0 Symplectic reflection algebras , Calogero - Moser system , and deformed Harish - Chandra homomorphism

To any finite group Γ ⊂ Sp(V ) of automorphisms of a symplectic vector space V we associate a new multi-parameter deformation, Hκ, of the algebra C[V ]#Γ, smash product of Γ with the polynomial algebra on V . The parameter κ runs over points of CP , where r =number of conjugacy classes of symplectic reflections in Γ. The algebra Hκ, called a symplectic reflection algebra, is expected to be related to the coordinate ring of a universal Poisson deformation of the quotient singularity V/Γ. If Γ is the Weyl group of a root system in a vector space h and V = h ⊕ h, then the algebras Hκ are certain ‘rational’ degenerations of the double affine Hecke algebra introduced earlier by Cherednik. Let Γ = Sn, the Weyl group of g = gln. We construct a 1-parameter deformation of the Harish-Chandra homomorphism from D(g)g, the algebra of invariant polynomial differential operators on the Lie algebra g = gln, to the algebra of Sn-invariant differential operators on the Cartan subalgebra C with rational coefficients. The second order Laplacian on g goes, under our deformed homomorphism, to the Calogero-Moser differential operator on C, with rational potential. Let Hκ be the symplectic reflection algebra associated to the group Γ = Sn, acting diagonally on V = C ⊕ C. We reinterpret the deformed Harish-Chandra homomorphism as a homomorphism: D(g)g ։ spherical subalgebra in Hκ. In the limit κ → ∞, our construction gives an isomorphism between the spherical subalgebra in H∞ and the coordinate ring of the Calogero-Moser space. We prove that all simple H∞-modules have dimension n!, and are parametrised by points of the Calogero-Moser space. The family of these modules forms a distinguished vector bundle on the Calogero-Moser space, whose fibers carry the regular representation of Sn. Moreover, we prove that the algebra H∞ is isomorphic to the endomorphism algebra of that vector bundle.