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

• 成果类型: Article
• 署名作者: Etingof, P; Ginzburg, V
• 署名单位: Massachusetts Institute of Technology (MIT); University of Chicago
• 刊物名称: INVENTIONES MATHEMATICAE
• ISSN/ISSBN: 0020-9910
• DOI: 10.1007/s002220100171
• 发表日期: 2002
• 页码: 243-348
• 关键词: quantum integrable systems; affine hecke algebras; kac-moody algebras; dualizing complexes; quiver varieties; weyl algebra; lie-algebras; Operators; THEOREM; rings

摘要

To any finite group Gamma subset of Sp(V) of automorphisms of a symplectic vector space V we associate a new multi-parameter deformation, H-kappa of the algebra C[V]#Gamma, smash product of Gamma with the polynomial algebra on V. The parameter kappa runs over points of P-r, where r = number of conjugacy classes of symplectic reflections in Gamma. The algebra H-kappa called a symplectic reflection algebra, is related to the coordinate ring of a Poisson deformation of the quotient singularity V/Gamma. This leads to a symplectic analogue of McKay correspondence, which is most complete in case of wreath-products. If Gamma is the Weyl group of a root system in a vector space h and V = h circle plus h*, then the algebras H-kappa are certain 'rational' degenerations of the double affine Hecke algebra introduced earlier by Cherednik. Let Gamma = S-n the Weyl group of g = gl(n) We construct a I-parameter deformation of the Harish-Chandra homomorphism from D(g)(g), the algebra of invariant polynomial differential operators on gl(n) to the alge-bra of S-n-invariant differential operators with rational coefficients on the space C-n of diagonal matrices. The second order Laplacian on g 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: D(g)(g) --> spherical subalgebra in H-kappa where H-kappa is the symplectic reflection algebra associated to the group Gamma = 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 --> infinity, our construction gives an isomorphism between the spherical subalgebra in H-infinity and the coordinate ring of the Calogero-Moser space. We prove that all simple H-infinity-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-infinity is isomorphic to the endomorphism algebra of that vector bundle.