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

Venue: | Invent. Math |

Citations: | 182 - 35 self |

### BibTeX

@ARTICLE{Etingof_symplecticreflection,

author = {Pavel Etingof and Victor Ginzburg},

title = {Symplectic reflection algebras, Calogero-Moser space, and deformed Harish-Chandra homomorphism},

journal = {Invent. Math},

year = {},

pages = {348}

}

### Years of Citing Articles

### OpenURL

### Abstract

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 r, 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 = 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 the Lie algebra g = gl n, to the algebra of Sn-invariant differential operators on the Cartan subalgebra C n with rational coefficients. The second order Laplacian on g goes, under our deformed homomorphism, to the Calogero-Moser

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