations of SO(3) and SU(2)) and mathematics to define and treat the semidirect product

~~G˜ := SL(2, C) R4.~~

For the definition of this product, one goes back to the proof of Theorem 7.6 in 7.3 where we fixed the surjection ρ : SL(2, C) −→ SO(3, 1): There we had an identification 4 4 ϕ : R −→ V = Herm2(C) and defined an action of SL(2, C)onV (and hence on R ) by ρ(A˜)X = AX˜ A˜∗ for all X ∈ V, A˜ ∈ SL(2, C).

0 4 We start with the discussion of G = G0 N,G0 = SO(3, 1) ,N= R (say, for aesthetical reasons) but soon shall be lead to take in also G˜. We follow the construction procedure from Theorem 7.7 in the last section.

Step 1: The dual Nˆ of N = R4 is again to be identiﬁed with R4. We treat N as consisting of (coordinate) columns x ∈ R4 and Nˆ as consisting of (momentum)rows t 4 p, p =(p1,p2,p3,p4) ∈ R . Hence we write

~~t iΣpj xj i px nˆ(n)=e = e =: χp(x).~~

~~(In the physics literature often minus signs appear in this identﬁcation of N and its dual, but we want to avoid the discussion of co- and contravariant vectors at this stage.)~~

~~0 4 Step 2: a ∈ G0, here given by A ∈ SO(3, 1) , acts onn ˆ ∈ Nˆ, given by p ∈ R ,by~~

~~−1 anˆ(n):=ˆn(ρ(A )n)=χpA−1 (x), i.e. A transforms p into pA−1. It is elementary calculus that the orbits for this action are subsets in the three types of quadrics~~

~~• Type 1: the two-sheeted hyperboloid given by~~

~~−| |2 2 − 2 − 2 − 2 2 ∈ p = p4 p3 p2 p1 = m for m R,m=0, 7.5 Unitary Representations of the Poincar´eGroup 155~~

~~• Type 2: the one-sheeted hyperboloid given by~~

~~−| |2 2 − 2 − 2 − 2 2 ∈ p = p4 p3 p2 p1 =(im) for m R,m=0,~~

~~• Type 3: the light cone given by~~

~~−| |2 2 − 2 − 2 − 2 p = p4 p3 p2 p1 =0.~~

~~More precisely, one veriﬁes using the explicit description of SO(3, 1) in 7.3 that R4 decomposes exactly into the following orbits:~~

• + ˆ+ ˆ Type 1 : Om := Onˆ0 = G0nˆ0 forn ˆ0 =(0, 0, 0,m), • − ˆ− ˆ − Type 1 : Om := Onˆ0 = G0nˆ0 forn ˆ0 =(0, 0, 0, m), • ˆ ˆ Type 2 : Oim := Onˆ0 = G0nˆ0 forn ˆ0 =(m, 0, 0, 0), • + ˆ+ ˆ − Type 3 : O0 = Onˆ0 = G0nˆ0 forn ˆ0 =( 1, 0, 0, 1), • − ˆ− ˆ − − Type 3 : O0 := Onˆ0 = G0nˆ0 forn ˆ0 =( 1, 0, 0, 1), • 0 ˆ0 ˆ { } Type 3 : O0 := Onˆ0 = G0nˆ0 = (0, 0, 0, 0) forn ˆ0 =(0, 0, 0, 0), + ˆ+ ˆ− The type 1 orbit Om is called the positive mass shell, Om the negative mass shell and ˆ± O0 the forward resp. backward light cone. The situation is sketched in the picture:

~~p 6 4 Oˆ+ @@ m @ @ Oˆ+ (0,m ) @ 0 @ (m, 0)- @ p1 Oˆim @ Oˆ− @ 0 @ @ ˆ− @ Om~~

~~The regularity of the semidirect product can be veriﬁed analogously as in our example 2 in the last section (see [BR] p.517/8).~~

~~Now, for each orbit O,ˆ we have to determine its little group, i.e. the stabilizing group~~

~~H0 := G0nˆ0 .~~

~~Step 3: We get the following four cases.~~

~~ ± ± H0 := G0nˆ0 SO(3) for the type 1 , i.e. nˆ0 =(0, 0, 0, m),~~

~~ SO(2, 1) for the type 2, i.e. nˆ0 =(m, 0, 0, 0), E ± G (2) for the type 3 , i.e. nˆ0 =(−1, 0, 0, ±1), 0 0 SO(3, 1) for the type 3 , i.e. nˆ0 =(0, 0, 0, 0). 156 7. Induced Representations~~

~~Proof: i) As in 7.3 we write the elements of G0 = SO(3, 1) as Bx A =( ),B∈ SO(3),x,y ∈ R3,a∈ R. tya~~

For m = 0, the condition (0, 0, 0,m)A =(0, 0, 0,m) enforces y = 0 and a = 1. Moreover, from A ∈ SO(3, 1) one deduces (via (7.21)) x =0 and, hence, the first assertion G0nˆ0 SO(3). ii) The other assertions can be proved similarly by direct computations. But as an- ˜ nounced above, it is more agreable to use the fact from 7.3 that one has SL(2, C)=:G0 via the surjection ρ : G˜0 −→ G0 as a double cover of the Lorentz group. In the proof of Theorem 7.6 in 7.3 we identified R4 via a map ϕ with the space V of hermitian 2 × 2−matrices −a + db+ ic t(a, b, c, d) −→ ( ), b − ic a + d and defined the map

~~ρ : SL(2, C) −→ GL(V ), A˜ −→ ρ(A˜) with ρ(A˜)X = AX˜ A˜∗ for X ∈ V.~~

~~Hence for m = 0, the condition~~

(m, 0, 0, 0)A =(m, 0, 0, 0),A∈ G0 can be translated into the condition −m ρ(A˜)X = AX˜ A˜ = X for X := ϕ(m, 0, 0, 0) = ( ), 0 0 0 0 m i.e. A˜ ∈ SU(1, 1). It is not diﬃcult to see (Exercise 7.24) that the homomorphism ρ : SL(2, C) −→ SO(3, 1)0 restricts to a surjection SU(1, 1) −→ SO(2, 1)0.

~~It will be useful later to take also mi X := ϕ(0, 0,m,0) = ( ). 0 −mi~~

By a small calculation, one veriﬁes that the stabilizing group {A˜; AX˜ 0A˜ = X0} here i is SL(2, R): For σ =( ) one has σ = σ−1 and tA−1 = σAσ−1. Hence from −i AσA∗ = σ we deduce σ−1Aσ = A∗−1 = tA−1, i.e. A = A¯.

~~2 iii) One has ϕ(−1, 0, 0, 1) = ( )=:X0, and AX˜ 0A˜ = X0 calls for~~

~~ab A˜ =( ), with a, b ∈ C, | a |=1. a−1~~

~~We put a =: eiϑ/2,b=: e−iϑ/2z, ϑ ∈ [0, 4π],z ∈ C, and get the multiplication law~~

~~a b i(ϑ+ϑ)/2 −i(ϑ+ϑ)/2 iϑ A˜A˜ =( − ), with a = e ,b = e (z + e z ). a 1 7.5 Unitary Representations of the Poincar´eGroup 157~~

This shows that in this case the stabilizing group H˜0 ⊂ G˜ is isomorphic to the semidirect product of a rotation group {eiϑ} and a translation group R2 C with the composition law (eiϑ/2,z)(eiϑ /2,z)=(ei(ϑ+ϑ )/2,z+ eiϑz). The homomorphism (eiϑ/2 −→ (r(ϑ)) makes S˜1 := {eiϑ/2;0<ϑ<4π} a double cover E of SO(2) and this extends to a surjection of H˜0 to the Euclidean group G (2). iv) In the last case there is nothing to prove.

~~Step 4: We already determined the unitary dual of covering groups H˜0 of all the little groups coming in here:~~

± ˜ i) In the ﬁrst case (type 1 ), for H0 = SU(2), the elements of the unitary dual are represented by the representations πj,j ∈ (1/2)N0 discussed in 4.2. ii) In the second case (type 2), one has H˜0 = SU(1, 1) and from 6.2 we know that SU(1, 1) is isomorphic to SL(2, R) and that the representations appear in three series ± ∈ ∈ πk ,k N; πis,±,s R; πs, 0 0. iv) For the type 30, one has the representations of the Lorentz group resp. its covering SL(2, C) discussed in 7.3, which come in two series πk,iv,k ∈ Z,v ∈ R; π0,u, 0

Step 5 and 6: Finally, we have to extend the representations of the little group to its product with N = R4 and, if this product is not the whole group, proceed to an induced representation. This way we get the following representations (as usual, we list those of the covering group G˜ of the Poincar´egroup):

~~The Unitary Dual of the Poincar´eGroup i) In the ﬁrst case, one has H˜ = SU(2) R4. For the type 1+ with m>0, we have for this group the representation π0 given by~~

~~imx4 t 4 π0((B,x)) = πj(B)e for B ∈ SU(2),x= (x1,x2,x3,x4) ∈ R .~~

~~We write for the induced representation~~

~~G˜ indH˜ π0 =: πm,+;j. Similarly, the type 1− leads to~~

~~−imx4 t 4 π0((B,x)) = πj(B)e for B ∈ SU(2),x= (x1,x2,x3,x4) ∈ R and we denote the induced representation by~~

~~πm,−;j,m>0,j ∈ (1/2)N0. 158 7. Induced Representations ii) In the second case, one has H˜ = SL(2, R) R4 and its representation~~

imx3 t 4 π0((A, x)) = π(A)e for A ∈ SL(2, R),x= (x1,x2,x3,x4) ∈ R where π here is one of the representations of SL(2, R) listed above in part ii) of Step 4 resp. in Section 7.2. For the corresponding induced representation we write

G˜ ± ind π =: π ± for π = π ,k∈ N; m>0, H˜ 0 im;k, k =: πim;is,± for π = πis,±,s∈ R; m>0, =: πim;s for π = πs,s∈ R, 0 0. iii) In the third case, for type 3±, one has H˜ =(S˜1 R2) R4 and its representation

i(x1±x4) 1 2 t 4 π0((A,˜ x)) = π(A˜)e for A˜ ∈ (S˜ R ),x= (x1,x2,x3,x4) ∈ R where π here is one of the representations of (S˜1 R2) indicated in Step 4. For the corresponding induced representation we write

G˜ ind π =: π ± for π = χ ,j ∈ (1/2)Z, H˜ 0 0, ;j j =: π0,±;r for π = πr,r∈ R,r >0. iv) In the forth case (for type 30) the little group is the whole group and we get the representations of G˜ by the trivial extension of the representations π of SL(2, C), i.e. one has π((A,˜ x)) = π(A˜) for (A,˜ x) ∈ G˜0. For the corresponding representation we write

~~π =: π0;j,iv,± for π = πj,iv,±,v ∈ R =: π0;0,u for π = π0,u,u∈ R, 0~~

+ Among these, the representations πm,+;j,m>0,j ∈ (1/2)N0, coming from the type 1 ± orbit and π0,±;j,j ∈ (1/2)Z, coming from the type 3 orbit have a meaning for physics since they are interpreted as describing elementary particles of spin j and mass m resp. of zero mass.

~~Example 7.14: As an illustration we discuss an explicit model for the representation πm,+;j following [BR] p.522:~~

From the preceding discussion we take over ˜ ˜ ˜ 4 G = G0 N, G0 = SL(2, C),N= R and write G˜ g =(A,˜ x), A˜ ∈ SL(2, C),x∈ R4 (a column). A bit more explicitely than ˜ at the beginning of this section, the action of G0 on N is described as follows. We have the identiﬁcation

~~4 ϕ : R −→ V = Herm2(C),~~

~~t x4 − x1 x2 + ix3 x = (x1,x2,x3,x4) −→ X =( ). x2 − ix3 x4 + x1 7.5 Unitary Representations of the Poincar´eGroup 159~~

~~∗ Via ϕ the action of G˜0 on V given by (A,˜ X) −→ ρ(A˜)X = AX˜ A˜ is translated into the 4 action of G˜0 on R given by~~

~~ρ(A˜)x := ϕ−1(ρ(A˜)ϕ(x)).~~

By a slight abuse of notation, as we already did above, we also take ρ(A˜) to be a matrix 0 from G0 = SO(3, 1) and understand ρ(A˜)x as multiplication of the column x by the t matrix ρ(A˜). We identify the unitary dual Nˆ with R4 by writingn ˆ(n)=ei px, i.e.n ˆ is t 4 coordinatized by (the row) p, p ∈ R . Then the action of G˜0 on Nˆ is simply given by the multiplication of the row tp by the matrix ρ(A˜−1), i.e.

~~tp −→ tpρ(A˜−1)=:A˜ · p.~~

~~+ The construction of the representation πm,+;j is based on the type 1 orbit~~

~~Oˆm = G˜0 · p0,p0 =(0, 0, 0,m),m>0.~~

~~As we remarked already, p0 corresponds to X0 = mE2 and has the stabilizing group H˜0 = SU(2). We take the representation πj of SU(2) and extend it to H˜ = H˜0 N to get π0 given by~~

~~t imx4 i p0x 4 π0((R, x)) = e π(R)=e πj(R) for all R ∈ SU(2),x∈ R .~~

To get some more practice and since it is important for application in physics, we work out the “ Second Realization” of the induced representation indG˜ π , i.e. with a repre- H˜ 0 sentation space consisting of functions u living on the homogeneous space

~~˜ ˜ ˜ ˜ ˆ+ G/H G0/H0 Om.~~

(At this moment, one usually does not bother with any ﬁniteness or integrability condition and simply takes smooth functions. We shall have occasion to do better in the next chapter.) The general theory of the Iwasawa decomposition or a direct calculation say that we have a decomposition

~~λz G˜ = SL(2, C) A˜ = A R ,R ∈ SU(2),A =( ),λ∈ R∗,z ∈ C. 0 p p p p λ−1~~

~~t Here it is customary in physics to coordinatize Ap by coordinates p =(p1,p2,p3,p4) such that p = Ap · p0. Then one has the Mackey decomposition~~

~~G˜ g = s(p)h, s(p)=(Ap, 0),h=(Rp,x) ∈ H˜ and the Master Equation~~

−1 −1 ∗ ∗ ∗ ∗ ∈ ˜ g0 s(p)=s(g0 (p))h ,h =(R ,x ) H with −1 −1 − −1 g0 s(p)=(A0 , ρ(A0 )x0)(Ap, 0) −1 − −1 =(A0 Ap, ρ(A0 )x0) −1 −1 −1 −1 =(A −1 , 0)((A −1 ) A Ap, −ρ((A −1 ) A )x0). g0 (p) g0 (p) 0 g0 (p) 0 160 7. Induced Representations

~~This shows that one has~~

~~∗ −1 −1 ∗ −1 −1 R =(A −1 ) A Ap,x = −ρ((A −1 ) A )x0. g0 (p) 0 g0 (p) 0~~

−1 ˆ+ One can verify that dµ(p)=p4 dp1dp2dp3 is an invariant measure on Om and that the −1 Radon-Nikodym derivative is dµ(g0 (p))/dµ(p) = 1. Hence by (7.10), in this case the representation is given by the prescription

~~∗−1 −1 π˜(g0)u(p)=π0(h )u(g0 p) with t ∗−1 i px0 ∗−1 π0(h )=e πj(R )~~

t i p0x as one can verify by a small calculation starting from π0((R, x)) = e πj(R): Namely, one has − − − h∗ 1 =(R∗ 1, x˜) withx ˜ = −ρ(R∗ 1)x∗ and hence t t ∗−1 ∗ t ∗−1 −1 p0x˜ = − p0ρ(R )x = p0ρ(R (A −1 ) A0)x0 g0 (p) i.e. t t −1 t p0x˜ = p0ρ(Ap )x0 = px0

~~∗ −1 −1 since we have R =(A −1 ) A Ap. g0 (p) 0~~

~~The functions u we are treating here are vector valued functions~~

~~ˆ+ −→ 2j+1 u : Om Vj C ,~~

j i.e. can be written as u(p)=(un(p))n=−j,−j+1,...,j . Here we write Dnn for the matrix elements of the representation πj from our discussion in 4.2 and then come to t i px0 j ∗−1 −1 ˜ ˜ · (7.24) (π((A0,x0))u)n(p)= e Dnn (R )un (A0 p). n From here one derives the following interpretation that we have a free particle of spin j and mass m :

In the rest system p = p0 and under rotations (R, 0),R∈ SU(2) (7.24) restricts to j (˜π((R, 0))un)(p0)= Dnn (R)un (p0), n i.e. the elementary expression of SU(2)-symmetry belonging to the representation πj. And for the inﬁnitesimal generators Pµ = ∂µ,µ =1, .., 4 of the translations in the µ-th direction x −→ xµ(t):=x +(tδµµ )µ one obtains from (7.24)

~~Pµun(p)=pµun(p).~~

~~Hence we have Mun = mun, i.e. un is an eigenfunction with eigenvalue m for the mass operator 2 − 2 − 2 − 2 M := P4 P3 P2 P1 . 7.6 Induced Representations and Vector Bundles 161~~

~~7.6 Induced Representations and Vector Bundles~~

The classic approach to the inducing mechanism we followed up to now, in the meantime has found a more geometric version using the notion of bundles. As Warner remarks in [Wa] p.376, this does not simplify in any essential way the underlying analysis but has the advantage that important generalizations of the entire process immediately suggest themselves (e.g. cohomological induction). Bundles play an important rôlein most parts of contemporary mathematics and we shall have to use them also in our next chapter concerning the beautiful orbit method. Hence, we present here some rudiments of the theory, which nowadays appears in most books on differential geometry, analytic and/or algebraic geometry etc. For instance, for a thourough study we recommend the book [AM] by Abraham and Marsden or 5.4 in [Ki] or, more briefly, the appendices in [Ki1] or [Be]. Whoever is in a hurry to get to the applications in Number Theory may skip this at first lecture. Before we try to shed some light on the notion of line, vector and Hilbert bundles, we look again, but from a different angle, at the construction of the representations of SU(2) from 4.2 and Example 7.3 in 7.1.5. We do this in the hope to create some motivation for the bundle approach. ζ We take G := SU(2) ⊃ H U(1) and π (h):=χ (ζ)=ζk for h = ∈ H, k ∈ N . 0 k ζ¯ 0 As we discussed in Example 7.3 in 7.1.5, we have the homogeneous space ab X = G/H P1(C),g= −→ (b :¯a) −¯b a¯

~~u uG and the unnormalized induction πk = ind H χk given by left (inverse) translation on the space Hπ of smooth functions φ : G −→ C with functional equation~~

~~−1 φ(gh)=χk(h )φ(g) for all g ∈ G, h ∈ H.~~

(In this example everything is unimodular, hence it does not matter whether we treat un- or normalized induction, but in general several authors stick to unnormalized induction though one can also treat normalized induction as we shall see below.) For V = C we introduce as our ﬁrst example of a line bundle

~~Bk := G ×H V := {[g, t]:=(g, t)∼; g ∈ G, t ∈ V } where the equivalence ∼ of the pairs is deﬁned by~~

~~(gh, t) ∼ (g, π0(h)t) for g ∈ G, h ∈ H, t ∈ V.~~

~~G acts on Bk by~~

~~g0[g, t]:=[g0g, t] for all g0 ∈ G, [g, t] ∈Bk . One has a projection~~

~~pr : Bk −→ G/H = X , [g, t] −→ x := gH.~~

~~Obviously the projection and the action are well deﬁned and one has the ﬁber over x~~

~~−1 (Bk)x := pr (x)={[g, t]; gH = x}V C. 162 7. Induced Representations~~

This leads to the picture that Bk is a bundle standing over the base space X and consisting of the ﬁbers (Bk)x over all base points x ∈X. Our main object is the space Γ(Bk)ofsections of Bk, i.e. maps

~~ϕ : X−→Bk with pr ◦ ϕ = idX .~~

These sections are of the form ϕ(x)=[g, φ(g)] for x = gH with a function φ : G −→ V . Since the coset gH may be represented also by gh, h ∈ H, the welldefinedness of the section asks for −1 [g, φ(g)] = [gh, φ(gh)] = [gh, π0(h )φ(g)] where the last equation is expression of the equivalence introduced in the definition of the bundle Bk. This leads to the fact that the function φ has to fulfill the functional equation −1 φ(gh)=π0(h )φ(g) for all h ∈ H, g ∈ G, which we know from the construction of the representation space for the induced representation. We define an action of G on the space Γ(Bk) of sections by

· −1 −1 ∈ ∈ B (g0 ϕ)(gH):=g0ϕ(g0 g)=[g, φ(g0 g)] for all g0 G, ϕ Γ( k), and this leads to an action of G on the space H of functions φ : G −→ V (satisfying the functional equation) deﬁned by · −1 g0 φ(g):=φ(g0 g)

Hence at least formally, we recover the prescription for the deﬁnition of the induced representation π. This can be completed to a precise statement if we provide our bundle with an appropriate topology. And this appears to be largely independent of the special example G = SU(2), which we proposed at the beginning. In the following, we will try to show how this works by giving some rudiments of the theory of bundles. We start by following the treatment of Warner [Wa] p.377, which is adapted to the application in representation theory, and proceed to another more general approach afterwards.

~~Bundles, First Approach~~

~~Definition 7.2: Let X be a fixed Hausdorff topological space. A bundle B over X is a pair (B,p) where B is a Hausdorff topological space and p is a continuous open map of B onto X .~~

~~One calls X the base space, B the bundle space, and p the bundle projection. By abuse of notation the pair (B,p) is often abbreviated by the letter B to indicate the bundle.~~

~~−1 For x ∈X, the set p (x) is called the ﬁber over x and often denoted by Bx.Across section or simply section of the bundle is a map s : X−→B with p ◦ s = idX .~~

If b ∈ B and s(p(b)) = b, then we say that s passes through b. If to every b ∈ B there is a continuous cross section s passing through b, then the bundle is said to have enough cross sections. 7.6 Induced Representations and Vector Bundles 163

~~A Hilbert bundle B over X is a bundle (B,p)overX together with operations and a norm making each ﬁber Bx,x ∈X into a complex Hilbert space and satisfying the following conditions:~~

1.) The map B −→ R,b−→ b is continuous, 2.) The operation of addition is a continuous map of {(b1,b2) ∈ B × B, p(b1)=p(b2)} into B, 3.) The operation of scalar multiplication is a continuous map of C × B into B, 4.) The topology of B is such that its restriction to each ﬁber Bx is the norm topology of the Hilbert space Bx. (This condition is formulated more precisely in [Wa] p.377.)

Example 7.15: Let E be any Hilbert space. The trivial bundle with constant ﬁber E is given by B := X×E equipped with the product topology and the projection p given by p(x, a)=x for x ∈X,a∈ E. The cross sections are given by the functions s : X−→E.

Let B =(B,p) be a Hilbert bundle over X and Γ(B) the set of continuous cross sections of B . Then it is clear that Γ(B) is a complex vector space under pointwise addition and scalar multiplication. In the application to representation theory this space shall provide the representation space of a representation. So we need a topology on it. And here things start to be a bit technical. Following [Wa], we assume that X is locally compact and for Γ(B) define the uniform on compacta topology. For our purposes it is sufficient that in particular this leads to the consequence that each sequence (sn)inΓ(B) converges to a continuous cross section ϕ iff for each compact subset C ⊂X one has − −→ −→ ∞ supx∈C sn(x) s(x) 0 for n . (In [Wa] the sequence is replaced by the more general notion of net but we will not go into this here.)

The integration theory in this context is still more delicate. So for details, we also refer to [Wa]. For our purpose it will suﬃce to ﬁx a measure µ on X and think of Lq(B,µ) B q 1/q ∞ as the space of sections s of with s q := ( X s(x) dµ(x)) < . Then the inner product on the space of sections is given by := dµ(x) for all s, t ∈ L2(B,µ). X

Let G be a linear group, H a closed subgroup, δ the quotient of the modular functions of G and H treated in 7.1 (normalized by the condition δ(1) = 1) and π0 a unitary representation of H on the Hilbert space V .

~~We deﬁne a continuous right action of H on the product space G × V as follows:~~

~~1/2 −1 (g, a)h := (gh, δ(h) π0(h )a) for all g ∈ G, a ∈ V.~~

The orbit under H of a pair (g, a) ∈ G × E will be denoted by [g, a]:=(g, a)H. The space of all such orbits equipped with the quotient topology will be called E(G, H, π0) or simply E. This space is Hausdorﬀ and the natural projection map

p : E −→ G/H =: X , [g, a]=(g, a)H −→ gH =: x is a continuous and open surjection. Therefore E := (E,p) is a bundle over X = G/H, which becomes a Hilbert bundle by the following natural assignments: 164 7. Induced Representations

~~We deﬁne addition and scalar multiplication in each ﬁber Ex := {[g, a]; a ∈ V } by~~

~~c1[g, a1]+c2[g, a2]:=[g, c1a1 + c2a2] for all c1,c2 ∈ C and the scalar product by~~

−1 < [g, a1], [g, a2] >Ex := δ(g) . Obviously, there are some verifications to be done. Whoever does not want to do these on his own can look into [Wa] p.380/1. The bundle is called the Hilbert bundle induced by π0 (and δ) and denoted by E(π0) or again E. G acts continuously from the left on E by g0[g, a]:=[g0g, a]. G Finally, we rediscover the normalized induced representation π = indH π0 of G as a representation by unitary operators on the space of square integrable sections of E:To this end, one first observes that there exists a one-to-one correspondence between cross sections ϕ for E = E(π0) and functions φ : G −→ V fulfilling the functional equation

~~1/2 −1 φ(gh)=δ(h) π0(h )φ(g) for all g ∈ G, h ∈ H.~~

~~The correspondence is given by~~

~~ϕ(gH)=(g, φ(g))H.~~

If φ and ϕ are connected in this way, one says that φ is the unitary form of ϕ and ϕ is the bundle form of φ. Evidently a cross section ϕ is continuous on X = G/H iﬀ its unitary form φ is continuous on G. It is a bit less evident but true (see [Wa] p.381) that E(π0) admits enough cross sections. Therefore it makes sense to consider the Hilbert space H 2 E π0 := L ( ,µ) where µ is the quasi-invariant measure on X associated with δ and the ρ−function on X as in 7.1. (It can be shown that the construction is independent of the choice of the function δ.) By a (small) abuse of notation, one denotes the space of bundle forms φ of H elements ϕ in the space of sections as well by π0 . This space inherits the Hilbert space structure and has as scalar product <φ,φ>˜ := δ(s(x))−1 <φ(s(x)), φ˜(s(x)) >dµ(x) X where s denotes the Mackey section we introduced in 7.1.

~~The bundle theoretic deﬁnition of the unitarily induced representation π is as follows: ∈ ∈H For g0 G and ϕ π0 , put~~

~~−1 π(g0)ϕ(gH):=g0ϕ(g0 gH) and correspondingly for the unitary form φ of ϕ~~

−1 π(g0)φ(g)=φ(g0 g). Some routine veriﬁcations parallel to those we did in 7.1 (see [Wa] p.382) show that we have got a unitary representation equivalent to the one constructed in 7.1. And one 7.6 Induced Representations and Vector Bundles 165 should get the feeling that all this is realized by the example of the (complex!) bundle 1 E = Bk over X = P (C) from the beginning of this section. We will say more when we come back to this at the end of the section.

As Warner remarks ([Wa] p.382), the preceding bundle-theoretic construction makes no use of the local triviality of the constructed bundle. Moreover, “the deﬁnition of Hilbert bundle is, strictly speaking, not in accord with the customaty version - however, it has the advantage of allowing to place Hilbert space structures on the ﬁbers without having to worry about transition functions”. As we shall need this later, we present some elements of this other way to treat bundles.

~~Bundles, Second Approach~~

Here we need the notion of a diﬀerentiable manifold, which already was alluded to in 6.2 and is to be found in most text books on advanced calculus. We recommend in particular the Appendix II in [Ki1] (or the Appendix A in [Be]).

Deﬁnition 7.3 : A p−manifold M (or a p-dimensional manifold) is a Hausdorﬀ topological space that admits a covering by open sets Ui,i ∈ I, endowed with one-to-one p continuous maps ϕi : Ui −→ Vi ⊂ R . M is called separable if it can be covered by a countable system of such open sets. A smooth resp. analytic p-manifold is a separable p-manifold so that all maps

~~◦ −1 | ϕi,j := ϕi ϕj ϕj (Ui∩Uj ) are inﬁnitely diﬀerentiable (resp. analytic).~~

~~We use the following terminology:~~

~~– the pairs (Ui,ϕi) (and often also the sets Ui) are called charts;~~

~~p – for m ∈ Ui =: U and ϕi =: ϕ the members x1,...,xp of the p−tupel ϕ(m)=:x ∈ R are called local coordinates;~~

~~– the functions ϕi,j are called transition functions;~~

~~– the collection (Ui,ϕi)i∈I is called an atlas on M.~~

It has to be clarified which atlases are really different: We say that two atlases (Ui)i∈I and (U j)j∈J are called equivalent if the transition functions from any chart of the first atlas to any chart of the second one are smooth (resp. analytic). The structure of a smooth (resp. analytic) manifold on M is an equivalence class of atlases.

~~When dealing with a smooth manifold, we use only one atlas keeping in mind that we can always replace it by an equivalent one.~~

In parallel to the just given notions for real manifolds, one deﬁnes a smooth (or holomorphic) complex p-manifold where Rp is replaced by Cp and the transition functions are required to be holomorphic, i.e. complex diﬀerentiable. 166 7. Induced Representations

~~Examples:~~

~~Example 7.16: M = Rp and open subsets are real p−manifolds.~~

~~Example 7.17: M ⊂ Rq is a p-dimensional submanifold if for any m ∈ M there are an q open set V ⊂ R and diﬀerentiable functions fi : V −→ R,i=1,...,q− p, such that~~

~~–a)M ∩ V = {x ∈ V ; f1(x)=···= fq−p(x)=0}, ∂f − – b) the functional matrix ( ∂x) has rank q p for all x.~~

~~The hyperboloids appearing as orbits in our discussion of the representations of the Poincar´egroup in 7.5 are three-dimensional smooth submanifolds of R4.~~

Example 7.18: M = P1(C) (as in Example 7.3 in 7.1.5 and the beginning of this section) is a smooth one-dimensional complex manifold: We cover P1(C) by two open subsets U := {(u : v); u, v ∈ C,v =0 } and U := {(u : v); u, v ∈ C,u=0 }. We have the homeomorphisms

~~ϕ : U −→ C, (u : v) −→ u/v =: w ∈ C, (7.25) ϕ : U −→ C, (u : v) −→ v/u =: w ∈ C.~~

~~And the map ϕ ◦ ϕ |{w=0 } (w)=1/w = w is complex diﬀerentiable. So here one has an atlas consisting of two charts U and U .~~

~~Certainly, at the same time, P1(C) is an example for a two-dimensional real manifold.~~

~~Example 7.19: Show as Exercise 7.25 that M is a smooth p−dimensional (real) manifold where~~

~~p p+1 M = P (R):={[u1 : ···: up+1]=(u1,...,up+1)∼;(u1,...,up+1) ∈ R \{0}~~

~~∗ with (u1,...,up+1) ∼ (u 1,...,u p+1) iﬀ there exists a λ ∈ R with u i = λui for all i.~~

If M is a smooth p−manifold and M a smooth p−manifold, there is a natural notion of a smooth map F : M −→ M . Namely, for any point m ∈ M with F (m)=m and any charts U m and U F (m)=m, the local coordinates of m must be smooth functions of local coordinates of m ∈ U. A smooth map from one p−manifold to another, which has a smooth inverse map is called a diffeomorphism. Two manifolds are called diffeomorphic if there is a diffeomorphism from one to another.

