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

TL;DR

This paper introduces a new multi-parameter deformation called symplectic reflection algebra, linking it to Calogero-Moser space and providing a deformed Harish-Chandra homomorphism, with applications to representation theory and algebraic geometry.

Contribution

It constructs a one-parameter deformation of the Harish-Chandra homomorphism using symplectic reflection algebras and relates it to Calogero-Moser space, extending classical results.

Findings

Deformation relates to Calogero-Moser space

Simple modules have dimension n!

H_infinity is isomorphic to endomorphisms of a vector bundle

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.