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 PeterWeyl 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 KunzeStein 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.

Provides a self-contained treatment of the unitary representation theory of Lie groups in low dimensions Explores how spectral gap and effective decay of matrix coefficients can be used in dynamical systems Includes over 250 exercises, many with hints

Autorentext

Manfred Einsiedler studied at the University of Vienna, with a Ph.D. in 1999 under Klaus Schmidt. He held research positions at the University of East Anglia, Penn State University, the University of Washington, and Princeton University as a Clay Research Scholar. After becoming a Professor at Ohio State University he joined ETH Zürich. In 2004 he won the Research Prize of the Austrian Mathematical Society, in 2008 he was an invited speaker at the European Mathematical Congress in Amsterdam, and in 2010 he was an invited speaker at the International Congress of Mathematicians in Hyderabad. He works on ergodic theory and its applications to number theory (especially dynamical and equidistribution problems on homogeneous spaces). He has collaborated with Grigory Margulis and Akshay Venkatesh. With Elon Lindenstrauss and Anatole Katok, Einsiedler proved that a conjecture of Littlewood on Diophantine approximation is "almost always" true. Thomas Ward studied at the University of Warwick, with a Ph.D. in 1989 under Klaus Schmidt. He held research positions at the University of Maryland, College Park and at Ohio State University before joining the University of East Anglia in 1992. Between 2008 and retirement in 2023 he served on university executives, as Pro-Vice-Chancellor for Education at the Universities of East Anglia, Durham, and Newcastle and as Deputy Vice-Chancellor (Student Education) at the University of Leeds. He worked on the ergodic theory of algebraic dynamical systems, compact group automorphisms, and number theory. A long collaboration with Graham Everest on links between number theory and dynamical systems included the book Heights of Polynomials and Entropy in Algebraic Dynamics and a paper on Diophantine equations that won the 2012 Lester Ford Prize for mathematical exposition. With Einsiedler he has written the books Ergodic Theory with a view towards number theory in 2011 and Functional Analysis, Spectral Theory, and Applications in 2017.

Inhalt

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.