Unitary Representations and Unitary Duals / by Manfred Einsiedler, Thomas Ward.

Format:

Book

Author/Creator:

Einsiedler, Manfred.

Contributor:

Ward, Thomas.

Series:

Graduate Texts in Mathematics, 2197-5612 ; 308

Language:

English

Subjects (All):

Topological groups.

Lie groups.

Harmonic analysis.

Dynamics.

Functional analysis.

Topological Groups and Lie Groups.

Abstract Harmonic Analysis.

Dynamical Systems.

Functional Analysis.

Local Subjects:

Topological Groups and Lie Groups.

Abstract Harmonic Analysis.

Dynamical Systems.

Functional Analysis.

Physical Description:

1 online resource (578 pages)

Edition:

1st ed. 2025.

Place of Publication:

Cham : Springer Nature Switzerland : Imprint: Springer, 2025.

Summary:

This graduate textbook introduces the unitary representation theory of groups, emphasizing applications in fields like dynamical systems. It begins with the general theory and motivation, then explores key classes of groups. Abelian and compact groups are treated through Pontryagin duality and the Peter–Weyl theorem. Metabelian groups illustrate links to ergodic theory and lead to the Mackey machine. Weak containment and the Fell topology are introduced through examples. The final chapters apply the theory to special linear groups in dimensions two and three, covering smooth vectors, spectral gaps, and decay of matrix coefficients. The two-dimensional case is examined in depth, including the Kunze–Stein phenomenon, spectral decomposition on the hyperbolic plane, and the Weil representation. The book concludes with a full description of the unitary dual of SL(2,R) and its Fell topology, applying the theory to prove effective equidistribution of horocycle orbits. With its focus on key examples and concrete explanations, this textbook is aimed at graduate students taking first steps in unitary representation theory. It builds the theory from the ground up, requiring only some familiarity with functional analysis beyond standard undergraduate mathematics.

Contents:

1 Unitary Representations

2 Abelian Groups

3 Compact Groups

4 Lie Algebras and Unitary Representations of SU2(R)

5 Normal Abelian Subgroups and Unitary Duals

6 Weak Containment and the Fell Topology

7 Smooth Vectors and Decay for SL3(R)

8 Discrete Series Representations and Temperedness

9 Unitary Representations of SL2(R)

Appendix A: Linear Algebra

Appendix B: Analysis

Appendix C: Topological Groups.

Notes:

Description based on publisher supplied metadata and other sources.

ISBN:

3-032-03899-5

9783032038999

OCLC:

1548325051