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, Hk, of the algebra ℂ[V]#Γ, smash product of Γ with the polynomial algebra on V. The parameter k runs over points of ℙr, where r = number of conjugacy classes of symplectic reflections in Γ. The algebra Hk, 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 h and V = h ⊕ h*, then the algebras Hk are certain 'rational' degenerations of the double affine Hecke algebra introduced earlier by Cherednik. Let Γ = Sn, the Weyl group g = gln. We construct a 1-parameter deformation of the Harish-Chandra homomorphism from D(g)g, the algebra of invariant polynomial differential operators on gln, to the algebra of Sn-invariant differential operators with rational coefficients on the space ℂn of diagonal matrices. The second order Laplacian on g goes, under the deformed homomorphism, to the Calogero-Moser differential operator on ℂn, with rational potential. Our crucial idea is to reinterpret the deformed Harish-Chandra homomorphism as a homomorphism: D(g)g ↠ spherical subalgebra in Hk, where Hk is the symplectic reflection algebra associated to the group Γ = Sn. 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 → ∞, 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 Sn. Moreover, we prove that the algebra H∞ is isomorphic to the endomorphism algebra of that vector bundle.