From symmetric spaces to DAHA
Symmetric spaces are Riemannian manifolds possessing isometric reflection in each its point. Careful analysis of this definition leads to the structure theories of compact and real semisimple Lie groups, inluding the remarkable duality between compact and noncompact symmetric spaces. Harmonic analysis on symmetric spaces discovers multidimensional spherical functions, and the ring of invariant differential operators is the basic point of famous family of commuting difference – differential operators by Dunkl. Now these operators are incorporated into affine and double affine Hecke algebras (DAHA) -fundamental objects of modern mathematics. The families of zonal spherical functions are generalized to Macdonald polynomials from one hand and wave functions of hyperbolic Ruijsenaars system and it nonrelativistic degenerations – Sutherland hyperbolic model. Moreover, the latter satisfy dual difference equations found by J. van Diejen and E.Emsiz.
The goal of the course is to trace, starting from the natural geometric object, the development of several fields of mathematics: Lie groups, spherical functions, and integrable systems.
The goal of the course is to trace, starting from the natural geometric object, the development of several fields of mathematics: Lie groups, spherical functions, and integrable systems.
讲师
日期
2026年03月06日 至 04月10日
位置
| Weekday | Time | Venue | Online | ID | Password |
|---|---|---|---|---|---|
| 周五 | 10:40 - 12:15 | Shuangqing-C746 | Zoom 15 | 204 323 0165 | BIMSA |
| 周五 | 13:30 - 15:05 | Shuangqing-C746 | Zoom 15 | 204 323 0165 | BIMSA |
课程大纲
1. Symmetric spaces as Riemannian manifolds: Killing fields, curvature, classification, duality.
2. Compact and noncompact symmetric spaces, Cartan decomposition, structure of real reductive groups, Iwasawa decomposition
3. Duality between compact and noncompact symmetric spaces
4. Invariant differential operators. Harish- Chandra homomorphisms.
5. Zonal spherical functions. General properties and constructions.
6. Spherical functions on symmetric spaces of rank 1
7. Dunkl operators and Heckmann – Opdam hypergeomeric function.
8. Harish – Chandra series. Dual difference equations.
9. Asymptotics. Harish – Chandra c- function. Gindikin-Karpelevich formula
10. Double affine Hecke algebras. Ruijsenaars and Macdonald operators
11. Macdonald polynomials and Hallnas-Ruijsenaars wave functions.
2. Compact and noncompact symmetric spaces, Cartan decomposition, structure of real reductive groups, Iwasawa decomposition
3. Duality between compact and noncompact symmetric spaces
4. Invariant differential operators. Harish- Chandra homomorphisms.
5. Zonal spherical functions. General properties and constructions.
6. Spherical functions on symmetric spaces of rank 1
7. Dunkl operators and Heckmann – Opdam hypergeomeric function.
8. Harish – Chandra series. Dual difference equations.
9. Asymptotics. Harish – Chandra c- function. Gindikin-Karpelevich formula
10. Double affine Hecke algebras. Ruijsenaars and Macdonald operators
11. Macdonald polynomials and Hallnas-Ruijsenaars wave functions.
参考资料
1. Helgason, Sigurdur. Differential geometry and symmetric spaces. Vol. 341. American Mathematical Society, 2024.
2. Helgason, Sigurdur. Geometric analysis on symmetric spaces. Vol. 39. American Mathematical Society, 2024.
3.Vilenkin N. I. A. Special functions and the theory of group representations. – American Mathematical Soc., 1978. – Т. 22.
4. Onishchik, Arkadij L., and Ernest B. Vinberg. Lie groups and algebraic groups. Springer Science & Business Media, 2012.
5. Heckman, Gerrit J. Dunkl operators. Astérisque, tome 245 (1997), Séminaire Bourbaki,exp. no 828, p. 223-246
6. Kirillov Jr, Alexander. "Lectures on affine Hecke algebras and Macdonald’s conjectures." Bulletin of theAmerican Mathematical Society 34.3 (1997): 251-292.
7. Van Diejen, J. F., & Emsiz, E. (2015). Difference equation for the Heckman–Opdam hypergeometric function and its confluent Whittaker limit. Advances in Mathematics, 285, 1225-1240.
8. Chalykh, Oleg. "Dunkl and Cherednik operators." arXiv preprint arXiv:2409.09005 (2024).
2. Helgason, Sigurdur. Geometric analysis on symmetric spaces. Vol. 39. American Mathematical Society, 2024.
3.Vilenkin N. I. A. Special functions and the theory of group representations. – American Mathematical Soc., 1978. – Т. 22.
4. Onishchik, Arkadij L., and Ernest B. Vinberg. Lie groups and algebraic groups. Springer Science & Business Media, 2012.
5. Heckman, Gerrit J. Dunkl operators. Astérisque, tome 245 (1997), Séminaire Bourbaki,exp. no 828, p. 223-246
6. Kirillov Jr, Alexander. "Lectures on affine Hecke algebras and Macdonald’s conjectures." Bulletin of theAmerican Mathematical Society 34.3 (1997): 251-292.
7. Van Diejen, J. F., & Emsiz, E. (2015). Difference equation for the Heckman–Opdam hypergeometric function and its confluent Whittaker limit. Advances in Mathematics, 285, 1225-1240.
8. Chalykh, Oleg. "Dunkl and Cherednik operators." arXiv preprint arXiv:2409.09005 (2024).
视频公开
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笔记公开
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语言
英文
讲师介绍
Sergey Khoroshkin is a professor of Higher School of Economy in Moscow. His research interests include representation theory, quantum integrable systems and special functions.