GL(2,R)
This is an introductory course on representations of real reductive groups. Also, the course can be considered as complementary/preparatory for studying automorphic forms.
这是一门关于实约化群表示和朗兰兹分类的入门课程。此外，该课程亦可视为研究自守形式的补充/预备课程。
这是一门关于实约化群表示和朗兰兹分类的入门课程。此外，该课程亦可视为研究自守形式的补充/预备课程。
Lecturer
Date
24th February ~ 30th May, 2025
Location
| Weekday | Time | Venue | Online | ID | Password |
|---|---|---|---|---|---|
| Monday,Wednesday | 14:20 - 16:05 | A3-3-201 | ZOOM 11 | 435 529 7909 | BIMSA |
Prerequisite
Basic representation theory
Syllabus
1. Group actions and linear representations.
2. Induced representations.
3. Compact groups.
4. Haar measure and Weyl's integration formulas.
5. Geometry of SL(2,R).
6. Lie algebras. Differential of a linear group representation.
7. Universal enveloping algebra. Invariant differential operators.
8. Spherical harmonics. The Legendre equation.
9. Additive and multiplicative characters of R. Fourier transform.
10. Classification of representations of sl(2,C).
11. The group algebra and its representations.
12. Harish-Chandra modules. Constructing representations of SL(2,R).
13. Spherical functions. Differential operators.
2. Induced representations.
3. Compact groups.
4. Haar measure and Weyl's integration formulas.
5. Geometry of SL(2,R).
6. Lie algebras. Differential of a linear group representation.
7. Universal enveloping algebra. Invariant differential operators.
8. Spherical harmonics. The Legendre equation.
9. Additive and multiplicative characters of R. Fourier transform.
10. Classification of representations of sl(2,C).
11. The group algebra and its representations.
12. Harish-Chandra modules. Constructing representations of SL(2,R).
13. Spherical functions. Differential operators.
Reference
R.Howe, E.-C.Tan, Non-Abelian harmonic analysis: Applications of SL(2,R), Springer 1992
A.Knapp, Representation theory of semisimple groups: An overview based on examples, PUP 2001
S.Lang, SL(2,R), Springer 1985
A.Knapp, Representation theory of semisimple groups: An overview based on examples, PUP 2001
S.Lang, SL(2,R), Springer 1985
Audience
Undergraduate
, Advanced Undergraduate
, Graduate
Video Public
Yes
Notes Public
Yes
Language
English
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.