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Foundations of Harmonic Analysis on the Heisenberg Group DIPLOMARBEIT Titel der Diplomarbeit Foundations of Harmonic Analysis on the Heisenberg Group Verfasser David Rottensteiner zur Erlangung des akademischen Grades Magister der Naturwissenschaften (Mag.rer.nat.) Wien, M¨arz2010 Studienkennzahl lt. Studienblatt: A 405 Studienrichtung lt. Studienblatt: Mathematik Betreuer: ao. Univ.-Prof. Dr. Roland Steinbauer Contents Introduction1 1. Some Prerequisites from Lie Group Representation Theory3 1.1. One-Parameter Groups of Operators.....................3 1.1.1. Definitions and Estimates.......................3 1.1.2. The Infinitesimal Generator......................5 1.1.3. Unitary One-Parameter Groups.................... 13 1.1.4. Smooth and Analytic Vectors..................... 17 1.2. Representations of Lie Groups......................... 22 1.2.1. General Groups and Measure..................... 22 1.2.2. Translations Revisited......................... 25 1.2.3. The Integrated Representation.................... 26 1.2.4. Smooth Vectors on G......................... 27 2. Foundations of Harmonic Analysis on the Heisenberg Group 30 2.1. The Heisenberg Group Hn ........................... 30 2.1.1. Motivation............................... 30 2.1.2. Lie Algebras and Commutation Relations.............. 30 2.1.3. The Heisenberg Algebra........................ 32 2.1.4. Construction of the Heisenberg Group................ 33 2.1.5. The Automorphisms of the Heisenberg Group............ 36 2.2. Representations of Hn ............................. 38 2.2.1. The Schr¨odingerRepresentation................... 38 2.2.2. Integrated Representation and Twisted Convolution........ 42 2.2.3. Matrix Coefficients and the Fourier-Wigner Transform....... 50 2.2.4. The Stone-von Neumann Theorem.................. 55 2.2.5. The Group Fourier Transform..................... 58 A. The Bochner Integral 65 A.1. Motivation................................... 65 A.2. Measure Space................................. 65 A.3. Measurability.................................. 65 A.4. Integrability................................... 68 A.5. Main Results for the Bochner Integral.................... 70 A.6. Lp(Ω;X; µ)-spaces............................... 74 B. On the representations of Banach algebras 77 I Contents B.1. Banach Algebras................................ 77 B.2. Algebra Representations............................ 80 C. Spectral Theory and Applications 83 C.1. Spectral Theorem................................ 83 C.2. Stone's Theorem................................ 87 C.3. Schur's Lemma................................. 89 List of Symbols 92 Bibliography 93 Abstract 95 Zusammenfassung 96 Curriculum Vitae 98 Acknowledgements 99 Danksagung 100 II Introduction This text focuses on the representation theory of the Heisenberg group Hn and some corresponding basic concepts of harmonic analysis on Hn. The Heisenberg group is of particular interest since it plays an important role in several branches of mathematics and theoretical physics. It is a nilpotent Lie group and it is the "simplest" non-compact, non-commutative example. Consequently, the most important representations in the context of harmonic analysis, namely the irreducible unitary representations, are no longer scalar-valued (as in the commutative case) neither finite-dimensional, that is to say matrix-valued (as in the compact case). In particular, the resulting notion of Fourier transform on Hn is therefore an operator-valued function acting on the separable Hilbert 2 n space L (R ). The first chapter gives a brief introduction to some basic concepts of representation theory. As a first step we introduce the notion of strongly continuous one-parameter groups, which turn out to be a simple example of a Banach space-valued Lie group repre- sentation. We will see that each of these one-parameter groups possesses an infinitesimal generator, which in turn induces a notion of smoothness and analyticity of vectors in the representation space. In that context we meet an instance of Banach space-valued integration (discussed in AppendixA), which turns out to be one of the technical main tools throughout this text. We furthermore focus on skew-adjoint operators in Hilbert space in order to study the special relation between unitary one-parameter groups and their skew-adjoint infinitesimal generators. The second part of Chapter1 is dedicated to general (strongly continuous) Lie group representations on Banach spaces. After recalling the differences between finite- dimensional and infinite-dimensional representations, we construct the Haar measure, a left-invariant Borel measure on the Lie group G, since (vector-valued) integration contin- ues to be an essential tool for many results. We shall use it, e.g., to show that the family of left