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401-3227-DRL 2 Credits DR D-MATH

Unitary Representations of Lie Groups

Lecturers & Examiners: Prof. Dr. Manfred Einsiedler
Only for ETH D-MATH doctoral students and for doctoral students from the Institute of Mathematics at UZH. The latter need to send an email to Jessica Bolsinger ( ) with the course number. The email should have the subject „Graduate course registration (ETH)“.
VVZ CR n/a

Last Updated: 2026-02-05 16:07:26

Abstract

This course will introduce unitary representations of Lie groups, discuss spectral gap in general, and discuss concrete unitary representations of SL(2,R).

Objective

The goal is to acquire familiarity with the basic formalism and results concerning Lie groups and their unitary representations. In the second part we will consider concrete representations of SL(2,R) and decompose these into irreducible representations or at least understand whether spectral gap is present.

Content

The course will start with the general framework of unitary representations of locally compact groups, which is in some sense a general theory of Fourier analysis related to groups. For this some functional analysis (in particular spectral theory of bounded selfadjoint operators, Krein-Milman and Choquet) will be important. In the interest of time we will only summarise the case abelian groups and use the abelian theory to understand some metabelian groups. After this we will discuss some more general theory. Some of the general phenomena will be discussed for the concrete group of SL(2,R). Moreover, we will understand the unitary dual of SL(2,R), discuss the notion of spectral gap for SL(2,R), and decompose the unitary representation of SL(2,R) arising from the hyperbolic plane into irreducible representation.

Resources

Lecture Notes

Unitary Representations and Unitary Duals,book project joint with Tom Ward, seehttps://tbward0.wixsite.com/books/unitary

Literature

Bekka, de la Harpe and Valette: "Kazhdan's Property (T)", Cambridge University Press.

Learning Materials (Links)

General Information

Language
English
Levels
DR

Examination

Type
ungraded semester performance

Registration & Places

Priority: Registration for the course unit is only possible for the primary target group

Course Components

Type Title Time & Place Hours
lecture with exercise Unitary Representations of Lie Groups
  • Tue 12:15-14:00 (HG E 1.1)
  • Wed 10:15-12:00 (HG E 1.1)
4 h weekly

Offered In

  • Doctorate Mathematics (More Information at: )
    • Subject Specialisation (The list of courses (together with the allocated credit points) eligible for doctoral students is published each semester in the newsletter of the ZGSM.)
      • Graduate School (Official website of the Zurich Graduate School in Mathematics: In addition to the 401-....-DRL course units, adapted versions for doctoral students of the following course units: 263-4400-00L Advanced Graph Algorithms and Optimization 401-3902-21L Network & Integer Optimization: From Theory to Application 401-3904-22L Convex Optimization 401-3629-00L Quantitative Risk Management 401-3652-00L Numerical Methods for Hyperbolic Partial Differential Equations 151-0530-00L Nonlinear Dynamics and Chaos II 227-0434-10L Mathematics of Information 401-4490-22L Topology Optimization of Engineering Systems ... (continued ))