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

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To any finite group G of automorphisms of a symplectic vector space V we associate a new multi-parameter deformation, Hₖ, of the smash product of G with the polynomial algebra on V. The algebra Hₖ, called a symplectic reflection algebra, is related to the coordinate ring of a universal Poisson deformation of the quotient singularity V/G. If G is the Weyl group of a root system in a vector space h and V=h⊕ h*, then the algebras Hₖ are `rational' degenerations of Cherednik's double affine Hecke algebra. Let G=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 glₙ, to the algebra of Sₙ-invariant differential operators with rational coefficients on Cⁿ. The second order Laplacian on g goes, under the deformed homomorphism, to the Calogero-Moser differential operator with rational potential. Our crucial idea is to reinterpret the deformed homomorphism as a homomorphism: D(g)ᵍ → spherical subalgebra in Hₖ, where Hₖ is the symplectic reflection algebra associated to Sₙ. This way, the deformed Harish-Chandra homomorphism becomes nothing but a description of the spherical subalgebra in terms of `quantum' Hamiltonian reduction. In the classical limit k -> ∞, 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 algebra H∞ is isomorphic to the endomorphism algebra of a distinguished rank n! vector bundle on this space.