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

Abstract

To any finite group G of automorphisms of a symplectic vector space V we associate a new multi-parameter deformation, H_k, of the smash product of G with the polynomial algebra on V. The algebra H_k, 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\oplus h^*, then the algebras H_k are `rational' degenerations of Cherednik's double affine Hecke algebra. Let G=S_n, 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 gl_n, to the algebra of S_n-invariant differential operators with rational coefficients on C^n. 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 \to {spherical subalgebra in H_k}, where H_k is the symplectic reflection algebra associated to 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 -> \infty, our construction gives an isomorphism between the spherical subalgebra in H_\infty and the coordinate ring of the Calogero-Moser space. We prove that all simple H_\infty-modules have dimension n!, and are parametrised by points of the Calogero-Moser space. The algebra H_\infty is isomorphic to the endomorphism algebra of a distinguished rank n! vector bundle on this space.

arXiv comment95pp. Final version, to appear in Inventiones Math

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Cited by in corpus (10)

• Completions of symplectic reflection algebras
• Hochschild Cohomology and Deformations of Clifford-Weyl Algebras
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• Axiomatic framework for the BGG category O
• Certain Examples of Deformed Preprojective Algebras and Geometry of Their *-Representations
• Wreath Macdonald polynomials at q=t as characters of rational Cherednik algebras
• Generalized nil-Coxeter algebras over discrete complex reflection groups
• Ideals generated by traces or by supertraces in the symplectic reflection algebra $H_{1,\nu }(I_2(2m+1))$ II
• The Bernstein-Sato b-function for the complement of the open $S{L}_n$-orbit on a triple flag variety
• Macdonald polynomials as characters of Cherednik algebra modules