A manifold is called oriented iﬀ it has an oriented atlas, i.e. an atlas where for any two ∂x charts with coordinates x and y the determinant of the functional matrix ( ∂y ) is positive whenever it is deﬁned.

As already said, there is much more to be discussed in this context (and we refer again to e.g. [Ki1]) but, we hope, this rudimentary presentation suﬃces to grab an understanding of our second approach to bundles: 7.6 Induced Representations and Vector Bundles 167

Definition 7.4: E is called a (fiber) bundle over the base X with fiber F iff E,F and X are smooth manifolds and there is given a smooth surjective map p : E−→Xsuch that locally E is a direct product of a part of the base and the fiber F .

More precisely, we require that any point x ∈Xhas a neighbourhood Ux such that −1 × ∈ EUx := p (Ux) can be identiﬁed with Ux F via a smooth map hx so that, for m EUx , one has hx(m)=(x ,v)ifx = p(m), i.e. the following diagram (where p1 denotes the projection to the ﬁrst factor) is commutative

~~−1 hx p (Ux) −→ Ux × F~~

~~↓ p ↓ p1~~

~~∼ Ux −→ Ux.~~

−1 From the definition it follows that all sets Ex := p (x),x∈X, called fibers, are smooth submanifolds diffeomorphic to F . One often writes E −→F X to denote a bundle with fiber F and base X . There is the natural notion of a map between bundles: Let E −→F X and E −→F X be two bundles. Then a bundle map is given by the pair of smooth maps

~~Φ:E−→E,f : X−→X such that the following diagram is commutative~~

~~E −→Φ E~~

~~p ↓↓p~~

~~f X −→ X .~~

~~Two bundles over the same base X are called equivalent if there is a bundle map with f = idX and Φ a diﬀeomorphism.~~

~~A bundle E −→F X is called a trivial bundle if it is equivalent to the bundle X×F −→F X .~~

We are mainly interested in vector or line bundles where the ﬁber is F := V a vector space (with dim V = n = 1 for the case of a line bundle) and the maps hx are linear in −1 each ﬁber p (x ),x ∈ Ux. Primarily, we think of real vector spaces and real manifolds but we also will treat complex cases.

~~In any case, the dimension of the ﬁber as a vector space is called the rank of the vector bundle. 168 7. Induced Representations~~

To make this more explicit and show a way to construct bundles, we take an atlas n of the real manifold X given by open sets Ui (i ∈ I), and V = R . Then we have homeomorphisms −1 −→ × n h := hi : EUi = p (Ui) Ui R , which a) respect the ﬁbers, i.e. with

~~−1 h(m)=(x, v) for all m ∈ Ex = p (x),x∈ Ui, and b) restrict to vector space isomorphisms~~

~~| −→ n h Ex : Ex R .~~

~~We denote by~~

−1 n n ◦ | n ∩ × −→ ∩ × hi,j := hi hj (Ui∩Uj )×R :(Ui Uj) R (Ui Uj) R the bundle transition maps and introduce (diﬀerentiable) functions, which are called bundle transition functions

~~ψi,j : Ui ∩ Uj −→ GL(n, R) Aut (V ) by n hi,j(x, v)=(x, ψi,j(x)v) for all (x, v) ∈ (Ui ∩ Uj) × R .~~

~~It is not diﬃcult to see that these ψi,j are uniquely determined and for i, j, k with~~

~~Ui ∩ Uj ∩ Uk = ∅ fulﬁll the following cocycle condition~~

~~(7.26) ψi,j ◦ ψj,k = ψi,k.~~

Here, our main point is that we can construct bundles with ﬁber Rn over a manifold n with atlas (Ui)i∈I by pasting together columns Ui × R via such a given cocycle system ψi,j. We have the following construction principle, whose validity is not too diﬃcult to prove.

~~Proposition 7.4: Let X be a manifold with atlas (Ui)i∈I and ψi,j : Ui ∩Uj −→ GL(n, R) be a system of smooth functions with~~

~~ψi,i =1 in Ui,~~

~~ψi,j ◦ ψj,i =1 in Ui ∩ Uj,~~

~~ψi,j ◦ ψj,k ◦ ψk,i =1 in Ui ∩ Uj ∩ Uk.~~

n n Let E˜ be the union of the manifolds Ui × R . One says that points (xi,vi) ∈ Ui × R n and (xj,vj) ∈ Uj × R are equivalent if xi = xj and vi = ψi,j(xj)vj. Then the factor space E of E˜ defined by this equivalence relation is a vector bundle with fiber Rn over X with projection p induced by the natural projections (xi,vi) −→ xi. And each such fiber bundle is equivalent to a bundle of this type. 7.6 Induced Representations and Vector Bundles 169

Proposition 7.5: Let E and E be vector bundles of the same rank n over X defined by the systems of bundle transition functions ψi,j and ψi,j (without loss of generality, by a refinement argument, one can assume that both bundles are defined by the same atlas (Ui)i∈I ). Then both bundles are equivalent iff there is a family of matrix functions fi : Ui −→ GL(n, R) such that one has −1 ∈ ψi,j = fi ψi,jfj for all i, j I. As in our first approach, for open sets U ⊂X we introduce (vector) spaces Γ(U, E)of sections s over U, i.e. smooth maps s : U −→ E with p ◦ s = idU .ForU = X we talk of global sections. Parallel to our treatment of real bundles, one can look at the complex case by replacing appropriately R by C.

~~Examples:~~

Example 7.20: The most prominent example, which led to the introduction of the notion of bundles, is the tangent bundle TM, which comes about if for an n-dimensional manifold M with charts (Ui,ϕi),i∈ I, as matrix of bundle transition functions one takes the Jacobian J of the manifold transition functions ϕi,j

~~ψi,j(m):=Jϕi,j (ϕj(m)).~~

~~n Then TM can be thought of as consisting of classes of pairs (m, a)∼,m ∈ M,a ∈ R , with (m, a) ∼ (m, b)iﬀa and b are related by the contravariant transformation property~~

n ∂y b = i (y)a i ∂x j j=1 j when m appears in the chart (U, ϕ) with coordinates x =(x1,...,xn) and simultaneously in the chart (U ,ϕ) with coordinates y =(y1,...,yn). This has a geometrical background: in first approximation, we take the manifold M and to each point m ∈ M associate as a fiber Em the tangent space TmM to M at this point m. From higher calculus one knows that there are different ways to describe the elements Xm ∈ TmM i.e. the tangent vectors to a manifold. We follow the one already prepared in 6.2 while discussing the definition of the Lie algebra associated to a linear group and think of the tangent space of M at m as the space of all tangent vectors of all smooth curves γ inM passing through m. More precisely, a tangent vector Xm to M at the point m is defined as the equivalence class of smooth parametrized curves γ = γ(t) passing through m.Two curves γ andγ ˜ in M with γ(0) =γ ˜(0) = m are equivalent iff their tangent vectors in m coincide in the coordinates of a chart (U, ϕ) with m ∈ U, i.e. one has γ ∼ γ˜ if there is a chart (U, ϕ) with m ∈ U and coordinates ϕ(m)=x such that

d d ϕ(γ(t)) | = ϕ(˜γ(t)) | . dt t=0 dt t=0 With respect to this chart, a tangent vector can be symbolized by the n-tupel of real d | numbers ai whereai is just the i-th component of dt ϕ(γ(t)) t=0 or by the differential n operators Xm = i=1 ai∂xi acting on smooth functions f in the coordinates x. Addition and scalar multiplication of these operators resp. tuples define a vector space structure, which can and will be carried over to the first given description of the space TmM. 170 7. Induced Representations

If (U ,ϕ) with coordinates y is another chart for m and (b1,...,bn) are the components of the tangent vector to γ expressed in this coordinate system, by the chain rule we have with ϕ(ϕ−1(x)) = y(x)

~~d d ∂y b =( ϕ(γ(t)) | ) =( (ϕ(ϕ−1(ϕ(γ(t)))) | ) = i a , i dt t=0 i dt t=0 i ∂x j j j i.e. the contravariant transformation property from our general construction above.~~

~~Deﬁnition 7.5: Global sections X : M −→ TM of the tangent bundle TM are called vector ﬁelds on M.~~

Let (U, ϕ) be a chart with coordinates x =(x1,...,xn). With a (now usual) slight misuse of notation we write n | X U =: ai(x)∂xi i=1 where the coeﬃcients ai are smooth functions deﬁned in U. For a chart (U ,ϕ) with coordinates y =(y1,...,yn), one has as well

n | X U = bj(y)∂yj j=1 and on U ∩ U = ∅ we have the contravariant transformation property ∂y b (y)= j (y) a (x(y)). j ∂x i i i V (M) denotes the vector space of smooth vector ﬁelds X on M.

∗ Example 7.21: The cotangent bundle to M has as its ﬁbers the cotangent spaces TmM, i.e. the duals to the tangent spaces TmM. It turns up if, in the same situation as above in Example 7.20, as matrix of bundle transition functions we take the transpose of the ◦ −1 Jacobian of the map ϕj ϕi =: ϕj,i, i.e.

~~t ψi,j(m)= Jϕj,i (ϕi(m)).~~

The global sections α : M −→ T ∗M of the cotangent bundle T ∗M are called differential forms of first degree or simply one-forms on M. As a parallel to the notation for vector ∗ ∗ fields above, with functions ai and b j defined in U resp. U we write ∗ ∗ | | α U = ai (x)dxi, resp.αU = bj (y)dyj, i j and here have the covariant transformation property ∂x (7.27) b∗(y)= i (x(y))a∗(x(y)). j ∂y i j j

~~Ω1(M) denotes the vector space of one-forms α on M. 7.6 Induced Representations and Vector Bundles 171~~

Example 7.22: The ﬁrst two examples concerned real bundles but similarly one can treat the complex case. Here we take the one-dimensional complex manifold M = P1(C) covered by the two charts (U, ϕ) and (U ,ϕ) ﬁxed above in (8.10) while discussing M −k as a manifold. For k ∈ Z we get a line bundle Lk by choosing ψ(w)=w as bundle transition function between these two charts. Let us work out the construction of this bundle along the line of the general construction principle given above: We have two “columns”, one over U and one over U , namely

~~L := {(w, t),w,t∈ C},L := {(w,t),w,t ∈ C}.~~

~~Using ψ we paste them together over U ∩ U = C∗ by identifying (w, t) ∼ (w,t)iﬀ~~

~~w =1/w, t = w−kt.~~

~~Remark 7.9: It can be shown that up to (holomorphic) equivalence there are no other 1 holomorphic line bundles over P (C) than these Lk.~~

~~Proposition 7.6: For the space of holomorphic sections one has~~

~~dimC Γ(Lk)=k + 1 for k ≥ 0, = 0 for k<0.~~

~~Proof: Let s be a section of Lk with~~

s :(u : v) −→ ∈Lk, where h()=(w, t) ∈ L for w = u/v, v =0 , and h()=(w,t) ∈ L for w = v/u, u =0. The section is holomorphic iﬀ t is given by a holomorphic function in w and t by a holomorphic function in w and we have the compatibility condition t = w−kt.As examples and prototypes for such a holomorphic function we take the monomials t = wp p and t = w with p, p ∈ N0. The compatibilty condition says for w =0

~~t(w)=wp = wkt(w)=w−k+p~~

and, since one has w =1/w, we get w−p = wp−k and see that everything is consistent for p =0, 1,...,k if k is not negative. And the holomorphicity of the relation breaks down for negative k. Hence we have k + 1 essentially diﬀerent holomorphic sections for non-negative k and none for negative k. One needs a bit more complex function theory to be sure that there are no more (see for instance [Ki] p.80).

~~Bundles and Representations~~

After this short digression into the world of bundles, which will be useful later, we return to the problem of the construction of group representations. As we know from our first approach to bundles, the space of sections is meant to provide a representation space and, hence, should get the structure of a Hilbert space. In general this needs some work, which we shall attack in the next chapter. But if the space of sections is a finite- dimensional vector space, this is no problem. To get an idea we return to our example ⊃ G G = SU(2) H U(1) and the unitary representation πj = indH π0 induced from the representation π0 = χk,k =2j ≥ 0, of H U(1). In our first approach we constructed 172 7. Induced Representations

~~1 the bundle space Bk = {[g, t]; g ∈ G, t ∈ C} with projection pr onto X = G/H = P (C) given by ab pr([g, t]:=gH =(u : v)=(b :¯a) for g = . −¯b a¯~~

We deﬁne a map of Bk to the line bundle Lk constructed by pasting together the sets L = {(w, t); w, t ∈ C} and L = {(w.t); w,t ∈ C} via the bundle transition function ψ(w)=wk as follows: For g with a = 0 we put

¯k Bk [g, t] −→ (w = b/a¯; b t) ∈ L and for g with b =0 k Bk [g, t] −→ (w :=a/b, ¯ a t) ∈ L . ζ Both maps are well deﬁned since for h = one has ζ−1 aζ−1 bζ gh−1 = −¯bζ−1 aζ¯ and k π0(h)=ζ . −1 Hence application of the mapping prescription to the element [gh ,π0(h)t], which as well represents [g, t], leads to the same images. And the images in L and L are consistent with the pasting condition (w,t)=(1/w, w−kt) for w = 0, since with w = b/a¯ one has −k k k w (b t)=¯a t. Thus both maps induce a map of Bk to Lk, which obviously is a bijection.

Here we have a complex line bundle over the one-dimensional complex manifold P1(C) whose space of holomorphic sections is a complex vector space and the representation space for the representation πj, 2j = k of G = SU(2). In principle, the base spaces for the bundles in our ﬁrst approach are real manifolds and the section spaces real vector spaces. This example shows that we may get the wanted irreducible representation by some more restrictions, namely we put more structure on the real construction to get the more restrictive complex case (as in holomorphic induction). We shall see more of this under an angle provided from the necessities of physics in the next chapter. Chapter 8

~~Geometric Quantization and the Orbit Method~~

The orbit method is an object of pure mathematics, but it got a lot of impetus from physics under the heading of geometric quantization. Historically it was proposed by Kirillov in [Ki2] for the description of the unitary dual of nilpotent Lie groups. By further work of Kirillov and many others, in particular B. Kostant, M. Duﬂo, M. Vergne and D. Vogan, it grew into an important tool for explicit constructions of representations of various types of groups. The construction mechanism uses elements that also appear in theoretical physics when describing the transition from classical mechanics to quantum mechanics. We therefore start by recalling some of this background, even though we cannot give all the necessary deﬁnitions.

~~8.1 The Hamiltonian Formalism and its Quantization~~

The purpose of theoretical mechanics is to describe the time development of the state of a physical system. In classical mechanics such a state is identfied with a point in an n-dimensional manifold Q, the configuration space. The point is usually described by local coordinates q1,...,qn, called position variables. The time development of the system then is described by a curve γ given in local coordinates as t −→ (qi(t)), which is fixed by an ordinary differential equation of second order. In the Hamiltonian formalism one adds to the position variables q =(q1,...,qn) the momentum variables p =(p1,...,pn) (related to the velocity of the particle or system, we do not need to be more exact here) and the time development t −→ (q(t),p(t)) is described by the hypothesis that the curve t −→ (q(t),p(t)) =: γ(t) has to fulfill Hamilton’s equations, i.e. the first order system of partial differential equations

dqi ∂H dpi ∂H (8.1) q˙i := = , p˙i := = − dt ∂pi dt ∂qi where the function H = H(q, p, t)istheHamiltonian of the system, i.e. encodes all the relevant information about the system. For instance, if our system is simply a mass one (point) particle in R3 moving under the inﬂuence of the force with a radial symmetric 2 2 2 2 2 2 potential V (q)=1/r, r = q1 + q2 + q3, one has H = p1 + p2 + p3 + V (q). 174 8. Geometric Quantization and the Orbit Method

~~This Hamilton formalism has several interpretations:~~

~~1. The Geometric Picture~~

A curve γ = γ(t) is called an integral curve for the vector field ··· X = a(q, p)∂q + b(q, p)∂p := a1(q, p) ∂q1 + ...an(q, p) ∂qn + b1(q, p) ∂p1 + + bn(q, p) ∂pn iff at each point of the curve its tangent vector is a vector from the vector field. We write this as γ˙ (t)=X(γ(t)). The statement that γ is an integral curve for the Hamiltonian vector field ∂H ∂H X = XH with ai = ,bi = − , ∂pi ∂qi is an expression of the assertion that γ fulfills Hamiltons equations (8.1).

~~2. The Symplectic Interpretation~~

To state this, we have to provide some more general notions about differential forms, which we will need anyway in the following sections. Hence we enlarge our considerations about the tangent and cotangent bundle and their sections from the last section: We look again at a (real) smooth n-manifold M. –Avector field X is defined as a section of the tangent bundle TM and in local coordinates x =(x1,...,xn) written as ··· X = a(x)∂x := a1(x)∂x1 + + an(x)∂xn . –Adifferential form of degree one or one-form α is defined as a section of the cotangent bundle T ∗M and in local coordinates written as

~~α = a(x)dx := a1(x)dx1 + ···+ an(x)dxn.~~

As one knows from calculus, there also are higher order differential forms of different types. We shall need the symmetric and the antisymmetric or exterior forms. –Asymmetric r-form β is written in the local coordinates from above as ····· β := bi1...ir (x)dxi1 dxir where the coefficients b are (smooth) functions of the coordinates x and the differentials dxi are commuting symbols, i.e. with dxi · dxj = dxj · dxi for all i, j, the sum is meant over all r-tuples i1,...,ir ∈{1,...,n} and the coefficients b are invariant under all permutations of the indices. For instance, if n = 2, one has the two-form

2 2 β = b11dx1 + b12dx1dx2 + b21dx2dx1 + b22dx2,b12 = b21. Certainly, there is the possibility to introduce a reduced form where one restricts the index-tuples by taking exactly one representative from each orbit of the symmetric group Sr acting on the index-tuples. Hence, as reduced form one has in the example above 2 ˜ 2 ˜ β = b11dx1 + b12dx1dx2 + b22dx2, b12 =2b12. 8.1 The Hamiltonian Formalism and its Quantization 175

If one wants to be more sophisticated (and precise), one says that a symmetric r−form is a (smooth) section of the r−th symmetric power of the cotangent bundle. But for our treatment here and for the antisymmetric case below the naive description above should be suﬃcient.

An antisymmetric or exterior r−form α or form of degree r is given in local coordinates by ∧···∧ α = ai(x)dxi1 dxir where here the diﬀerentials are anticommuting, i.e. with dxi ∧ dxj = −dxj ∧ dxi for all i, j. The summation is again over all r−tuples i =(i1,...,ir) and the coeﬃcients have the special property of antisymmetry

~~ai = sgn σaσi for all i =(i1,...,ir),σ∈ Sr, hence, in particular one has ai = 0 if two of the indices coincide. If we take in our example n = r =2, we get~~

~~α = a12dx1 ∧ dx2 + a21dx2 ∧ dx1,a12 = −a21.~~

~~It is very convenient to introduce here the reduced form where the sum is taken only over all r−tuples (i)=(i1,...,ir) with 1 ≤ i1 < ···~~

The set of these exterior r−forms on U ⊂ M is denoted by Ωr(U). By the operations on the coefficients one has a natural manner to define a vector space structure on Ωr(U) and one has the possibility to do scalar multiplication with functions. Moreover one has an associative multiplication if one multiplies a p−form with a q−form distributively re- garding the anticommutativity of the dxi. It is decisive for the application of differential forms that we have still two more operations, the differentiation d augmenting the degree by one and the inner multiplication i(X) with a vector field X lowering the degree by one (moreover there is the Lie derivative, which we will not consider for the moment). We will have to be content again with the description in local coordinates. ∧···∧ For α = a(i)dxi1 dxir we define the exterior differentiation dα by ∧ ∧···∧ dα := da(i) dxi1 dxir where da denotes the usual total differential of a function a, i.e. is given by

~~n ∂a da := dx . ∂x i i=1 i To handle the exterior diﬀerential, one usually reorders it to its reduced form. 176 8. Geometric Quantization and the Orbit Method~~

A simple but most important example appears if we look at the coordinates given by n x =(q1,...,qn,p1,...,pn) and take the one-form α = i=1 qidpi. We get n (8.2) ω0 := dα = dqi ∧ dpi i=1 and, obviously, have dω0 =0. We note that the relation ddα = 0 holds in general for all α.

~~A form α is called closed iﬀ one has dα = 0 and it is called exact iﬀ there is a form β such that α = dβ.~~

Moreover, we point to the fact that exterior differentiation commutes with substitution of the variables and, if Φ : M −→ M is a smooth map of manifolds and α a form on M , then one can draw back α to a form Φ∗α on M. Without being more pecise here, for practical purposes this comes out by substitution of the local coordinates x −→ y = y(x) ∂yi and using the well known formula dyi = dxj. We do not have enough space to go j ∂xj into this more deeply but later on shall see in examples how this works. ∧···∧ For α = a(i)(x)dxi1 dxir and the vector field X = bj(x)∂j, we have an inner product i(X)α, which comes out (this certainly is not the most elegant definition, but avoids more generalities) by applying successively the prescriptions

~~i(X)dxi = bi(x) and i(X)(α ∧ β)=(i(X)α) ∧ β +(−1)rα ∧ (i(X)β) for α ∈ Ωr,β ∈ Ωq.~~

To show how this works, we give an example, which is the most interesting for us: As n ∧ above, we take coordinates x =(q1,...,qn,p1,...,pn) and α = ω = i=1 dqi dpi and n X = i=1(ai(q, p) ∂qi + bi(q, p) ∂pi ). Then we have n i(X)ω = (aidpi − bidqi). i=1

Finally, we use this for the symplectic formulation of the Hamilton formalism: We go from the configuration space Q with coordinates q to its cotangent bundle M = T ∗Q, called the phase space, with coordinates (q, p). We provide this space with the two-form n ∧ ω = i=1 dqi dpi (and hence get the protype of a symplectic manifold, which will be analysed later more thoroughly). Now we can say that a curve γ provides a solution to Hamilton’s equations (8.1) if it is an integral curve of the vector field X, which fulfills the equation

~~(8.3) dH = i(X)ω.~~

~~Namely, in this case one has ∂H ∂H = −bi, = ai, ∂qi ∂pi i.e. X is a Hamiltonian vector ﬁeld XH . 8.1 The Hamiltonian Formalism and its Quantization 177~~

~~3. The Poisson Interpretation~~

In 6.1 we introduced the Poisson bracket for two smooth functions f and g with coordinates (q, p) n ∂f ∂g ∂f ∂g {f,g} := − . ∂q ∂p ∂p ∂q i=1 i i i i Hence Hamilton’s equations (8.1) can also be expressed as q˙ = {q, H}, p˙ = {p, H}.

And more generally, if one replaces the special observables positions qi and momenta pi by a more general observable f, i.e. a smooth function of (q, p), as equation of time development one has (8.4) f˙ = {f,H}. Here we ﬁnd an appropriate starting point to do quantization, i.e. to go over to a quantum mechanical description of the system. A quantum mechanical system is modeled by a complex Hilbert space H. The state of the system is represented by a one-dimensional subspace L = <ψ>of H,ψ ∈H\{0}, which can be seen as a point ψ∼ in the projective space ∗ P(H):={ψ∼; ψ ∈H\{0}},ψ∼ ψ iﬀ ψ = λψ for a λ ∈ C .

Then the time development reﬂects in a curve γ(t)=ψ(t)∼ in P(H) and the physical law determining this development is encoded into a one-parameter group G of unitary operators U(t),t∈ R, such that γ is a G−orbit in P(H). A physical observable f should correspond to an operator B = Bf acting on or in H (as there are subtle problems, we will not be more precise here) and provides the information about the probability to measure the observable in the state <ψ>. Similar to our treatment of inﬁnitesimal generators of one-parameter subgroups in section 6.2, one can look for a skew-adjoint operator A such that (in some precise sense) one has U(t) = exp tA and, hence, the counterpart of the classical equation of motion will be dψ = Aψ. dt In the physics literature, usually one introduces the (self-adjoint) Hamilton operator Hˆ by another normalization, namely

U(t)=:eitHˆ (we use units such that Planck’s constant is one) and then has the Schrödingerequation dψ = iHψ.ˆ dt In our standard example of a system consisting of a particle with mass m moving in 3 R in a central force field with potential V, one translates the position variables qj from the classical world into operators Bqj acting by multiplication with iqj, the momentum variables pj go to the differential operators Bpj = ∂qj acting on (a subspace of smooth functions in) the Hilbert space H = L2(R3) and one has the Schrödingerequation

~~dψ 3 = iHψˆ = i((1/2m) ∂2 − V (q))ψ. dt qj j=1 178 8. Geometric Quantization and the Orbit Method~~

In general, it can be a serious problem to ﬁnd a Hilbert space H and operators corresponding to the symplectic space and the observables from the classical description. For some discussion of this we refer to Chapter 5 in [Be]. For our purposes here, we retain the following observation even though it is somewhat nebulous:

In important cases the behaviour of a physical system is guided by its symmetry group G, i.e. a group of transformations, which do not change the physical content of the system. If we determine unitary representations of G we will be provided with a Hilbert space and a lot of unitary operators.

In the next sections we shall see that important groups, like the Heisenberg group, the Euclidean and the Poincar´egroup, SU(2) and SL(2, R), give rise to symplectic manifolds and line bundles living on these such that the spaces of sections of these bundles are representation spaces of unitary representation of G.

~~8.2 Coadjoint Orbits and Representations~~

We start by analysing the construction of bundles on symplectic manifolds. Beside the already mentioned sources [Ki] and [Ki1], for further studies we recommend the book by Woodhouse [Wo]. We return to the realm of mathematics though we still use some terms from physics, which should help to give some motivation.

~~8.2.1 Prequantization~~

In Section 8.1 we already established the cotangent bundle T ∗Q to a manifold Q and in n particular the space R with coordinates (q1,...,qn,p1,...,pn) and provided with the standard two-form (8.2) n ω0 := dqj ∧ dpj j=1 as prototypes of a symplectic manifold. In general, one has the fundamental notion:

~~Deﬁnition 8.1: A smooth (real) p-manifold M is called a symplectic manifold if it is provided with a closed nondegenerate symplectic two-form ω deﬁned on M, that is an ω ∈ Ω2(M), such that~~

~~–i)dω =0, and – ii) on each tangent space TmM,m ∈ M, the condition holds: if for X ∈ TmM one has~~

~~ωm(X, Y ) = 0 for all Y ∈ TmM, then X =0. 8.2 Coadjoint Orbits and Representations 179~~

Slightly paraphrasing the condition ii), we remark that M neccessarily has to have even dimension p =2n and that for ω in m ∈ M with local coordinates (q1,...,qn,p1,...,pn) given by (8.2) n ω0 = dqj ∧ dpj j=1 and for vector ﬁelds X, Y given by n n X = (a j(q, p) ∂qj + a j(q, p) ∂pj ),Y= (b j(q, p) ∂qj + b j(q, p) ∂pj ), j=1 j=1

~~from the general duality formula dxi(∂xj )=δij and the antisymmetry of the forms one has the relation n ω0(X, Y )= (a j(q, p) b j(q, p) − a j(q, p) b j(q, p)). j=1~~

~~At ﬁrst sight, this form ω0 may look very special. But there is a beautiful theorem stating that this really a general situation:~~

~~Theorem 8.1 (Darboux): For every point m of a symplectic manifold (M,ω) of dimension 2n, there is an open neighbourhood U of m and a smooth map~~

~~2n ∗ ϕ : U −→ R with ϕ ω0 = ω|U ,~~

~~2n where ω0 is the standard symplectic form (8.2) on R .~~

The examples of symplectic manifolds, which are of interest for us, are the coadjoint orbits belonging to the action of a group G on the dual g∗ of its Lie algebra g.For instance the projective space P1(C) will come out as such an orbit. Before we go into this, we elaborate a bit on the way to construct bundles on symplectic manifolds. As we see it, this construction essentially goes back to Weil’s book VariétésKählériennes [We].

~~Deﬁnition 8.2: A symplectic manifold (M,ω) is called quantizable if ω is integral.~~

~~The integrality of ω can be deﬁned in several equivalent ways. Perhaps the most elementary one is to require (as in [Wo] p.159]:~~

~~(I.) The integral of ω over any closed oriented surface in M is an integral multiple of 2π.~~

~~More elegant is the requirement~~

~~(II.) ω determines a class [ω]inH2(M,Z).~~

To understand (II.), one has to have at hand the notions of the second cohomology group H2(M,R) and its integral subgroup H2(M,Z). Though we are not too far from the possibility to deﬁne these, we renounce to do it here but discuss the construction of the prequantum line bundle where we can see how this condition works (later we shall be led to a more practical criterion to decide the quantizability of the manifolds showing up as coadjoint orbits): 180 8. Geometric Quantization and the Orbit Method

Recalling our deﬁnitions from 7.6, this bundle can be constructed as follows. We take a contractible open cover U =(Uj)ofM. Then Poincar´e’sLemma assures that on each Uj, there exists a real one–form βj such that

~~ω = dβj.~~

~~The same way, one knows that on each contractible Uj ∩ Uk = ∅ a smooth real function fjk = −fkj exists such that~~

~~df jk = βj − βk and, for each contractible Uj ∩ Uk ∩ U = ∅, there is a constant ajk such that~~

~~2πajk = fjk + fkl + fj on Uj ∩ Uk ∩ U.~~

~~It is the decisive point that the integrality of ω says that the ajk may be assumed to be integers (this is really the essence of the deﬁnition above). If we put~~

~~cjk := exp ifjk, we have~~

~~dcjk = icjk(βj − βk) and for ajkl ∈ Z on each non–empty Uj ∩ Uk ∩ U~~

~~cjkcklclj = exp(2πiajk)=1.~~

Hence Proposition 7.4 in 7.6 shows that the cjk are transition functions for a line bundle B over M. This bundle has a compatible Hermitian structure <, > (see [Wo] p.264), which we deﬁne below in 8.2.3. Since in Poincar´e’sLemma a form β with dβ = α is not unique (it can changed by an exact form), this construction is not unique. The general statement is that these constructions of prequantum bundles B are parametrized by H1(M,R)/H1(M,Z) ([Wo] p.286).

As it is our goal to construct unitary group representations, we need a Hilbert space. The Hilbert space H in our prequantization construction (the prequantum Hilbert space) is the space of all s ∈ Γ(M,B), i.e. smooth global sections of B for which in some sense an integral of (s, s)εω over M exists and is ﬁnite. Here the measure εω is associated to the Liouville form 1 (−1)n(n−1)/2 ωn n! known from calculus. For s1,s2 ∈Hthe inner product on H is 1 ~~:= (s ,s )ε . 1 2 (2π)n 1 2 ω M 8.2 Coadjoint Orbits and Representations 181~~

8.2.2 Example: Construction of Line Bundles over M = P1(C) The topic of line bundles on projective space is treated in each text book on algebraic geometry (for instance, see Griﬃths-Harris [GH] p.144) and we already constructed holomorphic line bundles over P1(C) at the end of section 7.6. But to illustrate the procedure proposed above, for those who – like the author – enjoy explicit descriptions, we add here the following very elementary constructions (to be skipped if one is in a hurry).

