Difference-elliptic operators and root systems

Abstract

Recently a new technique in the harmonic analysis on symmetric spaces was suggested based on certain remarkable representations of affine and double affine Hecke algebras in terms of Dunkl and Demazure operators instead of Lie groups and Lie algebras. In the classic case it resulted (among other applications) in a new theory of radial part of Laplace operators and their deformations including a related concept of the Fourier transform. In the present paper we demonstrate that the new technique works well even in the most general difference-elliptic case conjecturally corresponding to the $q$-Kac-Moody algebras. We discuss here only the construction of the generalized radial (zonal) Laplace operators and connect them with the difference-elliptic Ruijsenaars operators generalizing in its turn the Olshanetsky-Perelomov differential elliptic operators.

References in corpus (1)

• Elliptic quantum many-body problem and double affine Knizhnik-Zamolodchikov equation

Cited by in corpus (7)

• Duality in Integrable Systems and Gauge Theories
• Macdonald's Evaluation Conjectures and Difference Fourier Transform
• Macdonald's Evaluation Conjectures, Difference Fourier Transform, and Applications
• Quantum Lax pairs via Dunkl and Cherednik operators
• Discrete harmonic analysis on a Weyl alcove
• Bethe Equations for a g_2 Model
• A discrete Fourier transform associated with the affine Hecke algebra