By Pavel Etingof

Calogero-Moser platforms, which have been initially came upon by means of experts in integrable platforms, are at the moment on the crossroads of many parts of arithmetic and in the scope of pursuits of many mathematicians. extra particularly, those structures and their generalizations grew to become out to have intrinsic connections with such fields as algebraic geometry (Hilbert schemes of surfaces), illustration idea (double affine Hecke algebras, Lie teams, quantum groups), deformation idea (symplectic mirrored image algebras), homological algebra (Koszul algebras), Poisson geometry, and so on. The aim of the current lecture notes is to provide an advent to the idea of Calogero-Moser structures, highlighting their interaction with those fields. considering those lectures are designed for non-experts, the writer provides brief introductions to every of the themes concerned and offers a few workouts. A ebook of the eu Mathematical Society (EMS). allotted in the Americas by means of the yankee Mathematical Society.

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**Example text**

V/ D 0 for almost every, hence for every p. This is a contradiction, which implies that x0 does not exist, and hence S is a polynomial. 5 is proved. 4. g / D 1 (the Bernstein filtration), and the associated graded map gr HC W CŒg g g ! CŒhreg h W , which attaches to every differential operator the symbol of its radial part. It is easy to see that this map is just the restriction map to h ˚ h g ˚ g , so it actually lands in CŒh ˚ h W . Moreover, gr HC is a map onto CŒh ˚ h W . Indeed, gr HC is a Poisson map, so the surjectivity follows from the following lemma.

Hreg /W . Therefore, the map ‚ t;c restricts to an hreg W for t D 0. embedding B t;c ! hreg /W for t ¤ 0, and B0;c ! 7. (i) The spherical subalgebra B0;c is commutative and does not have zero divisors. (ii) Bc is an algebraic deformation of B0;c . Proof. (i) Follows immediately from the fact that B0;c CŒh (ii) Follows since Hc is an algebraic deformation of H0;c . hreg W . 7 implies that the spectrum Mc of B0;c is an irreducible affine algebraic variety. Moreover, Mc has a natural Poisson structure, obtained from the deformation Bc of B0;c .

3. Let X be a smooth affine algebraic variety with a free action of a connected reductive algebraic group G. X/, and W g ! Vect X ! X/ be the usual action map. X=G/. 3 The Levasseur–Stafford theorem In general, similarly to the classical case, it is rather difficult to compute the quantum reduction A==g. g/ is the algebra of differential operators on a reductive Lie algebra g, and g acts on A through the adjoint action on itself. This description is a very nontrivial result of Levasseur and Stafford.