## Calogero-Moser Systems and Representation TheoryCalogero-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, the author gives short introductions to each of the subjects involved and provides a number of exercises. |

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### Contents

Introduction | 1 |

Deformation theory | 21 |

Moment maps Hamiltonian reduction and the LevasseurStafford | 29 |

CalogeroMoser system | 39 |

CalogeroMoser systems associated to finite Coxeter groups | 47 |

The rational Cherednik algebra | 53 |

Symplectic reflection algebras | 59 |

Deformationtheoretic interpretation of symplectic reflection algebras | 65 |

### Common terms and phrases

action of G action-angle variables algebraic deformation algebraic variety associated graded associative algebra Azumaya algebra bivector Calogero Calogero–Moser space Calogero–Moser system classical mechanics coadjoint orbit Cohen–Macaulay commutative algebra construction coordinates corresponding D(bres defined Definition deformation quantization degree denote differential operators Dunkl operators eigenvalue elements example Exercise filtration finite dimensional flow formal deformation formula Frobenius G-invariant Hamilton's equations Harish-Chandra Hice Hºc Hochschild cohomology homological dimension homomorphism ideal implies irreducible representation isomorphism Koszul algebras Lecture Lemma Lie algebra linear matrix module moment map Notes open set Poisson algebra Poisson bracket Poisson manifold Poisson structure polynomial Proposition quantum Hamiltonian reduction quantum integrable system quantum moment map quantum reduction quotient rational Cherednik algebra representation of Hoc smooth spherical subalgebra symmetric symplectic manifold symplectic reflection algebra universal deformation vector space Weyl Weyl(V x e b zero