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

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

2000-11-16

arxiv.org/abs/math/0011114v6

Mathematics

Algebraic Geometry

95pp. Final version, to appear in Inventiones Math

Scientific paper

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.

Affiliated with

Also associated with

No associations

Rating

Symplectic reflection algebras, Calogero-Moser space, and deformed Harish-Chandra homomorphism does not yet have a rating. At this time, there are no reviews or comments for this scientific paper.

If you have personal experience with Symplectic reflection algebras, Calogero-Moser space, and deformed Harish-Chandra homomorphism, we encourage you to share that experience with our LandOfFree.com community. Your opinion is very important and Symplectic reflection algebras, Calogero-Moser space, and deformed Harish-Chandra homomorphism will most certainly appreciate the feedback.

Rate now

Comments { 0 }

Profile ID: LFWR-SCP-O-93862