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**Representations of nilpotent Lie groups and their applications. Part 1: Basic theory and examples.**
*(English)*
Zbl 0704.22007

Cambridge Studies in Advanced Mathematics, 18. Cambridge etc.: Cambridge University Press. viii, 269 p. £35.00; $ 69.50 (1990).

This book is the first volume of a text in two volumes on representation theory and harmonic analysis for nilpotent Lie groups. L. Pukanszky’s book [Leçons sur les représentations des groupes (Dunod, Paris, 1967; Zbl 0152.012)] has been a basic introduction to the subject, whose development has enlarged its domain and made it include many new topics. Thus one has been expecting some more manuals which offer an insight into the actual phase of the field. Taking this situation into account and in order to supply the reader with knowledge and interest necessary for current research, the authors attempt to give a self-contained modern treatise of the subject.

In this volume they explain the basic theory and tools, with particular emphasis on: Kirillov’s orbit method, algorithms for parametrizing all coadjoint orbits, C-vectors for irreducible representations, the Plancherel formula, square integrable representations and discrete subgroups. We find there, for example, Penney’s canonical objects. Moore- Wolf’s flat orbits, Malcev bases and rationality. They present an up-to- date treatment of these various topics and, among many recent results first covered in a text, scatter a number of computed examples to illustrate the theory.

The table of contents of the present volume is: §1. Elementary theory of nilpotent Lie groups and Lie algebras; §2. Kirillov theory; §3. Parametrization of coadjoint orbits; §4. Plancherel formula; §5. Discrete cocompact subgroups. This book will succeed Pukanszky’s one and serve us as a standard exposition of this branch of modern mathematics. The second volume will concern the major applications of the theory in the forefront of current research.

In this volume they explain the basic theory and tools, with particular emphasis on: Kirillov’s orbit method, algorithms for parametrizing all coadjoint orbits, C-vectors for irreducible representations, the Plancherel formula, square integrable representations and discrete subgroups. We find there, for example, Penney’s canonical objects. Moore- Wolf’s flat orbits, Malcev bases and rationality. They present an up-to- date treatment of these various topics and, among many recent results first covered in a text, scatter a number of computed examples to illustrate the theory.

The table of contents of the present volume is: §1. Elementary theory of nilpotent Lie groups and Lie algebras; §2. Kirillov theory; §3. Parametrization of coadjoint orbits; §4. Plancherel formula; §5. Discrete cocompact subgroups. This book will succeed Pukanszky’s one and serve us as a standard exposition of this branch of modern mathematics. The second volume will concern the major applications of the theory in the forefront of current research.

Reviewer: H.Fujiwara

### MSC:

22E27 | Representations of nilpotent and solvable Lie groups (special orbital integrals, non-type I representations, etc.) |

22E25 | Nilpotent and solvable Lie groups |

22-02 | Research exposition (monographs, survey articles) pertaining to topological groups |

22E30 | Analysis on real and complex Lie groups |

22E40 | Discrete subgroups of Lie groups |

43A65 | Representations of groups, semigroups, etc. (aspects of abstract harmonic analysis) |