This book presents a consistent development of the Kohn-Nirenberg type global quantization theory in the setting of graded nilpotent Lie groups in terms of their representations. It contains a detailed exposition of related background topics on homogeneous Lie groups, nilpotent Lie groups, and the analysis of Rockland operators on graded Lie groups together with their associated Sobolev spaces. For the specific example of the Heisenberg group the theory is illustrated in detail. In addition, the book features a brief account of the corresponding quantization theory in the setting of compact Lie groups. The monograph is the winner of the 2014 Ferran Sunyer i Balaguer Prize.

This book presents a consistent development of the Kohn-Nirenberg type global quantization theory in the setting of graded nilpotent Lie groups in terms of their representations. It contains a detailed exposition of related background topics on homogeneous Lie groups, nilpotent Lie groups, and the analysis of Rockland operators on graded Lie groups together with their associated Sobolev spaces. For the specific example of the Heisenberg group the theory is illustrated in detail. In addition, the book features a brief account of the corresponding quantization theory in the setting of compact Lie groups.The monograph is the winner of the 2014 Ferran Sunyer i Balaguer Prize. This work was published by Saint Philip Street Press pursuant to a Creative Commons license permitting commercial use. All rights not granted by the work's license are retained by the author or authors.

In this book we'll be using results and technics from deformation quantization of Poisson manifold theory in the sense Kontsevich and Cattaneo-Felder. The goal is to make suitable adaptations in order to use them in the Lie algebra case. This way we confront old problems of Lie theory and non commutative harmonic analysis. The first chapter is a detailed introduction to the part of the theory on (nilpotent) Lie groups and Lie algebras that we need. The second one is also a detailed introduction on deformation (bi)quantization and tools that we'll use in the sequence. Towards the end of chapter 2 we explain how these results will be used to prove theorems in the Lie case and introduce some central objects of study. Chapter 3 contains a detailed proof of a non-canonical isomorphism between a well known algebra of invariant differential operators and the corresponding to these data reduction algebra from deformation quantization. In chapter 4 the question of equivalence between characters from deformation quantization and harmonic analysis on Lie groups is answered positively. Finally in chapter 5 a central worked out example provides an overview of the above put in action.