Ali Baklouti - Representation Theory of Solvable Lie Groups and Related Topics

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Representation theory of Lie groups and noncommutative harmonic analysis on Lie groups and homogeneous spaces have witnessed a significant growth starting from the 1940s when research work in these areas began. Taken jointly, they constitute a pivotal and compelling field within mathematics, due to the tight relationships with so many other areasthink of number theory, algebraic geometry, differential geometry, operator algebras, PDEs, to name but a fewand also with farther fields like physics.

Within the setting of exponential solvable Lie groups, the so-called orbit method is a fundamental tool to associate through an elegant way unitary duals to the space of coadjoint orbits. Still, the harmonic analysis on the corresponding homogeneous spaces remains a difficult subject.

The purpose of this book is to discuss the latest advances in this area, prove novel results on noncommutative harmonic analysis on solvable homogeneous spaces, and provide many applications. The text offers the most recent solutions to a number of open questions posed over the last decades, presents the newest research results on the matter, and provides an alluring platform for progressing in this research area.

Throughout the text, unless otherwise explicitly stated, G will always denote a connected and simply connected exponential solvable Lie group with Lie algebra . This means the exponential mapping is a diffeomorphism. The first chapter aims to build the branching laws of mixed representations (or representations of mixed type), showing that they obey the orbital spectrum formula. Let A and H be two closed connected subgroups of G, and and two unitary irreducible representations of G and H, respectively. Mixed representations are unitary representations of the form (up-down representation) or