~~1. Real Analytic Bundles B −→ P1(C)~~

1 We look at P (C) as a real two-dimensional manifold and take the covering U =(Uj)j=1,...,6 with U1 = {(z : 1); z = x + iy ∈ C; |z| < 1}, 2 2 U2 = {(1 : w); w = u + iv ∈ C; |w| < 1},w=1/z =(x − iy)/(x + y ), U3,4 = {(z : 1); z ∈ C;Rez = x ≷ 0}, U5,6 = {(z : 1); z ∈ C;Imz = y ≷ 0}. This covering is chosen such that all intersections of the covering neighborhoods are simply connected (this is not the case for the covering by U and U , which we used in Example 7.18 in 7.6 as one has U ∩ U C∗). With c = /π ∈ R we take as a symplectic form on M dx ∧ dy i dz ∧ dz ω := c = c for (z :1)∈ U ,j =2 , (1 + x2 + y2)2 2 (1 + |z|2)2 j du ∧ dv i dw ∧ dw := c = c for (1 : w) ∈ U . (1 + u2 + v2)2 2 (1 + |w|2)2 2 Exercise 8.1: Verify that this two–form ω in the appropriate coordinates has potential forms θ and θ with dθ = dθ = ω given by z w z w c xdy − ydx i zdz − zdz θ = = −c for (z :1)∈ U ,j=2 , z 2 2 | |2 j 2 1+x + y 4 1+ z c udv − vdu i wdw − wdw θ = = −c for (1 : w) ∈ U . w 2 1+u2 + v2 4 1+|w|2 2

On Uj ∩ U2(j = 2) we have i zdz − zdz 1 dz dz θ − θ = −c − − z w 4 1+|z|2 1+|z|2 z z i zdz − zdz = −c 4 |z|2 c xdy − ydx = 2 x2 + y2 c = df˜ 2 2j ˜ with a potential function f2j. If we ﬁx the tangens as a function on (−π/2,π/2) and f = arctan as its inverse, we can take ˜ −f32(x, y)=f(y/x) = arctan(y/x) and moreover, for instance, ˜ ˜ −f52(x, y)=−f(x/y)+π/2=− arctan(x/y)+π/2 and −f42(x, y)= π + arctan(y/x). 182 8. Geometric Quantization and the Orbit Method

Then, on U2 ∩ U4 ∩ U5, we have ˜ ˜ ˜ f24 + f45 + f52 = −π − arctan(y/x)+π/2 − arctan(x/y) = −π, and similarly on the other intersections. Thus, we see that ω is integral in the sense

~~i) ω = dθj on Uj~~

~~ii) θj − θk = df jk on Uj ∩ Uk~~

~~iii) fjk + fk + fj = ajk ∈ 2πZ in Uj ∩ Uk ∩ U exactly for c ∈ 4Z. This result coincides with the usual volume computation~~

∞2π dx ∧ dy rdrdδ ω = c = c (1 + x2 + y2)2 (1 + r2)2 P1(C) 0 0 ∞ ∞ rdr du =2πc ”s = πc = πc (1 + r2)2 u2 0 1 telling that the volume is an integral multiple of 4π = vol(S2) exactly for c =4n, n ∈ Z.

~~Moreoverer, we see that we can construct the C∞–prequantum bundle~~

~~B −→π M = P1(C) in the standard way as~~

~~B = {(m, v, j)∼; m ∈ M, v ∈ C,j∈ I}, where here I = {1,...,6} and~~

~~(m,v,j) ∼ (m, v, j) exactly for m = m, v = ψjj(m)v.~~

~~The bundle transition functions ψjk are ﬁxed (in the charts with coordinates x, y)by~~

~~ifjk(x,y) ψjk(x, y)=cjke , i.e. the diﬀerent trivializations are glued together by~~

~~ei argz .~~

~~By the way, the transition function z comes out if one does not take real potential forms 2 −1 as above but complex forms like θ z =(1+|z| ) zdz.¯~~

~~2. Holomorphic Bundles E −→ P1(C)~~

Following Weil ([We] p. 90), the same construction can be repeated by looking at P1(C) as a one-dimensional holomorphic manifold and searching for holomorphic transition functions ∗ Fjk : Uj ∩ Uk −→ C . 8.2 Coadjoint Orbits and Representations 183

Here we take the same covering U =(Uj)j=1,...,6 as above and the symplectic form dz ∧ dz ω := = ddΦ forj =2 , 2πi (1 + |z|2)2 2πi j with 2 Φj(z):= log(1 + |z| ) resp. for j =2 dw ∧ dw ω := = ddΦ 2πi (1 + |w|2)2 2πi 2 with 2 Φ2(w):= log(1 + |w| ).

On Uj ∩ U2,j =2,wehave 2 Φj(z) − Φ2(1/z)= log(|z| ). Fixing appropriate branches of the complex log-function, we take for j =2 f := log z. 2j 2πi Then we have − − − Φj(z) Φ2(1/z)= 2πi(f2j f 2j) and zdz η = − dΦ = d log(1 + |z|2)= j 2πi j 2πi 2πi 1+|z|2 with dz ∧ dz ω = . 2πi (1 + |z|2)2

Using Weil’s notation, for the construction of a holomorphic bundle E = E we have transition functions Fjk(z) = exp (2πi fjk(z)) = z exactly if ∈ Z. We see immediately that for ∈ N0 the holomorphic sections are given in the z-variable by s(z)=zλ,λ=0, 1,..., (as exactly in these cases we have (1/w)λw holomorphic in w), i.e. we find ∼ +1 Γ(E) = C . As described in a general procedure below, this space consists of the polarized sections of the prequantum bundle B = B, where the polarization is given by assigning to the point m with coordinate z the complex subspace spanned by ∂z in the complexification 1 Tm P (C) ⊗ C C ∂z ⊕ C ∂z 1 1 of the tangent space Tm P (C) for each m ∈ P (C), i.e. Γ(E) can be identified with the space of those sections of B, which are constant “along the direction of thez ¯−coordinate”.

We try to make this a bit more comprehensive. But to do it, again we need more general notions from complex and algebraic geometry. As seen above, these geometric notions can and shall be made later rather tangible by replacing them by algebraic ones in our concrete examples (and, hence, may be skipped at ﬁrst try). 184 8. Geometric Quantization and the Orbit Method

~~8.2.3 Quantization~~

The prequantization Hilbert space H built from C∞–sections of the prequantum bundle B over M - as we just have seen - often turns out to be too large to be useful: This is to be seen again in the case of the coadjoint orbits, as the representations of the group G on H as a representation space are in general far from being irreducible. From representation theory procedures are known to construct appropriate representation spaces, for instance by parabolic or holomorphic induction. But in this approach here, it is “natural” to use a notion from the Kählergeometry to single out a subspace of H, namely the notion of the Kählerpolarization. We take over from [Wo] p.92 ff:

Deﬁnition 8.3: A complex polarization on a 2n–dimensional symplectic manifold (M,ω) is a complex distribution P with ∈ C (CP1) For each m M, Pm is a complex Lagrangian subspace of Tm M, i.e. with ⊥ Pm = Pm,

~~(CP2) Dm := Pm ∩ Pm ∩ TmM has constant dimension for all m ∈ M, (CP3) P is integrable.~~

~~The polarization P is of type (r, s) if the Hermitian form <, > on Pm deﬁned by~~

:= −4iω(Z, W ) for Z, W ∈ Pm is of sign (r, s) for all m ∈ M. P is a K¨ahlerpolarization if it is of type (r, s) with r +s = n. Such a polarization induces a complex structure J on M and (M,ω,J)isaK¨ahlermanifold (in the broad sense that the real symmetric tensor g determined by

~~g(X, Y )=2ω(X, JY ),X,Y∈ V (M) may be indeﬁnite, see the “note 1” to p.93 in [Wo]).~~

~~Perhaps the reader wants some more explanations. We follow [Wo] p.269:~~

–Areal distribution on a real manifold M is a subbundle of the tangent bundle TM. The fiber Pm at m ∈ M is a subspace of TmM, which varies smoothly with m. – A vector field X is tangent to P if X(m) ∈ Pm for every m. The space of vector fields tangent to P is denoted by VP (M). A smooth function is constant along P if df vanishes on restriction to P . The space of smooth functions constant along P is denoted ∞ by CP (M). – An immersed submanifold N ⊂ M is an integral manifold of P if Pm = TmN for every m ∈ N. – A distribution is integrable if [X, Y ] is tangent to P whenever X and Y are tangent to P . –Acomplex distribution is similarly defined to be a subbundle of the complexified tangent bundle TCM. A complex distribution P is integrable if in some neighbourhood of each point, one has smooth complex functions fk+1,...,fl with gradients that are independent and annihilate all complex vector fields tangent to P , where l − k is the fiber dimension of P . 8.2 Coadjoint Orbits and Representations 185

–Acomplex structure on a real vector space V is a linear transformation J : V −→ V such that J 2 = −1. Complex structures exist only in spaces of even dimension. On a real manifold a complex structure J is given if one has a complex structure Jm on each tangent space TmM and if these Jm vary smoothly with m, i.e. the complex distribution P spanned by the vector ﬁelds X − iJX, X ∈ V (M), is integrable. A real manifold with a complex stucture comes out as a complex manifold. –Aconnection on a vector bundle B is an operator that assigns a one-form s with values in B to each (smooth) section s of B, i.e.,

~~ :Γ(B) −→ Γ(T ∗M ⊗B),~~

~~∞ such that for any function f and sections s, s ∈CB (M) one has~~

~~(s + s)=s + s and (fs)=(df )s + f s.~~

Vice versa, a given Kählermanifold (M,ω,J), i.e. a manifold which is simultaneously in a compatible manner a symplectic and a complex manifold, carries two Kählerpolariza- tions: the holomorphic polarization P spanned at each point by the vectors (∂za ) and its complex conjugate, the anti–holomorphic polarization P spanned by the (∂za )’s. Locally it is possible to find a real smooth function f such that

ω = i∂∂f and Θ=−i∂f resp. Θ= i∂f is a symplectic potential adapted to P resp. P . –AHermitian structure on a complex vector bundle B is a Hermitian inner product (·, ·) on the ﬁbers, which is smooth in the sense that B−→C,v −→ (v, v) is a smooth function. It is compatible with the connection if for all sections s, s and all real vector ﬁelds X i(X)d(s, s )=(X s, s )+(s, X s ), X s := i(X) s.

Now, let be given a K¨ahlerpolarization P on the quantizable (M,ω) with prequantum line bundle B over M with connection , Hermitian structure <, > and Hilbert space H of square integrable sections of B. Then we deﬁne a space of polarized sections C∞ { ∈C∞ ∈ } B (M,P):= s B (M); X s = 0 for all X VC(M,P) . Here, for U ⊂ M open,

VC(U, P ):={X ∈ VC(M),Xm ∈ Pm for all m ∈ M} denotes the vector ﬁelds tangent to P on U. Then H H∩C∞ P := B (M,P) comes out as our “new” Hilbert space. As remarked above, it consists of all integrable holomorphic sections of B. Alternatively, it can be said that the prequantum bundle B (via P ) is given a structure of a complex line bundle. Anyway, in important cases, this HP is the representation space of a discrete series representation. 186 8. Geometric Quantization and the Orbit Method

We leave aside the problem, which of the classical observables have a selfadjoint counterpart acting in HP , discussed to some length in [Wo] and the modiﬁcation by the introduction of half–densities and go directly to the application in the construction of unitary representations via line bundles on the coadjoint orbits.

8.2.4 Coadjoint Orbits and Hamiltonian G-spaces This is the central topic of this chapter. For background information we recommend the standard sources by Kirillov [Ki] §15, Kostant [Kos], (at least) the ﬁrst pages of Kirillov’s book [Ki1], or, for a comprehensive overview, the article by Vogan [Vo].

Let G be a (real) linear group with Lie algebra g and its dual space g∗. As usual, we realize all this by matrices and then have the adjoint representation Ad of G on g given by Ad(g)X := gXg−1. −1 (This is the derived map of the inner automorphism κg : G −→ G, g0 −→ gg0g .) Following our preparation in section 1.4.3, we can construct the contragredient represen- ∗ tation Ad deﬁned on the dual g∗ to g by Ad∗(g) := Ad(g−1)∗.

The asterisk on the right-hand side indicates a dual operator: If <η,X>denotes the value η(X) of the linear functional η applied to the vector X and A is an operator on g the dual A∗ is deﬁned by

:=<η,AX> for all X ∈ g,η ∈ g∗. Hence we have as our central prescription ∗ < Ad (g)η, X > = <η,Ad(g−1)X> for all g ∈ G, X ∈ g,η ∈ g∗. This representation Ad∗ is called the coadjoint representation. Often we will abbreviate g · η := Ad∗(g)η.

~~In 6.6.1 we already worked with the trace form for A, B ∈ Mn(C) given by (A, B) −→ Re Tr (AB)=:.~~

(Here, unfortunately, one has a double meaning of <.,.>.) If g is semisimple or reductive, one can use this bracket to identify g∗ with g (G-equivariantly) via ∗ g η −→ Xη ∈ g, deﬁned by η(Y )= for all Y ∈ g.

~~−1 Then the coadjoint representation is ﬁxed by the simple prescription Xη −→ gXηg .~~

~~Now we are prepared to deﬁne the coadjoint orbit O of η ∈ g∗ by~~

~~Oη := G · η.~~

~~If Gη denotes the stabilizing group Gη := {g ∈ G; g · η = η}, one has Oη G/Gη and if we can identify g∗ with a matrix space as proposed above, we have~~

~~−1 Oη = {gXηg ,g∈ G}. 8.2 Coadjoint Orbits and Representations 187~~

~~It is an outstanding and most wonderful fact that our coadjoint orbits are symplectic manifolds. We cite from [Vo] p.187:~~

~~∗ Theorem 8.2: Suppose M = Oη is the coadjoint orbit of η ∈ g and gη := Lie Gη. Then one has 1.) The tangent space to M at η is~~

~~TηM g/gη.~~

~~2.) The skew-symmetric bilinear form~~

~~ωη(X, Y ):=η([X, Y ]) for all X, Y ∈ g on g has radical exactly gη and so deﬁnes a symplectic form on TηM. 3.) The form ωη makes M into a symplectic manifold.~~

~~The bilinear form ωη is called Kirillov-Kostant form, sometimes also Kirillov-Kostant- Souriau form or any permutation of these names. It is also written as ([Ki] p.230 or [Wo] p.52)~~

ˆ ˆ ˆ ˆ ∈ ∈O (8.5) ωη(η )(ξXˆ (η ),ξYˆ (η )) = <η, [X,Y ] > for all X,Y g,η η ˆ where ξXˆ denotes the vector field associated to X in the form of a differential operator acting on smooth functions f on M by the definition d (8.6) (ξ f)(η):= (f(exp tXˆ · η))| . Xˆ dt t=0

For a proof we refer to [Ki] 15.2 or [Kos] Theorem 5.4.1. It is obvious that ωη is skew symmetric. Also it is, in principle, not deep that a form given for the tangent space of one point can be transported to the whole space if the space is homogeneous. But it may be an intricate task to give an explicit description of ω in local coordinates. We shall give an example soon how (8.5) can be used. The fact that ω comes out as a closed form can be proved by direct calculation or by more sophisticated means as in [Ki1] p.5-10.

~~The coadjoint orbits still have more power, they are Hamiltonian G-spaces:~~

Deﬁnition 8.4: Suppose (M,ω) is a symplectic manifold and f ∈ C∞(M) is a smooth function with its Hamiltonian vector ﬁeld Xf . Moreover suppose that G is a linear group endowed with a smooth action on M, which respects the form ω. We say that M is a Hamiltonian G-space if there is a linear map

µ˜ : g −→ C∞(M) with the following properties: –i)µ ˜intertwines the adjoint action of G on g with its action on C∞(M). – ii) For each Y ∈ g,Xµ˜(Y ) is the vector ﬁeld by which Y acts on M. – iii)µ ˜ is a Lie algebra homomorphism.

~~µ˜ can be reinterpreted as a moment map~~

~~µ : G −→ g∗,µ(m)(Y ):=˜µ(Y )(m) for all m ∈ M, Y ∈ g. 188 8. Geometric Quantization and the Orbit Method~~

~~Then the highlight of this theory is the following result to be found also in the above cited sources. We shall not have occasion to use it here but state it to give a more complete picture.~~

Theorem 8.3: The homogeneous Hamiltonian G-spaces are the covering spaces of coadjoint orbits. More precisely, suppose M is such a space, with mapµ ˜ : g −→ C∞(M) and moment map µ. Then µ is a G-equivariant local diﬀeomorphism onto a coadjoint orbit, and the Hamiltonian G-space structure on M is pulled back from that on the orbit by the map µ.

If G is reductive and we have identified g with its dual g∗ as n × n−matrices, one has the following classification of the coadjoint orbits, which later will be reflected in a mechanism to construct representations of G.

Deﬁnition 8.5: An element X ∈ g is called nilpotent if it is nilpotent as a matrix, i.e. if XN = 0 for N large enough. X is called semisimple if X regarded as a complex matrix is diagonalizable. X is called hyperbolic resp. elliptic if it is semisimple and all eigenvalues are real resp. purely imaginary. An orbit O = G · η inherits the name from X = Xη.

~~For instance, the real matrix 01 X = −10 i 0 is elliptic, since over C it is conjugate to . Now, this ﬁnally is the moment to 0 −i discuss an example.~~

~~Coadjoint Orbits of SL(2, R)~~

~~We take G = SL(2, R) ⊃ K = SO(2) and ∗ xy+ z g g = Lie G = U(x, y, z):= ; x, y, z ∈ R = R y − z −x with 1 1 1 X = ,Y= ,Z= −1 1 −1~~

[X, Y ]=2Z, [X, Z]=2Y, [Y,Z]=−2X and 2 2 2 X + Y − Z =3I2. 1 Obviously, X and Y are hyperbolic, Z is elliptic, and F =( ) as well as G =( ) 1 are nilpotent. 8.2 Coadjoint Orbits and Representations 189

~~One can guess the form of the orbits generated by these matrices, but to be on the safe side we calculate:~~

~~ab gU(x, y, z)g−1 =: U(˜x, y,˜ z˜) for g =( ) cd is given by~~

x˜ =(ad + bc)x +(bd − ac)y − (bd + ac)z, (8.7) y˜ =(dc − ab)x +(1/2)(a2 + d2 − b2 − c2)y +(1/2)(a2 + b2 − c2 − d2)z, z˜ = −(ab + cd)x +(1/2)(a2 − b2 + c2 − d2)y +(1/2)(a2 + b2 + c2 + d2)z.

~~Hence, for U = αX, α =0 , as a hyperbolic orbit OαX we have the one-sheeted hyperboloid in R3 given by x2 + y2 − z2 = α2.~~

~~For U = αZ, α ≷ 0, as elliptic orbits OαZ we have the upper resp. lower half of the two-sheeted hyperboloid given by~~

~~x2 + y2 − z2 = −α2,z ≷ 0.~~

~~For U = F and U = G, as nilpotent orbits OF and OG we have the upper resp. lower half of the cone given by x2 + y2 − z2 =0,z ≷ 0.~~

3 Finally, for U =0, there is the nilpotent orbit O0 consisting only of the origin in R . Hence we have a disjoint decomposition of R3 into the orbits in their realization as 3 subsets of R 3 R = OαX OαZ (OF ∪{0}∪OG). α>0 α≷0

~~It is an easy exercise to determine the stabilizing groups of the elements we chose to generate our orbits. We get (again with α =0) a G = ; a ∈ R∗ , αX 1/a~~

~~GαZ = SO(2), ±1 b G = ; b ∈ R , F ±1 ±10 G = ; c ∈ R . G c ±1~~

~~From the general theory we have the prediction that the coadjoint orbits are symplectic manifolds. We can verify this here directly: 190 8. Geometric Quantization and the Orbit Method~~

~~Let us take the elliptic orbit~~

~~OαZ = G · (αZ) G/SO(2),α>0.~~

We want to give an explict expression for the Kirillov-Kostant form ω in this case. Here one has to be rather careful. We used the trace form for matrices =TrAB to ∗ identify g with g via η −→ Xη with = η(U) for all U ∈ g. Now, we write ∗ g =R with 1 1 1 X∗ = ,Y∗ = ,Z∗ = , −1 1 −1 and then have as non-vanishing duality relations

~~=2,=2,= −2.~~

~~We put ∗ g η = x∗X∗ + y∗Y∗ + z∗Z∗ and Xˆ = xX + yY + zZ, Yˆ = xX + yY + zZ ∈ g. Hence, by the commutation relations for X, Y, Z, we have~~

~~[X,ˆ Yˆ ] = 2((xy − xy)Z +(xz − xz)Y − (yz − yz)X).~~

~~From the Kirillov-Kostant prescription we have~~

~~ωη(X,ˆ Yˆ )=<η,[X,ˆ Yˆ ] > = −2(yz − yz )x∗ +2(xz − x z)y∗ − 2(xy − x y)z∗, i.e. in particular for the elliptic orbit through η = kZ∗~~

~~ˆ ˆ − − ωkZ∗ (X,Y )= (xy x y)k.~~

~~There are several ways to go from here to an explicit diﬀerential two-form ω on~~

~~M :=G · kZ∗ 3 2 2 2 2 = {(x∗,y∗,z∗) ∈ R ; x∗ + y∗ − z∗ = −k }.~~

~~It is perhaps not the shortest one but rather instructive to proceed like this: One knows that dx˙ ∧ dy˙ ω = 1 y˙2 is a G-invariant symplectic form on M := H G/SO(2) where~~

~~τ˙ =˙x + iy˙ := g(i) with bd + ac 1 ab x˙ = , y˙ = for g = . c2 + d2 c2 + d2 cd 8.2 Coadjoint Orbits and Representations 191~~

~~The coadjoint G-action on M and the usual action of G on M = H are equivariant (Exer- cise 8.2: Verify this). From (8.7) we have the description of the elements (x∗,y∗,z∗) ∈ M as~~

~~x∗ = −(ac + bd)k, 2 2 2 2 y∗ =(1/2)(a + b − c − d )k, 2 2 2 2 z∗ =(1/2)(a + b + c + d )k, and, hence, a G-equivariant diﬀeomorphism~~

~~ψ : M −→ M = H, (x∗,y∗,z∗) −→ (˙x, y˙) given by x∗ k x˙ = , y˙ = . y∗ − z∗ z∗ − y∗~~

~~We use ψ to pull back ω1 to a symplectic form ω0, which in local coordinates (x∗,y∗) 2 2 2 with z∗ = x∗ + y∗ + k , i.e. z∗dz∗ = x∗dx∗ + y∗dy∗ comes out as~~

~~∗ dx∗ ∧ dy∗ ω0 = ψ ω1 = . kz∗~~

~~(Verify this as Exercise 8.3.)~~

Now we have to compare this with the Kirillov-Kostant prescription above and determine the right scalar factor to get the Kirillov-Kostant symplectic form ω. We shall see that one has dx∗ ∧ dy∗ ω = − . z∗ Perhaps the more experienced reader can directly pick this oﬀ the Kirillov-Kostant prescription. As we feel this is a nice occasion to get some more practice in handling the notions we introduced here, we propose to evaluate the general formula (8.5)

ˆ ˆ ˆ ˆ ∈ ∈O ωη(η )(ξXˆ (η ),ξYˆ (η )) := <η, [X,Y ] > for all X,Y g,η η in this situation. For the vector ﬁelds ξXˆ acting on functions f on M we have the general formula d (ξ f)(η)= f((exp tXˆ) · η)| . Xˆ dt t=0 ˆ Putting gXˆ (t) := exp tX one has et g (t)= , X e−t 1 t g (t)= , F 1 1 g (t)= . G t 1 192 8. Geometric Quantization and the Orbit Method

And, putting A(g) for the matrix of the coadjoint action η −→ g · η, one deduces from (8.7) ⎛ ⎞ 1 ⎝ 2t −2t 2t −2t ⎠ A(gX (t)) = (1/2)(e + e )(1/2)(e − e ) , (1/2)(e2t − e−2t)(1/2)(e2t + e−2t) ⎛ ⎞ 1 t −t ⎝ 2 2 ⎠ A(gF (t)) = −t (1/2)(2 − t )(1/2)t , −t (1/2)t2 (1/2)(2 + t2) ⎛ ⎞ 1 −t −t ⎝ 2 2 ⎠ A(gG(t)) = t (1/2)(2 − t ) −(1/2)t . −t (1/2)t2 (1/2)(2 + t2)

~~From (8.6) we get the diﬀerential operators (with Y = F + G, Z = F − G)~~

~~ξX =2(z∗∂y∗ + y∗∂z∗ ), − ξY = 2(z∗∂x∗ + x∗∂z∗ ), − ξZ =2(y∗∂x∗ x∗∂y∗ ).~~

~~When we treat the functions f as functions in the two local coordinates (x∗,y∗) with 2 2 2 2 z∗ = x∗ + y∗ + k , these operators reduce to~~

~~ξX =2z∗∂y∗ , − ξY = 2z∗∂x∗ , − ξZ =2(y∗∂x∗ x∗∂y∗ ).~~

And a vector ﬁeld in these local coordinates comes as ˆ X = α∂x∗ + β∂y∗ = xξX + yξY + zξZ with α =2(zy∗ − yz∗),β=2(xz∗ − zx∗) and analogously for ˆ Y = α ∂x∗ + β ∂y∗ = x ξX + y ξY + z ξZ . One has

2 αβ − α β = 4((zx − xz )y∗z∗ +(xy − yx )z∗ +(yz − y z)x∗z∗) and hence in (8.5) for ω =(1/z∗)dx∗ ∧ dy∗ the left hand side is ∧ ∧ (1/z∗)dx∗ dy∗(ξXˆ ,ξYˆ )=(1/z∗)dx∗ dy∗(α∂x∗ + β∂y∗ ,α∂x∗ + β ∂y∗ ) = 4((zx − xz )y∗ +(xy − yx )z∗ +(yz − y z)x∗). And this is exactly the negative of the outcome of the right hand side evaluated for η = x∗X∗ + y∗Y∗ + z∗Z∗ <η, [X,ˆ Yˆ ] >= −4(xy − x y)z∗ +4(xz − x z)y∗ − 4(xy − yx )x∗. 8.2 Coadjoint Orbits and Representations 193

~~Coadjoint Orbits for SU(2)~~

~~As already done several times, we take ab G := SU(2) = {g = ; a, b ∈ C, |a|2 + |b|2 =1}. −b a~~

Thus, as a (real) manifold, G is the unit 3–sphere S3 ⊂ C2. One has t g = su(2) = X ∈ M2(C); X = − X 3 3 = {X =(1/2) ajHj; a =(a1,a2,a3) ∈ R } j=1 with (see (6.5) in Section 6.5) −i −i −1 H = ,H = ,H = , 1 i 2 −i 3 1 and [H1,H2]=2H3, [H2,H3]=2H1, [H3,H1]=2H2 . su(2) is isomorphic to so(3) (see the exercises 6.4, 6.5 and 6.6 in Section 6.1) and, as 3 usual, we identify X ∈ su(2) and a =(a1,a2,a3) ∈ R with ⎛ ⎞ 0 −a3 a2 ⎝ ⎠ A = a3 0 −a1 =: a1A1 + a2A2 + a3A3 ∈ so(3) −a2 a1 0

~~Using this identiﬁcation the Lie bracket corresponds to the vector product in R3~~

~~[A, B]=a ∧ b~~

~~3 and the trace form <, > in M3(R) to the scalar product (·, ·)inR = −2(a, b)=−2 tab.~~

~~Since <, > is proportional to the euclidean scalar product on R3, we can use the trace form to identify so(3)∗ with so(3) via the prescription~~

∗ (8.8) so(3) η −→ Aη such that η(B)= for all B ∈ so(3) : ∗ ∗ ∗ ∗ ∈ ∗ We choose a basis (A1,A2,A3) with = δij and for an element η so(3) we ≡ ∗ ∗ ∗ ∗ ∗ ∗ write η Aη = a1A1 + a2A2 + a3A3.

As is well known (see the proof of the fact that SU(2) is a twofold covering of SO(3) in section 4.3), using these identiﬁcations the coadjoint operations of SU(2) on su(2)∗ resp. so(3)∗ are simply the rotations of R3. The orbits are the spheres of radius s, · ∗ Ms := G ηs for ηs := sA3 with s>0, centered at the origin and the origin itself. One has a symplectic structure on the sphere Ms of radius s given by the volume form, which is given by restriction to Ms of the 2–form ∗ ∗ ∧ ∗ ω0 := (1/s) ai daj dak. {i, j, k} cyclic 194 8. Geometric Quantization and the Orbit Method

∗ ∗ As local coordinates we use (a1,a2) such that Ms is given by the parametrization { ∗ ∗ ∗ 2 ∗ 2 ≤ 2}−→ ψ : J := (a1,a2); (a1) +(a2) s Ms, ∗ ∗ −→ ∗ ∗ ∗ ∗ 2 2 − ∗ 2 − ∗ 2 (a1,a2) (a1,a2,a3), (a3) = s (a1) (a2) .

~~Exercise 8.4: Verify that one has~~

~~∗ ∗ ∧ ∗ ω0 =(s/a3)da1 da2 and ∗ 2 vol2(Ms)=2 ψ ω0 =4πs . J We want to compare this with the Kirillov-Kostant recipe~~

ωη(A, B)=η([A, B]), resp. (8.5) ω(ξA,ξB)=<η, [A, B] >. We have A = a1A1 + a2A2 + a3A3,B= b1A1 + b2A2 + b3A3, i.e. [A, B]=(a1b2 − a2b1)A3 +(a2b3 − a3b2)A1 +(a3b1 − a1b3)A2, ∗ ∗ ∗ ∗ ∗ ∗ and with η = a1A1 + a2A2 + a3A3 − ∗ − ∗ − ∗ (8.9) ωη(A, B)=(a1b2 a2b1)a3 +(a2b3 a3b2)a1 +(a3b1 a1b3)a2.

~~As in the preceding example we determine the associated vector ﬁelds ξAj . Namely using here gj(t) = exp tAj,j =1, 2, 3,~~

~~d ξ f(η)= f(g (t) · η)| Aj dt j t=0 leads to~~

~~∗ ∗ − ∗ ∗ ξA1 = a3∂a2 + a2∂a3 , ∗ ∗ ∗ − ∗ ξA2 = a3∂a1 a1∂a3 , ∗ ∗ − ∗ ∗ ξA3 = a2∂a1 + a1∂a2 . and hence in the local coordinates ﬁxed above~~

~~ξA = a1ξA1 + a2ξA2 + a3ξA3~~

∗ ∗ = α∂a1 + β∂a2 , with ∗ − ∗ ∗ − ∗ α =(a2a3 a3a2),β=(a3a1 a1a3) and the corresponding expressions α ,β for ξB replacing the aj by bj. We get − − ∗ ∗ − ∗ ∗ − ∗ ∗ αβ α β =(a2b3 a3b2)a1a3 +(a1b2 a2b1)a3a3 +(a3b1 a1b3)a2a3. 8.2 Coadjoint Orbits and Representations 195

∗ By evaluation at η = sA3 one obtains ∗ − ω0(ξA,ξB)=(s/a3)(αβ α β) 2 = s (a1b2 − a2b1). We compare this with (8.9) and see that the Kirillov-Kostant form in this case is 2 ∗ ∗ ∧ ∗ ω = sω0 =(s /a3)da1 da2. The isotropy group of η is eit 0 G = h = ,t∈ R U(1). η 0 e−it

~~2 G/Gη is identiﬁed with the sphere Ms S via the important Hopf map~~

ψ : G = SU(2) −→ Ms given by (a, b) −→ s(ba¯ + a¯b, iba¯ − ia¯b, aa¯ − b¯b) By some more calculation using the relation ada¯ +¯ada + bd¯b + ¯bdb =0, which is a consequence of aa¯ + b¯b =1, we get for the drawback of the Kirillov-Kostant form by the Hopf map ∗ ∗ ∗ ∧ ∗ ∧ ∧ ¯ ψ ((s/a3)da1 da2)=2is(da da¯ + db db). As we already constructed bundles on P1(C) (in 7.6 Example 7.19 and in 8.2.2) it is quite fruitful to compare this description of the coadjoint SU(2)-orbits with the following alternative: We discussed (in 7.1 Example 7.3) the identiﬁcation SU(2)/U(1) P1(C)={(u : v); u, v ∈ C2 \{(0, 0)}} essentially given by g −→ z = u/v = b/a,¯ which is a consequence of g acting by multiplication from the left on the column t(u, v). The same way, one has an identiﬁcation ψ˜ via (8.10) g −→ z = v/u = b/a coming from multiplication from the right to the row (u, v). If we take the symplectic form ω studied in 8.2.2 dz ∧ dz¯ ω =2ni ,n∈ Z, (1 + |z|2)2 and via ψ˜ draw it back to SU(2), we get again the form obtained above 2ni(da ∧ da¯ + db ∧ d¯b) and see that we have quantizable orbits for half integral s resp. n.

