Syllabus

1. The DAWG, DAArt, and DAHA

(a) The double affine Weyl group (DAWG)

  • Presentations

  • Automorphisms

  • Examples and exercises

(b) The double affine Artin group (DAArt)

  • Presentations

  • Automorphisms

  • Examples and exercises

(c) The double ane Hecke algebra (DAHA)

  • Presentations

  • Automorphisms

  • Examples and exercises

2. The polynomial representation and Macdonald polynomials

(a) Review of DAHA

(b) Intertwiners

(c) Construction of Macdonald polynomials (and monomial expansions)

(d) Symmetric Macdonald polynomials (and monomial expansions)

(e) Examples

  • GLn case

  • Koornwinder case

  • Other classical types

  • Examples and exercises

3. Specializations and the Weyl character formula for Macdonald polynomials

(a) Specializations arising in other fields (first week)

  • Weyl characters

  • Demazure characters and Demazure atoms

  • Hall polynomials and Hall algebras

  • Spherical functions

  • Jack polynomials

  • Zonal polynomials

  • GUE, GOE, and GSE

  • characters of Level 0 modules

(b) Shift operators and Weyl character formula (2nd week)

4. Evaluations

(a) Symmetry

(b) c-functions and Principal specializations

5. Orthogonality

(a) Definition of the inner products

(b) Adjoints and orthogonality

(c) Norm formulas: the symmetric case

(d) Norm formulas: the non-symmetric case

  • GL_n case

  • Koornwinder case

  • Other classical types

  • Examples and exercises

6. Product formulas and recent developments