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

*(English)*Zbl 1061.16032Summary: To any finite group \(\Gamma\subset\text{Sp}(V)\) of automorphisms of a symplectic vector space \(V\) we associate a new multi-parameter deformation, \(H_\kappa\) of the algebra \(\mathbb{C}[V]\#\Gamma\), smash product of \(\Gamma\) with the polynomial algebra on \(V\). The parameter \(\kappa\) runs over points of \(\mathbb{P}^r\), 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 \(\mathfrak h\) and \(V={\mathfrak h}\oplus{\mathfrak h}^*\), 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 \({\mathfrak g}=\mathfrak{gl}_n\). We construct a 1-parameter deformation of the Harish-Chandra homomorphism from \({\mathcal D}({\mathfrak g})^{\mathfrak g}\), the algebra of invariant polynomial differential operators on \(\mathfrak{gl}_n\), to the algebra of \(S_n\)-invariant differential operators with rational coefficients on the space \(\mathbb{C}^n\) of diagonal matrices. The second order Laplacian on \(\mathfrak g\) goes, under the deformed homomorphism, to the Calogero-Moser differential operator on \(\mathbb{C}^n\), with rational potential. Our crucial idea is to reinterpret the deformed Harish-Chandra homomorphism as a homomorphism: \({\mathcal D}({\mathfrak g})^{\mathfrak g}\twoheadrightarrow\) 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 \(\kappa\to\infty\), 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 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 \(H_\infty\) is isomorphic to the endomorphism algebra of that vector bundle.

Let \(\Gamma=S_n\), the Weyl group of \({\mathfrak g}=\mathfrak{gl}_n\). We construct a 1-parameter deformation of the Harish-Chandra homomorphism from \({\mathcal D}({\mathfrak g})^{\mathfrak g}\), the algebra of invariant polynomial differential operators on \(\mathfrak{gl}_n\), to the algebra of \(S_n\)-invariant differential operators with rational coefficients on the space \(\mathbb{C}^n\) of diagonal matrices. The second order Laplacian on \(\mathfrak g\) goes, under the deformed homomorphism, to the Calogero-Moser differential operator on \(\mathbb{C}^n\), with rational potential. Our crucial idea is to reinterpret the deformed Harish-Chandra homomorphism as a homomorphism: \({\mathcal D}({\mathfrak g})^{\mathfrak g}\twoheadrightarrow\) 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 \(\kappa\to\infty\), 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 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 \(H_\infty\) is isomorphic to the endomorphism algebra of that vector bundle.

##### MSC:

16S10 | Associative rings determined by universal properties (free algebras, coproducts, adjunction of inverses, etc.) |

14L30 | Group actions on varieties or schemes (quotients) |

17B66 | Lie algebras of vector fields and related (super) algebras |

17B80 | Applications of Lie algebras and superalgebras to integrable systems |

20C08 | Hecke algebras and their representations |

16S38 | Rings arising from noncommutative algebraic geometry |

16S80 | Deformations of associative rings |

14A22 | Noncommutative algebraic geometry |