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Symplectic reflection algebras, Calogero-Moser space, and deformed Harish-Chandra homomorphism. (English) Zbl 1061.16032
Summary: 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.

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
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