Now that we have examples for coadjoint orbits and bundles living on them, we continue in the program to use the sections of these bundles as representation spaces and we have to discuss how the general polarization business works in practical cases. 196 8. Geometric Quantization and the Orbit Method

~~8.2.5 Construction of an Irreducible Unitary Representation by an Orbit~~

If we want to construct representations from a coadjoint orbit of a group, the orbit has to be admissible, i.e. “integral” and been provided with a “nice” polarization. Then one can construct a complex line bundle, whose polarized sections lead to the space for a unitary (and perhaps even irreducible) representation of the group. Or, in another formulation, this representation can be got by an induction procedure from a subgroup arising from the polarized orbit. Up to now, all this is not in a final form if one strives for maximal generalilty (which we do not do in this text). Though there are now several more refined procedures using and discussing Duflo’s version of admissibility as for instance in Vogan [Vo] p.193 ff or in Torasso [To], we will follow the procedure designed by Kirillov [Ki] p.235 ff and [Ki1] (but in some places adapting our notation to the one used by Vogan). Similar presentations can be found in [GS1] p.235 and [Wo] p.103.

~~∗ Step 1. For each orbit Oη ⊂ g we look for real subalgebras n ⊂ g which are subordinate to η. This means that the condition~~

~~<η,[X,ˇ Yˇ ] > = 0 for all X,ˇ Yˇ ∈ n holds or, equivalently (this looks quite diﬀerent but is not too hard to prove), the map~~

~~Yˇ −→ 2πi < η, Y>ˇ~~

~~deﬁnes a one-dimensional representation = η of n .~~

~~Step 2. We say that n is a real algebraic polarization of η if in addition the condition~~

2 dim(g/n) = dim g + dim gη is satisﬁed. The notion of a complex algebraic polarization is deﬁned in the same way: we extend η to gc = g ⊗ C by complex linearity and consider complex subalgebras n ⊂ gc that satisfy the corresponding conditions as in the real case. An algebraic polarization is called admissible if it is invariant under Ad Gη. In the sequel, we only look for admissible algebraic polarizations.

~~The relation of these “algebraic” polarizations to the “geometric” ones deﬁned earlier is explained in [Ki1] p.28/9 and will be made explicit in a simple example later.~~

Step 3. Moreover, we suppose (as in [Ki] p.238) that in the complex case mc = n + n is a subalgebra of gc such that we have closed subgroups Q and M of G with Lie algebras q and m fulﬁlling the relations

~~◦ L := Gη ⊂ Q = GηQ ⊂ M, n ∩ n = qc, n + n = mc.~~

~~In the real case, we have q = n. 8.3 The Examples SU(2) and SL(2, R) 197~~

Step 4. Finally, and this is our practical criterium to guarantee the integrality or quantizability of the orbit, we suppose that η can be integrated to a unitary character χQ of Q with dχQ = η. Then we get a unitary representation π of our group G on (the ∞ completion of) the space F+ := C (G, n,Q,,χQ) of smooth C–valued functions φ on G with −1/2 −1 φ(gl)=Q(l) χQ(l )φ(g) for all l ∈ L, g ∈ G and L ˇ ∈ Yˇ φ = 0 for all Y u, n =: gη + u, L d ˇ | where Q is the usual modular function and Yˇ φ(g)= dt φ(g exptY ) t=0. (Here we follow [Vo] p.194 and p.199. For an essentially equivalent approach see [Ki] p.199ﬀ).

Though this looks (and is) rather intricate, we shall see that the scheme can easily be realized in our simple examples, in particular for the Heisenberg group in the next but one section. As already said, there are a lot of subtle questions hidden here. For instance (see [Ki] p.236), as another condition Pukanszky’s condition, i.e. the additional property ⊥ ⊥ ∗ η + n ⊂Oη comes in where n denotes the annihilator of n in g . And, as discussed in [Ki] p.239, the fundamental group π1(O) of the orbit plays an important role in the integration of the representation η of the Lie algebra q to a representation of the group Q. ∗ This leads to the notion of rigged orbits, i.e. orbits in grigg, the set of pairs (η, χ) where ∈ ∗ | η g and χ is here a one-dimensional representation of Gη such that dχ =2πiη gη (see [Ki1] p.123 (and Vogan’s criticism of Kirillov’s book on his homepage)). Here we only want to give an impression, and (from [Ki] p.241) we cite a theorem by Kostant and Auslander:

Theorem 8.4: Assume G to be a connected and simply connected solvable Lie group. Then representations that correspond to diﬀerent orbits or diﬀerent characters of the fundamental group of the orbit are necessarily inequivalent.

~~And (a special version of) the famous Borel-Weil-Bott theorem says:~~

~~Theorem 8.5: All irreducible representations of a compact connected, and simply connected Lie group G correspond to integral G-orbits of maximal dimension in g∗.~~

~~In more detail, in our special examples the result of this procedure is as follows.~~

~~8.3 The Examples SU(2) and SL(2, R)~~

~~1. G = SU(2)~~

~~We begin by taking up again the here already often treated example G = SU(2). Using (6.5) in Section 6.5, we take~~

~~g = {X =ΣajHj; a1,a2,a3 ∈ R}~~

~~∗ ∗ and, similar to (8.8) in 8.2.4, we choose (Hj ) such that = δjk and ∗ { ∗ ∗ ∗ ∗ ∗ ∈ } g = η =Σaj Hj ; a1,a2,a3 R . 198 8. Geometric Quantization and the Orbit Method~~

~~O ∗ From (8.9) for the elliptic orbit η SU(2)/U(1) passing through to η = sH1 we have the Kirillov-Kostant prescription~~

~~ωη(X, Y )=s(a2b3 − a3b2).~~

One has gη = Lie U(1) H1 .Asg has no two-dimensional subalgebras n, there is no real polarization. But one has two complex polarizations given by the subalgebras n+ = H0,H+ and n− = H0,H− of the complexiﬁcation gc = H0,H± with

~~H0 = iH1,H± = H2 ± iH3.~~

~~As one easily veriﬁes, both algebras are subordinate to η in the sense deﬁned above in Subsection 8.2.5 and, for s = k ∈ N0, the character of gη~~

gη Y −→ 2πi < η, Y > integrates to a unitary character χk of H = U(1) identiﬁed as a subgroup of G = SU(2). F G In this case the space + is the space belonging to the induced representation indH χk restricted by the polarization condition

L H± φ =0 since one has n+ = gη+ or n− = gη+ . And this space can be understood exactly as the subspace of holomorphic resp. antiholomorphic smooth sections of the real bundle B over P1(C) from 8.2 and the end of 7.6: To show this, we determine the appropriate diﬀerential operators acting on the smooth sections f of B resp. the func- −1 tions φ on G with φ(gh)=χ2s(h )φ(g) where we denote φ(g)=:φ(a, b)=:φ(α, α ,β,β ) decomposing our standard coordinates for g into real coordinates

~~a =: α + iα,b=: β + iβ.~~

~~We put gj(t) := exp tHj and get e−it cos t −i sin t cos t − sin t g (t)= ,g(t)= ,g(t)= . 1 eit 2 −i sin t cos t 3 sin t cos t~~

~~L d | Application of the prescription Hj φ(g):= dt φ(ggj(t)) t=0 to φ as a function of the four real variables α, α,β,β leads to the operators~~

L − − H1 = α ∂α α∂α β ∂β + β∂β , L − − H2 = β ∂α β∂α + α ∂β α∂β , L − − H3 = β∂α + β ∂α α∂β α ∂β , and hence, using the Wirtinger relations ∂a =(1/2)(∂α − i∂α ),∂a¯ =(1/2)(∂α + i∂α ), we get the operators

L ¯ − L − − H2+iH3 =2i(b∂a¯ a∂b), H2−iH3 = 2i(b∂a a∂¯ ¯b) acting on the functions φ here written in the coordinates a, a,¯ b, ¯b.Ifφ comes from a section f of the bundle B over P1(C), for b = 0 it is of the form (see (8.10) in 8.2)

~~φ(a, a,¯ b, ¯b)=f(z,z¯),z= a/¯b. 8.3 The Examples SU(2) and SL(2, R) 199~~

~~Hence we get L 2 L ¯2 H2+iH3 φ =(2i/b )fz¯, H2−iH3 φ =(2i/b )fz, and we see that the polarization by n± provides that the section f has to be holomorphic resp. antiholomorphic.~~

Summary: For G = SU(2) (and SO(3)) all irreducible unitary representations can be constructed by the orbit method. Each integral orbit Oη carries a line bundle and has two complex polarizations. The correponding polarized sections provide equivalent representations.

~~2. G = SL(2, R)~~

~~We continue by completing the results obtained for the orbits from the SL(2)-theory in the last section. The orbits are all realized as subsets of~~

~~3 2 2 2 Mβ := {(x∗,y∗,z∗) ∈ R ; x∗ + y∗ − z∗ = β} with convenient β ∈ R.~~

~~2.1. Hyperbolic Orbits~~

~~For the hyperbolic orbit Oη = G · αX∗ = Mα2 G/L , L = Gη = MA, α =0 wehave Ad L-invariant real polarizations given by~~

~~q1 := X, Y + Z and q2 := X, Y − Z .~~

~~The associated characters ρ1 and ρ2 of q1 resp. q2 given by~~

~~ρ1 (Yˇ )=2πi αX∗,xX +ˇy(Y + Z) =4πiαx for Yˇ = xX +ˇy(Y + Z) and~~

~~ˇ ˇ ρ2 (Y )=2πi αX∗,xX +ˇz(Y − Z) =4πiαx for Y = xX +ˇz(Y − Z) integrate uniquely to unitary characters χj,s (j =1, 2) of the respective groups~~

~~Q1 = MNA and Q2 = MNA,¯ which are trivial on Qj ∩ K. For~~

~~b = εn(x) t(y) resp. = ε n¯ (x) t(y) these characters are given by~~

~~1/2 is χj,s(b):=(y ) with s =4πiα. 200 8. Geometric Quantization and the Orbit Method~~

Then by inducing these characters from Qj to G one gets irreducible unitary representations. This coincides quantitatively with the fact that one has the principal series ±,is representations P := πis,± , which we constructed in 7.2 and which have models consisting of (smooth) functions φ = φ (g)onG with ε ε for − φ (mn(x)t(y)g)=γ(m)(y1/2)(is+1) φ(g) ,γ = , ε 1 for + i.e. in a space spanned by functions

~~φs,2k ,k∈ Z for + . and~~

~~φs,2k+1 ,k∈ Z for − with~~

~~(is+1)/2 ijθ (8.11) φs,j(g)=y e ,j∈ Z.~~

The unitary equivalence P±,is P±,−is can be translated into the fact that both polarizations produce equivalent representations (see the discussion in [GS1] p. 299 f). Thus, by the orbit method, the construction above yields only one half of the representations. This arises from the fact that the orbit Mα2 is not simply connected but has here a fundamental group π1 (Mα2 ) Z , so that we have one representation (the even one) belonging to id ∈ π1 (Mα2 ) and the other one to a generator of π1 (Mα2 ) (see the discussion in [Ki3] p.463). This may serve as motivation to introduce the notion of rigged orbits already mentioned at the end of our Subsection 8.2.5.

~~2.2. Elliptic Orbits~~

± O · For the elliptic orbits M−α2 := η = G αZ∗ G/SO(2),α= 0 , consisting of the upper + 2 half M−α2 (i.e. with z∗ > 0) of the two-sheeted hyperboloid Mα for α>0 and the lower − half M−α2 (with z∗ < 0) for α<0, we have two purely complex polarizations : We take −i 1 ± i Z = −iZ = ,X± =(1/2) (X ± iY )=(1/2) 1 i ±i 1 with

~~[Z1,X±]=±2X± , [ X+,X− ]=Z1 . and have~~

~~gc = Z1,X+,X− . This establishes the complex bilinear form~~

~~Bη([ X,ˇ Yˇ ]) = iα(a+ b− − a− b+) for~~

~~Xˇ := a0Z1 + a+X+ + a−X− , Yˇ := b0Z1 + b+X+ + b−X− . 8.3 The Examples SU(2) and SL(2, R) 201~~

~~We have Bη ≡ 0 on the Ad Gη-invariant subalgebras~~

~~n := n± = Z1,X± C of gc and n + n¯ = gc , n ∩ n¯ = kc = Z1~~

~~n± ∩ g1 = Y − Z R = k .~~

~~At ﬁrst we see that a one-dimensional representation of k is deﬁned by α ρ (a Z ):=i ,a Z = i 2αa 0 1 −α 0 1 0~~

ikθ and this integrates to a representation χk of SO(2) (with χk(r(θ)) = e ) exactly for 2α =: k ∈ Z \ 0 . Thus the elliptic orbits are quantizable for 2α = k ∈ Z \{0} . Then we ± ± use n+ and n− to deﬁne a complex structure on M = Mα2 G1/SO(2) H . And ± ∈ by the usual procedure we get the discrete series representations πk ,k N , realized 2 L ± on the L -spaces of holomorphic sections of the holomorphic line bundles on M−α2 . With similar calculations as we did for SU(2), these can also be interpretated as spaces ± of polarized sections φ of the bundle belonging to χk on M−α2 , i.e. with L L X+ φ = 0 resp. X− φ =0.

~~These representations are also representations of K = SO(2) and we have + ˜ ∈ mult(χk˜,πk ) = 1 for k = k +2l, l N0 and = 0 else. 2.3. Nilpotent Orbits~~

~~O · + ¯ In the nilpotent case η = G (Y∗ + Z∗)=M0 = G/N we have one real polarization. ¯ The form ωY∗+Z∗ is zero on the Ad N- invariant subalgebra of g given by~~

~~n = .~~

~~The standard representation ρ of n is trivial~~

~~ρ(xX + y(Y − Z)) = =0.~~

ab The group B = MNA = {±( ); a>0,b ∈ R} has Lie B = n. We can (at 0 a−1 least formally) proceed as follows. If χ = id denotes the trivial representation of B, one G has here the representation indB id given by right translation on the space of functions φ : G −→ C with φ(bg)=y1/2φ(g) for b ∈ B,g ∈ G, (spanned by φ(g)=y1/2eiθ, ∈ 2Z), i.e. the representation P+,0. As in the previous case, here we have a fundamental group π1 Z, so there is a second representation P−,0 consisting of odd functions and spanned by φ(g)=y1/2eiθ, ∈ 1+2Z), which decomposes into irreducible halves P−,0 + ⊕ − = π1 π1 − · (see for instance [Kn] p.36). The lower half of the cone M0 = G1 Y∗ generates an equivalent situation. 202 8. Geometric Quantization and the Orbit Method

~~All this is a small item in the big and important theme of how to attach representations to nilpotent orbits (see in particular the discussion in Example 12.3 of Vogan [Vo1] p.386).~~

~~Summary~~

For the representations of G = SL(2, R), we get a correspondence between – elliptic orbits and discrete series representations. – (rigged) hyperbolic orbits and principal series representations, We do not get the complementary series. For the nilpotent orbits in the SL(2, R)-theory, it seems still not clear how the correspondence is to be ﬁxed (see the discussion in [Vo1] p.386). To me it seems most probable (see in particular [Re] p.I.110f) to associate to the nilpotent cone the principal series +,0 −,0 P and the (decomposing) P = D1 ⊕ D−1.

Let us ﬁnally point to a remark by Kirillov in [Ki3] 8.4 and [Ki1] 6.4 that he sees the possibility to extend the correspondence framework such that even the complementary series ﬁts in. This should be further elucidated. Up to now one has here an example of an imperfect matching between the unitary dual and the set of coadjoint orbits. But there is a conjecture that by the orbit method one gets all representations appearing in the decomposition of the regular representation of a group ([Ki1] p 204): “Indeed, according to the ideology of the orbit method, the partition of g∗ into coadjoint orbits corresponds to the decomposition of the regular decomposition into irreducible components”.

~~8.4 The Example Heis(R)~~

The treatment of the Heisenberg group is even easier than that of the two previous examples. But one has one additional complication as one can not any longer Ad G- equivariantly identify the Lie algebra with its dual via the trace or Killing form. So we use the original deﬁnition of the coadjoint representation to determine the set O(G) of coadjoint orbits and compare it with the unitary dual Gˆ of the Heisenberg group G = Heis(R), which we determined in Example 7.11 in 7.4 as the disjoint union of the plane R2 with a real line without its origin R \{0}. We shall get a perfect matching between Gˆ and O(G).

As the formulae are a bit more smooth, we work with the realization of the Heisenberg group by three-by-three matrices, i.e. we take ⎛ ⎞ 1 ac G := Heis(R)={g = ⎝ 1 b⎠ ; a, b, c ∈ R} 1 and ⎛ ⎞ 0 xz g := {Xˇ = ⎝ 0 y⎠ = xX + yY + zZ; x, y, z ∈ R}. 0 8.4 The Example Heis(R) 203

~~The Coadjoint Action~~

As in 8.2, we have the adjoint representation Ad of G given on g by ⎛ ⎞ ⎛ ⎞ ⎛ ⎞ 1 ac 0 xz 1 −a −c + ab (Ad g)Xˇ = gXgˇ −1 = ⎝ 1 b⎠ ⎝ 0 y⎠ ⎝ 1 −b ⎠ 1 0 1 ⎛ ⎞ 0 xz+ ay − bx = ⎝ 0 y ⎠ . 0

~~∗ The coadjoint representation Ad is the contragredient of Ad (see 1.4.3) and given on g∗ by Ad∗g = (Ad g−1)∗, i.e. if we write~~

~~η, Xˇ := η(Xˇ)~~

~~∗ for all η ∈ g∗ and Xˇ ∈ g, then Ad g is ﬁxed by the relation~~

~~∗ Ad gη,Xˇ = η,(Ad g−1)Xˇ for all Xˇ ∈ g.~~

~~We have ⎛ ⎞ 0 xz− ay + bx (Ad g−1)Xˇ = ⎝ 0 y ⎠ . 0~~

∗ And if we take X∗,Y∗,Z∗ as a basis for g , dual to X, Y, Z with respect to <, >, and write ∗ g η = x∗X∗ + y∗Y∗ + z∗Z∗, we have ∗ Ad gη=˜x∗X∗ +˜y∗Y∗ +˜z∗Z∗ with

~~(8.12) x˜∗ = x∗ + bz∗, y˜∗ = y∗ − az∗, z˜∗ = z∗.~~

There is a more elegant approach to this result: In [Ki1] p.61 Kirillov uses the trace form ∗ on M3(R) to identify g with the space of lower-triangular matrices of the form ⎛ ⎞ ∗∗∗ η = ⎝x ∗∗⎠ . zy∗

Here the stars remind us that one actually considers the quotient space of M3(R) by the subspace g⊥ of upper-triangular matrices (including the diagonal). This way, one gets ⎛ ⎞ ⎛ ⎞ ∗∗∗ ∗∗∗ Ad∗gη= g ⎝x ∗∗⎠ g−1 = ⎝x + bz ∗∗⎠ . zy∗ zy− az ∗ 204 8. Geometric Quantization and the Orbit Method

~~Exercise 8.5: Do the same for the realization of the Heisenberg group and its Lie algebra by four-by-four matrices as in 0.1 and 6.3. Verify that Ad∗g is given by~~

~~(p∗,q∗,r∗) −→ (p∗ +2µr∗,q∗ − 2λr∗,r∗) ∗ for g =(λ, µ, κ) and η = p∗P∗ + q∗Q∗ + r∗R∗, where P∗,Q∗,R∗ is a basis of g dual to P, Q, R.~~

~~The Coadjoint Orbits~~

From (8.12) it is easy to see that one has just two types of coadjoint orbits Oη = G · η G/Gη. i) For η = mZ∗,m= 0 we have a two-dimensional orbit O O { ∈ } 2 m := mZ∗ = (x∗,y∗,m); x∗,y∗ R R .

ii) For η = rX∗ + sY∗,r,s∈ R, as orbit we have the point O O { } rs := rX∗+sY∗ = (r, s, 0) . Obviously g∗ R3 is the disjoint union of all these orbits and the set of orbits is in one-by-one correspondence with the unitary dual Gˆ described above. Now, we construct representations following the general procedure outlined in 8.2:

~~Orbits as Symplectic Manifolds~~

2 For M := Om R ,m=0 , we have as symplectic forms all multiples of ω0 = dx∗ ∧ dy∗. As it is a nice exercise in the handling of our notions, we determine the factor for the Kirillov-Kostant form ω: Again we use (8.5) ˆ ˆ ˆ ˆ ∈ ∈ ∗ ωη(η )(ξXˆ (η ),ξYˆ (η )) = <η, [X,Y ] > for all X,Y g,η gη We put Xˆ := xX + yY + zZ, Yˆ := xX + yY + zZ, and have [X,ˆ Yˆ ]=(xy − xy) Z and <η, [X,ˆ Yˆ ] > =(xy − x y)z∗ for η = x∗X∗ + y∗Y∗ + z∗Z∗. We determine the vector ﬁelds ξ using again (8.6) d (ξ f)(η)= (f(exp tXˆ · η))| . Xˆ dt t=0 One has ⎛ ⎞ 1 t ⎝ ⎠ gX (t) := exp tX = 1 , 1 ⎛ ⎞ 1 ⎝ ⎠ (8.13) gY (t) := exp tY = 1 t , 1 ⎛ ⎞ 1 t ⎝ ⎠ gZ (t) := exp tZ = 1 , 1 8.4 The Example Heis(R) 205 and, by (8.12),

~~gX (t) · η = x∗X∗ +(y∗ − tz∗)Y∗ + z∗Z∗,~~

~~gY (t) · η =(x∗ + tz∗)X∗ + y∗Y∗ + z∗Z∗,~~

~~gZ (t) · η = x∗X∗ + y∗Y∗ + z∗Z∗. and, hence, − ξX = z∗∂y∗ ,ξY = z∗∂x∗ ,ξZ =0, that is~~

~~ξXˆ = xξX + yξY + zξZ ,~~

~~= α∂x∗ + β∂y∗ with α = yz∗,β= −xz∗.~~

~~Thus we have − 2 − ω0(ξXˆ ,ξYˆ )=(αβ α β)=z∗(xy x y). If we compare this with the Kirillov-Kostant prescription above, we see that one has the symplectic form ω on M = Om given by~~

~~ω =(1/m)dx∗ ∧ dy∗.~~

~~Polarizations of the Orbits Om~~

The algebraic approach to real polarizations asks for subalgebras n of g = X, Y, Z subordinate to η = mZ∗. Obviously one has the (abelian) subalgebras n1 = X, Z and n2 = Y,Z and more generally, for α, β ∈ R with αβ =0 , n = αX + βY,Z . In the 2 geometric picture these correspond to a polarization given by a splitting of Om R ∗ into the union of parallel lines αx∗ + βy∗ = const., which are the orbits of Ad Q where Q is the subgroup of G with Lie Q = n, i.e. Q = {(αt, βt, c); t, c ∈ R}.

A complex polarization of Om, which is translation invariant, is generated by a constant ∈ \ vector ﬁeld ξ = ∂x∗ + τ∂y∗ ,τ C R, in particular τ = i. The functions F satisfying ξF = 0 are simply holomorphic functions in the variable w := x∗ + τy∗. In the algebraic version the complex polarizations are given by the subalgebra n = (X + τY),Z .

~~Construction of Representations~~

We want to follow the scheme outlined at the end of 8.2. As up to now our representations of the Heisenberg group were realized by right translations and the recipe in 8.2.5 is given for left translations, we have to make the appropriate changes. For the zero-dimensional orbits Or,s,r,s∈ R there is not much to be done: One has η = rX∗ + sY∗ and Gη = G and hence the one-dimensional representation π given by π(a, b, c) = exp(2πi(ra + sb)). For the two-dimensional orbits Om,m=0wehave η = mZ∗, Gη = {(0, 0,c),c∈ R} = L and for each polarization given by n the representation Yˇ −→ 2πi<η,Y>ˇ =2πimz integrates to a character χm of the group Q belonging to n. 206 8. Geometric Quantization and the Orbit Method

Hence each orbit Om is integral and we get a representation space consisting of sections of the real line bundle over M = Om (constructed using the character χm of Gη with χm((0, 0,c)) = exp (2πimc)=e(mc)) which are constant along the directions ﬁxed by the polarization n. This leads to consider the following functions: We look at φ : G −→ C with

~~φ(lg)=χm(l)φ(g), i.e. φ is of the form φ(a, b, c)=e(mc)F (a, b) with a smooth function F : R2 −→ C restricted by the polarization condition, which in our case for n = is given by~~

~~RU φ =0 where RU is the right-invariant diﬀerential operator~~

d R φ(g)= φ(g (t)−1g)| ,g (t) := exp tU. U dt U t=0 U We have −1 gX (t)=(t, 0, 0), and gX (t) g =(a − t, b, c − tb), −1 gY (t)=(0,t,0),gY (t) g =(a, b − t, c), −1 gZ (t)=(0, 0,t),gZ (t) g =(a, b, c − t). and hence

~~(8.14) RX = −∂a − b∂c, RY = −∂b, RZ = −∂c.~~

~~– i) The polarization given by n =leads to~~

~~0=RY φ = −∂bφ = −Fbe(mc), i.e. φ is of the form φ(a, b, c)=f(a)e(mc). As to be expected, via~~

~~π(g0)φ(g)=φ(gg0)=f(a + a0)e(m(c + c0 + ab0))~~

~~= e(mc)e(m(c0 + ab0))f(a + a0), we recover the Schr¨odingerrepresentation πm =: πS~~

~~f(t) −→ πm(g0)f(t)=e(m(c0 + b0t))f(t + a0).~~

~~– ii) n =leads to~~

~~0=RX φ = −(∂a + b∂c)φ = −(Fa +2πimbF )e(mc), i.e. φ is of the form φ(a, b, c)=e(−mab)f(b)e(mc). We get the representation~~

~~−→ − f(t) e(mc0 a0(b0 + t))f(t + b0)=:πm(g0)f(t).~~

From the Stone-von Neumann Theorem one knows that this representation (and any other constructed this way via another polarization) is equivalent to the Schr¨odinger representation. In our context we can see this directly using the Fourier transform as an intertwining operator: 8.4 The Example Heis(R) 207

~~We abbreviate again e(u) := exp(2πiu) and write as Fourier transform fˆ(b):= f(t)e(mbt)dt. R~~

We get ˆ − πm(g0)f(b)=e(m(c0 a0(b + b0))) f(t)e(mt(b + b0))dt R and see that this coincides with the Fourier transform of the application of the Schr¨odinger representation πm(g0)f, namely with a + a0 =: t we get (πm(g0)f)(b)= e(m(c0 + ab0)f(a + a0)e(mab)da

~~= e(mc0)e(−m(a0(b + b0))) e(mtb0)f(t)e(mbt)dt.~~

– iii) As another example of this procedure we use a complex polarization to construct the Fock representation, which (among other things) is fundamental for the construction of the theta functions. The Fock representation has as its representation space the space Hm of holomorphic functions f on C subjected to the condition f 2 = | f(z)| 2dµ(z) < ∞,dµ(z):=(1/(2i))dz ∧ dz.¯ C

~~To adopt our presentation to the more general one given in Igusa [Ig] p.31ﬀ, we introduce complex coordinates for our Heisenberg group~~

~~z := −ia + b, z¯ = ia + b, i.e. a =(¯z − z)/(2i),b=(z +¯z)/2 and ∂z =(1/2)(i∂a + ∂b),∂z¯ =(1/2)(−i∂a + ∂b).~~

We use the complexiﬁcation gc = with Y± := ±iX + Y,Z0 =2iZ and the complex polarization given by n =. Then, in order to construct a representation space following our general procedure, we come to look at smooth complex functions φ on G, which are of the form

~~φ(a, b, c)=F (z,z¯)e(mc) and subjected to the polarization condition~~

~~R Y− φ =0.~~

~~From (8.14) we have RX = −(∂a + b∂c), RY = −∂b. We deduce R − Y− = 2∂z¯ + ib∂c R − − and, hence, that one has 0 = Y− φ =( 2Fz¯ 2πmbF )e(mc), i.e.~~

~~∂z¯ log F = −πmb = −πm(z +¯z)/2. 208 8. Geometric Quantization and the Orbit Method~~

~~This shows that F is of the form~~

2 F (z,z¯)=e−(πm/2)(zz¯+¯z /2)f˜(z) with a holomorphic function f˜. As we soon will see, to get a unitary representation on ˜ 2 the space Hm introduced above, we have to choose here f(z) = exp(πmz /4)f(z) with f ∈Hm, i.e. our φ is of the form

~~2 2 φ(a, b, c)=e(mc)e−(πm/2)(zz¯+¯z /2−z /2)f(z).~~

~~By a small computation we see that the representation by right translation~~

~~π(g0)φ(g)=φ(gg0) leads to the formula of the Fock representation πF~~

2 2 −πm((z+z0/2)¯z0+(¯z0 −z0 )/4) (8.15) f(z) −→ (πF (g0)f)(z)=e(mc0)e f(z + z0). with g0 =(a0,b0,c0),z0 = b0 − ia0. Now it is easy to verify that πF acts unitarily in Hm (Exercise 8.6).

~~Perhaps it is not a complete waste of time and energy to show here how this representation comes up also (as it must) by the induction procedure we outlined in 7.1: We have the standard projection~~

~~G = Heis(R) −→ H \ G X,g=(a, b, c) −→ Hg = x =(a, b) ↔ (z,z¯) with the Mackey section s given by s(x)=(a, b, 0), i.e. g =(0, 0,c)(a, b, 0), and the master equation (7.7)~~

~~∗ s(x)g0 =(a, b, 0)(a0,b0,c0)=h s(xg0) with~~

~~∗ s(xg0)=(a+a0,b+b0, 0),h =(0, 0,c0 +ab0)=(0, 0,c0 +(zz0 −zz¯0 +¯z0z0 −z¯z¯0)/(4i)). dµ(x)=dµ(z) is a quasi-invariant measure on X = C with Radon-Nikodym derivative~~

~~dµ(xg ) 2 0 = e−mπ(zz¯0+¯zz0+|z0| ) = δ(h∗). dµ(x)~~

~~Hence, injecting these data into the formula from Theorem 7.4 in 7.1.2 for the representation in the second realization~~

~~∗ 1/2 ∗ π˜(g0)f(x)=δ(h ) π0(h )f(xg0) provides the formula (8.15) for the Fock representation obtained above.~~

We know that the Fock representation πF is equivalent to the Schr¨odingerrepresentation πS but it is perhaps not so evident how the intertwining operator looks like. 8.5 Some Hints Concerning the Jacobi Group 209

~~Proposition 8.1: The prescription f(t) −→ F (z)=If(z):= k(t, z)f(t)dt R with the kernel 2 2 k(t, z)=21/4e−πt e(tz)e(π/2)z~~

~~2 provides an isometry I between the spaces L (R) and Hm for m = 1 and intertwines the corresponding Schr¨odingerand Fock representations.~~

The fact that I is an intertwining operator is veriﬁed by a straightforward computation (Exercise 8.7). For the veriﬁcation of the isometry we refer to [Ig] p.31-35 where one goes back to explicit Hilbert bases for the spaces.

