2000/11/16 by Pavel Etingof, Victor Ginzburg, Etingof, Pavel +1 · 40 citations
Mathematics · #Advanced Algebra and Geometry #Advanced Topics in Algebra #Algebra homomorphism #Algebra over a field #Algebraic structures and combinatorial models #Computer science #Homomorphism #Mathematics #Physics #Pure mathematics #Reflection (computer programming) #Space (punctuation) #Symplectic geometry #math.AG #math.QA #math.RT
paper · pdf · doi:10.48550/arxiv.math/0011114
published in arXiv (Cornell University) (Cornell University) · 95pp. Final version, to appear in Inventiones Math
openalex publication_date 2000/11/16 · arxiv created 2001/07/24 · arxiv updated 2009/11/30 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/05
To any finite group G of automorphisms of a symplectic vector space V we associate a new multi-parameter deformation, Hk, of the smash product of G with the polynomial algebra on V. The algebra Hk, 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 Hk are `rational' degenerations of Cherednik's double affine Hecke algebra. Let G=Sn, the Weyl group of 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 Cn. The second order Laplacian on g goes, under the deformed homomorphism, to the Calogero-Moser differential operator with rational potential. Our crucial idea is to reinterpret the deformed homomorphism as a homomorphism: D(g)g → spherical subalgebra in Hk, where Hk is the symplectic reflection algebra associated to 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 algebra H_∞ is isomorphic to the endomorphism algebra of a distinguished rank n! vector bundle on this space.