Speaker
Description
The phase space of the Ruijsenaars integrable system can be identified with (a Poisson reduction of) the moduli space of $GL_n$ local systems on a punctured torus. The latter admits a structure of a cluster Poisson variety. On the algebraic level, this leads to an injective homomorphism from a spherical subalgebra of the double affine Hecke algebra into the quantized algebra of global functions on the named cluster variety. From an analytic point of view, it allows for a unitary equivalence between Toda and Ruijsenaars quantum integrable systems. This in turn allows one to present the eigenfunctions of Macdonald operators as a matrix coefficient of an order 4 element in the mapping class group of a punctured torus. During this talk we will focus on the $n=2$ case when no Hamiltonian reduction is required. It will be based on a joint work with P. DiFrancesco, R. Kedem, S. Khoroshkin, and G. Schrader.
