  To any finite group G of automorphisms of a symplectic vector space V we
associate a new multi-parameter deformation, H_k, of the smash product of G
with the polynomial algebra on V. The algebra H_k, 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\oplus h^*, then the algebras H_k are
`rational' degenerations of Cherednik's double affine Hecke algebra.
  Let G=S_n, the Weyl group of g=gl_n. We construct a 1-parameter deformation
of the Harish-Chandra homomorphism from D(g)^g, the algebra of invariant
polynomial differential operators on gl_n, to the algebra of S_n-invariant
differential operators with rational coefficients on C^n. 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 \to {spherical
subalgebra in H_k}, where H_k is the symplectic reflection algebra associated
to S_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 k -> \infty, our construction gives an isomorphism
between the spherical subalgebra in H_\infty and the coordinate ring of the
Calogero-Moser space. We prove that all simple H_\infty-modules have dimension
n!, and are parametrised by points of the Calogero-Moser space. The algebra
H_\infty is isomorphic to the endomorphism algebra of a distinguished rank n!
vector bundle on this space.
