| ABSTRACT |
The quantum Ruijsenaars model is a q-analogue of the Calogero?Moser model, described by n commuting partial difference operators (quantum hamiltonians) h_1, …, h_n. Its trigonometric variant is closely related to Macdonald polynomials and DAHAs. It turns out that for each natural number \ell>1, there exists an integrable system whose quantum hamiltonians look, loosely speaking, as the \ell-th powers of h_1, …, h_n. I will discuss several ways of arriving at this generalisation. In the elliptic case, the deformation parameter (“twisting”) is an arbitrary \ell-torsion point c on the underlying elliptic curve; when c=0 one gets precisely the \ell-th powers of h_1, …, h_n. The construction works for all root systems, including the Van Diejen's system (BC_n case) - the existence of such a twisted version of the Van Diejen's system was previously conjectured by Eric Rains. |
