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Double Affine Hecke Algebras in Representation Theory, Combinatorics, Geometry, and Mathematical Physics

Graduate · Historical course

Fall 2009

Double affine Hecke algebras (DAHA), also called Cherednik algebras, and their representations appear in many contexts: integrable systems (Calogero-Moser and Ruijsenaars models), algebraic geometry (Hilbert schemes), orthogonal polynomials, Lie theory, quantum groups, etc. In this course we will review the basic theory of DAHA and their representations, emphasizing their connections with other subjects and open problems.

Readings

  • WEEK #: 1 · TOPICS: Classical and quantum Olshanetsky-Perelomov systems for finite Coxeter groups · READINGS: Chapter 2
    Reading assignment
  • WEEK #: 2 · TOPICS: The rational Cherednik algebra I · READINGS: Chapter 3, sections 3.1-3.13
    Reading assignment
  • Chapter 3, sections 3.14-3.17
    Required
  • Chapter 4, section 4.1
    Required
  • WEEK #: 4 · TOPICS: The Macdonald-Mehta integral · READINGS: Chapter 4, sections 4.2-4.4
    Reading assignment
  • WEEK #: 5 · TOPICS: Parabolic induction and restriction functors for rational Cherednik algebras · READINGS: Chapter 5
    Reading assignment
  • Chapter 7, sections 7.1-7.5
    Required
  • WEEK #: 7 · TOPICS: Hecke algebras for varieties with group actions · READINGS: Chapter 7, sections 7.6-7.15
    Reading assignment
  • WEEK #: 8 · TOPICS: Symplectic reflection algebras I · READINGS: Chapter 8, sections 8.1-8.7
    Reading assignment
  • WEEK #: 9 · TOPICS: Symplectic reflection algebras II · READINGS: Chapter 8, sections 8.8-8.13
    Reading assignment
  • WEEK #: 10 · TOPICS: Calogero-Moser spaces · READINGS: Chapter 9
    Reading assignment
  • WEEK #: 11 · TOPICS: Quantization of Calogero-Moser spaces · READINGS: Chapter 10
    Reading assignment
  • Bezrukavnikov, R., and P. Etingof. “Parabolic Induction and Restriction Functors for Rational Cherednik Algebras.” Selecta Math 14, nos. 3-5 (2009): 397-425.
    Listed reading
Sources