~~Summary~~

For the Heisenberg group (as for each solvable group), one has a perfect matching between coadjoint orbits and irreducible unitary representations. Everything goes through as well for the higher-dimensional Heisenberg groups if one changes the notation appropriately.

~~8.5 Some Hints Concerning the Jacobi Group~~

In Kirillov’s book [Ki1] one can ﬁnd a lot of examples and a thorough discussion of the merits and demerits of the orbit method. As orbits already turned up naturally in our presentation of Mackey’s method in 7.1 where we applied it to Euclidean groups and the Poincar´egroup, it is a natural topic to inspect the coadjoint orbits for semidirect products, in particular for the Euclidean groups.

~~Exercise 8.8: Determine the coadjoint orbits of the Euclidean group~~

~~GE(3) = SO(3) R3.~~

The reader will ﬁnd material for this in [GS] p.124ﬀ. As the author of this text is particularly fond of the Jacobi group GJ , i.e. – in its simplest version – a semidirect product of SL(2, R) with Heis(R), as another exercise we propose here to treat some questions in this direction concerning the Jacobi group. Most anwers to these can be found in [BeS] and [Be1] where we collected some elements of the representation theory of GJ and the application of the orbit method to GJ . But at a closer look there are enough rather easily accessible open problems to do some original new work:

~~The Jacobi group is in general the semidirect product of the symplectic group with an appropriate Heisenberg group. Here we look at~~

~~GJ (R) := SL(2, R) Heis(R). 210 8. Geometric Quantization and the Orbit Method~~

In this case we ﬁx the multiplication law by the embedding into the symplectic group Sp(2, R) given by ⎛ ⎞ 1 µ ⎜ λ 1 µκ⎟ Heis(R) (λ, µ, κ) −→ ⎜ ⎟ , ⎝ 1 −λ ⎠ 1 ⎛ ⎞ ab ab ⎜ 1 ⎟ SL(2, R) M = −→ ⎜ ⎟ . cd ⎝ cd⎠ 1

We write g =(p, q, κ)M or g = M(λ, µ, κ) ∈ GJ (R). J As in [BeS] or [Ya], we can describe the Lie algebra g as a subalgebra of g2 = sp(2, R) by ⎛ ⎞ x 0 yq ⎜p 0 qr⎟ G(x, y, z, p, q, r):=⎜ ⎟ ⎝z 0 −x −p⎠ 00 0 0 and denote X := G(1, 0,...,0),...,R:= G(0,...,0, 1). We get the commutators

~~[X, Y ]= 2Y, [X, Z]=−2Z, [Y, Z]=X, [X, P]=−P, [X, Q]= Q, [P, Q]=2R, [ Y,P]=−Q, [ Z, Q]=−P, all others are zero. Hence, we have the complexiﬁed Lie algebra given by~~

~~J gc = where as in [BeS] p.12~~

~~Z1 := −i(Y − Z),Z0 := −iR,~~

~~X± := (1/2)(X ± i(Y + Z)),Y± := (1/2)(P ± iQ) with the commutation relations~~

~~[Z1,X±]=±2X±, [Z0,Y±]=±Y±, etc.~~

Exercise 8.9 : Verify all this and compute left invariant diﬀerential operators L Z0 = i∂κ L −1/2 ±iθ − ± − Y± =(1/2)y e (∂p (x iy)∂q (p(x + iy)+q)∂κ) L ± ±2iθ ∓ − X± = (i/2)e (2y(∂x i∂y) ∂θ) L − Z1 = i∂θ 8.5 Some Hints Concerning the Jacobi Group 211 acting on diﬀerentiable functions φ = φ(g) with the coordinates coming from

g =(p, q, κ)n(x)t(y)r(θ) where 1 x y1/2 αβ n(x)= ,t(y)= ,r(θ)= 1 y−1/2 −βα with α = cos θ, β = sin θ, and g(i, 0) = (τ,z)=(x + iy, pτ + q). As usual, we put N := {n(x); x ∈ R},A:= {t(a); a ∈ R>0}, K := SO(2) = {r(θ); θ ∈ R},M:= {±E}. Elements of the representation theory of GJ (R) can be found in the work of Pyatetskii– Shapiro, Satake, Kirillov, Howe, Guillemin, Sternberg, Igusa, Mumford and many others. To a large part, they are summed up in [BeS], the essential outcome being that the representations π of GJ (R) with nontrivial central character are essentially products of projective representations of SL(2, R) with a fundamental (projective) representation m J πSW of G called the Schrödinger–Weilrepresentation in [BeS], as it is in turn the product of the Schrödingerrepresentation πS of the Heisenberg group Heis(R) and the m (Segal–Shale–)Weil or oscillator representation πW of SL(2, R). This representation πSW is a representation of lowest weight k =1/2 and index m. It is characterized by the fact that all its KJ –types are one-dimensional (KJ := SO(2) × C(Heis(R)) SO(2) × R), m which in turn is characterized by the fact that the lowest weight vector for πSW satisfies the heat equation. To be a bit more explicit, the irreducible unitary representations π of GJ (R) with central character ψm (with ψm(x):=em(x)) for m = 0 are infinitesimally equivalent to

πˆm,s,ν ,s∈ iR,ν= ±1(ˆπm,s,ν πˆm,−s,ν ), 2 πˆm,s,ν ,s∈ R,s < 1/4, + ∈ πˆm,k,k N, − ∈ πˆm,k,k N. Here and in the following, we restrict our treatment to m>0, but there is a “mirror image” for m<0 which, to save space, we will not discuss. These inﬁnitesimal representations are of the form

~~m ⊗ πˆm,s,ν =ˆπSW πˆs,ν , ± m ⊗ ± πˆm,k =ˆπSW πˆk0 ,k= k0 +1/2. m The Schr¨odinger–Weil representationπ ˆSW is given on the space 1/2 Vm := vj j∈N0 by (see [BeS] p.33)~~

~~Zvj := (1/2+j)vj,Z0vj := µvj,Y+vj := vj+1,X+vj := −1/(2µ)vj+2,.... 212 8. Geometric Quantization and the Orbit Method~~

~~+ And, for instance, the discrete series representationπ ˆk−1/2 of sl(2) is given on Wk−1/2 := wl l∈2N0 by~~

~~Zwl := (k − 1/2+l)wl,X+wl := wl+2,X−wl := (l/2)(k − 3/2+l)wl−2.~~

~~+ That is, we have as a space forπ ˆm,k~~

~~+ 1/2 ⊗ ⊗ Vm,k := Vm Wk−1/2 = vj wl j∈N,l∈2N0 with Z0(vj ⊗ wl)=µ(vj ⊗ wl),~~

~~Z(vj ⊗ wl)=(k + j + l)(vj ⊗ wl),~~

~~Y+(vj ⊗ wl)=vj+1 ⊗ wl,~~

~~X+(vj ⊗ wl)= −(1/(2µ))vj+2 ⊗ wl + vj ⊗ wl+2. In particular, for the spaces of Z-weight k + λ, λ ≥ 0,~~

~~(λ) { ∈ + } V := v Vm,k; Zv =(k + λ)v we have~~

~~dim V (0) =1, dim V (1) =1, dim V (2) =2, dim V (3) =2, dim V (4) =3,...~~

~~J If we denote by ρm,k˜ the (one-dimensional) representation of K given by~~

~~(ikθ˜ )+2πimκ ∈ ∈ ρm,k˜(r(θ), (0, 0,κ)) = e ,θ [0, 2π],κ R, we have + ˜ mult (ρm,k˜,πm,k) = 1 for k = k, ˜ =1+l for k = k +2l or = k +2l +1,l ∈ N0 = 0 for k~~

~~L L L L X− φ0 = Y− φ0 =0, Z0 φ0 =2πmφ0, Z φ0 = kφ0 .~~

~~The Jacobi group is not semisimple (and indeed a prominent example for a non-reductive group). We describe its coadjoint orbits using our embedding of GJ into Sp(2, R):~~

~~∗ We identify the dual sp of sp by the Sp(2, R) invariant isomorphism η → Xη given by~~

~~η(Yˇ )=Tr(XηYˇ )=:ˇ for all Yˇ ∈ sp. 8.5 Some Hints Concerning the Jacobi Group 213~~

~~J ∗ η1,...,η6 denotes a basis of (g ) dual to~~

~~(X1,...,X6):=(X,...,R).~~

~~We realize (gJ )∗ as a subspace of sp by the matrices ⎛ ⎞ xp z0 ⎜ 00 00⎟ M(x, y, z, p, q, r):=⎜ ⎟ ,x,...,r ∈ R, ⎝ yq−x 0 ⎠ qr−p 0 and put~~

~~X∗ := M(1, 0,...,0),...,R∗ := M(0,...,0, 1).~~

~~Then~~

~~Xη1 = X∗,...,Xη6 = R∗ is a basis of (gJ )∗ with~~

~~=2,=1,=1 =2,=2,=1.~~

~~By a straightforward computation one obtains the following result:~~

~~Lemma: If g−1 is denoted as ⎛ ⎞ a 0 baµ− bλ ⎜ λ 1 µκ⎟ g−1 = ⎜ ⎟ , ⎝ c 0 dcµ− dλ ⎠ 000 1 the coadjoint action of g on M(x,...,r) is given by~~

~~Ad∗(g)M(x,...,r)=M(˜x,...,r˜) with~~

~~x˜ =(ad + bc)x + bdy − acz +(2acµ − (ad + bc)λ)p~~

~~+((ad + bc)µ − 2bdλ)q + r(aµ − bλ)(cµ − dλ),~~

~~y˜ =2dcx + d2y − c2z +2(cµ − dλ)(cp + dq)+r(cµ − dλ)2,~~

~~(8.16) z˜ = −2abx − b2y + a2z − 2(aµ − bλ)(ap + bq) − r(aµ − bλ)2,~~

~~p˜ = ap + bq + r(aµ − bλ),~~

~~q˜ = cp + dq + r(cµ − dλ),~~

~~r˜ = r. 214 8. Geometric Quantization and the Orbit Method~~

~~Exercise 8.10: Verify this and determine the coadjoint orbits. For instance, show that for U∗ = mR∗ + αX∗,m=0 ,α=0 , O one has the 4-dimensional hyperbolic orbit U∗ contained in~~

~~6 2 2 2 2 Mm,−α2 := {(x, y, z.p, q, r) ∈ R ; m(x + yz − α )=2pqx − p y + q z,r = m}.~~

~~O 2 Do you get U∗ = Mm,−α ?~~

It is natural to try to apply the general procedure outlined in this section to realize representations of GJ by bundles carried by these orbits and to ask, which ones you get and, if not, why? As already said above, there are some answers in [Be1] and [Ya], but far from all. Chapter 9

~~Epilogue: Outlook to Number Theory~~

Representations of groups show up in many places in Number Theory, for instance as representations of Galois groups. There seems to be no doubt that the relationship of the two topics culminates in the Langlands Program, which (roughly said) seeks to establish a correspondence between Galois and automorphic representations. Many eminent mathematicians have worked and work on this program, and it is now even interesting for physicists; see Frenkel’s Lectures on the Langlands Program and Conformal Field Theory [Fr]. We cannot dare to go into this here, but we will try to introduce at least some initial elements by presenting some representation spaces for the special groups we treated in our examples, consisting of, or at least containing, theta functions, and modular and automorphic forms. And we will also introduce the notions of zeta and L-functions, which ultimately are the foundations of the bridge between the Galois and automorphic representations. There are many useful books available. We can only cite some of them, which by now are classic: Representation Theory and Automorphic Func- tions by Gelfand, Graev and Pyatetskii-Shapiro [GGP], Automorphic Forms on GL(2) by Jacquet and Langlands [JL], Automorphic Forms on Adele Groups by Gelbart [Ge], Analytic Properties of Automorphic L-Functions by Gelbart and Shahidi [GS], Automor- phic Forms and Representations by Bump [Bu], Theta Functions by Igusa [Ig], The Weil representation, Maslov index and Theta Series by Lion and Vergne [LV], Fourier Analysis on Number Fields by Ramakrishnan and Valenza [RV], and Mumford’s Tata Lectures on Theta I, II, and III [Mu].

The goal of these last sections will be to lead a way to automorphic representations: While we already encountered the decomposition of the regular representation on the space L2(G), we shall here take a discrete subgroup Γ of G and ask for the representations appearing in the decomposition of L2(Γ\G), i.e., we consider representation spaces consisting of functions with a certain periodicity, invariance or covariance property. All this is most easily understood for the case of the Heisenberg group and leads to the notion of the theta functions. So we will start by treating this case, and afterwards will look at the group SL(2, R) and introduce modular forms as examples for more general automorphic forms. Finally, we shall consider several kinds of L-functions appearing in connection with representation theory. 216 9. Epilogue: Outlook to Number Theory

Since this is an epilogue and an outlook and the material has such a great extension and depth, the reader will excuse (we hope) that we shall not be able to give more than some indications, and even less proofs, than in the preceding chapters.

~~9.1 Theta Functions and the Heisenberg Group~~

Whatever we do in this section can easily be extended to higher dimensional cases by conveniently adopting the notation. As already done before, we denote again z ∈ C and τ ∈ H and moreover, as usual in this context,

~~q := e(τ)=e2πiτ ,:= e(z)=e2πiz.~~

~~Then one has the classic Jacobi theta function ϑ given by~~

~~2 2 (9.1) ϑ(z,τ):= eπi(n τ+2nz) = qn /2n n∈Z n∈Z going back to Jacobi in 1828. One has to verify convergence:~~

~~Lemma: ϑ converges absolutely and uniformly on compact parts of C × H.~~

~~Proof: Exercise 9.1.~~

Moreover, today, one has variants, which unfortunately are differently normalized by different authors. For instance in [Ig] p.V, one finds the theta function with characteristics m =(m,m) ∈ R2 2 θm(τ,z):= e((1/2)(n + m ) τ +(n + m )(z + m )). n∈Z

In our text, we shall follow the notation of [EZ] p.58 where, for 2m ∈ N,µ=0, 1,.,2m−1, one has r2/(4m) r (9.2) θm,µ(τ,z):= q r∈Z,r≡µ mod 2m = em((n + µ/(2m))2τ +2(n + µ/(2m))z). n∈Z

Obviously, we have ϑ(z,τ)=θ1/2,0(τ,z). Theta functions have a rather evident quasiperiodicity property concerning the complex variable z and a deeper modular property concerning the variable τ ∈ H, which makes them most interesting for applications in number theory and algebraic geometry, and – concerning both variables – ϑ satisﬁes the “one- dimensional” heat equation 4πiϑτ = ϑzz. At ﬁrst, we treat the dependance on the complex variable where, as we shall see, the Heisenberg group comes in. But to give already here at least a tiny hint, for instance, theta functions are useful if one asks for the number of ways a given natural number can be written as, say, four squares of integers (see [Ma] p.354 or our Example 9.3 in 9.2). 9.1 Theta Functions and the Heisenberg Group 217

As we keep in mind that the group really behind theta functions is the Jacobi group introduced in 8.5, we take the version of the Heisenberg group, which comes from its realization by four-by-four matrices though in other sources diﬀerent coordinizations are used (like we did in the last chapter). Thus, we take

~~G := Heis(R)={g =(λ, µ, κ) ∈ R3} with gg := (λ + λ,µ+ µ,κ+ κ + λµ − λµ).~~

~~For ﬁxed τ ∈ H, G = Heis(R) acts on C by~~

~~C z −→ g(z):=z + λτ + µ, g =(λ, µ, κ) ∈ G.~~

~~Obviously, the stabilizer of z ∈ C is the center C(R)={(0, 0,κ); κ ∈ R}R of the Heisenberg group and one has~~

~~X := G(R)/C(R) C.~~

~~This action induces an action on functions F on C in the usual way~~

~~F −→ F g with F g(z):=F (g(z)) = F (z + λτ + µ).~~

~~Recalling the cocycle condition (7.8) in 7.1 and the ﬁrst example of an automorphic factor in 7.2, we reﬁne this action by introducing for m ∈ R as an automorphic factor for the Heisenberg group~~

~~m 2 jm(g, z):=e (λ τ +2λz + λµ + κ).~~

~~Exercise 9.2: Verify the relation~~

~~jm(gg ,z)=jm(g, g (z))jm(g ,z).~~

~~Then we deﬁne F |[g]by F | [g](z):=jm(g, z)F (g(z)) and introduce as our ﬁrst example of an automorphic form a holomorphic function F on C with~~

~~(9.3) F |[γ]=F for all γ =(r, s, t) ∈ Γ = Heis(Z)={γ =(r, s, t) ∈ Z3}.~~

~~We denote by Θ(m) the vector space of all these functions F .~~

~~Thus, an entire function F (i.e. holomorphic on C) is an element of Θ(m)iﬀ~~

~~F (z + s)=F (z) and F (z + rτ)=e−m(r2τ +2rz)F (z) for all r, s ∈ Z.~~

~~If we relate this to the theta functions deﬁned above, we are led to the following central result, which is not too diﬃcult to prove (using the periodicity and the functional equation (9.3)).~~

~~Theorem 9.1: For 2m ∈ N,Θ(m) is spanned by the theta functions~~

~~θm,α,α=0, 1,...,2m − 1 and hence has dimension 2m. 218 9. Epilogue: Outlook to Number Theory~~

To any function F on C we associate a lifted function φ = φF deﬁned on G = Heis(R) by m 2 φF (g):=(F |[g])(0) = F (λτ + µ)e (λ τ + λµ + κ). This prescription is to be seen on the background of the relation between the First and the Second Realization in the induction procedure (see 7.1). Then we come to understand immediately the ﬁrst two items in the

Proposition 9.1: Under the map F −→ φF , the space Θ(m) is isomorphic to the space H(m) of functions φ : G −→ C satisfying i) φ(γg)=φ(g) for all γ ∈ Γ, ii) φ(gκ)=φ(g)em(κ) for all κ := (0, 0,κ) ∈ C(R), L iii) φ is smooth and satisﬁes Y− φ =0.

The third statement is result of a discussion parallel to the one of the Fock representation in 8.4: here we use Y− := P − τQ to express the holomorphicity of F by a diﬀerential equation for φ:For

~~gP (t)=(t, 0, 0),~~

~~gQ(t)=(0,t,0),~~

~~gR(t)=(0, 0,t) one has~~

~~ggP (t)=(λ + t, µ, κ − µt),~~

~~ggQ(t)=(λ, µ + t, κ + λt),~~

~~ggR(t)=(λ, µ, κ + t) and for the left invariant operators LU acting on functions φ living on G deﬁned by d L φ(g):= φ(gg (t))| , U dt U t=0 we get~~

~~LP = ∂λ − µ∂κ,~~

~~LQ = ∂µ + λ∂κ,~~

LR = ∂κ, and L L − − Y− = P −τQ = ∂λ τ∂µ (λτ + µ)∂κ. In the prescription for the lifting from C to G we assume z = λτ + µ, z¯ = λτ¯ + µ and (via Wirtinger calculus) deduce

∂z¯ =(τ∂µ − ∂λ)/(τ − τ¯). If we have φ(λ, µ, κ)=F (z,z¯)em(κ + λ2τ + λµ), L a tiny computation shows that the condition Y− φ = 0 translates into the holomorphicity condition ∂z¯F =0. 9.1 Theta Functions and the Heisenberg Group 219

This leads us directly to the following representation theoretic interpretation. The conditions i) and ii) of the space H(m) show up if one looks at still another standard representation of the Heisenberg group, namely the lattice representation:

~~We take the subgroup H := {h =(r, s, t); r, s ∈ Z,t∈ R} and, for m ∈ Z \{0}, its character given by~~

~~m π0(h):=e (t).~~

~~Then we have X = H \ G Z2 \ R2. The action of G on C given by~~

z −→ z + λτ + µ restricts to the action of H on 0 ∈ C producing the points rτ + s forming a lattice L in C. By our general procedure from Mackey’s approach in 7.1, we get a representation by right translation on the space of functions φ on G, which satisfy the functional equation

~~φ(hg)=em(t)φ(g) for all h =(r, s, t),r,s∈ Z,t∈ R,g∈ G.~~

(As X is compact, we have no problem with the condition of the ﬁnite norm integral.) This induced representation is not irreducible and we proceed as follows. For Γ = Heis(Z) we denote by L2(Γ \ G) the Hilbert space of Γ−invariant measurable functions G −→ C with the scalar product φ, ψ := φ(g)ψ(g)dg. Γ\G This is a Hilbert space direct sum

~~2 2 L (Γ \ G)=⊕m∈ZL (Γ \ G)m~~

~~2 where L (Γ \ G)m denotes the subspace of functions φ that satisfy~~

~~φ(gκ)=em(κ)φ(g) for all κ =(0, 0,κ) ∈ Z(R),g ∈ G.~~

~~2 Now for m>1/2, this induced representation on L (Γ \ G)m is not irreducible. We ﬁnd here the ﬁrst example of an important general statement.~~

~~Theorem 9.2 (Duality Theorem for the Heisenberg group): For m ∈ N, the multiplicity 2 of the Schr¨odingerrepresentation πm in L (Γ \ G) is equal to the dimension of the space Θ(m).~~

~~Proof: There are several diﬀerent approaches. In the our context, we can argue like this: 2 The Schr¨odingerrepresentation πm transforms a function f ∈ L (R)into~~

~~m πm(g0)f(x)=e (κ0 +(2x + λ0)µ0)f(x + λ0) and is intertwined with the Heisenberg representation (see Example 7.1 at the beginning of 7.1) given by right translation on functions φ on G via~~

~~m f(x) −→ e (κ + λµ)f(λ)=φf (g). 220 9. Epilogue: Outlook to Number Theory~~

Obviously, one has φ(κg)=em(κ)φ(g) but as the Heisenberg representation is induced by the two-dimensional subgroup {(0,µ,κ); µ, κ ∈ R} of G, one also has invariance concerning the µ−variable. Now, to produce a subspace of the lattice representation, one has to create in- or covariance concerning the λ−variable: We deﬁne for n ∈ Z and α =0, 1,...,2m − 1 λn,α := (n + α/(2m), 0, 0) and get λn,αg =(λ + n + α/(2m),µ,κ+ µ(n + α/(2m)).

Hence for every α,wehaveamapϑα : f −→ φf,α, called theta transform, with φf,α(g):= φf (λn,αg) ∈Z n = em(κ + µ(n + α/(2m))+(λ + n + α/(2m))µ)f(λ + n + α/(2m)). n∈Z It is not diﬃcult to verify that one has for all γ ∈ Γ and κ ∈ R

~~φf,α(γg)=φf,α(g) and φf,α(gκ)=φf,α(g).~~

Thus we see that for α =0, 1,...,2m − 1 every ϑα intertwines the Schrödingerrepresen- tation with the lattice representation. We still have to show that these are all different and that there are no others. To do so, we realize the Schrödingerrepresentation πm by 2 choosing as vacuum vector f0 not f0(x) = exp(−2πmx ), as in Remark 6.13 in 6.5, but m 2 f0(x)=e (τx ).

Then this function is annihilated by the diﬀerential operator Yˆ− = ∂x − (4πim)τx from the derived representation dπm (as in Example 6.9 in 6.3) belonging to Y− = P − τQ introduced above. If one applies the theta transform ϑα to f0 one gets m 2 ϑαf0(g)=e (κ + τλ + λµ)θm,α(τ,λτ + µ), i.e. exactly the theta basis of the space H(m) appearing in Proposition 9.1 above, which characterizes the lifting of Θ(m) to functions on G.

The reader could try to ﬁnd another version of the proof using the bundle approach to induced representations. One can prove (Theorem of Appell-Humbert) that every line bundle on the torus X := L \ C,L= Zτ + Z is of the form L(H, β), i.e. the quotient of the trivial bundle C × C −→ C by the action of L given by (, (v, z)) −→ (β()eiH(z,)+(1/2)H(,)v, z + ) for all ∈ L, v, z ∈ C. Here H : C × C −→ C is a hermitian form such that E =Im(H) is integer valued on L and β : L −→ S1 a map with

~~iπE(1,2) β(1 + 2)=e β(1)β(2).~~

A computation shows that these hermitian forms are exactly the forms Hm,m∈ Z, given by Hm(z,w)=mzw/¯ Im(τ). Up to now, we only treated the theta functions as quasiperiodic functions in the complex variable z. There is also a remarkable modular behaviour concerning the variable τ ∈ H. Before we discuss this, we return to some more general considerations. 9.2 Modular Forms and SL(2,R) 221

~~9.2 Modular Forms and SL(2,R)~~

Besides theta functions and strongly intertwined with it, modular forms is another topic relating classical function theory to number theory and algebraic and analytic geometry. There are many competent introductions. A very brief one is given in Serre [Se1]. More- over we recommend the books by Sch¨oneberg [Sch] and Shimura [Sh] containing more relevant details.

~~Classical Theory (Rudiments)~~

We already met several times with the action of G = SL(2, R) on the upper half plane X = H := {τ = x + iy ∈ C; y>0} given by ab aτ + b (g = ,τ) −→ g(τ):= . cd cτ + d As g and −g produce the same action, it is more convenient to treat the group

~~G¯ = PSL(2, R) := SL(2, R)/{±E2}.~~

~~But since we worked so far with G and its representation theory can be rather easily adopted to G¯, we stay with this group.~~

~~For k ∈ R (but mainly k ∈ Z) we have (see 7.2.2) the automorphic factor~~

~~−1 −k j(g, τ):=(cτ + d) resp.jk(g, τ):=(cτ + d) fulﬁlling the cocycle relation (7.8) j(gg,τ)=j(g, g(τ))j(g,τ). Let f be a smooth function on H. Then one writes~~

~~f|k[g](τ):=f(g(τ))jk(g, τ)~~

~~−1 −1 (so that πk(g)f(τ):=f(g (τ))jk(g ,τ) at least formally is the prescription of a representation).~~

There are several important types of discrete subgroups Γ of G and the theory becomes more and more beautiful if one treats more and more of these. For our purpose to give an introduction, it is suﬃcient to restrict ourselves to the (full) modular group

~~Γ = SL(2, Z) and eventually to the main congruence group of level N~~

~~Γ=Γ(N):={γ ∈ SL(2, Z); γ ≡ E2 mod N},N∈ N,~~

~~Hecke’s special congruence subgroup ab Γ=Γ (N):={γ = ∈ SL(2, Z); c ≡ 0modN},N∈ N, 0 cd~~

~~2 and the theta group Γϑ generated by T and S, i.e. 0 −1 11 Γ=Γ := with S = and T = . ϑ 10 01 222 9. Epilogue: Outlook to Number Theory~~

~~There is a lot to be said about these groups. We mention only two items:~~

~~Remark 9.1: Γ = SL(2, Z) is generated by just two elements, namely S and T .~~

Exercise 9.3: Prove this and realize that T acts on H as a translation τ −→ T (τ)=τ +1 and S as a reﬂection at the unit circle τ −→ S(τ)=−1/τ. Determine the ﬁxed points for the action of Γ and their stabilizing groups.

~~Remark 9.2: Γ = SL(2, R) has the following standard fundamental domain~~

~~F := {τ ∈ H; |τ| > 1, −1/2 < Re τ<1/2}.~~

Here we use the definion from [Sh] p.15: For any discrete subgroup Γ of G = SL(2, R), we call F a fundamental domain for Γ \ H (or simply for Γ) if –i)F is a connected open subset of H, – ii) no two points of F are equivalent under Γ, – iii) every point of H is equivalent to some point of the closure of F under Γ. (Other authors sometimes use different definitions.)

~~Exercise 9.4: Verify this and show that the set of Γ−orbits on H is in bijection to~~

~~F := F∪{τ ∈ C; |τ|≥1, Re τ = −1/2}∪{τ ∈ C; |τ| =1, −1/2 ≤ Re τ ≤ 0}.~~

~~If you have trouble, search for help in [Sh] p.16 or any other book on modular forms.~~

~~We look at functions f : H −→ C with a certain invariance property with respect to the discrete subgroup. Here we take Γ = SL(2, Z) and k ∈ Z.~~

~~Deﬁnition 9.1: f is called (elliptic) modular form of weight k iﬀ one has~~

–i)f|k[γ]=f for all γ ∈ Γ, – ii) f is holomorphic, – iii) For Im τ>>0,fhas a Fourier expansion or (q−development) f(τ)= c(n)qn,q = e(τ)=e2πiτ ,c(n) ∈ C. n≥0 f is called cusp form (forme parabolique in French) if one has moreover c(0) = 0. We denote by Ak(Γ) the space of all elliptic modular forms of weight k and by Sk(Γ) the space of cusp forms. Both are C vector spaces and their dimensions are important characteristic numbers attached to Γ. From i) taken for γ = −E2, one sees immediately that for odd k there are no non-trivial forms. For even k =2 one has

dim A2(Γ) = [/6] if ≡ 1mod6,≥ 0 =[/6] + 1 if ≡ 1mod6,≥ 0 where [x] denotes the integral part of x, i.e. the largest integer n such that n ≤ x. For a proof see for instance [Se] p.88 or (with even more general information) [Sh] p.46.

Remark 9.3: One also studies meromorphic modular forms where in ii) holomorphic is − n replaced by meromorphic and in iii) one has a q development f(τ)= n>−∞ c(n)q . 9.2 Modular Forms and SL(2,R) 223

~~As Γ is generated by S and T , it is suﬃcient to demand the two conditions~~

~~f(τ +1)=f(τ),f(−1/τ)=τ kf(τ). n Hence a modular form of weight k is given by a series f(τ)= n≥0 c(n)q converging for |q| < 1 and fulﬁlling the condition~~

~~f(−1/τ)=τ kf(τ).~~

Remark 9.4: The condition iii) concerning the Fourier expansion can also be understood in several different ways, for instance as a certain boundedness condition for y −→ ∞ k/2 (in particular for f ∈ Sk(SL(2, Z)), |f(τ)|y is bounded for all τ ∈ H), or, if k is even, as holomorphicity of the differential form α = f(τ)dτ k/2 at ∞: one can compactify the upper half plane H by adjoining as cusps the rational numbers and the “point” ∞, i.e. one takes H∗ := H ∪ Q ∪ {∞}. Then one extends the action of SL(2, R)onH to an action on H∗ and identifies Γ \ H∗ with the Riemann sphere or P1(C). In this sense modular forms are understood as holomorphic k/2-forms on P1(C).

All this is made precise in books on the theory of modular forms. Here, for our purpose, the essential point is that, similar to the theta functions in the previous section, modular forms are realizations of dominant weight vectors of discrete series representations of G = SL(2, R). Before we go into this, let us at least give some examples of modular forms and the way they can be constructed.

~~Example 9.1: The most well-known elliptic modular form is the ∆−function given by its q−expansion~~

−12 ∞ − n 24 (2π) ∆(q):=q Πn=1(1 q ) ∞ n − 2 3 − 4 =Σn=1τ(n)q = q 24q + 252q 1472q + .... This is a cusp form of weight 12. The function n −→ τ(n) is called the Ramanujan function. It has many interesting properties, for instance there is the Ramanujan conjecture

τ(n)=O(n(11/2)+ε),ε>0, which has been proved by Deligne in 1974 (as consequence of his proof of the much more general Weil conjecture concerning Zeta functions of algebraic varieties (see Section 9.6)). But it is still an open question whether all τ(n) are non-zero.

