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

Abstract.

To any finite group Γ⊂Sp(V) of automorphisms of a symplectic vector space V we associate a new multi-parameter deformation, H_κ of the algebra ℂ[V]#Γ, smash product of Γ with the polynomial algebra on V. The parameter κ runs over points of ℙ ^r , where r=number of conjugacy classes of symplectic reflections in $Γ$. The algebra H_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=?⊕?^*, then the algebras H_κ are certain ‘rational’ degenerations of the double affine Hecke algebra introduced earlier by Cherednik.

Let $Γ=S _n , the Weyl group of ?=?? _n . We construct a 1-parameter deformation of the Harish-Chandra homomorphism from ?(?)^?, the algebra of invariant polynomial differential operators on ?? _n , to the algebra of S _n -invariant differential operators with rational coefficients on the space ℂ _n of diagonal matrices. The second order Laplacian on ? goes, under the deformed homomorphism, to the Calogero-Moser differential operator on ℂ _n , with rational potential. Our crucial idea is to reinterpret the deformed Harish-Chandra homomorphism as a homomorphism: ?(?)^? ↠ spherical subalgebra in H_κ, where H_κ is the symplectic reflection algebra associated to the group Γ=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 κ→∞, 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 family of these modules forms a distinguished vector bundle on the Calogero-Moser space, whose fibers carry the regular representation of S _n . Moreover, we prove that the algebra $H_∞ is isomorphic to the endomorphism algebra of that vector bundle.