A new take on spherical, Whittaker and Bessel functions (Spherical and Whittaker functions via DAHA I,II)

Abstract

This paper is based on the lectures given by the first author at Harvard in February and March, 2009. It begins with an introduction to the classical p-adic theory of the Macdonald, Matsumoto and Whittaker functions. Its major directions are as follows: 1) extending the theory of DAHA to arbitrary levels; 2) the affine Satake isomorphism and Hall functions via DAHA; 3) the spinor Dunkl operators for the Q-Toda equation; 4) applications to the nil-DAHA and q-Whittaker functions; 5) the technique of spinors in the differential theory and its applications to the AKZ-QMBP isomorphism theorem.

v4: further improvements of the theory of affine symmetrizers, editing; v5: improving refferences, editing; v6: a somewhat extended version of v5 essentially equivalent to the papers "Spherical and Whittaker Functions via DAHA I,II" to be published by Selecta Mathematica

References in corpus (5)

• The quantum dilogarithm and representations quantum cluster varieties
• Trigonometric Cherednik algebra at critical level and quantum many-body problems
• Equivariant Satake category and Kostant-Whittaker reduction
• Non-semisimple Macdonald polynomials
• Double affine Hecke algebra in logarithmic conformal field theory

Cited by in corpus (5)

• Kernel functions and Bäcklund transformations for relativistic Calogero-Moser and Toda systems
• A Relativistic Conical Function and its Whittaker Limits
• Affine Gindikin-Karpelevich formula via Uhlenbeck spaces
• Affine Hall-Littlewood functions for $A_1^{(1)}$ and some constant term identities of Cherednik-Macdonald-Mehta type
• Quantum affine Knizhnik-Zamolodchikov equations and quantum spherical functions, I