Representations of Solvable Lie Groups: Basic Theory and Examples (New Mathematical Monographs)

✍ Scribed by Didier Arnal, Bradley Currey

• Publisher: Cambridge University Press
• Year: 2020
• Tongue: English
• Leaves: 463
• Series: New Mathematical Monographs (Book 39)
• Edition: 1
• Category: Library

✦ Synopsis

This monograph answers the need for a unified account of the basic theory of unitary group representations, combined with new results, in a style that is broadly accessible for both graduate students and 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