Representations of Solvable Lie Groups: Basic Theory and Examples

✍ Scribed by Didier Arnal, Bradley Currey

• Publisher: Cambridge University Press
• Year: 2020
• Tongue: English
• Leaves: 463
• Series: New Mathematical Monographs, Band 39
• Category: Library

✦ Synopsis

The theory of unitary group representations began with finite groups, and blossomed in the twentieth century both as a natural abstraction of classical harmonic analysis, and as a tool for understanding various physical phenomena. Combining basic theory and new results, this monograph is a fresh and self-contained exposition of group representations and harmonic analysis on solvable Lie groups. Covering a range of topics from stratification methods for linear solvable actions in a finite-dimensional vector space, to complete proofs of essential elements of Mackey theory and a unified development of the main features of the orbit method for solvable Lie groups, the authors provide both well-known and new examples, with a focus on those relevant to contemporary applications. Clear explanations of the basic theory make this an invaluable reference guide for graduate students as well as researchers.

✦ Table of Contents

Contents
Preface
1 Basic theory of solvable Lie algebras and Lie groups
1.1 Solvable Lie algebras
1.2 Representations of a Lie algebra and weights
1.3 The Lie theorem and its first consequences
1.4 Adjoint and coadjoint representations
1.5 Ado theorem for a solvable Lie algebra
1.6 Lie groups
1.7 Nilpotent and solvable Lie groups
1.8 Exponential groups
1.9 Finite-dimensional group representations
2 Stratification of an orbit space
2.1 Matrix normal form
2.2 Layers for a representation
2.3 Orbit structure for a completely solvable action
2.4 Construction of rational supplementary elements
2.5 Orbit structure for an action of exponential type
2.6 Structure of the generic layer
3 Unitary representations
3.1 Unitary representations
3.2 Decomposition of representations
3.3 Induced representations
3.4 Elements of Mackey theory
4 Coadjoint orbits and polarizations
4.1 Coadjoint orbits
4.2 Polarizations
4.3 Admissibility and the Pukanszky condition
4.4 Isotropic subspaces associated to a flag
4.5 Fine layering and Vergne polarizations
4.6 Positivity and properties of polarizations
4.7 Integral orbits
5 Irreducible unitary representations
5.1 Holomorphic induction
5.2 Construction of irreducible representations
5.3 Orbit method for solvable groups
6 Plancherel formula and related topics
6.1 Invariant measure on a coadjoint orbit
6.2 Character formula
6.3 Semicharacters and the Plancherel formula
List of notations
Bibliography
Index