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

To any finite group ΓSp(V) of automorphisms of a symplectic vector space V we associate a new multi-parameter deformation, H^sub κ^ of the algebra [V]#Γ, smash product of Γ with the polynomial algebra on V. The parameter κ runs over points of ^sup r^, where r=number of conjugacy classes of symplectic reflections in $Γ$. The algebra H^sub κ^, called a symplectic reflection algebra, is related to the coordinate ring of a Poisson deformation of the quotient singularity V/Γ. This leads to a symplectic analogue of McKay correspondence, which is most complete in case of wreath-products. If Γ is the Weyl group of a root system in a vector space ? and V=??^sup *^, then the algebras H^sub κ^ are certain 'rational' degenerations of the double affine Hecke algebra introduced earlier by Cherednik. Let $Γ=S ^sub n^, the Weyl group of ?=??^sub n^. We construct a 1-parameter deformation of the Harish-Chandra homomorphism from ?(?)^sup ?^, the algebra of invariant polynomial differential operators on ??^sub n^, to the algebra of S ^sub n^-invariant differential operators with rational coefficients on the space ^sub n^ of diagonal matrices. The second order Laplacian on ? goes, under the deformed homomorphism, to the Calogero-Moser differential operator on ^sub n^, with rational potential. Our crucial idea is to reinterpret the deformed Harish-Chandra homomorphism as a homomorphism: ?(?)^sup ?^ spherical subalgebra in H^sub κ^, where H^sub κ^ is the symplectic reflection algebra associated to the group Γ=S ^sub n^. 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 κ[arrow right]∞, our construction gives an isomorphism between the spherical subalgebra in H^sub ∞^ and the coordinate ring of the Calogero-Moser space. We prove that all simple H^sub ∞^-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 S ^sub n^. Moreover, we prove that the algebra $H^sub ∞^ is isomorphic to the endomorphism algebra of that vector bundle.[PUBLICATION ABSTRACT]