Lectures on Calogero-moser Systems

Calogero-Moser systems, which were originally discovered by specialists in integrable systems, are currently at the crossroads of many areas of mathematics and within the scope of interests of many mathematicians. More specifically, these systems and their generalizations turned out to have intrinsic connections with such fields as algebraic geometry (Hilbert schemes of surfaces), representation theory (double affine Hecke algebras, Lie groups, quantum groups), deformation theory (symplectic reflection algebras), homological algebra (Koszul algebras), Poisson geometry, etc. The goal of the present lecture notes is to give an introduction to the theory of Calogero-Moser systems, highlighting their interplay with these fields. Since these lectures are designed for non-experts, we give short introductions to each of the subjects involved, and provide a number of exercises. We now describe the contents of the lectures in more detail. In Lecture 1, we give an introduction to Poisson geometry and to the process of classical Hamiltonian reduction. More specifically, we define Poisson manifolds (smooth, analytic, and algebraic), momemt maps and their main properties, and then describe the procedure of (classical) Hamiltonian reduction. We give an example of computation of Hamiltonian reduction in algebraic geometry (the commuting variety). Finally, we define Hamiltonian reduction along a coadjoint orbit, and give the example which plays a central role in these lectures – the Calogero-Moser space of Kazhdan, Kostant, and Sternberg. In Lecture 2, we give an introduction to classical Hamiltonian mechanics and the theory of integrable systems. Then we explain how integrable systems may sometimes be constructed using Hamiltonian reduction. After this we define the classical Calogero-Moser integrable system using Hamiltonian reduction along a coadjoint orbit (the KazhdanKostant-Sternberg construction), and find its solutions. Then, by introducing coordinates on the Calogero-Moser space, we write both the system and the solutions explicitly, thus recovering the standard results