~~A modular form in a slightly more reﬁned sense is the η−function, also appearing in the physics literature in several contexts,~~

~~1/24 ∞ − n η(τ):=q Πn=1(1 q ). Example 9.2: A more comprehensive construction principle leads to the Eisenstein series Ek resp. Gk.Fork =4, 6,..., Gk is deﬁned by~~

~~−k Gk(τ):=Σm,n(mτ + n) where the sum is taken over all m, n ∈ Z with exception of (m, n)=(0, 0). 224 9. Epilogue: Outlook to Number Theory~~

~~It is not too much trouble to verify that this is really a modular form of weight k and has a Fourier expansion (see [Sh] p.32)~~

~~k ∞ (2πi) n G (τ)=2ζ(k)+2 σ − (n)q k (k − 1)! k 1 n=1 where ζ is Riemann’s Zeta function deﬁned for complex s with Re s>1by ∞ ζ(s):= n−s n=1~~

k and σk(n) denotes the sum of d for all positive divisors d of n. These functions appear naturally in the discussion of lattices L = τZ + Z in C. They can also be introduced by a far reaching averaging concept similar to the theta transform we used in the previous section to understand the construction of theta functions: Every function ϕ on H formally gives rise to an object satisfying the covariance property i) of a modular form by the averaging ϕ|k[γ]. γ∈Γ

One has the problem whether this series converges (and has the properties ii) and iii)). In general this is not to be expected. But one can use the following fact: For g0 = T our automorphic factor has the property j(g0,τ) = 1 , and, hence, one has

~~−1 j(g0g, τ)=j(g, τ)=(cτ + d) .~~

~~We see that for a constant function ϕ0, say for ϕ0(τ) = 1, we have~~

~~ ϕ0|k[T ]=ϕ0.~~

~~Thus to get a covariant expression, one has only to sum up Ek(τ):= (ϕ0|k[γ])(τ)~~

~~γ∈Γ∞\Γ~~

with Γ∞ := {T ; ∈ Z}. The map ab Γ −→ Z2,g= −→ (c, d) cd induces a bijection 2 Γ∞ \ Γ {(c, d) ∈ Z ;(c, d)=1} where here (c, d) denotes the greatest common divisor. Hence, one can write −k Ek(τ)= (cτ + d) . c,d∈Z, (c,d)=1

~~This Ek has the properties we wanted and is related to Gk by~~

~~ζ(k)Ek(τ)=Gk(τ). 9.2 Modular Forms and SL(2,R) 225~~

~~The same procedure goes through for every Γ∞-invariant ϕ. For instance, ϕ(τ)=e(mτ) leads to the so called Poincar´eseries, which are used to construct bases for spaces of cusp forms.~~

~~Example 9.3: As already mentioned in 9.1, Jacobi’s theta function ϑ has a modular behaviour concerning the variable τ. Here we restrict the function to its value at z =0~~

~~2 ϑ(τ):=ϑ(0,τ)= eπin τ . n∈Z~~

~~One has the relations~~

~~ϑ(τ +2)=ϑ(τ),ϑ(−1/τ)=(τ/i)1/2ϑ(τ).~~

The ﬁrst one is easy to see and the second one requires some work (see for instance [FB] p.344/5). These relations entail that this ϑ is a modular form with respect to the theta group Γϑ if one generalizes the deﬁnition of modular forms appropriately. There are more general theta series associated to quadratic forms resp. lattices leading directly to modular forms:

~~Let S ∈ Mn(Z) be a symmetric positive matrix, i.e. with~~

~~S[x]:=txSx > 0 for all 0 = x ∈ Rn then one deﬁnes an associated theta series by ϑ(S; τ):= eπiS[x]τ . x∈Zn~~

We get a function with period 2 whose Fourier development ∞ ϑ(S; τ)= a(S, m)eπimτ m=0 encodes the numbers a(S, m):=#{x ∈ Zn; S[x]=m} of representations of the natural number m by the quadratic form, which belongs to S. −1 If we assume that S is moreover unimodular, i.e. S is invertible with S ∈ Mn(Z), even, i.e. S[x] ∈ 2Z for all x ∈ Zn, and if n is divisible by 8, then ϑ(S; τ) is an elliptic modular form of weight n/2. For a proof we refer to [FB] p.352 or [Se1] p.53.

~~In particular, for~~

n { ∈ n 2 } an(m):=a(En,m)=# x Z ; xj = m j=1 one has results obtained by Jacobi in 1829 resp. 1828, namely m−d 3 a8(m)=16 (−1) d and a4(m)=8 d. d|m d|m,4 d 226 9. Epilogue: Outlook to Number Theory

Proofs can be found for instance in [FB] or [Mu] I. The second relation can be veriﬁed using the nice identity 4 −2 ϑ (τ)=π (4G2(2τ) − G2(τ/2)). Example 9.4: As a ﬁnal example we present the j−invariant or modular invariant, which is a meromorphic modular form: We put

~~g2(τ):=60G4(τ),g3(τ) := 140G6(τ).~~

~~3 − 2 − Then, we have ∆(τ)=g2(τ) 27g3 and the j function is deﬁned by~~

~~3 3 j(τ):=12 g2(τ) /∆(τ). j is a modular function with Fourier expansion (see for instance [Sh] p.33) ∞ −1 n j(τ)=q (1 + cnq ),n∈ Z. n=1~~

This function is of special importance because it induces a bijection Γ\H∗ P1(C). And the coeﬃcients cn hide some deep mysteries (essentially assembled under the heading of monstrous moonshine (see [CN], [FLM])).

~~Modular Forms as Functions on SL(2,R)~~

Parallel to the relation between the Second and First Realization in the induction procedure one deﬁnes a lifting of functions f from the homogeneous space X = G/K Gx0 with K = Gx0 to the group using the automorphic factor j(g, τ) satisfying the cocycle condition (7.8) by f −→ φf where we put

~~φf (g):=f(g(x0))j(g, x0)=f|[g](x0).~~

Our aim is to translate the conditions i) to iii) from the deﬁnition of modular forms f to appropriate conditions characterizing the lifts of these forms among the functions φ on G. Classic sources giving all details for this are [GGP], [Ge] and [Bu].

~~Remark 9.5: If for a subgroup Γ of G and a function f on X one has~~

~~(9.4) (f|[γ])(x)=f(γ(x))j(γ,x)=f(x) for all γ ∈ Γ, the lifted function φ = φf satisﬁes ia) φ(γg)=φ(g) for all γ ∈ Γ and ib) φ(gκ)=φ(g)j(κ, x0) for all κ ∈ K.~~

~~Proof: By deﬁnition, (9.4), and (7.8), we have~~

~~φ(γg)=f(γg(x0))j(γg,x0), −1 = f(g(x0))j(γ,g(x0)) j(γg,x0),~~

~~= f(g(x0))j(g, x0). 9.2 Modular Forms and SL(2,R) 227~~

~~And by deﬁnition, (7.8), and K = Gx0 ,wehaveaswell~~

~~φ(gκ)=f(gκ(x0))j(gκ, x0)~~

~~= f(g(x0))j(g, x0)j(κ, x0)~~

~~= φ(g)j(κ, x0).~~

~~Remark 9.6: In our case G = SL(2, R),x0 = i ∈X = H = G/K, K = SO(2) and j(g, τ)=(cτ + d)−k, for the lifted function we have~~

~~φ(g)=f(g(i))(ci + d)−k = f(τ)yk/2eikϑ if g is given by the Iwasawa decomposition (7.12)~~

~~g = n(x)t(y)r(ϑ).~~

~~Proof: In this case one has~~

~~ci + d = y−1/2(−i sin ϑ + cos ϑ)=y−1/2e−iϑ.~~

The translation of the holomorphicity condition ii) to a condition restricting a function φ on G follows the scheme we already used several times: For sl(2, C)c = one has the left invariant diﬀerential operators (7.16) L ± ±2iϑ ∓ − X± = (i/2)e (2y(∂x i∂y) ∂ϑ),

~~LZ = −i∂ϑ,~~

~~k/2 ikϑ and, hence, for a function φ = φf with φf (g)=ck(g)f(x, y),ck(g):=y e~~

~~LZ φ = kφ, L ˜ X± φ = ck±2D±f where 2 D˜ + = i(∂x − i∂y − ik/y), D˜ − = −iy (∂x + i∂y). Furthermore, using Wirtinger calculus with~~

~~∂τ =(1/2)(∂x − i∂y),∂τ¯ =(1/2)(∂x + i∂y) we get~~

~~LZ φ = kφ, L X± φ = ck±2D±f where 2 D+ =2i(∂τ + k/(τ − τ¯)),D− =(i/2)(τ − τ¯) ∂τ¯. Hence in particular, we have found:~~

~~Remark 9.7: Let φ = φf be the lift of f. The condition ii) that f is holomorphic on H is equivalent to~~

~~L ii’) X− φ =0. 228 9. Epilogue: Outlook to Number Theory~~

~~The last condition iii) is easily seen to be equivalent to the boundedness condition~~

~~|f(τ)| >0.~~

~~With a bit more work, one translates the cuspidality of a modular form, i.e. f ∈ Sk(Γ), to the following~~

~~Lemma: f ∈ Ak(Γ) is in Sk(Γ) iﬀ for every y0 > 0 there is an M>0 such that~~

~~k/2 y |f(τ)| y0.~~

~~Hence one has for the lifted functions φ = φf the following property.~~

~~Remark 9.8: The condition iii) translates to~~

~~|φ(x, y, ϑ)y−k/2| >0 and the cusp condition c(0) = 0 to~~

~~|φ(g)|~~

~~The remarks 9.5 to 9.8 together give proof to a fundamental statement.~~

Theorem 9.3: Via the lifting ϕk, the space Ak(Γ) of modular forms on H is isomorphic to the space Ak of smooth functions φ on G with ia) φ(γg)=φ(g) for all γ ∈ Γ = SL(2, Z), ib) φ(gr(ϑ)) = φ(g)eikϑ for all r(ϑ) ∈ K = SO(2), L ii’) X− φ =0, iii’) |φ(g)y−k/2| >0.

~~Moreover, the space of cusp forms Sk(Γ) is isomorphic to~~

~~A0 { ∈A | | } k = φ k; φ~~

~~Remark 9.9: The cusp condition in Remark 9.8 and in the Theorem above is equivalent to ∈ (cusp) Γ∩N\N φ(n(x)g)dx = 0 for all g G.~~

~~This follows by identifying N = {n(x); x ∈ R} with R and computing 1 φ(n(x)g)dx = φ(n(x)g)dx Γ∩N\N 0 1 = f(τ + x) eikϑyk/2dx 0 = eikϑyk/2c(0).~~

~~Now we can reﬁne Theorem 9.3 above: 9.2 Modular Forms and SL(2,R) 229~~

A0 Theorem 9.4: The space of cusp forms k is characterized by the following conditions ia) φ(γg)=φ(g) for all γ ∈ Γ = SL(2, Z), ib) φ(gr(ϑ)) = φ(g)eikϑ for all r(ϑ) ∈ K = SO(2), ii) ∆φ = LΩφ = k((k/2) − 1)φ, iii) φ is bounded, ∈ iv) Γ∩N\N φ(n(x)g)dx = 0 for all g G.

Here L 2 2 2 − ∆= Ω =2y (∂x + ∂y ) 2y∂x∂ϑ is the Laplacian (7.17) from 7.2 and, by a small calculation, it is easy to see that the lift φf of a cusp form f ∈ Sk(Γ) satisﬁes the condition ii) and hence all the conditions from the theorem. The proof of the fact that the lifting map ϕk is a surjection onto the space ∈A0 given by the conditions ia) to iv) reduces to the veriﬁcation that for φ k the function f with

(9.5) f(τ):=φ(g)y−k/2e−ikϑ is well deﬁned for each g with g(i)=τ (which is quite obvious) and that f is a cusp form. This assertion is based on a considerably deeper theorem, the Theorem of the discrete spectrum, which we shall state below.

~~Decomposition of L2(Γ \ G)~~

For G = SL(2, R) ⊃ Γ = SL(2, Z), we put H = L2(Γ \ G) with respect to the biinvariant measure dg˙ on X =Γ\ G ﬁxed by 1 dxdy dg = dϑ. 2π y2 We want to discuss the decomposition of the representation ρ of G given by right translation on H and show how the cusp forms come in here. A ﬁrst indication is given by the

~~Remark 9.10: For f ∈ Sk(Γ), the lift φf = ϕk(f)isinH.~~

~~This follows from 2 2 k−2 |φf (˙g)| dg˙ = |f(τ)| y dxdy < ∞ X F by the boundedness condition iii’) for the cusp form f.~~

~~We need another tool from functional analysis (for background see for instance [La1] p.389ﬀ):~~

Remark 9.11: The minimal closed extension of the Laplacian ∆ is a selfadjoint operator ∆.ˆ As ∆ does by constuction, the extension ∆ˆ also commutes with the right translation | and hence, by Schur’s Lemma (“the unitary Schur” mentioned in 3.2) ∆ H0 is scalar for each G-invariant subspace H0 of H. From functional analysis we take over that ∆ˆ and ρ have a discrete spectrum only for the subspace of cuspidal functions H0 := {φ ∈ L2(Γ \ G); φ(n(x)g)dx = 0 for almost all g ∈ G}. Γ∩N\N 230 9. Epilogue: Outlook to Number Theory

~~The central fact is as follows.~~

~~Theorem 9.5 (Theorem of the discrete spectrum): The right regular representation ρ of SL(2, R)onH0 is completely reducible~~

~~0 H = ⊕j∈ZHj and each irreducible component has ﬁnite multiplicity.~~

As this leads too much into (most fascinating) functional analysis, we can not go into the proof of this theorem and refer to [La1] and [GGP] p.94ﬀ. A main point is a multiplicity- one statement going back to Godement, which we already mentioned (without proof) in 6.4. But we try to give an indication of the power of this theorem: For the proof of characterization of cusp forms by the Laplacian in the theorem above, we have to show that the function f from (9.5)

~~f(τ)=φ(g)y−k/2e−ikϑ is holomorphic. If one decomposes φ as in Theorem 9.5 φ = φj, j the components φj have to fulﬁll the same relation ib) as φ does. And one has~~

~~∆φj = λkφj,λk = k((k/2) − 1), because by the last remark above, ∆ is scalar on Hj, i.e. one has a λj with~~

~~∆φj = λjφj.~~

~~And then using the inner product in H one has~~

~~<φ,∆φ>= λj <φ,φj > and, as ∆ˆ is self adjoint,~~

<φ,∆φj > = < ∆φ, φj > = λk <φ,φj > and therefore λj = λk. Hence, for the restriction of ρ to its irreducible subrepresentations the components φj are vectors of lowest weight k for the discrete series representation + L πk . Such vectors are annihilated by X− and, hence, f is holomorphic.

~~As another consequence we come to the statement parallel to the one we proved in 9.1 for theta functions.~~

Theorem 9.6 (Duality Theorem for the discrete series of SL(2, R)): The multiplicity − H0 of πk in the right regular representation ρ on the space of cuspidal functions is equal to the number of linear independant cusp forms of weight k

~~− mult(πk ,ρ) = dim Sk(Γ). 9.2 Modular Forms and SL(2,R) 231~~

Proof: There is a nice and comprehensive classic proof in [GGP] p.53-57. As we already have at hand the necessary tools, we reproduce its highlights changing only the parametrization of G = SL(2, R) used by these authors to the one used here all the time.

~~− a)At ﬁrst, we show that mk := mult(πk ,ρ) is equal to the numberm ˜ k of linear independent functions Ψ on X =Γ\ G in H0 with~~

−ikϕ (9.6) ∆Ψ = λkΨ,λk := k((k/2) − 1),ρ(r(ϕ))Ψ = e Ψ for all ϕ ∈ R. H H0 − ai) For each irreducible subspace j of , which is a model for πk , there is - up to a nonzero factor - exactly one highest weight vector where ρ restricted to Hj has highest weight −k and, hence ∆ has eigenvalue λk. aii) Every Ψ ∈H0 satisfying (9.6) is a linear combination of vectors of highest weight H0 − from irreducible subspaces of equivalent to a representation space of πk . b) In the second step, we show thatm ˜ k is equal to the number of cusp forms: The space of functions Ψ as above is in bijection to the space of functions Φ on G with

~~−ikϕ −1 (9.7) ∆Φ = λkΦ, Φ(gr(ϕ)) = e Φ(g), Φ(γ g)=Φ(g) for all ϕ ∈ R,γ∈ Γ. We describe G by our parameters τ,ϑ with τ = g(i),e−iϑ =(ci + d)/ | ci + d | .~~

−1 −1 Hence, for g0 with components a0,b0,c0,d0, the matrix g0 g has parameters τ1,ϑ1 with −1 − τ1 = g0 (τ),ϑ1 = ϑ arg(c0τ + d0). And gr(ϕ) has the parameters τ,ϑ + ϕ. Thus the −ikϑ second equation in (9.7) shows that Φ(g) is a function of type f1(τ)e . Guided by the proof of our Remark 9.7, in order to also fulﬁll the other conditions in (9.7) we try the ansatz Φ(g)=e−ikϑyk/2f(τ). −k Using our previous notation, this can also be written as Φ(g)=j(g, i) f(τ). We see that using this ansatz, the third condition in (9.7) is equivalent to the functional equation for modular forms

~~f(γ−1(τ))j(γ−1,τ)−k = f(τ) for all γ ∈ Γ.~~

~~For the real coordinates x, y of τ the ﬁrst equation in (9.7) comes down to~~

~~2+(k/2) −ikϑ ¯ ¯ 1+(k/2) −ikϑ ¯ ¯ y e (fxx + fyy)+iky e (fx− ify)=0 resp. y(fxx + fyy) − ik(fx + ify)=0~~

If f is a modular form, ergo holomorphic, one has fτ¯ = 0 and this equation is fulﬁlled. On the other hand, if Φ fulﬁlls (9.7), by the Decomposition Theorem, it is a linear combination of highest weight vectors with higest weight −k, which are annihilated by L 2iϑ − − X+ = e (2y(∂x i∂y) ∂ϑ). And this again leads just to the condition fτ¯ = 0 showing that Φ corresponds to a linear combination of cusp forms.

The reader again is invited to watch very closely whether all signs are correct. It has to be emphasized that the notions highest and lowest weight depend on our parametrization of K = SO(2) and interchange if one uses r(−ϑ) in place of our r(ϑ) (as Gelbart does in [Ge]). [GGP] wisely use the term vector of dominant weight. 232 9. Epilogue: Outlook to Number Theory

Remark 9.12: It is a natural question to ask under which conditions the lift φ = φf of a cusp form f is a cyclic vector, i.e. generates an irreducible representation. There is a nice answer to this: if f is an eigenfunction to the Hecke operators T (p). We will introduce these operators later in the context of associating L−functions to modular forms and here only mention that the space of cusp forms has a basis consisting of such eigenfunctions.

The big diﬀerence between this duality statement and the one for the Heisenberg group in 9.1 is that in this case we have no way to an explicit construction of the modular forms starting from their property of being a dominant weight vector. Apparently, this reﬂects the fact that the structure of the group at hand is considerably more complicated than that of the Heisenberg group. Some progress here should give a deep insight into the modular group.

The characterization theorem is the starting point for several generalizations leading to the notion of automorphic forms. It can be extended to other groups, even adelic ones as to be found for instance in [Ge]. There it is also pointed out that the right habitat for the modern theory of modular forms is not SL(2) but GL(2). Here we can not be too ambitious and we stay with G = SL(2, R), K = SO(2) and Γ = SL(2, Z) and only treat the following generalization.

Definition 9.2: The smooth function φ on G is called an automorphic form for Γ iff φ satisfies the following conditions ia) φ(γg)=φ(g) for all γ ∈ Γ, ib) φ is right K−finite, i.e. dim φ(gr(ϑ)); r(ϑ) ∈ K < ∞, ii) φ is an eigenfunction of ∆, iii) φ fulfills the growth condition that there are C, M with |φ(g)|≤CyM for y −→ ∞ .

~~φ is called cusp form if moreover one has (cusp) Γ∩N\N φ(n(x)g)dx =0.~~

− Besides the cusp forms realizing highest weight vectors of πk we just treated, as their + counterparts there come in forms realizing the lowest weight vectors of πk , corresponding to antiholomorphic modular forms on H and the famous Maass wave forms: these are bounded right K-invariant eigenfunctions of the Laplacian ∆φ = λsφ with 2 λs = −(1 − s )/2,s∈ iR. In analogy to the duality theorem stated above, one has here the fact that the dimension of the space Ws(Γ) of these wave forms is equal to the multiplicity of the principal series representation πis in ρ (for this and in general more precise information see [GGP] in particular p.50). In [Ge] p.29 we find the statement: one knows that Ws(Γ) is non-trivial for infinitely many values of s, but almost nothing is known concerning the specific values of s for which Ws(Γ) is non-trivial. Still now, Maass wave forms are an interesting subject. For information on more recent research we recommend the paper by Lewis and Zagier [LZ].

We ﬁnish this section by some remarks concerning the continuous spectrum of the Lapla- cian, resp. the orthocomplement Hc of the subspace Hd of H = L2(Γ \ G) consisting of 9.2 Modular Forms and SL(2,R) 233 the cuspidal and the constant functions. One has rather good control about this space as it can be spanned by the so called non-holomorphic Eisenstein series. As sources for this we go back to [Ku] and [La1] p.239ﬀ.

~~The classical object is ﬁxed as follows. One takes~~

~~µ fµ(τ):=y ,µ∈ C and for k ∈ N0 Ek(τ,µ):= (f(µ−k)/2|k[γ])(τ),~~

γ∈Γ∞\Γ y(µ−k)/2 1 = |cτ + d|µ−k (cτ + d)k c,d∈Z, (c,d)=1 1 b with Γ∞ = {γ = ; b ∈ Z}. 1 This prescription formally has the transformation property of a modular form of weight k and it can be veriﬁed that for µ with Re µ>2 it deﬁnes a (real analytic) function on H, which is non-cuspidal. Using the standard lift ϕk, this function can be lifted to a function on G

~~Ek(g, µ):=ϕkEk(τ,µ), cτ¯ + d 1 = yµ/2eikϑ ( )k/2 , cτ + d |cτ + d|µ ∈Z c,d , (c,d)=1 = φs,k(γg),~~

~~γ∈Γ∞\Γ where for µ = is + 1 the functions (8.11) from 8.3 φs,k,k ∈ Z, with~~

~~(is+1)/2 ikϑ φs,k(g)=y e come in, which span the space of the principal series representations. We have~~

~~2 ∆Ek(., µ)=−((s +1)/2) Ek(., µ) and these functions are further examples for the deﬁnition of automorphic forms above.~~

Already in our (brief) chapter on abelian groups, we met with a continuous decomposition of the regular representation for G = R by throwing in elements of Fourier analysis without really explaining the background from functional analysis. Now, this gets worse here: There is the central statement ([Ge] p.33):

~~Theorem 9.7: The restriction ρc of ρ to Hc is the continuous sum of the principal series representations πis,+, i.e. ∞ c ρ (g)= πis,+(g)ds. 0 234 9. Epilogue: Outlook to Number Theory~~

Using the inner product for functions f,f˜ on H given by (f,f˜):= f(τ)f˜(τ)y−2dxdy F one can say that any slowly increasing function f on H, which is orthogonal to all cusp forms, may be expressed by a generalized Fourier integral ∞ y(is)/2 f(τ)= fˆ(s)E(τ,s)ds, fˆ(s):=(f,E(., s)),E(τ,s):= . |cτ + d|2s 0 c,d∈Z;(c,d)=1

~~Though we will not try to get to the bottom of the proofs, we indicate the following also otherwise useful tools:~~

~~The Theta Transform θ~~

We already used the averaging procedure while introducing Eisenstein series: If ϕ is a function on N \ G, one gets, at least formally, a function θϕ on Γ \ G by the prescription θϕ(g):= ϕ(γg).

~~γ∈Γ∞\Γ~~

~~It is part of nice analysis to discuss under which conditions this leads to convergent objects and it is no wonder that it works at least if ϕ is a Schwartz function.~~

~~The Constant Term θ∗~~

~~Let φ be an element of H := L2(Γ \ G). To this φ we associate θ∗φ (again at least formally) by θ∗φ(g):= φ(ng)dn. Γ∞\N~~

~~As φ can be assigned a periodic function, which one denotes again by φ, θ∗φ may be understood as the constant term in a Fourier expansion φ(n(x)g)= φj(g)e(jx). j~~

~~θ and θ∗ are adjoint operators if we introduce an inner product~~

~~<ϕ,ϕ>˜ N\G := ϕ(˙g)˜ϕ(˙g)dg,˙ N\G i.e. we have for functions assuring the convergence of our prescriptions the relation~~

~~∗ <θϕ,φ>Γ\G = <ϕ,θ φ>N\G .~~

~~Exercise 9.5: Verify this. 9.2 Modular Forms and SL(2,R) 235~~

~~The Zeta Transform Z~~

This is a prescription to produce elements in the representation space His,± of a principal series representation πis,± :Forµ ∈ C and ϕ a function on N \ G, we put (again at least formally) ∞ dy Z(ϕ, g, µ):= ϕ(t(y)g)y−µ . 0 y For µ =(is +1)/2, we get

~~∞ −µ dy (is+1)/2 Z(ϕ, n(x0)t(y0)g, µ)= ϕ(t(y0y)g)y = y0 Z(ϕ, g, µ), 0 y i.e. an element fulﬁlling the equation (7.11) in 7.2 characterizing the elements of His,+.~~

~~The following statements are easy to prove.~~

~~Remark 9.13: For ϕ in the Schwartz space S(N \G) and Re µ>0, the integral deﬁning Z(ϕ, g, µ) converges absolutely and Z(ϕ, g, µ) is an entire function in µ if moreover ϕ has ﬁnite support.~~

~~With some more work composing the Zeta and the Theta transform, one obtains a Γ−invariant function:~~

~~Remark 9.14: For ϕ ∈S(N \ G) and Re µ>1, the Eisenstein series E(ϕ, g, µ):=θZ(ϕ, g, µ)= Z(ϕ, γg, µ)~~

~~γ∈Γ∞\Γ is absolutely convergent.~~

~~By a small computation, we get the relation between this Eisenstein series and the one we deﬁned above.~~

~~Remark 9.15: For~~

~~− −π/y ikϑ ϕ(g):=ϕ,k(g)=y e e ,∈ N0,k ∈ Z, one has −(+µ) E(ϕ, g, µ)=π Γ( + µ)Ek(g, µ).~~

2 The functions ϕ,k,∈ N0,k ∈ Z are a Hilbert basis of the space L (N \ G) with respect −2 to the measure dν = y dydϑ. For every ﬁxed and µ = is +1, the images of these ϕ,k by the Zeta transform are a basis for the representation spaces His,± if one takes all even resp. odd k ∈ Z.

There is still much to be said here. We shall come back to the Zeta transform in the context of a discussion of ζ− and L−functions. But before doing this, we shall ﬁnish our outlook to automorphic forms by returning to theta functions. 236 9. Epilogue: Outlook to Number Theory

~~9.3 Theta Functions and the Jacobi Group~~

In 9.1 we introduced theta functions and studied their behaviour as functions of the complex variable z. But as we already indicated and partially explored, there also is an important dependance on the modular variable τ ∈ H. This leads to the ultimate interpretation of (lifts of) theta functions as automorphic forms for the Jacobi group GJ , which we introduced in 8.5. We deﬁned

~~q := e(τ)=e2πiτ ,:= e(z)=e2πiz. and the classic Jacobi theta function ϑ (9.1) given by~~

2 2 ϑ(z,τ):= eπi(n τ+2nz) = qn /2n, n∈Z n∈Z the theta function with characteristics m =(m,m) ∈ R2 2 θm(τ,z):= e((1/2)(n + m ) τ +(n + m )(z + m )), n∈Z and its variant (9.2) (following the notation of [EZ] p.58) with 2m ∈ N,µ=0, 1,...,2m − 1 r2/(4m) r θm,µ(τ,z):= q r∈Z,r≡µ mod 2m = em((n + µ/(2m))2τ +2(n + µ/(2m))z). n∈Z

~~One has ϑ(z,τ)=θ1/2,0(τ,z). We already explored the quasiperiodicity property concerning the complex variable z~~

~~2 ϑ(z + rτ + s, τ)=e−πi(r τ+2rz)ϑ(z,τ) and (in Example 9.3 in 9.2) the modular property of the value ϑ(τ):=θ(0,τ) for z =0 (also called by the German word Thetanullwert)~~

~~ϑ(τ +2)=ϑ(τ),ϑ(−1/τ)=(τ/i)1/2ϑ(τ).~~

~~To be consistent with [EZ] and the usual Jacobi Theory, we have to change (from Jacobi’s and Mumford’s) notation and put~~

~~θ(τ,z):=ϑ(z,τ).~~

~~Then one has the fundamental modular transformation relation.~~

~~Theorem 9.8: For a, b, c, d ∈ Z with ad − cd = 1 and ab, cd ∈ 2Z, there is an eighth root of unity ζ such that one has~~

~~aτ + b z (9.8) θ( , )=(cτ + d)1/2 ζe((1/2)cz2/(cτ + d))θ(τ,z). cτ + d cτ + d The unit root ζ can be determined as follows: 9.3 Theta Functions and the Jacobi Group 237~~

We may assume c>0orc = 0 and d>0 (otherwise one would multiply the matrix ab by −1) and hence have (cτ + d)1/2 in the ﬁrst quadrant, i.e. with real- and cd imaginary part > 0. a We take the Jacobi symbol ( b ), i.e. the multiplicative extension of the Legendre symbol, which for a, p ∈ Z,pprime is given by

~~a | ( p )=0 if p a, =1 p a and there is an x ∈ Z with x2 ≡ a mod p, = −1 p a and there is no x ∈ Z with x2 ≡ a mod p.~~

~~Then one has (d−1)/2 c ζ = i ( |d| ) for c even and d odd, −πic/4 d = e ( c ) for c odd and d even.~~

~~The proof of this theorem is a very nice piece of analysis but too lengthy for this text. So we refer to the books of Eichler [Ei], in particular p.59-62, Lion-Vergne [LV] p.145ﬀ, or Mumford [Mu] I.~~

~~Another most remarkable fact is the following characterization of θ, which comes about as an exercise in complex function theory.~~

Theorem 9.9: Let f : H × C −→ C be a holomorphic function satisfying a) f(τ,z +1)=f(τ,z), b) f(τ,z + τ)=f(τ,z)e(−(1/2)τ − z), c) f(τ +1,z+1/2) = f(τ,z), d) f(−1/τ, z/τ)=f(τ,z)(−iτ)1/2e((1/2)z2/τ) for all (z,τ) ∈ C × H, e) limImτ −→ ∞ f(τ,z)=1. Then one has f = θ.

