Introduction to representations of real groups
This is an introductory course with a special emphasis given on examples.
Lecturer
Date
18th September ~ 18th December, 2026
Location
| Weekday | Time | Venue | Online | ID | Password |
|---|---|---|---|---|---|
| Friday | 13:30 - 16:55 | Qiuzhen | ZOOM 11 | 435 529 7909 | BIMSA |
Syllabus
1. Representations of one-dimensional Lie groups. Basic Fourier analysis.
2. Finite Fourier analysis. Applications to number theory.
3. Group algebra of a Lie group. Haar measure.
4. Regular representation of a Lie group. Unitary representations.
5. The Lie algebra of a Lie group. Universal enveloping algebra.
6. Weyl's theory: integration formulas and the Peter-Weyl theorem.
7. Examples: SU(2) vs SU(1,1). Semidirect products.
8. Hecke algebra H(G,H)
9. Geometry of GL(2,R). Maurer-Cartan form. Integration on GL(2,R).
10. Universal enveloping algebra of gl(2). Casismir elements.
11. Principal series representations: various realisations.
12. Admissible representations. Harish-Chandra modules.
13. Discrete series representations.
14.Applications to automorphic forms.
2. Finite Fourier analysis. Applications to number theory.
3. Group algebra of a Lie group. Haar measure.
4. Regular representation of a Lie group. Unitary representations.
5. The Lie algebra of a Lie group. Universal enveloping algebra.
6. Weyl's theory: integration formulas and the Peter-Weyl theorem.
7. Examples: SU(2) vs SU(1,1). Semidirect products.
8. Hecke algebra H(G,H)
9. Geometry of GL(2,R). Maurer-Cartan form. Integration on GL(2,R).
10. Universal enveloping algebra of gl(2). Casismir elements.
11. Principal series representations: various realisations.
12. Admissible representations. Harish-Chandra modules.
13. Discrete series representations.
14.Applications to automorphic forms.
Reference
1) A. Deitmar, A first course in harmonic analysis, 2d edition, Springer 2005
2) Bill Casselman, Representations of SL(2,R), version of November 1 2020, available on the personal homepage
3) S. Lang, SL(2,R), Springer 1985
2) Bill Casselman, Representations of SL(2,R), version of November 1 2020, available on the personal homepage
3) S. Lang, SL(2,R), Springer 1985
Video Public
Yes
Notes Public
No
Lecturer Intro
Sergey Oblezin received his PhD at Moscow Institute of Physics and Technology in 2004. Education in Moscow and work experience at the Alikhanov Institute for Theoretical and Experimental Physics shaped his intra-disciplinary vision in mathematics, based on a unique and mutually transformative synthesis of quantum physics and mathematics. At early stage, his research achievements were recognized by several awards including two Russian Federation President Fellowships for young mathematicians (in 2007-2008 and 2008-2009). In 2009-2012, Sergey's research was awarded by the Pierre Deligne Prize (supported by P.Deligne's Balzan Prize, 2004). In 2013-17 Sergey's project "Topological field theories, Baxter operators and the Langlands programme" was supported by the Established Career EPSRC grant (UK). During 2015-2023, Sergey was an Associate Professor in Geometry at the University of Nottingham (UK), before taking his current full-time Professor position at BIMSA in 2024. Sergey Oblezin is working on a long term research project devoted to transferring and developing methods and constructions of quantum physics to the Langlands Program. His research interests include representation theory, harmonic analysis and their interactions with number theory and mathematical physics.