translations form a Lie group representation of G on Lp(G); p 2 [1; 1); called the left regular representation of G. It also permits the definition of the integrated repre- sentation, where an integral over G is used to assign an operator to certain functions on G. This procedure furthermore leads to the notion of Banach algebra representations, which we discuss in AppendixB. Finally, we return to the concept of smooth vectors and show their denseness in the representation space. The second chapter is the core of this text. We start by constructing the Heisenberg Lie algebra hn, motivated by the commutation relation from quantum mechanics. Using the exponential map on hn, we discuss two approaches to the Heisenberg group, which display slightly different group laws. We then determine the automorphisms of Hn and, in view of future applications, its center Z. In the second and major part of this chapter we construct the most important rep- 1 INTRODUCTION n 2 n resentation of H , the so-called Schr¨odingerrepresentation on L (R ). We then make use of the results from Chapter1 in order to prove that the Schr¨odinger representation is in fact a strongly continuous unitary representation of Hn, generated by an essen- tially skew-adjoint Lie algebra representation of hn. Subsequently, we learn that the ∗ Schr¨odingerrepresentation can be parameterized with h 2 R , giving a family of in- equivalent unitary representations of Hn. Next we turn to the Stone-von Neumann theorem and the classification of the irreducible unitary representations of Hn. To this end, we study twisted convolution, a non-commutative convolution product of functions n ∼ 2n n defined on the quotient H =Z = R , which preserves the structure of H . Using its 2n compatibility with the integrated Schr¨odingerrepresentation (restricted to R ) we de- rive the technical tools required for proving the Stone-von Neumann theorem. Together with Schur's lemma it leads to a classification of the irreducible unitary represenations of Hn, asserting that each one is unitarily equivalent either to some Schr¨odingerrepre- ∗ sentation of parameter h 2 R or some one-dimensional representation with values in S1. All the results collected that far eventually culminate in the final part of this text, where we define a Fourier transform for Hn in analogy to the ordinary Fourier transform n on R . As a consequence of the Stone-von Neumann theorem the Fourier transform is either one-dimensional (agreeing with the ordinary Fourier transform) or given by an 2 n operator-valued function acting on L (R ). The two main statements of this section are the Plancherel theorem for the Fourier transform on L2(Hn), which states the existence of a measure µ that turns the Fourier transform into a unitary isomorphism. The second statement provides an explicit inversion formula of the Fourier transform for all Schwartz functions. Some material on essential concepts used throughout the text is collected in three appendices. AppendixA provides a compilation of facts on Bochner integration - one possible approach to Banach space-valued integration. All definitions and statements are given in such a way that they are directly applicable to the statements in Chapters 1 and 2 without any further modification. After characterizing the basic notions of Bochner measurability and Bochner integrability we prove Bochner versions of three fundamental theorems of integration theory: Fubini's theorem, the dominated convergence theorem and the fundamental theorem of calculus. We furthermore show that Bochner inte- gration interchanges with the application of bounded linear mappings and, under some assumptions, closed linear mappings as well. Finally, we introduce the natural Bochner generalization of the Lebesgue spaces Lp to functions with values in some Banach space. AppendixB gives a brief account on the notion of Banach algebras and their repre- sentations. Its main purpose is the illustration of the way integrated representations act on their natural domains, namely convolution algebras of integrable functions. Last but not least, AppendixC provides some important facts on the spectral theory of self-adjoint operators in Hilbert space. We present the main aspects of functional calculus, prove Stone's theorem, which gives the relation between unitary one-parameter groups and their infinitesimal generators, and Schur's lemma. 2 1. Some Prerequisites from Lie Group Representation Theory In Chapter1 we introduce some frequently used concepts of infinite-dimensional Lie group representation theory and show some basic results that will later be used in Chapter2 to treat the Heisenberg group. To this end, we first concentrate on strongly continuous one-parameter groups on R, a quite particular case of representations that yet possesses a vast field
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