By a closer look one sees that the quasi-periodicity and the modular transformation property of θ can be combined to a transformation property under an action of the Jacobi group GJ , which is a semi-direct product of SL(2, R) and Heis(R) and in 8.5 was realized as a subgroup of Sp(2, R) given by four-by-four matrices. As in 8.5, we use two notations ab ab GJ g = (λ, µ, κ)=(p, q, r) . cd cd By a short calculation one can verify that for (τ,z) ∈ H × C

aτ + b z + λτ + µ (τ,z) −→ g(τ,z):=( , ) cτ + d cτ + d deﬁnes an action of GJ on H × C. The stabilizing group of (i, 0) is KJ := SO(2) × C(R), C(R) the center of the Heisenberg group. One has

~~GJ /KJ H × C,g(i, 0) = (τ = x + iy, z = pτ + q) where g =(p, q, r)n(x)t(y)r(ϑ). 238 9. Epilogue: Outlook to Number Theory~~

~~And for k, m ∈ Q one has (see the ﬁrst pages of [EZ]) an automorphic factor~~

~~c(z + λτ + µ)2 (9.9) j (g;(τ,z)) := (cτ + d)−kem(− + λ2τ +2λz + λµ + κ). k,m cτ + d~~

~~This leads to an action of GJ on functions f living on H × C via~~

~~f −→ f|k,m[g] with f|k,m[g](τ,z):=f(g(τ,z))jk,m(g;(τ,z)).~~

Similar to modular forms and theta functions as automorphic forms for SL(2, R) resp. Heis(R), for the Jacobi group GJ , its discrete subgroup ΓJ := SL(2, Z) Z3, and k, m ∈ N0 one deﬁnes (as in [EZ]):

Deﬁnition 9.3: A function f : H × C −→ C is called a Jacobi form of weight k and index m if one has J –i)f|k,m[γ](τ,z)=f(τ,z) for all γ ∈ Γ , – ii) f is holomorphic, – iii) for Im τ>>0,f has a Fourier development f(τ,z)= c(n, r)qnεr,q= e(τ),ε= e(z),c(n, r) ∈ C. n,r∈Z, 4mn−r2≥0 f is called a cusp form if it satisﬁes moreover – iii’) c(n, r) = 0 unless 4mn > r2.

cusp The vector spaces of all such functions f are denoted by Jk,m resp. Jk,m . They are ﬁnite dimensional by Theorem 1.1 of [EZ]. As usual, one lifts a function f on H × C to J a function φ = φf on G by

~~m ikϑ k/2 φf (g):=f(g(i, 0))jk,m(g;(i, 0)) = f(x, y, p, q)e (κ+pz)e y ,g=(p, q, κ)n(x)t(y)r(ϑ).~~

~~Similarly to the analysis in the former cases, one can characterize the images of Jacobi forms.~~

∞ J Proposition 9.2: Jk,m is isomorphic to the space Ak,m of complex functions C (G ) with –i)φ(γg)=φ(g) for all γ ∈ ΓJ , – ii) φ(gr(ϑ)(0, 0,κ)) = φ(g)em(κ)eikϑ, L L – iii) X− φ = Y− φ =0, −k/2 – iv) φ(g)y is bounded in domains of type y>y0.

~~cusp A0 A Jk,m is isomorphic to the subspace k,m of k,m with – iv’) φ(g) is bounded.~~

~~For a proof of this and the following statement see [BeB] or [BeS] p. 76ﬀ. 9.4 Hecke’s Theory of L−Functions Associated to Modular Forms 239~~

∈ cusp Theorem 9.10 (Duality Theorem for the Jacobi Group): For m, k N, dim Jk,m is equal to the multiplicity of the appropriate discrete series representation of GJ in the 0 H0 right regular representation on the space m of cuspidal functions of type m, which consists of the elements φ ∈ L2(ΓJ \ GJ ) satisfying the functional equation

~~φ(g(0, 0,κ)) = e(mκ)φ(g), for all κ ∈ R and fulﬁlling the cusp condition J φ(gλ/(2m)g)dn = 0 for almost all g ∈ G and λ =0,...,2m − 1 N J ∩ΓJ \N J~~

~~J where N := {(0,q,0)n(x); q, x ∈ R} and gλ := (λ, 0, 0).~~

~~Further information concerning the continuous part of the decomposition of L2(ΓJ \ GJ ) is collected in [BeS].~~

At a closer look, one can see that the classic ϑ has the transformation property of a Jacobi form of weight and index 1/2 for a certain subgroup of ΓJ . As already the inﬁnitesi- mal considerations indicated, one has to multiply with modular forms of half-integral weight to get Jacobi forms in the sense of the deﬁnition above. This is very nicely exploited in [EZ]. The appearance of the half-integral weights can be explained by the fact that, somewhat similar to the relation of the representation theory of SO(3) and SU(2) resp. SO(3, 1) and SL(2, C), here one has coming in a double cover of SL(2, R) called the metaplectic group Mp(2, R). But this group is no more a linear group and hence, unfortunately, outside the scope of this book.

9.4 Hecke’s Theory of L−Functions of Modular Forms ∞ n It is a classical topic of analysis to study the relation of a power series f := n=1 c(n)q to a Dirichlet series ∞ c(n) L (s):= . f ns n=1

In our context, we take a cusp form f ∈ Sk(Γ) and get an associated Dirichlet series convergent for s with Re s>kas the coefficients c(n) of the Fourier expansion of f fulfill an appropriate growth condition. The most important Dirichlet series have an Euler product similar to the prototype of a Dirichlet series, namely the Riemann Zeta series ∞ ζ(s):= n−s, n=1 which is convergent for s with Re s>1 and has the Euler product (taken over all prime numbers p) 1 ζ(s)=Π . 1 − p−s Moreover, it is known that ζ fulfills a functional equation under s −→ 1 − s and can be continued to a meromorphic function on the whole plane C with a simple pole at s =1. 240 9. Epilogue: Outlook to Number Theory

~~Roughly said, Hecke’s theory analyses how these and more general properties are related to properties of modular forms. The main tools are the following operators.~~

~~Hecke Operators~~

Deﬁnition 9.4: For a natural number n one introduces the Hecke operator T (n)asan operator acting on functions f on the upper half plane H by aτ + b (9.10) T (n)f(τ):=nk−1 d−kf( ). d a≥1,ad=n,0≤b

The origin of this prescription can be understood in several ways. There is the more general background asking for the decomposition of double cosets into cosets for Γ: For instance, for a prime p one has 1 1 b Γ Γ= Γ p p 0≤b

(as exploited in [Sh] p.51ﬀ). But we follow the presentation in [Se] p.98 based on the notion of correspondences: Let E be a set and let GE be the free abelian group generated by E.Acorrespondence on E with integer coeﬃcients is a homomorphism T of GE to itself. We can describe T by its values on the elements x of E: T (x)= ny(x)y, ny(x) ∈ Z,ny(x) = 0 for almost all y. y∈E

A C-valued function F on E can be extended linearly to a function on GE, which is again denoted by F . The transform of F by T , denoted TF, is the restriction to E of the function F ◦ T . With the notation introduced above, it is given by TF(x)=F (T (x)) = ny(x)F (y). y∈E

Let R be the set of lattices L in C and n ∈ N. We denote by T (n) the correspondence on R which transforms a lattice to the sum (in GR) of its sublattices Ln of index n. Thus we have for L ∈R T (n)L = Ln.

~~[L:Ln] The sum on the right side is ﬁnite. If n is prime, one sees that one has n + 1 such sublattices. For the general case one has~~

~~Lemma: There is a bijection between the set of matrices ab M = ; ad = n, a ≥ 1, 0 ≤ b~~

~~Exercise 9.6: Prove this. 9.4 Hecke’s Theory of L−Functions Associated to Modular Forms 241~~

~~∗ One also uses homothety operators Rλ,λ∈ C deﬁned by~~

~~RλL = λL.~~

~~It makes sense to compose the correspondences T (n) and Rλ, since they are endomor- phisms of the abelian group GR.~~

~~Proposition 9.3: The correspondences T (n) and Rλ verify the identities~~

~~∗ RλRµ = Rλµ,λ,µ∈ C , ∗ RλT (n)=T (n)Rλ,n∈ N,λ∈ C , T (m)T (n)=T (mn), if (m, n)=1, n n+1 n−1 T (p )T (p)=T (p )+pT (p )Rp,pprime,n∈ N.~~

~~Proof: We refer to [Se] p.98.~~

~~n Corollary 1: The T (p ) are polynomials in T (p) and Rp. Corollary 2: The algebra generated by the Rλ and the T (p),pprime, is commutative; it contains all the T (n).~~

~~For the description of the action of these operators on modular forms of weight k,we associate to f a corresponding function F deﬁned on the set R of lattices by~~

−k F (L):=(τ1/τ2) f(τ1/τ2) if (τ1,τ2) is a basis of L. We deﬁne T (n)f as the function on H associated to the function nk−1T (n)F on R, i.e. T (n)f(τ):=nk−1T (n)F (L(τ,1)) where L(τ,1) denotes the lattice with basis (τ := τ1/τ2, 1) (it can be arranged and is silently understood that the description of the lattices is chosen such that this τ is in H). Using the Lemma above, we get the formula (9.10) in the deﬁnition at the beginning. Certainly, one has to and can verify that this T (n) transforms the spaces of modular forms of weight k into itself (and similarly for cusp forms). We leave this to the reader and only note the relations, which are immediate consequences of Proposition 9.3

~~(9.11) T (m)T (n)f = T (mn)f, if (m, n)=1, (9.12) T (pn)T (p)f = T (pn+1)f + pk−1T (pn−1)f, p prime n ∈ N.~~

~~The central point of all this is the action of the Hecke operators expressed by the coeﬃ- cients of the q−development f(τ)= c(n)qn:~~

~~Proposition 9.4 We have T (n)f(τ)= γ(m)qm m∈Z with (9.13) γ(m)= ak−1c(nm/a2). a∈N,a|(n,m) 242 9. Epilogue: Outlook to Number Theory~~

Using the relations obtained earlier, the proof is straightforward. Again we refer to [Se] p.101. In particular, we come to k γ(0) = σk−1(n)c(0),σk(n):= d , d∈N,d|n γ(1) = c(n), γ(m)=c(pm)ifn = p prime and m ≡ 0modp, = c(pm)+pk−1c(m/p) m ≡ 0modp.

~~Since the Hecke operators commute, one can hope for simultaneous diagonalizability.~~

~~Hecke Eigenforms~~

~~Let f be a Hecke eigenform of weight k, i.e. a modular form of weight k, which is not identically zero and an eigenfunction for all T (n),n∈ N,~~

~~T (n)f(τ)=λ(n)f(τ).~~

~~Examples are the Eisenstein series Gk and the discriminant ∆. One has~~

~~12 T (n)Gk = σk−1(n)Gk,T(n)∆ = (2π) τ(n)∆, for all n ∈ N. The veriﬁcation is non-trivial but standard ([Se] p.104).~~

~~Our central statement is as follows:~~

Theorem 9.11: The coeﬃcient c(1) of a Hecke eigenform f is non-zero. If f is normalized by c(1) = 1, one has λ(n)=c(n) for all n ∈ N. Proof: The formula (9.13) in Proposition 9.4 above shows that the coeﬃcient of q in T (n)f is c(n). On the other hand, as f is a T (n)−eigenfunction, it is also λ(n)c(1). Thus we have c(n)=λ(n)c(1). If c(1) were zero, all the c(n),n > 0 would be zero, and f would be a constant, which is absurd.

~~Applying the ﬁrst statement of the Theorem to the diﬀerence of two normalized Hecke eigenforms, we get~~

~~Corollary 1: Two modular forms of weight k, k > 0, which are both normalized Hecke eigenforms with the same eigenvalues, coincide.~~

~~By the second statement of the Theorem the relations (9.11) for the Hecke operators translate immediately to relations for the Fourier coeﬃcients. We get~~

~~Corollary 2: For the Fourier coeﬃcients of a normalized Hecke eigenform f of weight k one has the relations~~

~~(9.14) c(m)c(n)=c(mn), if (m, n)=1, (9.15) c(pn)c(p)=c(pn+1)+pk−1c(pn−1), if p is prime and n ∈ N. 9.4 Hecke’s Theory of L−Functions Associated to Modular Forms 243~~

n Now, these relations for the Fourier coeﬃcients of a Hecke eigenform f = c(n)q trans- −s late to the statement that the associated Dirichlet series Lf (s)= c(n)n (convergent, as already mentioned, for Re s>k) has an Euler product.

~~Corollary 3: The Dirichlet series Lf of a Hecke eigenform f of weight k has the Euler product~~

~~1 (9.16) L (s)=Π ∈P . f p 1 − c(p)p−s + pk−1−2s~~

Proof: From Corollary 2 we know that n −→ c(n) is a multiplicative function, so with t := p−s we can write ∞ n n Lf (s)=Πp∈P( c(p )t ). n=0 Hence it is suﬃcient to show that we have ∞ n n k−1 2 −1 c(p )t = Lf,p(t) for Lf,p(t):=(1− c(p)t + p t ) . n=0

~~We compute ∞ ψ(t):=( c(pn)tn)(1 − c(p)t + pk−1t2)=: d(n)tn n=0 and as coeﬃcient of t get d(1) = c(p) − c(p) = 0 and, for n ≥ 1, as coeﬃcient of tn+1~~

~~d(n +1)=c(pn+1) − c(p)c(pn)+pk−1c(pn−1)=0 in consequence of the second relation in Corollary 2. I.e., we have ψ(t)=c(1) = 1.~~

~~Remark 9.16: The statement that Lf has an Euler product as in Corollary 3 is equivalent to the fact that the Fourier coeﬃcients of f fulﬁll the relations in Corollary 2.~~

Remark 9.17: Hecke proved that Lf can be analytically continued to a meromorphic function on the whole complex plane, resp. to a holomorphic function if f is a cusp form, and that the completed L-function

~~∗ −s Lf (s):=(2π) Γ(s)Lf (s) satisﬁes the functional equation~~

~~∗ − k/2 ∗ − (9.17) Lf (s)=( 1) Lf (k s).~~

The proof is based on the classical Mellin transform: Let f be a normalized Hecke cusp form of weight k, i.e. we have ∞ f(τ)= c(n)qn with f(−1/τ)=τ kf(τ). n=1 244 9. Epilogue: Outlook to Number Theory

Using the Γ−function ∞ Γ(s):= e−tts−1dt 0 we determine the Mellin transform of f, namely ∞ ∞ dy f(iy)ys−1dy = ( c(n)e(niy)) ys 0 0 y ∞ = c(n) e−2πnyys−1dy 0 = c(n)(2πn)−sΓ(s)

~~−s ∗ =(2π) Γ(s)Lf (s)=Lf (s).~~

In the integral on the left hand side, we put u =1/y and use the transformation formula for f to get ∞ 0 du f(iy)ys−1dy = − f(i/u)u−s u 0 ∞ ∞ du = f(iu)ikuk−s 0 u k ∗ − = i Lf (k s).

Remark 9.18: The whole theory has been extended to other groups than Γ = SL(2, Z), in particular to Γ0(N) (see for instance [Ge]). There is also a converse to the statement in Remark 9.17: Every Dirichlet series L, which fulfills a functional equation of the type just treated and has certain regularity and growth properties, comes from a modular form of f of weight k. This was first remarked by Hecke and then refined considerably by Weil in [We1] and is now cited under the heading of the famous Converse Theorem.

~~Remark 9.19: The Hecke operators are hermitian with respect to the Petersson scalar product for forms f1,f2 of weight k given by k−2 := f1(τ)f2(τ)y dxdy, F as in Remark 9.2, F i.e. one has~~

~~= . Arithmetic Modular Forms~~

Though it does not directly belong to our topic, we can not resist the temptation to complete this section by indicating a concept of arithmetically distinguished modular forms, which has been exploited by Shimura and many others (see [Sh]): We look at modular forms (for the modular group Γ, but this is open to wide generalizations) with integral Fourier coeﬃcients n Ak(Γ, Z):={f = c(n)q ∈ Ak(Γ); c(n) ∈ Z}. 9.4 Hecke’s Theory of L−Functions Associated to Modular Forms 245

~~One can show that one can ﬁnd a Z-basis of Ak(Γ, Z), which is also a C-basis of Ak(Γ). Prototypes of these forms are, up to factors, forms we already met with, namely~~

~~∞ F (τ):=q (1 − qn)24, n=1 ∞ n E4(τ) := 1 + 240 σ3(n)q , n=1 ∞ n E6(τ):=1− 504 σ5(n)q . n=1~~

~~For k ∈ 4N Ak(Γ, Z) has as Z-basis (see [Se1] p.105)~~

~~α β ∈ E4 F ,α,β N0 with α +3β = k/4, and for k ∈ 2(2N +1)~~

~~α β ∈ − E6E4 F ,α,β N0 with α +3β =((k/2) 3)/2.~~

Our analysis of the action of the Hecke operators above shows that Ak(Γ, Z) is stable under all T (n). Hence the coeﬃcients of the characteristic polynomial of T (n) acting on Ak(Γ, Z) are integers and one can deduce that the eigenvalues are algebraic integers. There are explicit formulas for the traces of the T (n), the Eichler-Selberg trace formulas, starting by [Ei1] and [Sel].

~~Summary~~

In this section we looked at L-functions L(s)=Lf (s) coming from modular cusp forms f. Via a functional equation these functions have analytic continuation to the whole complex plane. This functional equation is a consequence of the covariant transformation property of the modular form. The modular form is related to an (inﬁnite-dimensional) automorphic representation of SL(2, R) (or GL(2, R)). We call such L-functions automorphic L-functions. If f is a Hecke eigenform, Lf has an Euler product with factors of degree 2, i.e. with

~~(1 − c(p)p−s + pk−1−2s)−1 =(1− c(p)t + pk−1t2)−1,t:= p−s.~~

There is a general procedure to associate such functions to automorphic representations of other groups leading to Euler products with other degrees. In particular, one has a functional equation and factors of degree 1 in the theory of Hecke’s L-series with grössencharacters (from the German word “Grössencharakter”). This is fairly easy to be understood ([Ge] p.99f) as a theory of automorphic L-functions for the group GL(1) if one uses the more functional analytic idelic approach initiated by Tate’s thesis (reproduced in [CF] or described in [La2] and, provided with the necessary analytic background, in [RV]). Hecke’s classic approach is carefully displayed in [Ne] p.515ff. Here we shall indicate the definition of this function later when we have more number theory at hand. For an introduction to more general automorphic L-functions we refer to [Ge] and [GeS]. 246 9. Epilogue: Outlook to Number Theory

~~9.5 Elements of Algebraic Number Theory and Hecke L-Functions~~

~~Any book on number theory is good as a background to the following survey, in particular we recommend the books by Lang [La2] and by Neukirch [Ne] (now also in English).~~

~~Number Fields and their Galois Groups~~

~~Definition 9.5: An algebraic number field is a finite extension K of the field Q of rational numbers, i.e. a field containing Q, which is a finite-dimensional vector space over Q.~~

Such a field is obtained by adjoining to Q roots of polynomials with coefficients in Q. For example, the field Q(i):={a + bi; a, b ∈ Q} is obtained by adjoining the roots of the polynomial x2 + 1, denoted as usual by i and −i. We obtain a field, which has dimension 2 as a vector space over Q, the field of Gauss numbers. This is the most elementary√ example of an imaginary quadratic field, which has the general form K = Q( −d),d∈ N,dsquarefree.

Similarly, for N ∈ N, adjoining to Q a primitive N−th root of unity ζN we obtain the N-th cyclotomic ﬁeld QN := Q(ζN ). Its dimension as a vector space over Q is ϕ(N), the Euler function of N, i.e.

~~ϕ(N):=#{m ∈ N;1≤ m~~

~~We can embed Q(ζN )intoC in such a way that ζN −→ e(1/N ). But this is not the only possible embedding of QN into C, since we could also send ζN −→ e(m/N) where (m, N)=1.~~

Suppose now that K is an algebraic number field and F a finite extension of K, i.e. another field containing K, which has a finite dimension as a vector space over K. This dimension is called the degree of F over K and denoted by degK F, in particular deg F for K = Q.

Definition 9.6: The group of all field automorphisms σ of F, preserving the field structure and such that σ(x)=x for all x ∈ K, is called the Galois group of F over K and denoted by Gal(F/K), in particular Gal(F) for K = Q.

Example 9.5: The Galois group Gal(QN ) is naturally identiﬁed with the group (Z/N Z)× := {n¯ := n + NZ ∈ Z/N Z;(n, N)=1} with respect to multiplication. The elementn ¯ ∈ (Z/N Z)× gives rise to the automor- n m mn phism of QN sending ζN to ζN , and hence ζN to ζN for all m.IfM divides N, then QM is contained in QN , and the corresponding homomorphism of the Galois groups Gal(QN ) −→ Gal(QM ) coincides under the above identiﬁcation with the natural surjective homomorphism

~~(Z/N Z)× −→ (Z/M Z)×;¯n −→ n mod M.~~

~~A central notion from number theory is the notion of a Galois extension: 9.5 Elements of Algebraic Number Theory and Hecke L-Functions 247~~

~~Definition 9.7: An algebraic field extension F/K is called a Galois extension iff the order of the Galois group G = Gal(F/K) is equal to the relative degree degK F.~~

It is a main part of the theory to give different versions of this definition. Then, Galois theory consists in establishing for a Galois extension a (contravariant) bijection between subgroups of the Galois group and subfields of F containing K.

Next, one has class field theory, which tries to collect information about a Galois field extension F/K, in particular the structure of the Galois group, only from the structure of the basefield K. We shall need some of this and have to recall some more notions and facts from algebra:

~~The Ideal Class Group~~

Let R be an integral domain, i.e. a commutative ring with unit e and without zero- divisors. A subring a in R is an ideal in R iff one has ra ∈ a for all r ∈ R and a ∈ a. An ideal a is a principal ideal iff it can be generated by one element, i.e. is of the form a = aR for an element a ∈ R. An ideal p in R is a prime ideal iff from ab ∈ p one can conclude a or b ∈ p, or, equivalently, iff the factor ring R/p has no zero-divisors.

For instance, every normalized irreducible polynomial f of degree n in the polynomial ring R = Q[x] generates a prime ideal p = fR. This ideal is maximal and the factor ring R/p is an algebraic number ﬁeld of degree n. Every algebraic number ﬁeld of degree n can be described like this.

~~If for two ideals we have a ⊃ b,wesaya divides b. And (a, b) denotes the greatest common divisor of a and b (corresponding to the usual notions for integers if R = Z and a = aZ, b = bZ).~~

~~From two ideals a and b one can build new ideals, the sum a + b and the product a · b (the last one consisting of ﬁnite sums of products ab, a ∈ a,b∈ b).~~

In number theory, the main occasion to use these notions is the maximal order of an algebraic number field K, i.e. the ring o of integral elements of K: an element a ∈ K is called integral iff it is the root of a normalized polynomial f ∈ Z[x] (one has to show that the set of these elements is a ring). The determination of the maximal orders is a fundamental task of number theory. For instance, for the field K = Q(i) of Gauss numbers one has the Gauss integers o = Z[i]={a + bi; a, b ∈ Z} (this is not difficult, try to prove it as Exercise 9.7). For the cyclotomic field K = Q(ζN ) one has also the nice result o = Z[ζN ].

~~Main tools in this context are the maps norm and trace:Forx ∈ F one has~~

~~λx : F −→ F,a−→ xa for all a ∈ F, the (additive) trace map~~

~~TrF/K : F −→ K,x−→ Tr λx and the (multiplicative) norm map~~

~~NF/K : F −→ K,x−→ det λx. 248 9. Epilogue: Outlook to Number Theory~~

An ideal in o is also called an ideal of K and this notion is extended to the notion of a fractional ideal of K: This is an o-submodule a =0of K, which has a common denominator, i.e. there is an element 0 = d ∈ o with da ⊆ o.

All this is done with the intention to associate another finite group to an algebraic number field: It is not difficult to see that the set JK of fractional ideals provided with multiplication as composition has the structure of an abelian group. It is called the ideal group of K. The (almost unique) decomposition of an integer in Z into a product of powers of primes is extended to the statement (see for instance [Ne] p.23):

Every fractional ideal a ∈ JK has a unique representation as a product of powers of prime ideals νp a =Πp p ,νp ∈ Z, almost all νp =0. ∗ The principal fractional ideals PK := {(a)=ao; a ∈ K } form a subgroup of JK and, ﬁnally, one introduces the factor group

~~ClK := JK/PK.~~

~~This comes out as a ﬁnite abelian group and is called the ideal class group or simply class group of K.~~

Class field theory tries to relate the Galois group of a field extension to the ideal class group of the ground field. An extension F/K with abelian Galois group G is called a Hilbert class field iff one has G(F/K)=ClK.

~~Example 9.6: The cyclotomic ﬁeld QN is a Hilbert class ﬁeld over Q.Wehave × ClQN (Z/N Z) .~~

~~Dirichlet Characters and Hecke’s L-Functions~~

For 0 = m ∈ Z, the classical Dirichlet character mod m is a character of the group (Z/mZ)× χ :(Z/mZ)× −→ S1. The character χ is called primitive if there is no genuine divisor m of m such that χ factorizes via a character of (Z/mZ)×. Given such a χ, we get a (multiplicative) function for all natural numbers n, which we denote again by χ, by mapping n to χ(¯n) if (n, m) = 1 andn ¯ is the class of n mod m, and to zero if (n, m) = 1. Hence, to such a primitive character χ, one can associate an L-series ∞ χ(n) L(χ, s):= . ns n=1 Obviously, for the trivial character χ0 mod 1 with χ0(n) = 1 for all n, we obtain the Riemann Zeta series. Hecke proved that these series have an Euler product (with degree one factors) 1 L(χ, s)=Π ∈P , p 1 − χ(p)p−s analytic continuation to C (resp. C\{1} for χ0), and a functional equation for s −→ 1−s. (The proof relies again on the transformation formula of a (generalized) theta series.) 9.5 Elements of Algebraic Number Theory and Hecke L-Functions 249

These are examples of Hecke’s L-functions, which can be deﬁned more generally: Let K be an algebraic number ﬁeld with maximal order o, m ⊂ o an ideal, and J m the group of fractional ideals a in K, which are prime to m

~~J m := {a ⊂ K;(a, m)=1}.~~

~~Let χ : J m −→ S1 be a character. Then we deﬁne an L-series~~

~~χ(a) L(χ, s):= . N(a)s a⊂o~~

Here the sum is taken over all ideals a in o, N(a) denotes the number of elements of the residue class ring o/a, and again we put χ(a) = 0 for (a, m) = 1. The assumption that χ is a character translates easily to the existence of an Euler product taken over all prime ideals p in o 1 L(χ, s):=Π . p 1 − χ(p)N(p)−s

~~If we take χ = χ0 with χ0(a) = 1 for all a, we get the usual Dedekind Zeta function of the number ﬁeld K 0 −s L(χ ,s)= N(a) =: ζK(s). a⊂o~~

The question asking for the biggest class of characters, for which one can prove a functional equation (and hence has analytic continuation) leads to Hecke’s deﬁnition of the notion “Gr¨ossencharakter”:

~~Definition 9.8: A character χ : J m −→ S1 is called a grössencharacter mod m iff there is a pair of characters~~

~~∗ 1 ∗ 1 χf :(o/m) −→ S ,χ∞ : R −→ S such that~~

~~χ((a)) = χf (a)χ∞(a) for all a ∈ o with ((a), m)=1.~~

We have remarked (in 5.1) that the set of characters of the additive group R is simply in bijection to R. A sign that multiplicative structure is more intricate than additive ∗ structure is again given by the fact that the set of these characters χ∞ of R is more complicated (for a complete statement see [Ne] p.497).

As already said, a more elegant way to understand these constructions is based on the theory of adèles and idèles introduced into number theory by Tate and then pursued by many others. Here one has the central notion of a Hecke character as a character of the idel class group. The relation to the notions presented here is also to be found in [Ne] p.501. There are further refinements using ray class groups and their characters but we will stop here and change the topic to get again more concrete examples. 250 9. Epilogue: Outlook to Number Theory

~~9.6 Arithmetic L-Functions~~

In classical number theory and in arithmetic geometry there are many occasions to introduce Zeta- and L-functions as Dirichlet series encoding diophantine or arithmetic information. By their nature as Dirichlet series, these series converge in right half planes. Then, as the main task, one has to find a way to extend the function given by such a series to a function on the whole plane, eventually with some poles, and study the values of this function at special, in particular, integral points. This may turn out to be the expression of a mysterious regularity property inherent in the arithmetic or diophantine problem we started with. A way to prove this comes up if this arithmetic L-function can be seen as an automorphic L-function. For instance the proof of the Fermat conjecture by Wiles (and Taylor) incorporates a proof that the L-function of an elliptic curve can be interpreted as an L-function of a modular form. Here we are far from making this sufficiently precise but to give at least some flavour and define the L-functions for number fields Emil Artin discussed around 1923, we borrow heavily from an expository article on the role of representation theory in number theory by Langlands [L].

~~Two Examples:~~

~~We look at diophantine equations, i.e. polynomial equations with integral coeﬃcients to which integral solutions are sought. A famous example is the Fermat equation~~

~~xm + ym = zm.~~

For m = 2 there are inﬁnitely many solutions, the Pythagorian triples, e.g. (3, 4, 5), and for m>2, as Wiles (and Taylor) proved, there are no non-trivial integral solutions. But if there are no integral solutions to a diophantine problem, one can look for solutions mod p, p prime, try to count their number and encode these numbers into a function.

~~Example 9.7: Probably the simplest example is the equation~~

~~(9.18) x2 +1=0.~~

The primes p for which the congruence x2 +1 ≡ 0modp can be solved are 2, 5, 13,... all of which leave the remainder 1 upon division by 4, whereas primes like 7, 19, 23,... that leave the remainder 3 upon division by 4 never do so. In 9.3 we already presented the Legendre symbol a ( )=±1 for all a ∈ Z, with (p, a)=1, p which indicates whether the congruence x2 ≡ a mod p is solvable or not. Having at hand the notion of a Dirichlet character, we see that here we have an example with a multiplicative function χ on Z given by

~~−1 ≡ χ(p)=(p ) = 1 for p 1mod4, = −1 for p ≡ 3mod4, = 0 for (p, 4) =1 , i.e. with χ(1) = 1,χ(2) = 0,χ(3) = −1,χ(4) = 0 ... . 9.6 Arithmetic L-Functions 251~~

~~We have ∞ χ(p) χ(n) 1 1 1 1 L(χ, s)= (1 − )−1 = =1− + − + + ... . ps ns 3s 5s 7s 9s p n=1~~

Like every Dirichlet series, this series defines a function in a right half plane. It is an expression of the mysterious symmetry of our problem that one can bring into play a theta function, which, by its transformation property, produces a function valid for all s ∈ C. Now, having prepared some material from algebraic number theory, we can look at the function L(χ, s) from another point of view: The polynomial f(x)=x2 + 1 defines the Gauss field K = Q(i)={a + bi, a, b ∈ Q} and this field has the Galois group G = G(K) of automorphisms elementwise fixing Q consisting√ just of the identity map and the map ϕ, uniquely fixed by changing the root θ1 := −1=i into θ2 := −i. Hence, one has G {±1} and this group has exactly two one-dimensional representations, the trivial one and a representation π with π(ϕ)=−1. One can use this to define another type of L-function, the Artin L-function. To explain this (following Langlands’ article), we go back some steps: The solvability of the congruence f(x)=x2 +1≡ 0modp is equivalent to the possibility to factorize f mod p into a product of two linear factors, e.g.

