Difference-elliptic Operators and Root Systems

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 (see [O,H,C5]) 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 (see [DJO]). Some observations indicate that there can be connections with the so-called W∞-algebras. In papers [C1,C2,C4] the analogous difference methods were developed to generalize the operators constructed by Macdonald (corresponding to the minuscule and certain similar weights) and those considered in [N] and other works on q-symmetric spaces. It is quite likely that the Fourier transform is self-dual in the difference setting (in contrast to the classical theory). Paper [C3] is devoted to the differential-elliptic case presumably corresponding to the Kac-Moody algebras. Presumably because the ways of extending the traditional harmonic analysis to these algebras are still rather obscure although there are interesting projects. 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 considered at the critical level. We discuss here only the construction of the generalized radial (zonal) Laplace operators. These operators are closely related to the so-called quantum many-body problem (Calogero, Sutherland, Moser, Olshanetsky, Perelomov), the conformal field theory (KnizhnikZamolodchikov equations), and the classic theory of the hypergeometric functions. They are expected to have applications to the characters of the Kac-Moody algebras and in Arithmetic. The natural problem is to extend the Macdonald theory [M1,M2,C2] to the elliptic case.