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 + h^*, then the algebras
H_k are `rational' degenerations of Cherednik's double affine Hecke algebra.
We develop the theory of Harish-Chandra
bimodules over the rational Cherednik algebra $H(W)$
associated to a finite Coxeter group $W$
in a vector space h.
We apply representation theory of Cherednik algebras to study
$W$-quasi-invariant} polynomials on h,
an algebra $C[h]^W\subset Q_c\subset C[h],$
introduced by Chalykh, Feigin, and Veselov
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