~~x2 +1≡ x2 +2x +1=(x +1)2 mod 2 x2 +1=x2 − 4+5≡ (x − 2)(x +2) mod5.~~

~~For the primes p = 7 and 11 and, in general, for all primes, for which the congruence has no solutions, the polynomial x2 + 1 stays irreducible mod p.~~

~~Example 9.8: Another famous example is~~

~~(9.19) f(x)=x5 +10x3 − 10x2 +35x − 18.~~

~~The polynomial is irreducible mod p for p =7, 13, 19, 29, 43, 47, 59,... and factorizes into linear factors mod p for p = 2063, 2213, 2953, 3631,....~~

~~In general, we can look at a polynomial with integral coeﬃcients~~

~~n n−1 f(x)=x + an−1x + ···+ a1x + a0 with roots α1,...,αn, which we assume to be pairwise diﬀerent. There will be various relations between these roots with coeﬃcients that are rational,~~

~~F (α1,...,αn)=0. √ √ 3 For example, the roots of x −1 = 0 are α1 =1,α2 =(−1+ −3)/2,α3 =(−1− −3)/2 and two of the valid relations for these roots are~~

~~α1 =1,α2α3 = α1.~~

To the equation f(x) = 0 we can associate the group of all permutations of its roots that preserve all valid relations. In the last example the sole possibility in addition to the 252 9. Epilogue: Outlook to Number Theory

trivial permutation is the permutation that fixes α1 and interchanges α2 and α3. The group G that we get is called the Galois group of the equation f(x) = 0. It is identical with the Galois group of the algebraic number field K defined by the polynomial f. We introduce the discriminant ∆ of the equation (resp. the number field) as defined by the equation ∆:= (αi − αj). i= j

It is an integer and one of the most important characteristic numbers to be associated to a number field. Now, one can attach to any prime p that does not divide ∆ an element Fp ∈ G, called the Frobenius automorphism, that determines among other things how the equation factors mod p. More precisely, it is the conjugacy class of Fp within G that is determined (we will indicate below how this is done). To construct an L-series we need a prescription associating numerical information to the primes p. We can come to this by choosing a (finite-dimensional) representation ρ of G and taking the trace or the determinant of the matrix ρ(Fp). Things are particularly easy if we have a one- dimensional representation ρ as for instance for the group G in our first example (9.18) above: We can define an Euler product −s −1 L(ρ, s):= (1 − ρ(Fp)p ) p where the primes dividing ∆ are omitted. By closer analysis, one can realize that in the example we get just the Hecke L-function L(χ, s) if we take for ρ the non-trivial character χ of G {±1}. This is another expression of (a special case of) the main theorem of abelian class field theory.

In Chapter 2 we studied very briefly representations ρ of finite groups. Now, here, we find an occasion to expand the subject slightly. Beyond one-dimensional representations, the next possibility is that ρ is a two-dimensional representation, which we may suppose to be unitary. We know that SU(2) is a double cover of SO(3). It is customary to classify finite subgroups of the unitary group by their image in the group of rotations. Taking for an irreducible ρ the finite subgroup to be ρ(G), we obtain dihedral, tetrahedral, octa- hedral, and icosahedral representations if G has as its image in SU(2) the corresponding group. Dihedral representations can be treated by the classical theory, and also the other representations were treated at the time of Langlands’ article with the exception of a complete treatment of the icosahedral case. An example of an equation with an icosahedral representation ρ is given by our equation (9.19). For all primes, which do not divide 800, the conductor (related to the discriminant in a way we do not explain here), one can form the matrix ρ(Fp), which has two eigenvalues, λp and µp. Artin’s L-function is then 1 1 L(ρ, s):= 1 − λ p−s 1 − µ p−s p p p the primes dividing the conductor being omitted from the product. This product converges for Re s>1 and it is a special case of fhe famous Artin Conjecture from 1923 to show that it can be continued to an analytic function for all s ∈ C. For the equation (9.19) this was done in 1970 by J. Buhler. The general case is still open. 9.6 Arithmetic L-Functions 253

~~Decomposition Theory and Artin’s L-Function~~

To start with, we look again at the example K = QN = Q(ζN ): Here we have the abelian × Galois group G = Gal(QN ) (Z/N Z) . This isomorphism relates every irreducible representation ρ of G uniquely to a character χ of (Z/N Z)×. To each prime number p, which does not divide N, we associate the classp ¯ := p mod N in (Z/N Z)× and then ∈ −→ p the Frobenius automorphism Fp G given by ζN ζN . Now we can deﬁne the Artin L-function 1 L(ρ, s):= −s 1 − ρ(Fp)p pN and it is clear that it coincides with the Hecke L-function 1 L(χ, s):= . 1 − χ(p)p−s pN

For the general case, we look at an algebraic field extension F/K with Galois group G = Gal(F/K) and rings of integral elements O ⊃ o. For a prime ideal P in O and a prime ideal p in o we say that P lies over p if P contains p and hence P ∩ o = p.We say that p decomposes into the primes Pi if p generates in O an ideal (p), which has the decomposition h ei (p)= Pi i=1 into the primes Pi lying over p with the ramification indices ei ∈ N. We denote by k(P) the residue field O/P and similarly k(p):=o/p. Both fields are finite fields. If Pi lies over p, k(Pi) is a finite extension of k(p) and we denote fi := degk(p) k(Pi). These degrees and the ramification indices are related by the central formula of Hilbert’s ramification theory h eifi = n. i=1 From now on, we assume that the extension F/K is a Galois extension. Then the decomposition simplifies to h e (p)=( Pi) i=1 and all degrees fi coincide (=: f). The Galois group G acts on O and moreover even transitively on the primes Pi lying over a given p ⊂ o. We introduce the decomposition group

GP := {σ ∈ G; σP = P}. The index of the decomposition group in G is equal to the ramiﬁcation index, which is deﬁned as e =[G : GP]. One has the two extreme cases: i) If GP contains only the identity, we say that p is totally decomposed, and ii) GP = G where p stays indecomposed. 254 9. Epilogue: Outlook to Number Theory

~~Every σ ∈ GP induces an automorphism of the residue class ﬁeld k(P):=O/P by a mod P −→ σa mod P~~

Since this automorphism preserves the elements of the subfield k(p), we get a homomorphism ϕP : GP −→ Gal(k(P)/k(p)), which is surjective. We define the inertia group as its kernel, IP := ker ϕP.Ifp is unramified, IP consists only of the identity, one has GP Gal(k(P)/k(p)) and, hence, this group may be seen as a subgroup of G. In this case we can find exacly one automorphism FP ∈ G such that q FPa ≡ a mod P for all a ∈ O with q := N(p). This is the Frobenius automorphism, it generates the cyclic group GP. If IP is not trivial, we take as Frobenius FP an element of G such that its image in Gal(k(P)/k(p)) has again the action a mod P −→ aq mod P.

Now we are ready to define Artin’s L-function for this general situation. We take a representation ρ of G in the finite-dimensional vector space V and denote by V P the subspace of elements fixed by ρ(IP). The characteristic polynomial

~~P det(E − ρ(FP)t; V )~~

~~P for the action of ρ(FP)onV depends only on the prime p and not on the prime P lying over p.~~

Definition 9.9: Let F/K be a Galois extension of algebraic number fields and (ρ, V )a representation of its Galois group G. Then Artin’s L-series for ρ is defined by 1 L(ρ, s):=L(F/K,ρ,s):= det(1 − ρ(F )N(p)−s; V IP ) p P where the product is taken over all prime ideals p in K (i.e. in o).

This series converges in the right half plane Re s>1 to an analytic function. For K = Q and the trivial one-dimensional representation ρ0 with ρ0(σ) = 1 for all σ ∈ G, Artin’s L-function is just the Dedekind Zeta-function, one has

~~0 L(F/Q,ρ ,s)=ζF(s).~~

In [Ne] Kapitel VII, we ﬁnd a lot of results concerning the Artin L-function. To ﬁnish with this topic, we only mention Theorem (10.6) in [Ne], clarifying the relation between Artin’s and Hecke’s L-function in the abelian case.

Theorem 9.12: Let F/K be an abelian extension, f the conductor of F/K,ρa nontrivial character of Gal(F/K) and χ the associated gr¨ossencharacter mod f. Then we have the relation 1 { | } L(F/K,ρ,s)= −s L(χ, s),S:= p f,χ(IP)=1 . 1 − ρ(FP)N(p) p∈S Here we should still explain the notion of a conductor, but again we have to refer to [Ne]. 9.6 Arithmetic L-Functions 255

~~L-Functions of Elliptic Curves~~

~~Again we follow Langlands’ article [L] and look at another diophantine problem: We ask for integral or at least rational solutions of the equation~~

(9.20) y2 = x3 + Dx, with D ∈ Z,D =0 , say D = −1. This is a special case of an affine equation defining an elliptic curve E = ED. More generally, one looks at a smooth projective curve defined by the homogeneous equation

~~y2z = x3 + axz2 + bz3,a,b∈ Q, 4a3 +27b2 =0 .~~

A curve like this is distinguished among all smooth curves because it can be given the structure of an abelian group. Elliptic curves now are very popular even outside of number theory because of their importance for cryptography. On the way to deﬁne an L-function via Euler factors for the primes p, it seems a good idea to count again the number Np of solutions of (9.20) mod p (or of the associated homogeneous equation, but we stay here with Langlands in his article). For instance, for D = −1 and p =5, we get N5 = 7. We deﬁne αp and βp by the conditions

Np = p + αp + βp; βp =¯αp; αpβp = p. √ (These conditions will deﬁne αp and βp only if |Np − p|≤2 p, but it can be proven that this is true.) Then one should be tempted to deﬁne as L-function for the elliptic curve

~~1 1 L(E ,s)= D 1 − α p−s 1 − β p−s p p p omitting perhaps some primes p related to D.~~

~~For D = (6577)2, one omits p = 2 and 3 and gets a function for any s and not only for Re s>3/2 where the Euler product converges.~~

At this point it is to be emphasized that the value of an elliptic L-function at s = 1 has a particular signiﬁcance: It is a (special case of a) conjecture of Birch and Swinnerton-Dyer that the equation (9.20) must have a solution if L(ED, 1) = 0. This conjecture from 1965 is based on numerical calculations and in general is still open.

The L-functions of elliptic curves are special cases of the Hasse-Weil Zeta functions associated to projective varieties over number ﬁelds. Starting with work by Hasse, Weil, and Dwork, and up to work by Deligne, Langlands and many others, there are deep theorems and conjectures for these relating them again to automorphic L-functions when the varieties are Shimura varieties, i.e. come as homogeneous spaces from a linear group, as for instance (roughly said) the completion of the modular curve Γ \ H =Γ\ SL(2, R)/SO(2). There was a famous conjecture, connected mainly with the names of Taniyama, Shimura, and A. Weil, indicating (again roughly) that each elliptic L-function is an automorphic L-function belonging to some modular form, which has been proved by Wiles and Tay- lor in the framework of their proof of the Fermat conjecture already mentioned at the beginning of this section. 256 9. Epilogue: Outlook to Number Theory

~~9.7 Summary and Final Reﬂections~~

There are still much more L- and Zeta functions than we mentioned up to now (Dwork once remarked to the author that he had the impression that every mathematician feels obliged to deﬁne his own one). As in this text we restricted our treatment to the discussion of real and complex groups, we have to leave aside the true story, which happens using the groups G(Qp) deﬁned over Qp, the non-archimedian completions of Q, and the adelic groups G(A), which come up as the restricted direct product of the groups over all archimedian and non-archimedian completions of Q and/or as groups of matrices with adelic entries.

Hence we ﬁnish our story by giving a rudimentary survey of what Langlands says in [L] concerning the archimedian case, in particular as several notions, which we treated in our chapters 6 and 7, reappear in a language using intuition from elementary particle physics.

~~On the Way to (Archimedian) Langlands Parameters~~

We look at representations π of a subgroup G of GL(n, R), in particular G = GL(n, R). There are essentially two ways to analyse or construct representations: i) the inﬁnitesimal method using the Lie algebra of G, as exploited in Chapter 6, and ii) the method, which came up several times in Chapter 4, 7, and 8 and which starts with the action of G on a manifold M, passes to an associated action on functions living on the manifold, and ﬁnally decomposes this action into irreducibles.

If G is compact, as for instance G = SO(n), one takes M = G and decomposes the right-regular representation on H = L2(G). This way, one comes to representations π with square integrable matrix coeﬃcients

~~<π(g)v, w >, v, w ∈H, and discrete parameters characterizing π.~~

If G is not compact, the situation changes considerably as we saw in our discussion of the example SL(2, R) in 7.2. Langlands proposes that the system being treated is best com- pared to a Schr¨odingerequation for which both asymptotically independent and bound states appear (for instance, in the case of the hydrogen atom, for the electron one has discrete orbits and a continuous spectrum):

Since every element g ∈ GL(n, R) can be written as a product k1ak2 where k1 and k2 are orthogonal matrices and a := D(α1,...,αn) a diagonal matrix with positive eigenvalues αj, the simplest representations should have n freely assignable parameters, and they are analogous to a system of n interacting but asymptotically independent particles to which arbitrary momenta can be assigned. In addition, the presence of the factors k1 and k2 may entail the presence of discrete quantum numbers. We exemplified this in 7.2.1 while constructing the principal series representations of SL(2, R) where we had the purely imaginary “momentum” is and the discrete parameter ∈{0, 1}. In the GL(n, R) case, the general induction procedure from 7.1 proposes to look at first at the group B of 9.7 Summary and Final Reflections 257 superdiagonal matrices ⎛ ⎞ α1 ∗∗... ∗ ⎜ ⎟ ⎜ α2 ∗ ... ∗ ⎟ ⎜ ⎟ ⎜ α3 ... ∗ ⎟ (9.21) b = ⎜ ⎟ . ⎜ ... ⎟ ⎝ ⎠

αn

~~Given n real parameters s1,...,sn and n numbers k = ±1 (the supplementary discrete quantum numbers), one introduces the characters~~

~~k isk χk : α −→ sgn(α) |α| of R∗, as well as the character χ of B that sends the matrix b to~~

n δ−k χk(αk)|αk| . k=1 Here we have δ =(n +1)/2 and the supplementary exponent comes from the modular function and makes things unitary: Similar to our treatment in the special case in 7.2 we G associate to χ the induced representation indBχ given by right translation on the space of functions φ on G that satisfy the functional equation

~~φ(bg)=χ(b)φ(g) for all b ∈ B,g ∈ G.~~

~~The representations associated to two sequences of parameters χk are equivalent iﬀ one sequence is a permutation of the other.~~

If one replaces the exponents isk by arbitrary complex numbers, one gets representations, which in general will be neither irreducible nor unitary. But there is a well-determined process for choosing a speciﬁc irreducible factor of the representation that is then taken as the representation associated to the characters χ1,...,χn. In the intuitive language, this means that one allows complex momenta.

In our study of SL(2, R) in 7.2 we met with the discrete series representations. Keeping this in mind, one can deduce that for GL(2, R) there are representations associated to one continuous parameter (or, if the analogy is pursued, one momentum) and one discrete parameter (or quantum number).

~~In the treatment of GL(n, R) the case n = 2 plays a special role because there is just one algebraic extension of the ﬁeld R, namely C, and this is of degree 2 over R.~~

For the group GL(n, C) of complex matrices, the representation theory is simpler. One has no discrete series and therefore no bound states. In Langlands’ language, for K = R or C the general irreducible representation of GL(n, K) is analogous to r interacting but asymptotically bound states with n1,...,nr particles, Σnk = n and the nk being subject to constraints appropriate to the ﬁeld. To construct the representation, one introduces the group P of matrices of the form (9.21) where n is replaced by r, each αk is a square nk–matrix, and the asterisks represent block matrices of the appropriate size. From here, we can again apply the induction process but we have to replace the character χ by 258 9. Epilogue: Outlook to Number Theory

the tensor product of (in general infinite-dimensional) representations χk of GL(n, K), so that the functions φ with their functional equation as above take their values in an infinite-dimensional space. If the momenta are all real, then the representation is irreducible. Otherwise it is again necessary to pass to a specific factor. As before, the order of the χk is irrelevant. After some more consideration, finally one can conclude that the classification of irreducible representations of GL(n, K) has an extreme formal simplicity. For K = C,to specify representations we specify the n-dimensional representations of GL(1, C)=C×. As Langlands emphasizes, here we compare two very different objects: on one hand, irreducible and in general infinite-dimensional representations of the group GL(n, C), and on the other, finite-dimensional but in general reducible representations of GL(1, C)=C×.

A similar statement for K = R requires a group that is not commutative but whose irreducible representations are of degree at most two, in order to accomodate the existence of two particle bound states. The appropriate group is the Weil group of R, WR.Itis obtained by adjoining to the group C× an element w such that

~~w2 = −1, and wz =¯zw for all z ∈ C×.~~

This is a kind of dihedral group and, in our text, it appeared as an example already in the Chapters 0 and 1. It has as non-trivial irreducible representations the character π0 given by z −→ zz,¯ w −→ − 1 and the two-dimensional representations πm, for 0 = m ∈ Z given by zm 0 01 z −→ ,w−→ . 0¯zm (−1)m 0

× The Weil group WC of the complex number ﬁeld C is just C . One has the nice result that for K = R and C the irreducible representations of GL(n, K) are parametrized by n-dimensional representations of the Weil group WK. The Weil group can be attached to many of the ﬁelds appearing in number theory. It can be rather complicated because, as Langlands assures, it incorporates many of the deepest facts about the theory of equations known at the present.

To get some more flavour, we add a remark concerning orthogonal subgroups G of GL(2n +1, R) defined as stability groups of quadratic forms. As we know, there are several types of groups depending on the index of the quadratic form and the representation theory of these will be quite different. But one can get a unified point of view by a parametrization of the representations via homomorphims of the Weil group WR into the L-group LG of G. The general definition of the L-group is given in terms of the theory of equations and of root systems. For G = GL(n, R) it is simply LG = GL(n, C) and for G = SO(2n + 1) it is the symplectic group of 2n × 2n-matrices, LG = Sp(n, R).

The principle that the irreducible representations of the real group G are classified by L the (continuous) homomorphisms of WR into G is valid, but this has consequences. A reducible n-dimensional representation of WR is a homomorphism of WR into GL(n, C) that is isomorphic to C× and not in the center of GL(n, C). Hence the notion of irre- L ducibility has an obvious analogue for the homomorphisms of WR into G, and it can be verified that the homomorphisms which are irreducible in this sense correspond to 9.7 Summary and Final Reflections 259 representations whose matrix coefficients are square-integrable over the group. If the quadratic form chosen is Euclidean, then G is compact and all representations have this L property, so that any homomorphism of WR into G that is not irreducible in this sense will have to be excluded from the list of parameters. A similar but less restrictive condition must also be imposed for the groups with other indices. L It also turns out that the classification provided by WR and G is coarse. Some homomorphisms correspond to several irreducible representations of G. This finally leads to the notion of L-packets which indicates representations with the same L-functions.

~~L-Groups and Functoriality~~

For Langlands the introduction of the L-group is an essential step towards his famous principle of functoriality. A homomorphism from a group H to a group G does not provide any way to transfer irreducible representations from one of these groups to another, unless G is abelian, nor are homorphisms between H and G usually related to homomorphisms between LH and LG. On the other hand, a homomorphism from LH to LG,by composition, L L WR −→ H −→ G yields a map from the set of parameters for the irreducible representations of H to the irreducible representations of G, and thus implicitly a way to pass from irreducible representations of H to irreducible representations of G. Langlands calls this passage the principle of functoriality in the L-group. A possibility to realize this principle in general would have deepest consequences.

~~The Standard L-Function~~

In 9.6 we already insisted on the possibility to associate to a given problem or object an L-function via an Euler product by encoding information about the object in such a way that one gets a reasonable “Euler polynomial” in the variable t = p−s, which can be put in the denominator for an Euler factor. While constructing representations, say of GL(n, K), via induction in Chapter 7, the main idea was that at least some representations are ﬁxed by choosing characters for the group H of diagonal matrices D(α1,...,αn), which can be understood as distinguishing matrices

~~is1 isn D(α ,...,α ),s1,...,sn ∈ R.~~

Even if we still avoid to go into the representation theory of the non-archimedian ﬁeld Qp, it should not be too hard to accept that the analogon of the matrix above in the p-adic theory is the matrix

~~is1 isn A(πp):=D(p ,...,p ),s1,...,sn ∈ R.~~

~~To this matrix one can attach the function 1 L(πp,s):= −s det(En − A(πp)p ) and then form the Euler product L(π, s):= L(πp,s). 260 9. Epilogue: Outlook to Number Theory~~

In this product perhaps some “special” primes are left out or are represented by special factors and one has to ﬁnd an appropriate “factor at ∞” associated to the archimedian representation. Such a product has the chance to converge for, say, Re s ≥ a + 1, pro- a vided that the eigenvalues of each A(πp) are less than p in absolute value. The outcome is called a standard L-function. Whoever wants to see that this really happens (and even more...) should be motivated to take a look into the non-archimedian theory, as for instance in [Ge] or [GeS].

~~L-Groups and Automorphic L-Functions~~

To finish our fantastic and magical story (the German word Märchen appears well in the title of one of Langlands’ articles ([L1])), we look at an (automorphic) representation π of a group G (in particular G = GL(n)), take a finite-dimensional representation ρ of the L-group LG belonging to G. Then we have again a finite Euler polynomial given by det(E − ρ(A(πp)t) and one can introduce as automorphic L-function 1 L(π, ρ, s):= −s det(E − ρ(A(πp))p ) where again for some primes p we have to put in special factors, which we do not discuss here. In particular, one has to find a completing factor belonging to the archimedian theory. The final goal is to generalize what we sketched in the last section by showing that to given π and ρ there is an automorphic representation Π such that

~~A(Πp)=ρ(πp) for almost all p. Then one could have~~

~~L(π, ρ, s)=L(Π,s) and use properties of one type of functions to be transfered to the other.~~

There are many other important topics coming up in this context. We only mention the topic of the trace formula but can not touch this here. The only theme whose formulation is within our reach is the following: Given a representation π of a group G = GL(n) and a representation π of G = GL(n), by the methods from of our text one can construct representations π + π of GL(n + n) and π ⊗ π of GL(nn). It is natural to ask for a similar construction for automorphic representations. This is related to functoriality and up to now there are very few results.

We repeat our hope that the vagueness of the presentation of some of the material in the last sections would make the reader curious to know more about all this and to be stimulated to more intense studies of this most fascinating part of mathematics. All the time, we were driven by J. R. Jim´enez’words:

~~¡Voz m´ıa,canta, canta; que mientras haya algo que no hayas dicho t´u, t´u nada has dicho! Bibliography~~

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~~[Ba] Bargmann, V.: Irreducible Unitary Representations of the Lorentz Group. An- nals of Math. 48 (1947) 568-640.~~

~~[Be] Berndt, R.: Einf¨uhrung in die Symplektische Geometrie. Vieweg, Braun- schweig/Wiesbaden 1998. Now also translated: An Introduction to Symplectic Geometry. GSM 26, AMS 2001.~~

~~[Be1] Berndt, R.: The Heat Equation and Representations of the Jacobi Group. Con- temporary Mathematics 389 (2006) 47 – 68.~~

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algebraic number field, 246 of Heis(R), 83 analytic subgroup, 89 of a group, 5, 83 arithmetic modular form, 244 of a Lie algebra, 87 Artin L-function, 252, 254 centralizer, 5 Artin conjecture, 252 character automorphic L-function, 245 of a representation, 21 automorphic factor, 136 of an abelian group, 59 for SL(2, R), 221 coadjoint orbit, 186 for the Heisenberg group, 217 cocycle condition, 122 for the Jacobi group, 238 complementary series automorphic form, 217, 232 of SL(2, C), 143 of SL(2, R), 142 Borel section, 121 complex structure, 185 Borel space, 38 configuration space, 173 Borel subgroup, 130 conjugacy class, 4 bundle, 162 conjugation, 4 cotangent bundle, 170 connection on a vector bundle, 185 equivalent, 167 continuous decomposition, 60 fiber, 167 continuous sum, 62 Hilbert bundle, 163 correspondence, 240 holomorphic line bundle over P1(C), cusp form, 222 171, 183 cyclotomic field, 246 line bundle, 161 prequantum line bundle, 179 Darboux theorem, 179 real analytic line bundles over P1(C), decomposition group, 253 182 Dedekind Zeta function, 249 section of a bundle, 162 degree of a field extension, 246 tangent bundle, 169 delta function, 223 transition function, 168 derivation, 65 vector bundle, 167 derived representation, 70 bundle map, 167 differential form, 174 closed, 176 Cartan decomposition of g,95 exact, 176 Cartan involution, 85 exterior differentiation, 175 Cartan matrix, 92 inner product with a vector field, 176 Cartan subalgebra, 89, 93 of first degree, 170 Casimir element, 101, 135 direct integral, 62 Cauchy Riemann differential equation, 136 Dirichlet character, 248 Cayley transform, 137 Dirichlet series, 239 CDW diagram, 92 discriminant, 252 center disrete series of SL(2, R), 133 INDEX 267 distribution on a manifold, 184 Poincarégroup GP (n), 148 duality theorem reductive, 85 for the discrete series representation of simple, 85 GJ , 239 solvable, 86 for the discrete Series of SL(2, R), 230 standard Borel, 2 for the Schrödingerrepresentation of standard unipotent, 2 Heis(R), 219 symmetric group, 5 Dynkin diagram, 92 symmetry group, 178 symplectic, 85 Eisenstein series, 223 theta group, 221 non-holomorphic, 233 topological, 31 elliptic curve, 255 unimodular, 39 eta function, 223 unitary group, 1 Euler angles, 42, 52 Weil group of R,2,258 Euler function, 246 Weyl group, 95 Euler product, 239, 245, 249 group action exponential function for matrices, 67 effective, 3 from the left, 3 factor representation, 36 from the right, 5 fractional ideal, 248 transitive, 3 Frobenius automorphism, 254 group ring, 23 Frobenuis reciprocity, 127 fundamental domain, 222 Haar measure, 39 fundamental functional equation, 117 hadron, 110 half plane model, 136 G-orbit, 3 Hamilton operator, 177 G-space, 33 Hamilton’s equations, 173 Galois extension, 247 Hamiltonian G-space, 187 Galois group, 246 Hamiltonian formalism, 173 Gamma function, 244 Hecke L-function, 249 Gauss integer, 247 Hecke eigenform, 242 Gell-Mann’s eightfold way, 110 Hecke operator, 232, 240 general linear group, 1 Heisenberg algebra, 66 generalized weight space, 97 Heisenberg commutation relations, 70, 72 Grössencharakter, 249 Heisenberg representation, 118 group Highest Weight Theorem, 98, 103 alternating, 6 Hilbert class field, 248 classical, 2 homogeneous space, 4 congruence group, 221 homothety, 18 derived, 85 Hopf map, 195 euclidean group GE(n), 148 Heisenberg group, 2 ideal Jacobi group, 209 fractional, 248 linear group, 32 of a commutative ring, 247 Lorentz group GL, 145 of a Lie algebra, 86 modular group, 221 ideal class group, 248 nilpotent, 86 induced representation, 119 of exponential type, 68 compact picture, 125 orthogonal group, 1 first realization, 123 Poincarégroup GP , 154 induced picture, 125 268 INDEX

noncompact picture, 126 Lorentz normalized, 120 boost, 146 second realization, 124 rotation, 146 induction Lorentz group GL, 145 holomorphic, 141 parabolic, 130 Maass wave forms, 232 induction in stages, 127 Mackey decomposition, 121 infinitesimal generator, 69 Mackey Theory for semidirect products, 150 Int g,89 manifold, 165 integral curve, 174 analytic, 165 integral element, 247 diffeomorphic, 166 internal symmetry group, 110 Kähler,185 intertwining operator, 9 oriented, 166 isospin, 110 quantizable symplectic, 179 isotypic subspace, 74 smooth, 165 Iwasawa decomposition, 69, 96, 130 symplectic, 176, 178 master equation, 122 Jacobi form, 238 maximal order, 247 Jacobi group, 148, 209, 236, 237 maximal torus, 94 Jacobi identity, 64 measure Jacobi symbol, 237 invariant, 39 Jacobi theta function, 216, 236 projection valued, 62 Jordan-Höldertheorem, 18 quasi-invariant, 121 spectral, 60 Killing form, 88 Mellin transform, 244 Kirillov-Kostant form, 187 Minkowski (pseudo)norm, 154 Minkowski metric, 145 ladder operator, 73 mock discrete series of SL(2, R), 142 Langlands decomposition, 125 modular form, 222 Laplace operator arithmetic, 244 euclidean, 56 modular function ∆, 40 Laplacian on SL(2, R), 135, 229 modular group, 221 left translation, 4 modular invariant, 226 Legendre symbol, 237 moment map, 187 Lie algebra, 64 multiplicity, 9 classical, 89 multiplier system, 55, 125 exceptional, 89 nilpotent, 86 neutron, 110 of SL(2, R), 65, 69 normal subgroup, 4 of SO(3), 66 normalizer of SU(2), 65 of h in g,89 of a linear group, 68 of H in G,4 of the Heisenberg group, 66, 70 radical of a Lie algebra, 87 one-parameter subgroup, 68 semisimple, 87 operator simple, 87 adjoint, 34 solvable, 86 bounded, 33 linear fractional transformation, 133 continuous, 33 linear group, 32 norm, 33 little group, 151 self-adjoint, 34 INDEX 269

symmetric, 34 continuous, 34 orbit contragredient, 16 admissible, 196 cyclic, 37 coadjoint, 186 decomposable, 16, 36 elliptic, 188 derived, 70 hyperbolic, 188 discrete series, 142 nilpotent, 188 discrete series of SL(2, R), 133 rigged, 197 disjoint, 9 eigenspace, 56 parabolic induction, 130 equivalent, 9 parabolic subgroup, 130 factor representation, 16 partition, 6 Fock representation, 207 Pauli matrices, 53 induced, 119, 164 permutation matrix, 6 irreducible, 8, 36 Peter–Weyl Theorem, 48 lattice, 219 Petersson scalar product, 244 Lie algebra, 66 phase space, 176 linear, 7 Poincarégroup, 154 permutation representation , 11 two-dimensional, 127 primary, 20 Poincaréhalf plane, 133 principal series, 130 Poincaréseries, 225 projective representation, 55 Poincaré-Birkhoff-WittTheorem, 100 regular representation of a compact group, Poisson algebra, 64 47 Poisson bracket, 64, 177 regular representation of a finite group, polarization 27 admissible, 196 Schrödingerrepresentation, 13, 118, 152, algebraic approach, 196 206 geometric approach, 184 Schrödinger-Weil representation, 211 Pontrjagin dual, 59 smooth, 70 prequantum Hilbert space, 180 unitary, 9 prime ideal, 247 representations principal series of SL(2, C), 143 of SL(2, C), 143 principal series of SL(2, R), 130, 200, 235 principle of functoriality, 259 of SL(2, R), 130 projection operator, 20 of SO(3), 54 projective representation, 55, 145 of SU(2), 48 P projective space, 55, 129, 166 of the Poincarégroup G , 157 proton, 110 of the two-dimensional Poincarégroup, 127 q-development, 222 unitarily integrable representations of quantization, 177 sl(2, R), 78 quasi-measure relation, 121 unitarily integrable representations of su(2), 82 Radon-Nikodym derivative, 121 Riemann Zeta function, 224, 239 Ramanujan function, 223 right translation, 4 representation root, 90 adjoint, 186 positive, 92 admissible, 143 reduced, 92 coadjoint, 186 restricted, 96 completely reducible, 16, 36 simple, 92 270 INDEX root reflection, 91 Weyl character formula, 103, 104 root space, 90 Weyl denominator, 103 root system, 91 Weyl group, 95 reduced, 91 Weyl operator, 44 Weyl’s dimensionality formula, 107 Schrödingerequation, 177 Schrödingerrepresentation, 118 zeta transform, 235 Schur’s Lemma, 18 finite-dimensional unitary Schur, 19 the unitary Schur, 37 semidirect product, 147 regular, 150 separable Hilbert space, 33 special linear group, 1 spectral measure, 61 spherical harmonics, 56 spin, 55 standard L-function, 260 Stone-von Neumann Theorem, 84 subrepresentation, 8, 36 subrepresentation theorem, 143 symplectic form ω, 178 integral, 179 tangent space, 169 tensor algebra, 99 tensor product of representations, 14 theta function with characteristics, 216 theta group, 221 theta transform, 234 total differential, 175 trace form, 88, 186 unit disc model, 138 unitary (linear) dual, 9 unitary dual of SL(2, R), 80, 130 of the Heisenberg group, 152 of the Poincarégroup GP , 157 universal enveloping algebra, 99 vacuum vector of Heis(R), 83 vector field, 170, 174 Hamiltonian, 174 vector of lowest weight, 75 Verma module, 101 weight, 74, 97 algebraically integral, 98 Weil group of R, 2, 13, 258