**Mathematical Surveys and Monographs**

Volume: 71;
1999;
128 pp;
Hardcover

MSC: Primary 22;

**Print ISBN: 978-0-8218-1088-0
Product Code: SURV/71**

List Price: $77.00

AMS Member Price: $61.60

MAA Member Price: $69.30

**Electronic ISBN: 978-1-4704-1298-2
Product Code: SURV/71.E**

List Price: $72.00

AMS Member Price: $57.60

MAA Member Price: $64.80

# Characters of Connected Lie Groups

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*Lajos Pukánszky*

This book adds to the great body of research that extends back to
A. Weil and E. P. Wigner on the unitary representations of locally
compact groups and their characters, i.e. the interplay between
classical group theory and modern analysis. The groups studied here
are the connected Lie groups of general type (not necessarily
nilpotent or semisimple).

Final results reflect Kirillov's orbit method; in the case of
groups that may be non-algebraic or non-type I, the method requires
considerable sophistication. Methods used range from deep functional
analysis (the theory of \(C^*\)-algebras, factors from
F. J. Murray and J. von Neumann, and measure theory) to differential
geometry (Lie groups and Hamiltonian actions).

This book presents for the first time a systematic and concise
compilation of proofs previously dispersed throughout the
literature. The result is an impressive example of the deepness of
Pukánszky's work.

#### Readership

Graduate students and research mathematicians working in topological groups and Lie groups; theoretical physicists.

#### Reviews & Endorsements

The material is very similar to that found in some papers of the author from the early 1970s … but … perhaps more structured thanks to the perspective from which the author viewed these matters later … gives an easier access to the topic than the original papers.

-- Mathematical Reviews

Apart from many important results which appear for the first time in book form, the present book is a very valuable source for many techniques in the representation theory of general Lie groups. These techniques are beautiful combinations of methods in abstract harmonic analysis and others which are more specific to Lie theory and related coadjoint orbits. [The book is recommended] to everyone interested in general and abstract aspects of the representation theory of Lie groups.

-- Fachbereich Mathematik

A very valuable source for many techniques in the representation theory of general Lie groups. These techniques are beautiful combinations of methods in abstract harmonic analysis and others which are more specific to Lie theory and related to coadjoint orbits. I can recommend the book to everyone interested in general and abstract aspects of the representation theory of Lie groups.

-- Zentralblatt MATH

#### Table of Contents

# Table of Contents

## Characters of Connected Lie Groups

- Contents vii8 free
- Preface ix10 free
- Foreword xi12 free
- Introduction xiii14 free
- Some notation used throughout the book xvii18 free
- Chapter I. Unitary Representations of Locally Algebraic Groups 120 free
- Chapter II. Representations of Elementary Groups 3150
- Chapter III. Existence of Characters 5574
- 3.1. Some subgroups of G 5574
- 3.2. The orbits of J 6382
- 3.3. Proof that J is surjective 7089
- 3.4. Technical tools 7493
- 3.5. Existence of normal representations with given kernels 86105
- 3.6. The type I case 90109
- 3.7. The non-type I case 90109
- 3.8. Proof of principal result 98117
- 3.9. The theorem of Poguntke 99118

- Chapter IV. Generalized Kirillov Theory 101120
- 4.1. Preliminary facts 102121
- 4.2. Construction of holomorphic representations 103122
- 4.3. Extension of an irreducible representation 105124
- 4.4. Computation of U[sub(π)] and K[sub(π)] 108127
- 4.5. Holomorphically induced representations 112131
- 4.6. Proof that ind is independent of the polarization 115134
- 4.7. Condition for unitary equivalence 116135
- 4.8. Regularized orbits 117136
- 4.9. Generalized orbits 119138
- 4.10. Auxiliary facts 121140
- 4.11. Proof that J is surjective 123142
- 4.12. Construction of a normal representation with kernel J 124143
- 4.13. Type-one primitive ideals 125144

- References 127146